woodruff school of mechanical engineering georgia

Woodruff School of Mechanical Engineering Georgia Institute of Technology

Atlanta, Georgia, USA October 1-2, 2010

Volume 9

© Copyright 2010 by The American Society of Mechanical Engineers Southeast District F, Three Park Avenue, New York, NY 10016 All rights reserved. Printed in the United States of America. Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or any means, or stored in a data base or retrieval system, without the prior written permission of the publisher. ASME shall not be responsible for statements or opinions advanced in papers or printed in its publications (B7.1.3). Statement of ASME Bylaws. ASME District F encompasses the states of Alabama, Delaware, District of Columbia, Florida, Georgia, Maryland, Mississippi, North Carolina, South Carolina, Tennessee and Virginia. For more information go to http://districts.asme.org/DistrictF.htm.

ISBN 978-1-4507-9222-6

ii

http://districts.asme.org/DistrictF.htm�

ASME EARLY CAREER TECHNICAL JOURNAL

Volume 9

Presented at ASME District F Early Career Technical Conference

Woodruff School of Mechanical Engineering Georgia Institute of Technology

Atlanta, Georgia October 1-2, 2010

Sponsored by

The ASME Old Guard Georgia Institute of Technology

HOLTEC International Unified Brands

ASME District F ASME Atlanta Section

ASME Mississippi Section

Technical Committee Dr. J. Donnell

Dr. P. Durbetaki Dr. S. Jeter

Chair of Coordinating Committee Dr. K. R. Rao

For more information about ECTC 2010 visit: http://districts.asme.org/DISTRICTF/ECTC/2010ECTC/

THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS SOUTHEAST DISTRICT F

THREE PARK AVENUE, NEW YORK, NY 10016

iii

http://districts.asme.org/DISTRICTF/ECTC/2010ECTC/�

FOREWORD Journal of ASME District F Early Career Technical Conference

October 1 – 2, 2010

One of the primary objectives of American Society of Mechanical Engineers (ASME) is the dissemination of technical information. Pursuant to this goal of ASME Southeastern Region (Region XI) had initiated the Technical Conference (RTC) that was in fact in the earlier years the Graduate Technical Conferences when Dr. P. Durbetaki was the Vice President of Region XI. In 2001 Regional Vice President Dr. K. R. Rao had with the help of Region XI Operating Board approved the creation of the Region XI Technical Conference (RTC). The first RTC was held on April 6, 2002, in Jackson, Mississippi with the participation of students across all five states (Florida, Georgia, Alabama, Tennessee and Mississippi) of the former ASME Region XI. The quality of the reviewed papers, published in the four volumes of the Regional Technical Journal from 2002 through 2006, called for this Conference to be renamed in 2006 as the Early Career Technical Conference (ECTC) indicating the changing demands of the times. On October 6 and 7, 2006 ECTC was held, in Jackson, MS with a robust support of Entergy Operations. Inc. In 2007, for the first time, ECTC was held in a University, Florida International University (FIU), in Miami, Florida. Invitations were extended to seventy eight universities and companies comprising of fourteen states to participate in this conference. These included the states within the ASME International Southeast District F - Alabama, Delaware, District of Columbia, Florida, Georgia, Maryland, Mississippi, North Carolina, South Carolina, Tennessee, Virginia. In addition ECTC extended invitations to Louisiana, Arkansas and Texas of the ASME International Southwest Rocky Mountain District E. ECTC 2008 was also held at FIU, Miami, Florida with invitations being extended to all of the Universities invited for ECTC 2007. Dr. Yong Tao was the Chair, Technical Committee and Editor for ECTC 2007 and Dr. Sabri Tosunoglu for ECTC 2008. The generous financial support of ASME Old Guard funded the registration and even offset a part of their commutation expenses of the paper presenters to attend these conferences. The success of the preceding ASME District F Early Career Technical Conferences prompted the organizers to look beyond the frontiers of the District. For the current ECTC 2010 invitations had been sent to all of the universities affiliated with all of the ASME Districts A to J, that were even beyond USA. It is gratifying to mention that we reached the cap of 35 paper presenter entries that we targeted. Of these we have 14 submittals from outside USA. Thus, the theme for this Conference we had “ECTC Opens the Window to Outside USA” had been amply demonstrated! The Department of Mechanical Engineering, University of Alabama, Tuscaloosa, Alabama hosted the ECTC 2009. The Editor of the ECTC 2009 Journal established an elaborate review process similar to the process used by ASME Technical Divisions. Papers were distributed to Associate Editors of the Editorial Board based on their areas of expertise and they obtained reviews for each paper by expert reviewers in the field. The Editorial Board made a significant effort to ensure the review process for each paper followed the criteria and deadlines. Because of this rigorous requirement, the papers submitted were accepted as either for presentation only at the conference, or for both presentation and journal publication. Authors whose papers were rated acceptable for publication in the Technical Journal were required to make corrections and enhancements, based on reviewers’ comments. The Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, Georgia became the host for the ECTC 2010. The review and selection process initiated with the ECTC 2009 Conference was followed for the ECTC 2010 Conference. This ECTC 2010 Journal is a compilation of thirty three (33) reviewed and accepted technical papers from USA, Canada and Asia. K. R. Rao, PhD, PE., Fellow ASME, FIE, CE Chair ECTC Coordinating Committee

P. Durbetaki, PhD, Life Fellow ASME Co-Chair ECTC Coordinating Committee

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ACKNOWLEDGEMENTS ASME District F Early Career Technical Journal

October, 2010 Many individuals contributed to the success of this Early Career Technical Conference and this ECTC 2010 Journal. The Coordinating Committee express gratitude and appreciation to Dr. K R. Rao, Chair, Dr. P. Durbetaki, Co-chair of the Conference Coordinating Committee, and members of the Conference Committee Dr. Bill Wepfer, Dr. Sheldon Jeter, Dr. Jeffrey Donnell, Michael D. Stewart, Dr. Yong Tao and John Mulvihill for their tireless and dedicated efforts in organizing the conference, directing the review process and overseeing the presentations. Their hard work has made this journal possible. In addition we thank all the members of the Editorial Board and invited reviewers, who spent countless hours of their own time reviewing and critiquing the papers to ensure a quality publication. ECTC 2010 Journal Editorial Board comprises of the Editors, Dr. J. Donnell, Dr. P. Durbetaki and Dr. S. Jeter. The advice and guidance of Dr. Yong Tao is much appreciated and as web master for ECTC 2010 his help is very much recognized. Like the previous year’s ECTC 2009, the ECTC 2010 had separate tiers of Associate Editors from the Faculty and from the Professionals. The names, contact details, bios and pictures of each of the reviewers are posted on the ASME ECTC website http://districts.asme.org/DISTRICTF/ECTC/2010ECTC/. ASME District F ECTC Editorial Board and the Coordinating Committee thank each of the reviewers for assuring a high standard of papers are accepted for presentation at the conference publication in this journal. ASME District F and the Coordinating Committee congratulate all of the authors who demonstrated a high degree of professionalism in producing an excellent collection of papers. ASME District F and the Coordinating Committee thank ASME staff, Burt Dicht Managing Director and his staff for the administrative support to the success of the conference and the publication of this journal. Finally, we all owe a great debt of gratitude to our principal sponsors Woodruff School of Mechanical Engineering of the Georgia Institute of Technology, the ASME Old Guard, HOLTEC International, Unified Brands, ASME District F, ASME Mississippi and Atlanta Sections and Faculty Support from the Faculty of Georgia Tech, ME.

-- 2010 ASME ECTC Coordinating Committee

v

http://districts.asme.org/DISTRICTF/ECTC/2010ECTC/�

The ASME Early Career Technical Conference Committee and Technical Journal Editorial Board Express their Appreciation

to the Following Sponsors for Supporting ECTC 2010

WWooooddrruuffff SScchhooooll ooff MMeecchhaanniiccaall EEnnggiinneeeerriinngg

Georgia Institute of Technology

ASME DISTRICT F

ASME ATLANTA SECTION

vi

Table of Contents

INSIDE COVER iii

FOREWORD iv

ACKNOWLEDGEMENTS v

SPONSORS vi

1. SANDEEP KULATHU and DAVID L. LITTLEFIELD 1 Development of a Biofidelic Material Model for Brain Tissue and its Application to the Impact Response of the Human Head

2. TAO JIANG 9 The Research of New Approaches to Coupling Windage Problem Based on Pressure Experiments and Analysis

3. OLANREWANJU ALUKO and BRIAN S. MOORE 19 The Failure Characterization of Mechanically Fastened Composite Joints

4. KEVIN STANLEY McFALL 27 Solving Coupled Systems of Differential Equations Using the Length Factor Artificial Neural Network Method

5. AHMAD ALMAGABLEH and P. RAJU MANTENA 35 Effects of Environmental Aging on the Thermal and Mechanical Properties of Vinyl Ester Nanocomposites

6. ROY DOWNS, DAKOTA SCRIVENER, SACHIN S. TERDALKAR and JOSEPH J. RENCIS 41 Molecular Dynamic Simulation to Determine Temperature Dependent Young’s Modulus of Monolayer and Bilayer Graphene Sheets

7. MANUSCRIPT REMOVED 49

8. MANUSCRIPT REMOVED 57

9. FARSHID ZABIHIAN, HSIAO-WEI D. CHIANG and ALAN S. FUNG 66 Performance Analysis of Micro Gas Turbine Fueled by Blends of Biodiesel and Petroleum-based Diesel

10. MOHAMMED ALI 75 In-silico Simulation of Electrostatic Charge Effects on Inhaled Aerosol Particle Deposition in the Human Lung

vii

11. REETA WATTAL and SUNIL PANDEY 80 Mathematical Models for Prediction of Weld Bead Geometry in GMAW of Aluminium Alloy 7005

12. GREGORY J. HICKMAN and KUANG-TING HSIAO 88

Finite Element Analysis of Bonded Composite Single-lap Joint as a Basis for Optimizing Future Nano-reinforcement Configurations

13. ZEAID F. HASAN and AHMAD BANI-YOUNES 96 Analyzing Nonlinear Flexible Structures Using Perturbation Approach

14. TEZESWI P. TADEPALLI and CHRISTOPHER L. MULLEN 105 Interactive Computational Tool for Simulation of Dynamic Response and Damage in Composite Structures

15. FAISAL ASFAND 112 CFD Simulation of Thermal Stratification in Pressurizer Surge Line

16. VIDYA K. NANDIKOLLA and SUHAS S. PHARKUTE 118 Flowchart Visual Programming in Mechatronics Course

17. VIDYA K. NANDIKOLLA, JENNA MATTHEWS, MARCO P. SCHOEN, SUHAS S. PHARKUTE UWE REISCHL and AJAY MAHAJAN 124 Adaptive Multi Airbag Shoe Insert for Diabetic Foot Care

18. BRADLEY L. ZUNDEL 130 Design Approach to Fire Life Safety in Tunnels Through the Use of Subway Environment Simulation Software

19. POONAM V. SAVSANI and HEMANT NAGARSHETH 138 Trajectory Planning of 3R Robotic Arm Using Genetic Algorithm (GA) and Artificial Bee Colony (ABC) Optimization Methods

20. VIMAL J. SAVSANI, R. V. RAO and D. P. VAKHARIA 146 Multi-objective Optimization of Mechanical Elements Using Artificial Bee Colony Optimization Technique

21. MICHAEL B. PRESNELL and ARUNACHALAM M. RAJENDRAN 153 A Comparative Study of Segmented Tungsten Rod Penetration into a Thick Steel Target Plate

22. TAMIR A. EMRAN, RYAN C. ALEXANDER, CHRIS T. STALLINGS, MARK A. DeMAY and MATTHEW J. TRAUM 158 Method to Accurately Estimate Tesla Turbine Stall Torque for Dynamometer or Generator Load Selection

23. DALE B. McDONALD 165 Efficient Continuous-Time Optimization

viii

24. BASIL I. FARAH, GAIL D. JEFFERSON and KUANG-TING HSIAO 173

Water Absorption of Carbon Nano-Fiber Enhanced Polyester/E-Glass Composite Materials

25. MICHAEL HEMPOWICZ, GAIL D. JEFFERSON, KUANG-TING HSIAO and N. GASTON 178 An Experimental Study for Flexural Properties of 22- and 32-Ply Honeycomb Composite Panels IM7/8552

26. STEVEN M. KRAFT, ALI P. GORDON and FRANK MARINACCI 181 The Elasto-Plastic Mechanical Properties of a Twill Dutch Woven Wire Mesh

27. IMRAN AKHTAR, ZHU WANG, JEFF BORGGAARD and TRAIAN ILIESCU 189 A Novel Strategy for Nonlinear Closure in Proper Orthogonal Decomposition Reduced-Order Models

28. MATTHEW TURK, RYAN MELSERT, SHELDON JETER and HANY AL ANSARY 198 Preliminary Investigation of a Chemical Heat Engine for Solar Energy Applications

29. MATTHEW GOLOB, SHELDON JETER and DENNIS SADOWSKI 203 Heat Transfer from Flat Surfaces to Moving Sand

30. BRIAN S. MUNN 208 DOE Investigation into the Diminishing Effect of Applied Mean Stress and Bending Stress on the High Cycle Fatigue Behavior of a Threaded Fastener

31. DAVID J. BRANSCOMB, DAVID G. BEALE and ROYALL M. BROUGHTON, Jr. 215 Computer-Aided Product and Process Development of Lace Braided Composites

32. DAVID J. BRANSCOMB and IDRIS CERKEZ 221 Damage Evaluation of Braided Kevlar®/Carbon/Epoxy Composite Beams in Three Point Bending With Infrared Thermography

33. MOHAMED G. GHORAB 226

A New Single Transient Technique to Measure Film Cooling Performance: Gas Turbine Application AUTHOR INDEX 235

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ASME Early Career Technical Journal

2010 ASME Early Career Technical Conference, ASME ECTC October 1 – 2, Atlanta, Georgia USA

DEVELOPMENT OF A BIOFIDELIC MATERIAL MODEL FOR BRAIN TISSUE AND ITS APPLICATION TO THE IMPACT RESPONSE OF THE HUMAN HEAD

Sandeep Kulathu, David L. Littlefield Univ. of Alabama at Birmingham

Birmingham, AL, U.S.A

ABSTRACT The development of a finite element model capable of predicting brain injury sustained as a result of an impact load to the head is of considerable importance for the development of adequate protection systems and design of treatment strategies. With advancements in imaging technologies, it is now possible to visualize the neuron fiber structures in the brain. In this study, a composite material model was developed from the individual isotropic behaviors of the Cerebro-Spinal Fluid (CSF) and brain tissue present in the brain such that the resultant behavior of the material is transverse isotropic. To capture the presence of neuron fibers in the brain, the data obtained from Diffusion Tensor MRI (DTI) was incorporated into the finite element model so that, based on the fiber direction in each voxel, the material model was suitably applied to behave as a transverse isotropic material. The use of DTI data in conjunction with the transverse isotropic material model is a novel method that has not been attempted so far because the material data available for brain tissue is mostly isotropic in character. A Lagrangian mesh based solver was developed so that it could be customized to incorporate DTI data and the developed constitutive material model over a simplified geometry of the human head. Loading conditions from literature where experiments have been performed on cadavers were used as a means of comparing computed strains and pressures with the observed values. INTRODUCTION

Traumatic Brain Injury (TBI) is a cause of death or lifelong disability in people of all ages around the world. From 1995 to 2001, an average of 1.4 million traumatic brain injuries occurred in the United States alone, each year. Of these, 79.6% were emergency department visits, 16.8% hospitalizations and 3.6% deaths [1]. Though many people recover from their injuries, each year an estimated 80,000 to 90,000 people sustain a TBI that results in permanent disability [2]. Among children aged 0-14 years, TBI results in an estimated 3000 deaths,

29000 hospitalizations and 400,000 emergency department visits each year [3]. In sports, an estimated 300,000 brain injuries of mild to moderate severity occur in the United States each year [4]. It has been observed that 75% of all TBI that occur each year are either concussions or a mild form of TBI [5]. In the military, advancements in body armor and helmets have reduced abdominal, thoracic, and penetrating head injuries for the soldier. Protective helmets made from Kevlar have greatly reduced penetrating head injuries from projectiles. But, helmets give limited protection for the brain against non-penetrating forces like explosive blasts, falls, etc., that cause Coup Contrecoup injury, and pressure wave trauma to the brain. Though the modern armor ensures the survivability of the soldier, it has resulted in an increase in the number of soldiers returning from the battlefield with some form of TBI. Of the wounded soldiers admitted to the Walter Reed Army Medical Center who suffered a blast injury, or motor vehicle accident, or a fall, 60% have sustained a TBI [6].

The response of the head-brain complex to impact loading has been studied for decades. Over the years, many computational models have been developed that can describe the stress/strain and pressure distributions within the brain when the head is subjected to an impact load [7]. Due to lack of sufficient experimental data to characterize the brain tissue, many approximations are made in the computational models, thus limiting their accuracy. Further, different visco-elastic material models are chosen by different authors to capture the behavior of the brain tissue with varying success [8]. All material models so far cited in literature utilize the assumption that the brain tissue is isotropic in behavior, but biological tissues are not isotropic. Most of the times, biological tissues are either transverse isotropic or anisotropic, visco-elastic materials assumed to be homogeneous in the continuum scale. Experimental studies [9] have shown that the response of brain tissue to an applied mechanical load has directional dependence. Elkin [10] has shown that different regions of the brain respond differently for

ASME 2010 Early Career Technical Journal, Vol. 9 1

identical stimuli. One possible reason is that the orientations of the axons vary in different regions of the brain, thus resulting in different responses to a similar load. Arbogast [11] used Hashin’s fiber reinforced composite material model with linear visco-elastic material constants to model the brainstem with the orientation of the fiber being assumed to be vertical. With the recent advances in imaging techniques, it is now possible to capture the direction of the neuron fiber bundles in the brain using Diffuse Tensor Imaging (DTI) [12]. These visualization techniques are used in the medical field to capture the presence of tumors and other abnormalities not only in the brain, but also in the other parts of the human body.

In this paper, a novel method was developed to utilize the DTI data to model the brain tissue as a transverse isotropic material. Due to the lack of experimental data that capture brain tissue behavior as anisotropic, a mathematical formulation was developed to derive transverse isotropic behavior of a composite from isotropic behavior of the individual constituents namely, fiber and matrix. The resultant Transverse Isotropic (TI) elastic material properties were used in conjunction with the DTI data to model the mid-sagittal section of the brain in two-dimensional space using an in-house solver called Arbitrary Lagrangian Eulerian Adaptive Solver (ALEAS). Based on the elastic TI model, a fluid-solid TI material model was developed by utilizing the isotropic material properties of CSF and brain tissue. In the literature reviewed so far, the authors have observed that the use of DTI data in conjunction with a TI material model to determine the response of the head-brain complex to a non-penetrating impact load has not been attempted so far. MATHEMATICAL FORMULATION

Since available experimental data only allow for characterization of the different constituents of brain tissue as isotropic, most researchers utilize the simplification that brain tissue is isotropic for computational purposes. Using isotropic properties from previous works, a TI material model was derived by extending the rule of mixtures to three dimensions.

Hypo-elastic formulation

A hypo-elastic formulation allows the material constitutive equations to be expressed in terms of objective stress rates. The objective stress rates are frame invariant and hence when a rigid body motion is superimposed, the stress rates remain identical before and after the transformation. Though the strain, rate of deformation, and Cauchy stress tensors are frame invariant, the Cauchy stress rate is not an objective measure. Hence, the Jaumann stress rate, which is objective, was used instead of the Cauchy stress rate to derive the material constitutive relations. The Jaumann

stress rate tensor σ ∇J and Cauchy stress tensor σ are related using the spin tensor W as shown in Eq. (1). The spin tensor is the skew symmetric part of the spatial velocity gradient tensor.

σ ∇J =dσdt

−W .σ −σ .W T (1)

Solid-Solid TI Model

By extending the rule of mixtures in three dimensions, such that, when two isotropic materials are combined to result in a transverse isotropic material, the stresses and strains carried by the matrix and fiber along the three perpendicular axes can be represented as in Eqs. (2) - (10). These equations represent the behavior of the resultant composite material, where along the direction of fiber axis, the strains are equal and stresses are additive, and transverse to the fiber axis, stresses are equal and strains are additive. Here, it is assumed that the fibers are aligned along direction 1 as shown in Figure 1.

σ 11∇J = Vmσ 11

m ∇J + Vfσ 11f ∇J (2)

σ 11m ∇J =

Em

(1+ υ m)(1− 2υ m)[(1− υ m)D11 + υ mD22

m + υ mD33m ] (3)

σ 11f ∇J =

E f

(1+ υ f )(1− 2υ f )[(1− υ f )D11 + υ f D22

f + υ f D33f ] (4)

D22 = VmD22m + Vf D22

f (5)

D22m =

−υ mσ 11m ∇J

Em +σ 22

∇J

Em −υ mσ 33

∇J

Em (6)

D22f =

−υ fσ 11f ∇J

E f +σ 22

∇J

E f −υ fσ 33

∇J

E f (7)

D33 = VmD33m + Vf D33

f (8)

D33m =

−υ mσ 11m ∇J

Em −υ mσ 22

∇J

Em +σ 33

∇J

Em (9)

D33f =

−υ fσ 11f ∇J

E f −υ fσ 22

∇J

E f +σ 33

∇J

E f (10)

Figure 1. Representation of the composite material

depicting the fiber orientation in the matrix

On solving the equations simultaneously, a transverse isotropic material tensor is obtained with some of the


terms in the compliance matrix as shown in Eqs. (11) – (13).

S11t =

1Emvm + E f vf

(11)

S22t =

Ef Em(Vm2 +Vf

2)−VmVf (Efν m −Emν f )2 +

VmVf (Em2 +E f 2)

EmEf (EmVm +E fVf )

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥⎥⎥⎥⎥

(12)

S32t =

E f Em (ν mVm2 + ν f V f

2) + VmVf (E f ν m −

Emν f )2 + VmVf (Em2ν f + E f 2ν m)EmE f (EmVm + E f Vf )

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥

(13)

where superscript ‘m’ and ‘f’ refer to matrix and fiber respectively, Vm and Vf refer to volume fractions of matrix and fiber respectively, Dii ,σ ii

∇J , i=1,2,3 refer to rate of deformation and Jaumann stress rate components in xx, yy, zz directions, St

ij, i, j=1,2,3 refer to the components of the resultant transverse isotropic compliance matrix, Em, Ef are the Young’s moduli of matrix and fiber respectively, υm, υ f are the Poisson’s ratio of fiber and matrix respectively. The compliance terms for the normal and shear stresses for the solid-solid TI model can be found in Appendix A.

Figure 2. Decomposition of the shear stress

behavior in direction 13 and direction 31

For the shear components, the same theory can be extended so that in direction 32 which corresponds to component S44 the shear strains are additive whereas, for direction 31, the components are broken down individually to account for the different behaviors that can be observed in direction 13 and direction 31 as shown in Figure 2 and the resultant value is averaged over the two directions. For σ 31 acting on the face along the fiber axis, the strains are equal and stresses are additive and for σ 13 which is transverse to the fiber axis, the stresses are equal and strains are additive. Similar behavior can be observed for the stresses acting in directions 1-2 and 2-1. The shear strains and stresses are related by the compliance terms as shown in Eq. (14) and Eq. (15).

S44t =

(1+ υ m)Em Vm +

(1+ υ f )E f Vf (14)

S55t =

12

(1+ υ m)(1+ υ f )Em (1+ υ f )Vm + E f (1+ υ m)Vf

+

E f (1+ υ m)Vm + Em (1+ υ f )Vf

EmE f

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

(15)

Table 1. Elastic, isotropic material properties for

different regions of the mesh [13]

Region Density Kg/m3

Young’s Modulus N/m2

Poisson’s ratio

Skull 2040 6.5x109 0.22

Brain Tissue 1004 5.4x105 0.48

CSF 1004 1.20 x 104 0.49

To obtain the material properties for the fiber and

matrix, the volume fractions are assumed to be equal and have a value of 0.5. Further, the fiber is assumed to be 75% stiffer than the matrix. This assumption is based on the fact that the load carrying capability of the axon is much higher than the extra-cellular matrix that is mostly a fluid. Using elastic, isotropic material properties for the brain tissue from Table 1, the transverse isotropic properties are obtained by equating the stresses and strains by utilizing the analogy of performing a uni-axial test.

Fluid-Solid TI Model

Since the brain is composed of about 70% water, a fluid-solid TI model is developed to incorporate the presence of such a large volume of water. Based on the TI material model for elastic behavior, a fluid-solid TI material model was developed by combining the material properties of CSF and brain tissue. An implicit assumption that the fluid and solid remain glued together as a composite material was made while developing this material model. The stress-strain constitutive relations for the elastic solid and fluid material are written as shown in Eq. (16) and Eq. (17) respectively.

σ ij

∇J = λδ ij Dkk + 2μDij (16) σ ij = −Pδ ij + λ*δ ij Dkk + 2μ*Dij (17)

where σ ij ,σ ij

∇J are the Cauchy stresses and Jaumann stress rates respectively, λ , μ are the Lame’s parameters for elastic solids, Dij is the rate of deformation, λ* , μ* are viscosity coefficients of the fluid, P is pressure, and δ ij is Kronecker delta function. By combining the equations based on the load carrying capability of the solid and fluid


based on the fiber direction, nine equations similar to that for TI solid elastic model can be written as shown in Eqs. (18) – (26).

σ 11 = Vmσ 11m + Vfσ 11

f (18) σ 11

m = −P + λ* + 2μ *( )D11 + λ*D22m + λ*D33

m (19)

σ 11f ∇J = λ + 2μ( )D11 + λD22

f + λD33f (20)

D22 = VmD22m + Vf D22

f (21) D22

m = K1σ 11m + K2σ 22 + K1σ 33 + K3P (22)

D22f = M 1σ 11

f ∇J + M 2σ 22∇J + M 1σ 33

∇J (23) D33 = VmD33

m + Vf D33f (24)

D33m = K1σ 11

m + K1σ 22 + K2σ 33 + K3P (25) D33

f = M 1σ 11f ∇J + M 1σ 22

∇J + M 2σ 33∇J (26)

where K1, K2, K3, M1, M2, and M3 are the terms from the compliance matrix for the fluid and solid respectively which are shown in Appendix B. On solving the 9 equations simultaneously, and after some algebraic manipulation as shown in Appendix B, equations for stress and stress rates relating strain and rates of deformation are obtained as shown in Eqs. (27) – (30).

σ 11f ∇J =

λ + 2μ( )D11 + λ (M 1 + M 2)(σ 22∇J +σ 33

∇J )1− 2λM 1

(27)

σ 22∇J = k21D11 + k22D22 + k23D33 + c22σ 22 + c23σ 33 + c24 P (28)

σ 33∇J = k21D11 + k23D22 + k22D33 + c23σ 22 + c22σ 33 + c24 P (29)

σ 32∇J =

2μVf

D32 −VmμVmμ * σ 32 (30)

where Kij and Cij are material constants which are functions of λ , μ , λ* , and μ * . Similar to the solid-solid TI material model, the shear stresses can be found by decomposing the stress strain behavior in directions parallel and transverse to the fiber axis. The resultant equation for shear stress in direction 12 is shown in Eq. (30). A time implicit, finite difference method was used to solve Eq. (28) and Eq. (29) simultaneously for the stresses in directions 2 and 3 respectively. The complete stress strain relations for the fluid-solid TI material are shown in Appendix B. MODEL DEVELOPMENT AND MATERIAL PROPERTIES

DT-MRI is an imaging technique that measures the diffusion of water molecules within the neural fiber network. The water molecules are constrained within the microstructure of the neural fibers. The direction of fastest diffusion is aligned along the fiber length and the slowest diffusion is at right angles to the fiber axis. This feature is exploited in capturing the orientation of neuron fiber bundles within the brain or muscle fibers in the body. The

theory behind DTI can be found in [12] and [14]. DTI data set consists of a tensor describing local water diffusion for each voxel. The tensor can be represented in the form of a symmetric 3 x 3 matrix A, the components of which capture directional variation in the diffusion rate. The visualization of DTI data sets can be achieved by reducing the tensor to scalar or vector values and using ellipsoids, color schemes or arrows to represent the values pictorially as shown in Figure 3(a). These visualization techniques allow the user to track the fibers in different parts of the body.

(a) (b)

Figure 3. Vector Glyphs show the orientation of neural fibers at (a) every nodal location in the mesh, and (b) averaged at the element centers of the mesh

By solving the eigenvalue problem, A − λI = 0 , the eigenvalues λ i , i=1,2,3 and corresponding eigenvectors can be obtained. The eigenvalues being a quantitative measure of diffusion are all positive and real because negative diffusion is physically not possible. The magnitude of the eigenvalue can be viewed as a measure of fiber alignment. From a mechanics standpoint, the largest eigenvalue can be interpreted as a measure of stiffness because when there are a larger number of fibers aligned closely in a particular direction, the stiffness in that direction increases. The eigenvector corresponding to the largest eigenvalue describes the direction of principal diffusion, which lies along the longitudinal axis of the neural fiber. Based on DTI data sets, it is possible to map the known fiber orientation at every voxel onto the meshed finite element model.

(a) (b)

Figure 4. (a) Mesh with outer dark region as skull, inner dark region as brain tissue and lighter region

as CSF, and (b) Applied Force


The simplified geometry of the brain-skull complex

at the mid-sagittal section was reconstructed from the visible human data set as shown in Figure 4(a). The DTI data set at the mid sagittal section was extracted from the data set of the whole brain and mapped onto the geometry of the brain being used in the model. With the known fiber orientation, the elastic, transverse isotropic material model was applied to every element comprising the brain tissue so that the fiber direction was oriented with the major direction obtained from the DTI data.

The material properties used for the different regions of the mesh, for isotropic case are shown in Table 1. The model consisted of 3623 quadrilateral elements and 3513 nodes. In order to simulate the presence of the neck region, a torsional stiffness of 80 N-m/rad was used at the surface of the elements that form the neck region. This particular stiffness was chosen based on experiments described in [14]. For the TI elastic fluid-solid material model, the brain tissue properties that were used for the fluid part were based on that for water which is similar to CSF, that is, dynamic viscosity μ * =1.00 CentiPoise and bulk viscosity k*=3.09 CentiPoise. For the solid, the lame parameter values were chosen so that the bulk modulus of the solid and fluid were within one order of magnitude resulting in the values for λ and μ as 1.488e8 N/m2 and 9.924e7 N/m2 respectively. The bulk moduli for water and solid were 2.15e9 N/m2 and 2.15e8N/m2 respectively. For simplicity of calculation, the volume fractions of fiber and matrix were taken to be 0.5. In determining the direction of fiber at each element, the average fiber directions from the four nodes that comprise the element were taken as shown in Figure 3(b). From the average fiber direction, the dominance of the x component or y component helped in deciding if the net resultant orientation of fiber for the particular element was going to be taken as either x or y only. A time dependent input force corresponding to experiment 37 listed by Nahum et al [13] as shown in Figure 4(b) was applied to the surface that comprises the frontal lobe of the model in an anterior-posterior direction. The solution was obtained using ALEAS, which is a finite element solver developed in-house. ALEAS utilizes the h-adaptive mesh update procedure, which increases the mesh density in regions of steep solution gradient based on an error indicator. RESULTS AND DISCUSSION

For computational purposes, to model the brain tissue as a soft and pliable material, the elastic isotopic material property is derived by lowering the Young’s modulus, and to mimic incompressibility due to the presence of water in the tissues, the Poisson’s ratio is set close to 0.5. This results in the physics being altered for the sound speed propagation through the brain tissue. The effective speed at which the pressure wave propagates through the brain

for the isotropic material model was calculated to be around 60 m/s whereas for the fluid-solid material model, the effective wave speed was about 800m/s. The transmission of a pressure wave through the brain tissue can be seen in Figure 5 where, for the TI fluid-solid model the waves propagate at a much higher rate than for either the isotropic or the TI solid-solid model.

(a) (b)

(c)

Figure 5. Pressure waves in the brain using different material models (a) Isotropic at 1.5ms, (b) solid-solid

TI at 3.5ms, and (c) Fluid Solid TI at 0.6ms The speed of sound in the brain tissue has been

experimentally found to be in the range of about 1570m/s [16], which is about the same as that observed for water. In the TI models, the presence of neuron fibers adds to the stiffness of the material in the direction of the fiber and hence there is added resistance to the propagation of pressure waves through the brain tissue along the fiber axis direction. In the fluid-solid TI model, the presence of CSF equivalent material property for the matrix material in the composite helps to increase the wave propagation speed through the brain tissue. In the fluid-solid TI model, wave speed along the fiber axis is limited by the sound speed in the fiber and transverse to the fiber axis, the impediment to the fluid sound speed is only because of the small cross section of the fiber. So, there is a higher sound speed in directions transverse to the fiber axis.

The distribution of pressure in the different regions of the brain is shown in Figure 6. From the pressure distribution it can be seen that, compared to the isotropic material, the TI fluid-solid model predicts higher pressure whereas the TI solid-solid has a damped response. The computed displacements in the fluid-solid TI is higher than that observed with the other two material models.


Due to the low compliance of the matrix in the fluid-solid material, it would be possible to capture the effect of localized shear strain on neural fibers because the directional dependence is incorporated into the model and this leads to the pressure waves travelling differently either along or transverse to the fiber axis as shown in Figure 5. The isotropic and TI solid-solid material models have similar behaviors with the response being damped for the TI solid-solid.

Figure 6. Pressure Distribution in different regions of

the brain Initial results show promise in the models ability to

transmit pressure waves according to the measured values. This pressure wave transmission becomes important especially in the modeling of brain injury observed in soldiers returning from active duty where the human body is subjected to blast waves. A three-dimensional geometric model would be able to account for the different orientations of the neuron fibers in the brain, thereby, improving the capability of the material model to capture the phenomenon of localized large shear strains that serves as the basis for causing DAI. Comparison with experimental data available from experiments performed on cadavers [17], [18], and [19] was not possible because the location of the transducers used in the measurement of pressures were in different locations from the mid sagittal plane where the simulation was conducted.

FUTURE WORK

The current material model was developed taking into account the effect of the underlying microstructure of the brain tissue. The initial results show that it is possible to utilize a TI material model in conjunction with DTI data to enhance the capability of computational models to predict the localization of injury in the brain. The current

work utilizing a TI material model developed by combining the isotropic properties of brain tissue and CSF, as detailed in this paper, serves as a foundation to build a transverse isotropic visco-elastic material model. Application of the current TI material on a three-dimensional geometry of the brain would greatly enhance the capability of the model and allow subsequent validation with experiments performed on cadavers. AKNOWLEDGEMENTS

The authors would like to thank Dr. Song Zhang for having provided the DTI data set that was used in the current computational model. REFERENCES [1] Langlois JA, Rutland-Brown W, Thomas KE., 2006, “Traumatic brain injury in the United States: emergency department visits, hospitalizations, and deaths”, Atlanta (GA): Centers for Disease Control and Prevention, National Center for Injury Prevention and Control, http://www.cdc.gov/ncipc/pub-res/tbi_in_us_04/tbi%20in% 20the%20us_jan_2006.pdf [2] Thurman D, Alverson C, Dunn K, Guerrero J, Sniezek J., 1999, “Traumatic Brain Injury in the United States: a Public Health Perspective”, J. Head Trauma Rehabil, 14(6), pp. 602–615. [3] Langlois J, Gotsch K., 2001, “Traumatic Brain Injury in the United States: Assessing Outcomes in Children”, Atlanta (GA): National Center for Injury Prevention and Control, Centers for Disease Control and Prevention (CDC), www.cdc.gov/traumaticbraininjury/pdf/TBI_assessing.pdf [4] Sosin D, Sniezek J, Thurman D., 1996, “Incidence of Mild and Moderate Brain Injury in the United States”, 1991, Brain Injury, 10(1), pp. 47-54. [5] Centers for Disease Control and Prevention (CDC), National Center for Injury Prevention and Control, 2003, “Report to Congress on Mild Traumatic Brain Injury in the United States: Steps to Prevent a Serious Public Health Problem, Centers for Disease Control and Prevention”, www.cdc.gov/ncipc/pub-res/mtbi/mtbireport.pdf [6] Lew, H.L., Poole, J.H., Alavrez, S., & Moore, W., 2005, “Soldiers with Occult Traumatic Brain Injury”, Am J Phys Med Rehabil, 84(6), pp. 393-398. [7] King H. Yang, Albert I. King, 2003, “A Limited Review of Finite Element Models Developed for Brain Injury Biomechanics Research”, Int. J. Vehicle Design, 32(1/2), pp. 116 - 129 [8] J.S.Ruan, T.Khalil, A.I.King, 1994, “Dynamic Response of the Human Head to Impact by Three Dimensional Finite Element Analysis”, J. Biomech. Eng., 116(1), pp. 44-50 [9] Michael T. Prange, Susan S. Margulies, 2002, “Regional, Directional, and Age-Dependent Properties of


the Brain Undergoing Large Deformation”, J. Biomech. Eng., 124(2), pp. 244-252 [10] Benjamin S. Elkin and Barclay Morrison III, 2007, “Region Specific Tolerance Criteria for the Living Brain”, Stapp Car Crash J., 51, pp. 27-38 [11] Arbogast KB, Margulies SS. 1998, “A Fiber-Reinforced Composite Model of the Viscoelastic Behavior of the Brainstem in Shear”, J Biomech., Vol 32(8), pp. 865-870. [12] Song Zhang, C¸ agatay Demiralp, and David H. Laidlaw, 2003, “Visualizing Diffusion Tensor MR Images Using Streamtubes and Streamsurfaces”, IEEE Trans. on Vis. and Comp. Graph., 9(4), pp. 454-462 [13] Nahum, Alan M., Randall Smith and Carley C. Ward, 1977, “Intracranial Pressure Dynamics During Head Impact”, SAE Paper Number 770922 [14] C.F. Westin, S.E. Maier, H. Mamata, A. Nabavi, F.A. Jolesz, R. Kikinis, 2002, “Processing and Visualization for Diffusion Tensor MRI”, Medical Image Analysis, 6(2), pp. 93–108 [15] Tanya Garcia, Bahram Ravani, 2003, “A Biomechanical Evaluation of Whiplash using a Multi-Body Dynamic Model”, J. Biomech. Eng., 125(2), pp. 254-265 [16] Matthew Hussey, 1975, “Diagnostic Ultrasound”, Blackie & Son Limited, London. [17] Nyquist, G.W., Cavanaugh, J.M., Goldberg, S.J., and King, A.I, 1986, “Facial Impact Tolerance and Response”, Proc. 30th Stapp Car Crash Conference, pp. 733–754. [18] Trosseille,X., Lavaste, F., Guillon,F., and Domont,A., 1992, “Development of a F.E.M of the Human Head According to a Specific Test Protocol”, Proc. 36th Stapp Car Crash Conference, pp. 235-253. [19] Hardy WN, Foster CD, Mason MJ, Yang KH, King AI, Tashman S., 2001, “Investigation of Head Injury Mechanisms using Neutral Density Technology and High Speed Biplanar X-ray”, Proc. 45th Stapp Car Crash Conference, pp. 337-368. APPENDIX A

This appendix contains the equations relating the rate of deformation to Jaumann stress rate for the solid-solid TI material model.

D11 =σ 11

∇J − (Vmν m + Vf νf )σ 22

∇J − (Vmν m + Vf νf )σ 33

∇J

EmVm + E f Vf

(31)

D22 =−(Vmν m + V f ν

f )σ 11∇J

E mVm + E f Vf

+

E f Em (Vm2 + V f

2) − VmVf (E f ν m −

Emν f )2 + VmVf (Em2 + E f 2)EmE f (EmVm + E f V f )

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥

σ 22∇J

−



E mν f )2 + VmVf (Em2ν f + E f 2ν m)EmE f (E mVm + E f Vf )

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥

σ 33∇J

(32)

D33 =−(Vmν m + Vf ν

f )σ 11∇J

EmVm + E f Vf

−



Emν f )2 + VmVf (Em2ν f + E f 2ν m)EmE f (EmVm + E f Vf )

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥

σ 22∇J

+

E f Em (Vm2 + Vf

2) − VmVf (E f ν m −

Emν f )2 + VmVf (Em2 + E f 2)EmE f (EmVm + E f Vf )

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥

σ 33∇J

(33)

D32 =1+ ν m

Em

⎛

⎝ ⎜

⎞

⎠ ⎟ Vm +

1+ ν f

E f

⎛

⎝ ⎜

⎞

⎠ ⎟ Vf

⎡

⎣ ⎢

⎤

⎦ ⎥ σ 32

∇J (34)

D31 = 0.5

1+ ν m

Em

⎛

⎝ ⎜

⎞

⎠ ⎟ Vm +

1+ ν f

E f

⎛

⎝ ⎜

⎞

⎠ ⎟ Vf

⎡

⎣ ⎢

⎤

⎦ ⎥ +

1Em

1+ ν m

⎛

⎝ ⎜

⎞

⎠ ⎟ Vm +

E f

1+ ν f

⎛

⎝ ⎜

⎞

⎠ ⎟ Vf

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

σ 31∇J (35)

D21 = 0.5

1+ ν m

Em

⎛

⎝ ⎜

⎞

⎠ ⎟ Vm +

1+ ν f

E f

⎛

⎝ ⎜

⎞

⎠ ⎟ Vf

⎡

⎣ ⎢

⎤

⎦ ⎥ +

1Em

1+ ν m

⎛

⎝ ⎜

⎞

⎠ ⎟ Vm +

E f

1+ ν f

⎛

⎝ ⎜

⎞

⎠ ⎟ Vf

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

σ 21∇J (36)

From the constitutive equations relating the rate of

deformations to Jaumann stress rates, the compliance matrix and hence the stiffness matrix can be suitably obtained. From the Jaumann rates, the Cauchy stresses can be solved for from equation 1. APPENDIX B

This appendix contains the relations for the rate of deformation to the Cauchy stresses and Jaumann stress rates for the fluid solid TI material (Eqs. (18) – (26)) where the coefficients K1, K2, K3, M1, M2 are


K1 =−λ*

4μ *2+ 6λ*μ*

, K2 =λ* + μ *

2μ *2+ 3λ*μ*

, K3 =−μ *

2μ *2+ 3λ*μ*

M 1 =−λ

4μ 2 + 6λμ, M 2 =

λ + μ4μ 2 + 6λμ

σ 11m =

−P(1− 2λ*K3) + λ* + 2μ*( )D11 +

λ* (K1 + K2)(σ 22 +σ 33)

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

1− 2λ*K1

(37)

σ 11f ∇J =

λ + 2μ( )D11 + λ (M 1 + M 2)(σ 22∇J +σ 33

∇J )1− 2λM 1

(38)

Using Eq. (1), Eq. (38) can be solved for σ 11f . Using

the stresses from above in Eq. (18) gives the stress-strain relation in direction 1-1.

σ 22∇J = k21D11 + k22D22 + k23D33 + c22σ 22 + c23σ 33 + c24 P (39)

σ 33∇J = k21D11 + k23D22 + k22D33 + c23σ 22 + c22σ 33 + c24 P (40)

where the material coefficients k21, k22, k23, c21, c22, c23 are

k11 = Vm

λ* + 2μ*

1− 2λ*K1

; k12 = Vf

λ + 2μ1− 2λM 1

c12 = Vm

λ* (K1 + K2)1− 2λ*K1

+ Vf

λ (M 1 + M 2)1− 2λM 1

c13 = Vm

λ* (K1 + K2)1− 2λ*K1

+ Vf

λ (M 1 + M 2)1− 2λM 1

c14 = −Vm

1− 2λ*K3

1− 2λ*K1

k21 =−a2 (a5 − a6)

(a5 − a6)2 ; k22 =

a5

(a5 − a6)2

k23 =−a6

(a5 − a6)2 ; c22 =

−(a3a5 − a4 a6)(a5 − a6)

2

c23 =−(a4 a5 − a3a6)

(a5 − a6)2 ; c24 =

a1 (a5 − a6)(a5 − a6)

2

(41)

a1 = Vm

(K1 − K3)1− 2λ*K1

; a2 = Vm

K1 (λ* + 2μ *)1− 2λ*K1

+ Vf

M 1 (λ + 2μ)1− 2λM 1

a3 = Vm

K1 + K1λ* (K2 − K1)

1− 2λ*K1

; a4 = Vm

K2 + K1λ* (K1 − K2 )

1− 2λ*K1

a5 = Vf

M 1 + M 1λ (M 2 − M 1)1− 2λM 1

a6 = Vf

M 2 + M 1λ (M 1 − M 2 )1− 2λM 1

The Jaumann stress rates σ 22

∇J , σ 33∇J can be expressed

in terms of the Cauchy stresses using equation 1. By substituting the appropriate terms for the Jaumann rates in terms of Cauchy stresses, Eqs. (39), (40) can be solved simultaneously using forward difference, time-implicit

finite difference scheme to obtain the Cauchy stresses at the current time level. The pressure P in the fluid is calculated using the bulk modulus of the fluid and the average of the strains sustained by the fluid part of the composite material during deformation. This can be expressed in the form of equation as,

P =−Km

3(ε11 +ε22

m +ε33m ) (42)

where Km is the bulk modulus of the fluid. For the calculation of stress in direction 1-2, the strain rates are averaged from the two directions (1-2, 2-1) and then the terms are rearranged to get the stress rate in direction 1-2 which is solved using time implicit method to get the stress in direction 1-2. The steps are shown in Eqs. (43) – (46).

D12 = Vm

σ 12

2μ* + Vf

σ 12∇J

2μ (43)

σ 21 = Vmσ 21m + Vfσ 21

f (44) σ 21

m = Vm 2μ*D21 (45) σ 21

f ∇J = 2μD21 (46)

Solve Eq. (46) using the relationship in Eq. (1) to get σ 21

f which can then be substituted into Eq. (44), along with Eq. (45). The resultant equation is averaged with Eq. (44) to obtain the net resultant stress or strain rate in direction 1-2. A similar equation can be obtained for the behavior in direction 3-1. For the direction 3-2, the stress is obtained by solving Eq. (47) using forward difference, time-implicit finite difference scheme.

σ 32∇J =

2μVf

D32 −VmμVmμ* σ 32 (47)


THE RESEARCH OF NEW APPROACHES TO COUPLING WINDAGE PROBLEM BASED ON PRESSURE EXPERIMENTS AND ANALYSIS

Tao Jiang Department of Mechanical Engineering

University of Cincinnati Cincinnati, OH, USA

ABSTRACT

According to OSHA requirements, all couplings with flexible elements which are mounted on steel shafts must be surrounded by a guard. For turbo machinery applications, this guard is typically an oil-tight enclosure.

Increasingly, turbo machinery users are setting upper limits on coupling enclosure temperatures. This is done mainly to protect personnel, by ensuring that the bearing oil intentionally drained through the coupling guard never exceeds the safety temperature, and also to make sure that coupling accessories (e.g. torque meter) can still work normally and safely at this level of temperature.

In some cases, the formula and programmed software are not necessarily useful in predicting an accurate temperature rise because most of the research was done under a laboratory environment based on limited configurations. Couplings are usually not considered as a part of the motor train for temperature measurement, which also serves as the reason there is not a direct correlation with “shop” situations.

This paper demonstrates in depth the reasons for pressure and temperature problems and reviews the conventional method of temperature calculation. Three approaches to solve windage problems will be compared and analyzed, leading to the conclusion that newly developed device can balance more than an 80% pressure differential at 4500 rpm. Some effective ways of accelerating enclosure heat dissipation will also be discussed. INTRODUCTION Windage Generation

Generally speaking, coupling windage problems can be divided into two aspects, temperature problems and pressure problems. When couplings rotate at high speeds in oil-tight enclosures they shear the air, and this results in a significant heat generation which is mainly dependent on the amount of air shearing and enclosure heat dissipation capability.

Another problem is the air pressure difference between the shaft vicinity and coupling diameter, causing oil to be absorbed in the guard. If it gets heated, this oil will be burned black and be detrimental to the whole train. Therefore, when using flexible element couplings, coupling guards can not be the simple oil-tight enclosures used in the past, but must now be configured with aerodynamics in mind.

Figure 1: Shear Effect of Coupling

The temperature problem, also regarded as the

cylinder effect in some cases, is produced by rotating a tube in a stationary guard. As figure 1 illustrates, the air in the guard is contained in a donut-like-shaped guard with the inner diameter equal to the coupling diameter and the outer diameter equal to the guard diameter. When the tube is spinning at high speed for a while, air near the tube will rotate with the same velocity while the outer part of the air may stick to the stationary guard. There will then be great friction between adjacent layers of air, since they have relative movements and this could generate massive heat.

The pressure problem, also known as the disk effect, is produced by coupling sides similar to rotating plates. Air is centrifuged from the smaller diameter to the larger, like a very inefficient centrifugal compressor (Figure 2).


The air turbulence is mainly due to the friction between the air and the face of the discs, and is also generated by the heads of bolts and nuts connecting the various components of the coupling. Consequently the oil in the equipment could be sucked out into the enclosure. Surrounded by heated air turbulence, the oil is apt to become mist and even burn out. A well designed coupling incorporates shrouds over bolt heads and nuts, particularly those which are of large diameters. According to Zimmermann and Firsching’s research [1], fully covered bolts, or arrangements with maximal number of bolts, are the only low loss configurations tested and should be used wherever possible. As a consequence, the generated heat and pressure differentials can be reduced, and this has a positive effect on the disc life.

Figure 2: Air Turbulence in Enclosure

δ = 27.0)(terplingDiameMaximumCoureanDiameteEnclosureM (2)

urfaceAreaEnclosureSactorrfaceAreaFCouplingSu

=α (3)

=β Coupling Circulation Factor

=2.0

terplingDiameMinimumCouterplingDiameMaximumCou (4)

=γ Air Flow Factor = 0.6

S = Application Speed (RPM) The coupling surface area factor (Κ ) used to determine α can be evaluated with the formula:

∑=

=Κc

iii ld

1

8.2 * (5)

where:Κ = Coupling Surface Area

c = Number of coupling cylindrical surfaces il , =id Length, diameter of cylinder i

The enclosure surface area is:

∑=

=Φn

iii LD

1* (6)

where: =Φ Enclosure surface area

Temperature Prediction When a high speed drive is retrofitted from an oil

lubricated coupling to a dry coupling, the temperature of the coupling can be estimated. However, there is no generally agreed upon method of predicting coupling temperature. The following updated summary of formulae is based on acquired data from actual field cases [2]. We first pay attention to the windage problem of dry couplings, which means there is no oil cooling in the enclosure.

Coupling Temperature without Cooling

2*)

1000(***

8.1sA

cTTST +

= δγβα (1)

Where cT = Coupling Temperature; AT = Ambient

Temperature; sT = Shaft Temperature

n= Number of enclosure cylindrical surfaces ,, LengthDL ii = diameter of cylinder i The enclosure diameter to be used in formula (2) is given by:

∑

∑

=

== n

i i

i

n

ii

en

DL

Ld

1

1 (7)

Where: end = Enclosure mean area

Temperature of the Enclosure

The temperature of the enclosure (Te) can be evaluated with the formula:

10)85.0*( += ce TT (8) Where: Tc= Coupling temperature, obtained from equations above.


EXPERIMENTAL OBSERVATION The solutions to coupling windage problems

typically have three aspects. The first two methods are special enclosure design and baffle plate employment, which has been put forward earlier [3] and the third one, an Anti-Windage attaching device. The third one is innovatively designed and developed. Here we will take a quick review of the first two methods, redo parts of the experiment and then offer a brief introduction to the WDD --- Windage-Defending Device. A succession of tests on this newly developed method will also be conducted and followed by elementary analysis. Enclosure Design

The author and his helper designed the enclosure with a special “mailbox” shape for the diaphragm, gear and disc pack couplings, enclosed on three sides with an open bottom. First, it was designed to prevent high coupling temperature during operation, which would cause electronic instrumentation to operate erratically. However, the high temperature problem still existed. So, additional tests were conducted over the last three months with the speed ranging from 1000 to 7000 rpm. The results and calculated temperatures, for a diaphragm coupling, which was tested in the research center, are shown in figure3. A single diaphragm coupling with 0.5 meters diameter was tested sequentially with a 0.6-meter-diameter enclosure and another 0.8 meters.

Figure 3: Test data on diaphragm coupling

The Bendix method [3], which was published in

1982 by Bendix Corporation for the calculation of enclosure temperature without any cooling[4] (figure 4), differs from the data obtained here. It was found that the enclosure’s diameter has a larger influence than the one resulting from the Bendix formula.

Figure 4: Bendix Temperature Calculation

According to the test data, for the same coupling

size and speed, the temperature values above ambient are higher than those calculated from the Bendix Method. High temperature was eventually reduced by drilling holes throughout the guard surface to create more air current and cooling effects.

Figure 5: “Mailbox” coupling with air holes

However, though the temperature problem was

solved temporarily, it was found that the surface area of this enclosure would not be enough to permit sufficient heat transfer to the outer environment to maintain acceptable coupling temperatures. For most of the time, coupling design is combined with the usage of baffle plates to achieve ideal pressure and temperature effects.


Baffle Plates

Baffle plates are widely used in the coupling industry, mainly to serve one purpose: the prevention of oil spills. As was mentioned above, oil could be sucked into an enclosure due to the vacuum effect. This happens because the dry couplings have larger outer diameters and thus generate more airflow and pressure differentials. The baffle plate is set up just in front of the seal to alternate the air path, which can increase pressure at the seal along the specific plate clearances, thereby reducing the pressure gap across the seal. (Figure 6).

Figure 6: Baffle plate setup

Several tests of the baffle plates were dedicated to

the study of pressure distribution in this critical seal area shown above [4]. The measurements were performed on a disc coupling with high speed. The air pressure at the shaft surface close to the seal spot was measured by pitot tubes. The pressure of the original configuration was also measured and recorded as a control. The result is shown in figure 7.

-8

-7

-6

-5

-4

-3

-2

-1

01

Figure 7: Actual pressure in different conditions

As was described previously, fully covered bolts

are the only low loss configurations tested and should be used wherever possible; so air pressure was also measured with and without shrouded bolts. The four bars in figure 7 are (from left to right) : original configuration, with shrouded bolts but no baffle plate, with baffle plate but no shrouded bolts and with both shrouded bolts and baffle plate.

The results show the employment of a baffle plate greatly reduced the negative pressure around shaft surface, with a more than 50% decrease. This effect would be even more apparent if anti-windage discs with shrouded bolts were also used. However, as we can see from figure 7, the air pressure, though very close to ambient pressure, is still in the negative realm. This means the baffle plate does alleviate the problem but does not eradicate it. Furthermore, this fix also reduced the air circulation, and hot guards sometimes resulted.

In the next part, a new method which may totally solve the problem will be introduced and relevant tests will also be analyzed. EXPERIMENTAL RESULTS Method Introduction

The reason oil is sucked into the enclosure is the existence of an air pressure differential, or the air pressure at the coupling outer diameter far exceeds the pressure around shaft. So, it would be of great help if there existed something which could “convey” part of the higher pressure to the lower spot and thus reduce the pressure difference. On top of that, the pressure at the shaft surface must be positive, larger than the ambient pressure, or else it would be meaningless because the baffle plate would have already attained extremely low negative pressure. The new design is called “WDD”, or windage—defense Device. It is a thin hollow plate attached to one end of coupling (Figure 8). The two arrows show the air entry and exit. The inner structure of the device is illustrated in Figure 9.

Figure 8: WDP attaches on the right end

In

ches

of w

ater


Figure 9: Inner Structure of the device

Figure 9 shows the basic structure and working

principle of the device. The arrow “C” means the rotating direction, and spots “A” and “B” show the air flow entry and exit. When running with the coupling, the spoke ends (A) capture plenty of air at high pressure and with the help of the spoke curvature and chamfer at the inner diameter (B), transfer them to the shaft surface. There is a small clearance between device inner diameter and shaft diameter to let the air exit through. This amount of axial air flows against the seal, increasing the pressure across it and preventing the oil from being sucked in. If looked at from the side, the whole setup will be like figure 10.

Figure 10: Side view of coupling with device (The curved arrow shows airflow direction)

Test Procedure

From earlier conclusions [5], the vacuum effect will be most obvious when the coupling is a reduced moment type, with an outer diameter at least 0.4 meters and spining at the rate of more than 10,000 rpm.

Although this operating condition is not critical for large-scale high performance disc couplings, there still exist a number of uncertainties if the specially-shaped device is mounted aside. These uncertainties include the balance of device, the radial or tangential growth of the device under centrifugal force, the thermal effect it may generate and so forth. Even small problems can eventually result in a catastrophic failure and be an extreme threat to personnel. So before beginning full scale testing, we simply came up with a model to test whether this idea would work or not.

Figure 11: General Setup

The test requires several main components: an enclosure, a coupling hub, the device, a shaft, a motor and bearings. As illustrated in figure 11, the bearings are set firmly onto a horizontal base, with a long shaft penetrating through. A V-belt rounds the shaft at one end and connects to a Leeson Variable Speed Motor under the base. A coupling hub is installed on the shaft in the enclosure, and the device is mounted and attached just beside it (figure 8). The hub and device share the same outer diameter at 0.15 meter and the enclosure diameter is 0.24 meter. Five small holes with the diameter 0.008 meter each are drilled along the radius of the guard panel on the side. The first hole is drilled just at the shaft and the other four are drilled with a constant distance of 0.03 meter between each other. After the motor turns on, a hand tachometer will be used to measure the rotating speed of the shaft and the probes of pitot tube will poke through these holes and measure the air pressure at different spots from shaft down to the coupling outer diameter. When the probe pokes into one certain spot, the other four should be covered by caps to avoid uncertainties and achieve more accuracy. The test model is shown in figure 12.


Figure 12: Testing model

As a control, a test without the device will be

conducted first and then data will be compared with new data with the device to check to what degree the device influences the coupling performance. As for the design of the device, the author came up with two choices. The first design is shown in figure 9. The second design is in figure 8 and the inner structure is listed in figure 13.

Figure 13: Inner structure of design 2

The most essential difference between the two

designs is the air flow path, or how the air is captured into the device and what path it follows to the exit. The first one, with several curved spokes, hopefully can achieve better balance than the second spiral-path design because it is still a circular shape. (The spiral design has some parts protruding out of the base circle.) For the same reason, the amount of air captured in the device may be much less for the first design because the two plates holding the spokes would be more inclined towards pushing air away than absorbing air in, thus reversely aggravating the pressure problem. It may be easier for the second design to capture more air in, but since the path is much longer and more complicated than the spokes, it would be more difficult to release the air axially.

Since they both have pros and cons, the only way to determine the best design is to put them to the test.

The tests related to air pressure were conducted over the last 2 weeks. Tests were performed, with and without the device, at speeds varying between 1000 and 5000 rpm. The data presented herein are based on the test results obtained. This data shows the influence the device brought. The author set up the running speed at 3800 rpm and monitored the air pressure under this speed at five different spots. For comparison, they are arranged and shown in table 1.

Table 1: Air pressure comparison (1)

Hole number

1 2 3 4 5

Without device (psi)

-0.007 -0.004 0 0.001 0.005

With device (psi)

-0.009 -0.004 0.022 0.04 0.05

The values shown in the table are the air pressure

difference between the certain spot and the ambient pressure (around 14.7 psi). The first row lists the numbers for the five measuring spots, the second row shows the pressure difference without the device and the third row with the device. It is apparent that the employment of the first design made the situation worse. The pressure at the shaft was lowered by 0.002 psi and, the biggest problem, pressure at the outer diameter was raised by 10 times, from 0.005 psi above ambient pressure to 0.05 psi. This is the opposite scenario of what we were looking for. With the pressure difference increased by five times from 0.012 psi to 0.059 psi, the first design was proclaimed a failure.

From trouble-shooting and analysis, the author

found that the outer diameter part of the device generates much more pressure than the inner part. So it might be the existence of two plates besides the spokes is the source of failure. Actually, what happened while running was that the contour of the plates pushed the air away from the device before it could be captured by the curved spokes. A small modification was done on the device, cutting part of the plates off at one side and leaving the ends of the spokes visible (Figure 14). The test was conducted again following the same process and the new results are listed in table 2.


Figure 14: Modified spoke design


Hole number

1 2 3 4 5

Without device (psi)

-0.007 -0.004 0 0.001 0.005

With Modified device (psi)

-0.007 -0.001 0.006 0.016 0.028

Clearly the pressure gap this time was narrowed,

from 0.059 psi of the original design to 0.035 psi. However, it still widened the gap dramatically from 0.012 psi and made the vacuum effect even more powerful. In order to increase the shaft pressure to close or above ambient pressure and largely reduce pressure at the outer diameter, the second spiral design is the only possibility. After following the same test process, we obtained this data:


Values recorded are all average values because of

measuring fluctuation. The pressure at the outer diameter decreased significantly, which also raised the pressure at the shaft. Compared with the former two designs, the spiral device could be a candidate method to pressure equilibrium, though outer pressure was still infinitesimally bigger than inner pressure.

To make the test on the spiral device closer to actual working conditions and the conclusion more convincing, the author, with the help of technician Rick Kuchera, made a full scale model out of cast aluminum (figure 15).

Figure 15: Full-scale spiral device

The total air input area was 26.860 centimeter

square and total output area was 20.253 centimeter square. Since this device is hand-made and lacked strict balance, the power was limited at 2230 watts and rotation speeds under 1800 rpm. Air pressure was measured by differential manometer and air velocity was also measured, by an “Omega” HH-F10 air speed indicator. The results will be in the next section.

RESULT ANALYSIS

The tests on the model in the laboratory ranged from speeds of 1000 rpm to 5000 rpm. For the sake of simplicity, three speed points, 2700, 3800, and 4500 rpm were picked out and comparison charts were made here (Figure 16, 17, 18):

Hole number 1 2 3 4 5 Without device (psi)

-0.007 -0.004 0 0.001 0.005

With Spiral device (psi)

-0.001 0 0.001 0.001 0.002


Figure 16,17,18: Pressure difference comparison

Firstly, it was apparent that the spoke design,

whether the original one or the modified one, can not ameliorate the vacuum performance. Instead, it made the situation worse. While both red and yellow lines were mostly above the deep blue line (Figure 17, 18, 19), the light blue line which represented the spiral design was more horizontal than the other three. This simply meant that the air pressure gradient was smaller than before by a large amount. The spiral design improved the coupling performance not only quantitatively but also qualitatively.

Secondly, the spiral device had more influence when the rotating speed was higher. For figure 17, air pressure increased 0.003 psi along the radius after the device was mounted on. Since the original air pressure difference was 0.008 psi, the device successfully balanced 62.5% of pressure difference; for figure 18, when rotating speed was raised up to 3500 rpm, the original pressure difference inched to 0.012 psi.

However, the device still kept the pressure

difference within 0.003 psi, which meant it balanced 75% of high pressure; a similar phenomenon also happened at the speed of 4500 rpm, the spiral device suppressed the pressure difference under 0.005 psi when its original peer was already 0.026 psi. 80.8% of the pressure difference was balanced.

As was mentioned in last section, the author also brought the spiral design test into full scale. With the help of an air speed indicator, the air speed and direction in the device was detected and recorded. A 3-D speed chart was obtained using simulation software (figure19):

Figure 19: Air speed and direction simulation

At least two facts can be confirmed from figure

20. The first one is that air really goes along the path it is supposed to. Air is hooked into the device and then released axially along the shaft surface.

Air direction arrows are gathered and aligned at the entry and evenly distributed at the circular exit, which is indicated by the intensive yellow arrows and sparse green arrows. The second fact, which can be seen in reference to the color bar at the upper left corner, is that the air at the entry flows quicker on average than the one near the center. This shows the device is better at decreasing high pressure at the outer diameter than in increasing negative pressure at shaft. From figures 17 through 19, a problem at the outer part is much more severe than at the shaft and this property of the spiral device is perfectly suited for solving this problem.

A pressure simulation was also made and the results are shown in figure 20. This figure roughly shows the air pressure in and around the guard. The part in the black circle was where we focused attention because this was where the oil got sucked in and where the device was attached.

Note that the ambient pressure is 14.7 psi and pressure at the spot near the shaft and outside the enclosure is 14.73 psi, a little bit larger than 14.7 and the spot near shaft, but inside enclosure it has the pressure as 14.75 psi. The pressure gradient here can generate a thrust against the equipment oil seal when in operation, preventing oil from leaking into the coupling chamber.


Figure 20: Air pressure simulation

One fact that must be emphasized here is this result

may not be the optimal one. The author and his helper made and tested the full-scale device from their experience, and not all the possibilities were taken into consideration and not all the parameters were supervised strictly. Among all of them, the ratio between entry area and exit area might be the most important one because it could directly determine the air velocity. Based on the Bernolli Equation we believe that air velocity can greatly affect the pressure. Besides this ratio, other parameters like the hood size and paddle size should also be taken into account. Other modifications like the removal of the rear wall or the undercut of the rear wall are also strongly suggested because any tiny change may cause tremendous effect. This is what the author is researching now and further results will be published later.

DISCUSSION Heat Dissipation

As mentioned above, both the baffle plate method and the windage-defending device can allay the vacuum problem and, at the same time, result in more temperature problems. A baffle plate can successfully block the circulation of airflow and reduce the pressure difference, but it also decelerates the heat being transferred out of the enclosure. The windage-defending device also generates more air friction when it guides airflow going along the path [6]. In order to alleviate this type of problem and achieve a better equilibrium between these two problems, we need to discuss some effective heat dissipation methods.

The heat generated within the enclosure is normally dissipated by its outside surface into the surrounding air. To increase the dissipation, common engineering sense should be used:

• Increase the surface area, either by making the

enclosure large or by adding radiator fins • Allow air circulation over the enclosure • Shade the enclosure from exposure to the sun or from

other hot components, such as turbine exhausts and steam lines

If the enclosure cannot be modified, and its

temperature is unacceptably high, spraying the coupling or the inside of the surface of the enclosure will provide more help in dissipating heat. To calculate the temperature drop that can be achieved by using oil cooling, the following information is needed:

• Cooling oil flow rate • Cooling oil temperature • Sprayed surface • Enclosure’s original temperature

If there is any kind of air replacement while the coupling is running, the enclosure pressure can be significantly reduced and the most convenient way to do that is to replace the enclosure air with cold air outside. However, the oil mist can also escape out of the enclosure with the air venting out. This problem can be eliminated through proper design, mostly according to these two points:

• Sizing the air ports • Separating the oil from the air

The exhaust port should be tangential to the outer diameter of the guard and set along with the centrifugal force direction because this can let the air go out without any kind of bending or “shunt”. Ports can be made in any shape but from past experience a cylinder port is the best one to use. The larger the ratio between exhaust port area and guard area, the lower the enclosure temperature will be. On the other hand, an excessively large port could have trouble in separating oil from air. This will also be discussed later.

If air replacement is employed, intake and exhaust ports must be carefully placed in case that there would be a “stuffy zone” in the chamber where no circulation exists and temperature keeps rising. In addition, the total intake port area should be half the total exhaust area to make sure that the air leaves through the exhaust and not the intake.

To prevent the oil in the enclosure from mixing with air, the centrifugal force may be one of the best helps to turn to. Since oil is by far heavier than air, the centrifugal force of the coupling can exhaust the oil out through port with only a little amount of air if well controlled.

A baffle plate is strongly recommended in that case. It should be installed, over the exit port, with the direction along with the exhaust port. This baffle plate can coerce the oil out of the enclosure without rotating more than one revolution before exiting. If the port area is too big, more air could be guided out of the enclosure with oil and oil mist could appear again. That’s why the exhaust port area must be accurately designed.


CONCLUSIONS

Based on the present study on the coupling enclosure used for power transmission, the following conclusions can be made.

• Windage problems can be divided into two types ---

pressure problems and temperature problems. • Without any methods employed, enclosure

temperature increases with the increase of coupling speed.

• Under the same speed, enclosure size has a large influence on the temperature.

• In some real cases, a baffle plate can be of great help, but it still doesn’t solve the pressure problem once and for all.

• A spoke design device actually made the problem worse, but a spiral design could provide improvement, though further study is needed.

• Typically, methods which relieve the pressure problem can cause the temperature problem to become worse. Good heat dissipation design should be considered.

REFERENCES [1].H.Zimmermann, A.Firsching, 1986, “Friction Losses and Flow Distribution for Rotating Disks with Shielded and Protruding Bolts”, ASME [2].Calistrat, M. M. and Munyon, R. E. 1985 ,”Design of Coupling Enclosures”, Proceedings of the Fourteenth Turbo machinery Symposium, The Turbo machinery Laboratory, Texas A&M University, College Station, Texas [3].Donald Carter, Martin Garvey and Joseph P. Corcoran. 1990 “The Baffling and Temperature Prediction of Coupling Enclosures”, ASME [4].Bendix Corporation, 1982 Fluid Power Division,, Catalog 67U-6-8211A [5].Gibbons. C.B.1983,“The Use of Diaphragm Couplings in Turbo machinery”, Professional Development Seminar, South Texas Section, ASME. [6].Mancuso, J.R., 1986 “Couplings and Joints”, New York: Marcel Dekker, Inc.


ASME Early Career Technical Journal 2010 ASME Early Career Technical Conference, ASME ECTC

October 1 – 2, Atlanta, Georgia USA

THE FAILURE CHARACTERIZATION OF MECHANICALLY FASTENED COMPOSITE JOINTS

O. Aluko Brian S. Moore Department of Computer Sc, Engrg and Physics US Department of Transportation University of Michigan-Flint Office of Hazardous Materials Tech., PHH-22 Flint, Michigan, USA Washington DC, USA

ABSTRACT An analytic model was developed to predict failure strength of pin loaded composite joints using two dimensional analysis and characteristic curve model. The characteristic length in compression and tension that defines the characteristic curve was obtained by stress analysis associated with no bearing and tensile tests on laminates with and without holes. The available experimental data in literature for different laminated composites was used to evaluate the joints strength. The method of analysis was confirmed to be adequate by the good agreement between the experimental data and the analytically obtained joint strength for all joint configurations evaluated. Additionally, the mode that characterizes the failure condition depends on the orientations of the laminas that constitute the laminate used for joint construction.

INTRODUCTION Composite materials have been extensively used in

aerospace structures, space vehicles and many other engineering structures in recent years due to their high specific strength and stiffness. However, the joints often required for fitting composite parts usually constitute a region of weakness which can lead to premature failure of structures. Based on the fact that the structural efficiency of a composite structure depends on its joints rather than the component members, the stresses associated with mechanical joining of composites have received much attention in recent years [1-17]. Mechanical joining includes bolted, riveted and pinned joints; these joints are prone to high stress concentration which occurs in the vicinity of the hole and is often the source of premature failure in composite structures containing joints. Therefore, adequate design of a mechanical joint requires correct determination of the stresses that can be used to evaluate load bearing capacity of the joint. De Jong [3] analyzed the extent of slip and no-slip regions within the contact boundary between the hole and the plate and found that the extent of no-slip regions is insignificantly small and in some

cases can be approximated to be zero. Wang et al.[4] investigated the possibility of changing the bolt and hole shapes from circular to elliptical in order to reduce bearing stress and thereby increase the joint strength, especially when it is not possible to increase the hole diameter due to insufficient edge distance. Their analysis showed that the bearing stress at the joint hole can be significantly reduced by changing the bolt shape to elliptical.

Chang et al. developed [11] a characteristic length method for the failure analysis of composite joints and it is still currently used. Kweon et al. [15] utilized finite element analysis with a no-bearing test method to determine the characteristic length used to evaluate the strength of composite joints and found the method to be very efficient in analyzing different joint configurations. In this investigation, an analytical method for determining the strengths of composite joints is presented based on Lekhnitskii’s [5] complex stress functions which satisfy the displacement boundary conditions along the hole contour. This study involves the influence of joint configurations on the strength and failure modes of the joint when no friction is considered. Further, characteristic length in tension and compression would be determined by stress analysis related to the results of no-bearing and tensile tests on notched and unnotched laminates.

CHARACTERICTIC LENGTHS METHOD Several strength prediction methods for composite joints have been proposed, including stress concentration coefficient, damage zone model based on fracture energy and progressive failure analysis, and failure area index (FAI). However, one of the most common and efficient methods of predicting the strength is the characteristic length method. This method was proposed by Whiney and Nuismer [1, 7], and it has been further developed by Chang et al. [11]. For this method, both the characteristics length in tension, Rt and compression, Rc must be determined by stress analysis associated with the results of bearing and tensile tests on notched and


unnotched plates before employing an appropriate failure theory along the characteristic curve, rc as shown in Figure 1. This characteristic curve which was first proposed by Chang et al. [11] can be expressed as

cos (1)

where x and y are equal to rccosθ and rcsinθ respectively.

By definition, the characteristic length (in tension or compression) is the distance from the edge of the hole boundary over which the plate must be critically stressed to initiate a flaw sufficient to cause failure.

Figure 1. SCHEMATIC DIAGRAM FOR THE CHARACTERISTIC CURVE

This analytical method employs an approach that entails the determination of stress distributions within the plate and at the pin-plate interface. In order to demonstrate the practical application of the characteristic length method, Whitney-Nuismer’s point stress criterion [7] and the Yamada failure criterion [12] are used to predict the joint’s strength.

In this study, a CPL1 plate [15] with layers [±453/90/±452/04/90/04//±452/90/±453] was utilized. The notation “±45” is used to represent a plain weave graphite/epoxy layer and the lamina thicknesses of the unidirectional and woven fabric layers are 0.114 mm and 0.198 mm, respectively. The unidirectional layers are from the USN125 graphite/epoxy prepreg by Hankook Fiber Glass. The DMS2288 graphite/epoxy woven fabrics are by Sunkyong. Another typical example utilized in this analysis is AS4/3502 graphite/epoxy laminate having the stacking sequence [(0/90/90/0)]s; lamina thickness h, of 0.127 mm [10]. A typical example of the joint configuration of this pin loaded plate is shown in Figure 2. The material properties of the laminates and joint configurations are as given in Tables 1-3. It should be noted that the ‘rs’ in joint identification

codes, WDrs shown in Tables 2 and 3 can be interpreted as w/d=rs.

Figure 2. THE CONFIGURATIONS OF JOINT

Table1. THE MATERIAL PROPERTIES OF COMPOSITE MATERIALS [10, 15]

Properties Type of Material USN 125 DMS

2288 AS43502-6

E1 (GPa) 131 65 139.97 E2 (GPa) 8.2 65 10.27 G12 (GPa) 4.5 3.6 5.72

υ12 0.281 0.058 0.3 Xc (MPa) 1400 692.9 1406.60 S12 (MPa) 70 65 102.05

Table 2. GEOMETRIES OF COMPOSITE JOINTS [15]

Joint ID

Hole-diameter, d

mm

Width, w (mm)

Edge-distance, e

mm

w/d e/d

WD20 9.53 19.00 13.40 2.0 1.4 WD25 9.53 23.80 13.40 2.5 1.4 WD28 9.53 26.80 13.40 2.8 1.4 WD35 9.53 33.40 13.40 3.5 1.4 WD40 9.53 38.00 13.40 4.0 1.4

COMPRESSIVE CHARACTERISTIC LENGTH As stated earlier, for the characteristic length method of predicting joint strength, compressive characteristic length is an important parameter that must be determined. In this analysis, the no-bearing test method introduced by Kweon et al. [15] is employed, using analytical techniques for the determination of compressive characteristic length.

θ

Characteristic curve

Rc

d Rt

rc

w

L

d

e


Table 3. GEOMETRIES OF COMPOSITE JOINTS

[10]

Joint ID Hole-diameter, d mm Width, w (mm) w/d WD31 15.5 47.5 3.1 WD33 7.62 25.4 3.3 WD40 6.35 25.4 4.0 WD50 2.54 12.7 5.0 This method utilizes any arbitrary load to compute the mean bearing stress. Within the context of this approach, the compressive characteristic length Rc, shown in Figure 3 for the point stress criterion [7] is the distance from the front hole-edge to a point where the local compressive stress exerted by the arbitrarily applied load is the same as the mean bearing stress, defined by

(2)

where, Fm, d and h are the arbitrary loads, diameter of hole and laminate thickness, respectively.

Figure 3. SCHEMATIC DIAGRAM OF THE

COMPRESSIVE CHARACTERISTIC LENGTH AND NORMAL STRESS DISTRIBUTION

PATTERN

In order to determine compressive characteristic length, the composite plate is considered to be homogeneous and semi-infinite with a circular hole loaded by a rigid pin that has the same diameter as the hole Figure 4. The load F applied in the x direction is assumed to cause a displacement of uo in its own direction and uo /c at points G1 and G2. It is assumed that the pin load is resisted by distributed load at infinity. The boundary conditions of this geometry can be expressed as [6]

and 0 θ 2 (3)

and 0 at 0 (4)

an between 2 2 (5)

where u and v are the displacements along the x and y axes, respectively.

Therefore, the displacements u and v along the hole can be expressed by the following trigonometric series,

cos2 cos4 (6)

sin2 sin4 (7)

where u1, u2, v1 and v2 are the unknowns to determine from boundary conditions.

The condition of stress at the free surface where x equals zero is

0 and 0 (8)

Similarly, for the case of a frictionless condition, the shear stress at the hole boundary vanishes, and then,

0 at /2 (9)

0 (10)

Substituting Equations 6 and 7 into Equations 3, 4 and 5, we obtain

12 ,

12 and 3 1

2 (11)

Figure 4. SCHEMATIC REPRESENTATION OF COMPRESSIVE LOADING The stress functions that satisfy the above displacement boundary conditions are

Aln 12

(12)

ln 12

(13)

Rc

σr=σmb

y

x

θ

G1 G2 F

Fm


where the coefficients A and B for a plate of arbitrary thickness h are given by the formulas

/

(14)

/ (15)

where bar implies conjugate and

and (16)

and (17)

/ 1,2

(18)

with aij,i,j =1,2,6 are the compliances, and μ1 and μ2 are the roots of the characteristic equation for an orthotropic plate defined by

2 0 (19)

where 1 and 2 are parallel and transverse to the loading direction respectively and r is the radius of the hole.

The stresses in polar co-ordinates are given by the expressions

2Re sin cos sincos (20a)

2Re sin cos cos sinsin cos cos sin (20b)

2Re sin cos sincos (20c)

The prime implies the derivative of the stress functions.

Substituting Equations 12 -18 into the expressions for radial and shear stress in Equation 20 yields

1 cos5 1 3

3 1 cos3 2

2 cos (21)

1 sin5 1

2 1 cos3

2 2

sin (22)

where the parameters k, n and g are the material constants of the plate expressed as

⁄ ⁄

2 ⁄ (23)

1 ⁄ ⁄

Using Equation 22 to satisfy the boundary conditions in Equations 9 and 10, we have

⁄ (24)

where

10 9 21 10

10 19 11 10

and

⁄ (25a)

where

10 9 21 10

20 28 52 48

12 40 28 5220 (25b)

The unknown coefficients ui and vi can be determined from Equations 11, 24 and 25; the results substituted into Equation 20a yield the values of the normal stress. In this approach, the normal stress distribution in Equation 20a was numerically evaluated by a symbolic computer code written in Mathematica. As mentioned before the point stress criterion, introduced by Whitney and Nuismer [7], was used to evaluate the compressive characteristic length.


TENSILE CHARACTERISTIC LENGTH As mentioned previously, joints evaluation by the characteristic length method also requires the determination of the tensile characteristic length. This involves the tensile test to failure of notched and unnotched laminate, since the tensile strength of notched laminate ( ) and unnotched laminate (σ0) must be known a priori. Figure 5 shows the stress distribution of an infinite plate with a through-to-thickness hole subjected to distributed tensile loads per unit area, p at infinity. The tensile stress would be highest at the side edges of the hole, and would decrease further away from the side edges of the hole as shown in this figure.

Figure 5. SCHEMATIC DIAGRAM OF THE TENSILE CHARACTERISTIC LENGTH AND

STRESS DISTRIBUTION PATTERN

In this analysis, the idea proposed by Konish and Whitney [16] is utilized for the approximate solution to the normal stress distribution σy (x,0) in an infinite orthotropic plate with an open hole loaded in tension. This solution can be expressed as

,1 5 7

(26)

where is the stress concentration factor expressed by the following

1 (27)

and

⁄ (28)

where Aij, i,j = 1,2,6 denote the effective laminate in-plane stiffness with 1 and 2 parallel and perpendicular to the loading direction, respectively.

The point stress criterion [7] can be applied to Equation 26 to compute the tensile characteristic length at which failure analysis of the joint should be evaluated. The criterion defines the tensile characteristic length as

the distance from the side-edge of the hole to a point where the tensile stress is the same as the strength of unnotched laminate. This can be expressed as

, 0 (29)

where,

; (30)

The criterion assumes that the stress at infinity which determines the characteristic length defines the strengths of the notched laminate. Substituting Equations 29 and 30 into 26, the ratio of unnotched to notched strength is obtained as

1 5 7 (31)

Therefore, value of tensile characteristic length can be determined if the data for tensile notched and unnotched laminate strength are known. The experimental values of these quantities as obtained in the literature are as shown in Tables 4 and 5. It should be noted that the unnotched failure strength (σ0) computed using classical lamination theory for CPL1 plate is 828 MPa.

Table 4. LOADS USED TO EVALUATE TENSILE CHARACTERISTIC LENGTHS FOR CPL1 PLATE

AS OBTAINED FROM [15]

Joint ID Tensile failure load, kN , MPa ⁄ WD20 24.3 788.45 1.051 WD25 33.8 731.13 1.1325 WD28 41.9 755.28 1.097 WD35 56.1 728.10 1.137 WD40 66.4 718.15 1.152

Table 5. LOADS USED TO EVALUATE TENSILE CHARACTERISTIC LENGTHS FOR AS43502

PLATE AS OBTAINED FROM [10]

Joint ID Tensile failure load, kN , MPa ⁄ WD31 18.92 581.88 1.678 WD33 8.90 492.45 1.980 WD40 11.87 613.04 1.59 WD50 7.71 746.67 1.305

JOINT STRENGTH AND FAILURE ANALYSIS The present study utilized a compact analytical technique that utilized the thickness of the plate as a parameter. The composite plate is considered to be homogeneous and infinite with a circular hole loaded by a rigid pin. In addition, the pin has the same diameter as the hole. The load Fr applied in the x direction is

Tensile stress distribution

Tensile

strength

Rt

p p


assumed to cause a displacement of co in its own direction and c1 at the ends of the contact boundary. It is assumed that the hole deforms into an ellipse under the action of the pin load Fr, which is also resisted by the distributed load at infinity as shown in Figure 6. For the linear case of zero clearance, contact between the plate and the pin spans through half of the hole’s circumference. Since the results from [3] have shown that the no-slip region within the contact boundary is zero degrees, and since this point is located at the point of symmetry, therefore the boundary conditions of the geometry shown in Figure 5 can be expressed as follows

Contact Region: 3π/2 ≤θ≤π/2

and 0 θ 0 (32)

and 0 at 32 (33)

cos sin (34)

where u and v are the displacements along the x and y axes, respectively.

In the case of the frictionless condition at the contact region within the hole boundary, the relationship can be expressed as

0 at /2 (35a)

0 (35b)

No-contact region: π/2 ≤θ≤3π/2

The condition for the no-contact surface can be expressed as

0 (36)

0 (37)

Similarly, using Equations 6 and 7 to also satisfy the above displacement boundary conditions, the unknown ui and vi can be expressed as

2 , 2 and

32 (38)

Therefore, the stress functions that satisfy the above displacement boundary conditions are

ln 14 3

(39)

ln 14 3

(40)

Additionally, the stresses in rectangular co-ordinates can be expressed as

2Re (41a)

2Re (41b)

2Re (41c)

The prescribed displacements co and c1 are obtained under the assumption of no-friction between the pin and plate, and these can be expressed as

9 10 21 10 /4 513 3 12 7 13 10

7 5 (42)

10 11 19 10 /4 513 3 12 7 13 10

7 5 (43)

Figure 6. SCHEMATIC REPRESENTATION OF COMPOSITE JOINT UNDER LOADING

The unknown coefficients ui and vi can be determined from Equations 38, 42 and 43 and have the results substituted into Equation 41 to yield the values of stresses. The stress results computed from the above analytical method along the characteristic curve shown in Figure 1 were used in failure analysis to predict the joint strength. Additionally, Yamada-Sun failure criterion was used to test condition for first ply failure at any point along the characteristic curve. This criterion can be expressed as [12]

q

y

x

θ

Fr


1 ⁄ 212⁄ 2 (44)

Where σ1 and τ12 are longitudinal compressive and shear stresses, Xc the ply longitudinal compressive strength and S the ply shear strength. In this model, failure is expected to occur when the value of e is either equal or greater than unity. The computer code that was used to test for the failure load was written in Mathematica. The failure of the joint can be characterized by three types of failure modes depending on the failure location, θf [11] namely:

0⁰≤θf≤15⁰: Bearing mode

30⁰≤ θf ≤60⁰: Shear-out mode

75⁰≤ θf ≤90⁰: Net-tension mode

Additionally, the concept of finite-width correction factor developed by Tan [9] was utilized to correct the infinite assumption of the plate. By definition, finite-width correction factor (FWC) is a scale factor which is applied to multiply the notched infinite plate solution to obtain the notched finite-plate result based on the assumption that the normal stress profile for a finite plate is identical to that for an infinite except for a FWC factor. This correction factor is given by the relationships

KT∞

KT

∞ (45)

where, σN and σN ∞ denote the notched strength for finite and infinite plate, respectively, and KT/KT

∞ is the FWC factor. KT and KT

∞ denote the stress concentration at the opening edge on the axis normal to the applied load for a finite plate and an infinite plate, respectively. This factor is expressed by the relationship [9]

KT∞

KT

22rW

2 2rW

4

2

2rW

6KT

∞ 3 12rW

2 (46)

RESULTS AND DISCUSSION Table 6 shows the computed characteristic lengths and failure loads as compared with the experimental results from CPL1 [15] for joints with various ratios of width-to diameter (w/d). As stated earlier, the Yamada-Sun failure criteria were used to evaluate the failure of the joints. Results from this table show that the predicted loads for different joint configurations are in good agreement with experimental data. It should be noted that this correlation shows that the present analytical model used to evaluate the strength proved adequate for the analysis of the joint strength. Also documented in Table 7 are the computed characteristic lengths and failure loads for AS43502 joints with different ratios of width-

to-diameter (w/d). However, there was no available experimental data for these joints. The prediction is assumed to be reasonable based on the present analytical method that has been found adequate for the evaluation of CPL1 joints. Tables 6 and 7 also showed the mode of failure for all the joint configurations analyzed in this study. In Tables 6 and 7, N-T (Net-tension), B (Bearing), B/S (Bearing-shear), and S (Shear-out) refer to mode of failure. As can be seen from these tables, the failure modes predicted by the present analysis are conservative with the change of joint geometry for both CPL1 and AS43502 joints. This might be caused by infinite plate assumption that was initially made in the analysis before using FWC factor. Nevertheless, the AS43502 joints that failed in shear regardless of joint dimensions showed the poor performance of cross-ply laminate under compressive loading. Further, comparison of results from CPL1 and AS43502 joints showed that the obtained mode of failure depend on the stacking sequence of the laminas that constitute the laminated joints. Even at higher width-to-diameter ratio for AS43502 joint, the mode of failure is by shear-out.

Table 6. CHARACTERISTIC LENGTHS, FAILURE LOADS AND MODES OF CPL1 JOINTS

Joint ID

Comp. Charact. length, Rc, m

Tensile Charact. length, Rt, m

Failure Load, KN

Failure Mode

Present

Exp

Present

Exp

WD20 0.00348 0.011 10.9 9.8 B/S N-T

WD25 0.00348 0.0054 10.9 10.1 B/S B

WD28 0.00348 0.0072 11.3 10.5 B/S B

WD35 0.00348 0.0053 11.3 10.5 B/S B

WD40 0.00348 0.0047 11.5 10.6 B/S B

Table 7. CHARACTERISTIC LENGTHS, FAILURE LOAD AND MODES OF AS43502 JOINTS

Joint ID

Compressive characteristic length, Rc m

Tensile characteristic length, Rt m

Failure load Present, KN

Failure mode Present

WD31 0.00348 0.00217 1.030 S

WD33 0.00165 0.000667 2.485 S

WD40 0.00142 0.001041 2.240 S

WD50 0.00057 0.000787 1.030 S


CONCLUSION Stress analysis was performed to predict the failure strength of pin loaded composite joints using Yamada-Sun failure criterion and on the characteristic curve model. The characteristic length in compression and tension that defines the characteristic curve was obtained by stress analysis associated with no-bearing and tensile tests on laminate with and without hole. The available experimental data in literature for different laminated composite joint configurations were used to evaluate the joints strength. The method of analysis was proven adequate by the good agreement between the experimental data and the analytically obtained joint strength evaluated. Additionally, the analysis was utilized to characterize the failure mode of joints. The failure mode was found to depend on orientations of the laminas that constitute the laminate used for the joint.

REFERENCES [1] Nuismer, R.J., 1979, “Application of the Average Stress Failure Criterion: Part II - Compression,” Journal of Composite Materials, 13, pp. 49-60. [2] Whitworth, H. A., Othieno, M., and Barton, O., 2003, “Failure analysis of Composite pin loaded Joints,” composite Structures, 59, pp. 261-266. [3] De Jong, T., 1982, “Stresses around pin-loaded holes in composite materials,”.Mechanics of composite materials recent advances. New York: Pergamon Press, pp. 339-353. [4] Wang, J.T., Lotts, C.G.,and Davis, D.D., 1993, “Analysis of bolt-loaded elliptical holes in laminated composite joints. Journal of reinforced plastics and composites,” 12, pp. 128-138. [5] Lekhnitskii, S.G., 1968, “Anisotropic Plates”, Translated from the 2nd Russian edition by Tsai, S.W. and Chevron,T., Gordon and Breach, London 1968. [6] Aluko, O., 2009, “An Analytical Method to Determine Compressive Characteristic Length,” ASME Early Career Journal, pp. 6.1-6.8. [7] Whitney, J.M., and Nuismer, R.J., 1974, “Stress Fracture Criteria for Laminated Composites Containing stress Concentrations,” Journal of Composite Materials,” 10, pp. 253-265. [8] Whitney, J.M., and Nuismer, R.J., 1975, “Uniaxial Failure of Composite Laminated Containing Stress Concentrations,” Fracture Mechanics of Composite, ASTM STP, pp. 117-142. [9]Tan, S.C., 1988, “Finite-width Correction Factor for Anisotropic Plates Containing Central Opening”, Journal of Composite Materials,” 22,, pp. 1080-1097. [10] Tan, S.C., 1987, “Laminated composites containing an opening elliptical II. Experiments and model modification,” J. of composite Materials, 21, pp. 949-968.

[11] Chang, F.K., and Scott, R.A., 1982, “Strength of mechanically Fastened Composite joints,” Journal of Composite Materials, 16, pp. 470-494. [12] Nahas, M.N., 1986, “Survey of Failure and Post-Failure Theories of Laminated Fiber-Reinforced Composites,” Journal of Composites Technology and Research, 8(4), pp.138-153. [13] Choi, J., and Chun, Y., 2003, “Failure Load Prediction of Mechanically Fastened Composite Joints,” Journal of Composite Materials, 37(24), pp. 2163-2177. [14] Zhang, K., and Ueng, C.E.S., 1984, “Stress Around a Pin Loaded Hole in Orthotropic Plates,” Journal of Composite Materials, 18, pp. 432-446. [15] Kweon, J.,Ahn, H., and Choi J., 2004, “A new Method to determine the Characteristic Lengths of Composite Joints without Testing,” Composite Structures, 66, pp. 305-315. [16]Konish, H.J., and Whitney, J.M., 1975, “Approximate Stresses in an Orthotropic Plate Containing a Circular Hole,” Journal of Composite Materials, 9, pp. 157-166. [17] Hart-Smith, L.J., 1980, “Mechanically Fastened Joints For Advanced composites phenomenological Considerations and Simple Analysis,” Fibrous Composite in Structural Design, Plenum Press, pp. 543-574.



October 1 – 2, Atlanta, Georgia USA

SOLVING COUPLED SYSTEMS OF DIFFERENTIAL EQUATIONS USING THE LENGTH FACTOR ARTIFICIAL NEURAL NETWORK METHOD

Kevin Stanley McFall

The Pennsylvania State University Lehigh Valley Campus Center Valley, PA, USA

ABSTRACT The length factor artificial neural network method for solving differential equations has previously been shown to successfully solve boundary value problems involving partial differential equations. This manuscript extends the method to solve coupled systems of partial differential equations, including accurate approximation of local Nusselt numbers in boundary layers and solving the Navier-Stokes equations for the entry length problem. With strengths including an explicit and continuous approximate solution, elimination of meshing concerns and simple implementation for nonlinear differential equations, this method is emerging as a viable alternative to traditional numerical techniques such as the finite element method. INTRODUCTION

Many problems in science and engineering involve differential equations (DEs) sufficiently complicated to require numerical techniques for approximating their solutions. Traditionally, the finite difference [1], finite element [2], and boundary element [3] methods have been employed to numerically solve DEs. Although powerful and widespread, these tools do have drawbacks including problematic discretization of the problem domain and complications in solving nonlinear DEs. Artificial neural networks (ANNs) have emerged as an alternative method for numerical solution of DEs [4-12]. Methods using ANNs generally avoid the drawbacks of traditional numerical techniques. For example, domain discretization often involves simple square grids where no special treatment is necessary for nonlinear DEs.

ANN methods begin with a trial approximate solution (TAS) continuous over the problem domain whose value is influenced by a number of ANN parameters initialized to random values. Those parameters are then optimized to most closely approximate a solution to the given DE equation. While most ANN methods follow this general strategy, they vary greatly in implementation. The length factor ANN method featured here was developed to be simple in order to improve accessibility to those less familiar with ANNs. The hallmark of this method is that the boundary conditions (BCs) associated with the DE are automatically satisfied during all stages of training the

ANN, including during initialization of network parameters with random values. Such an approach removes the BC constraint, allowing a simpler and more straightforward optimization stage compared with other ANN methods.

The length factor method for solving DEs has already been shown to successfully solve partial differential equations (PDEs) in two and three dimensions [8]. The main contribution of this manuscript is to expand the method to solve coupled systems of PDEs including the two-dimensional steady Navier-Stokes equations. The method is first demonstrated with a toy problem on an irregularly shaped domain, it produces an approximation for the local Nusselt number in the Blasius boundary layer, and it models the entrance region of flow between two infinite flat plates. FORM OF THE TRIAL APPROXIMATE SOLUTION

Consider the two-dimensional, second-order differential equation defined by

2 2 2

2 2, , , , , , 0fx y x yx y

x (1)

over the domain 2 with exact analytical solution

( ) x subject to the Dirichlet BCs g x x (2) for domain boundary given the position vector

2x

y

x (3)

For the length factor method, the TAS to the DE represented by Equation (1) is of the form ,t DA L N x x x x θ (4)


where the continuous function AD(x) is developed to satisfy the BCs such that

DA g x x (5) AD may take any value inside the domain. The length factor L quantifies a measure of distance from the domain boundary and satisfies

0 and 0 L L x x (6) The ANN output N(x,) depends of the spatial vector x as well as network parameters . Defined in such a manner, the TAS t(x) automatically satisfies the BCs in Equation (2) regardless of network parameter values. Conversely, the ANN output controls the value of the TAS inside the domain and is adjusted by minimizing the function

2 2 2

2 2, , , , , , , 0t t t t ttG f

x y x yx y

x θ x (7)

by iteratively updating network parameters with a gradient descent scheme. The exact functional values for AD, L, and N are developed in the following sections. GENERATING AD AND L

The only constraints on the function AD are that it must be continuous and return specific values on the domain boundary. Some problems lead to straightforward determination of an acceptable AD function, such as

1

2 2sin sin tanD

y yA

x x y

(8)

for BCs requiring zero value on the x-axis and unity value on the y-axis. For domain shapes and BCs without obvious AD functions, an appropriate function can be interpolated using thin plate splines (TPSs) [13] which are inspired by equilibrium of a plate exposed to forces at various locations called control points. The TPS for AD is defined as

2 21 2 3

1ln

n

D i i i n n ni

A F r r F F x F y

x (9)

where

2 2 2i i ir x x y y d (10)

A number of control points

for 1ii

i

xi n

y

x

(11)

are chosen for the TPS where the value of AD is to be correctly specified. The 1≤i≤n+3 TPS parameters Fi are determined by solving a linear system where the first n equations for 1D i iA g i n x x (12) ensure the desired values for AD, and the final three

1 1 1

0, 0, and 0n n n

i i ii i i

F F x F y

(13)

ensure force and moment balance. The parameter d is chosen as the small value 0.01 so that each force Fi represents approximately a point force with localized effect at each of the control points. Use of TPSs allows for straightforward creation of an appropriate AD function for arbitrary domain shapes with Dirichlet BCs.

Like AD, the length factor L can be determined ad hoc for a given problem. For example, L xy (14) is appropriate for a domain boundary including the x and y axes. Length factors for complicated domain shapes can be constructed using a two-dimensional TPS to map the boundary control points onto a circle and computing the distance of a point inside the domain from that circle [8]. ARTIFICIAL NEURAL NETWORK DEFINITION

ANNs are inspired by the brain where neurons send electric pulses to neighboring output neurons if the neuron in question receives sufficiently many pulses from the input neurons connected to it. Multilayer perceptron (MLP) networks as illustrated in Figure 1 are one of the most common architectures of ANNs. MLPs are characterized by H hidden nodes where the output of the jth hidden node

,0 ,

1

1 for 1

1

I

j j i ii

jw w x

j H

e

(15)

Figure 1: Illustration of a multilayer perceptron structure with two inputs (x and y), three hidden

nodes, and a single output (N).

1

2

3

x

yN


depends on the weighted sum of the I inputs to the ANN where wj,i is the weight for input xi for node j, and wj,0 is the jth node’s bias. The logistic sigmoid function in Equation (15) is chosen to approximate the binary decision of sending or not sending an electric pulse with a continuous function of range (0,1). Hidden nodes are so named since they are directly connected to neither input nor output. The final output of the ANN

,0 ,

1

0 01 1

,

1

I

j j i ii

H Hj

j jw w xj j

uN u u u

e

x θ (16)

is defined as a weighted sum of the hidden node values. The output N is an explicit function depending on the position vector x and parameter vector 1,0 , 0H I Hw w u u θ (17) which includes the weights and biases for the hidden and output nodes, with a total of (I+2)H+1 network parameters. All problems solved here have I=2 inputs (x and y) and H=30 hidden nodes. MLP networks have been shown to be universal approximators [14], ensuring that a exists so that N can approximate any function of x with an arbitrarily small error given sufficiently many hidden nodes. TRAINING THE ARTIFICIAL NEURAL NETWORK

One of the most common algorithms for optimizing, or training, the ANN is the gradient descent Levenberg-Marquardt technique [15]. Executing this technique involves selecting a discrete set of points S in the domain at which to evaluate the function in Equation (7). All problems solved here use a 40×40 square grid over the domain to define S. The sum-squared error is defined as

2 T

1

,S

ii

E G

θ x θ G G (18)

where

T

1, ,SG G G θ x θ x θ (19)

given that xiS for 1i||S|| and S.

The error E can be expanded as the Taylor series T T1

0 0 2 H.O.T.E d E d d d θ θ θ g θ θ H θ (20) around arbitrary point 0 where g is the error gradient and H is the Hessian matrix. The optimal change d is determined by differentiating Equation (20) with respect

to , setting the result to zero, and solving for

1T

1 T2 2d

Jθ H g J J G g

θ (21)

where the Hessian is replaced with an identity involving the Jacobian matrix

TT

T1

,, SGG

x θx θJ θ

θ θ (22)

In order to avoid computing mixed second derivatives, Equation (21) is approximated as 1T2d

θ J J I g (23)

where is a small number chosen to be 0.01 for all calculations here. ANN parameters are updated iteratively according to the scheme 1T

1 2i i

θ θ J J I g (24) where the parameter is adjusted every iteration by a line search algorithm [16] where the largest value is chosen such that =i+1–i is still a descent direction. This scheme converges rapidly to the (possibly locally) minimal error value once a region in the parameter space is found with sufficiently low error.

Practically implementing this learning algorithm involves computing the sensitivities of G to each ANN parameter (i.e. the weights and biases wj,i and uj) evaluated at the points selected in S. If the DE in question were, for example, the Laplace equation

2 2

2 2t tG

x y

(25)

then the partial derivatives

3 3 3 3

2 2 2 2, ,

, , , and t t t t

j i j i j jx w y w x u y u

(26)

would be required; they are straightforward to obtain from the TAS in Equation (4) and the explicit expression for ANN output in Equation (16). EXPANDING TRAINING TO COUPLED SYSTEMS

The learning algorithm described in the previous section is appropriately developed to solve a boundary


value problem whose DE is defined by f in Equation (1) and whose Dirichlet BCs are satisfied by appropriately choosing AD and L for the TAS in Equation (4). Expanding to solve systems of coupled DEs involves solving for the 1im variables (i) represented approximately by m DEs

12 2

1, , , , , ,m

mt tk k t tG f

x y x y

x (27)

with 1km. In general, each DE fk depends on the position vector as well as the value and partial derivatives of all the 1im variables (i). The sum-squared error

T

1

m

k kk

E

θ G G (28)

is developed by adding the error from approximating each DE. The Jacobian for each DE

TT

T1

,, k Skk

GG

x θx θJ θ

θ θ (29)

can be combined to correctly approximate the Hessian as

T

12

m

k kk

H J J I (30)

The learning rule for coupled systems of DEs is then

1

T1

12

m

i i k kk

θ θ J J I g (31)

which involves simply summing the contributions for each DE before inverting the approximate Hessian. DEMONSTRATION WITH A TOY PROBLEM

In order to demonstrate the technique, consider the functions

1 2

1 2

1 1 22

1 2

2 21

2

andA x A y

A x A y

B Be e

A A

BB e e

A

(32)

which satisfy the coupled system

1 2

1 1 2 22 2

2 2

1 2

1 2

0 and

0A x A y

x yx y

B e B ex y

(33)

The BCs for this problem are created by evaluating the analytical solutions along the cardioid boundary 3 3

2 2 + cosr (34) using constants 1 2 1 20.85, 0.2, 0.3, and 0.3A A B B (35)

The TPS generated length factor appears in Figure 2 where markers indicate the chosen control points. The functions AD for both unknowns are developed according to Equation (9) using the same control points in Figure 2 set to values according to Equation (32). The resulting TASs for (1) and (2) were optimized and compared with the analytical solutions to produce the percentage errors appearing in Figure 3. Approximations for both variables were excellent, with the largest local error at less than three thousands of a percent.

Figure 2: Length factor for the cardioid domain

generated using the marked control points.

Figure 3: Percentage errors in the variables (a) (1)

and (b) (2) as approximated by the ANN.

x

y

y

x x(a) (b)


MODELLING THE BLASIUS BOUNDARY LAYER The Blasius boundary layer involving laminar flow

over a semi-infinite plate of length L held parallel to a constant and uniform flow is subject to the continuity equation

* *

0u v

x y

(36)

momentum balance equation

2

* * 2 ** *

* * *

1Re

u u uu v

x y y

(37)

and energy balance equation

2

* * 2 ** *

* * *

1Re Pr

T T Tu v

x y y

(38)

The governing equations are normalized to limit the magnitude of values necessary for the ANN to represent. The three unknowns are the dimensionless x- and y-components of velocity u* and v*, as well as the dimensionless temperature T*. BCs on these three variables are

*

** * * * *

**

0 : 00 : 0

: 1 , 0 0, 0 : 1

0 : 1

yx

u y v y Ty

x

(39)

The TASs

2 2

2 2

*1* * *

1* *

2* *2

*3* * *

3* *

and

t

t

t

yu x y N

x y

v y N

xT x y N

x y

(40)

satisfy all the BCs except for the requirement that u* approach unity as y* approaches infinity. This requirement is intentionally omitted to illustrate that this ANN method can successfully solve DEs even with incomplete boundary information.

The local dimensionless heat transfer coefficient is represented by the local Nusselt number

* *

**

*, 0

Nu x

x y

Tx

y

(41)

which is straightforward to compute from the TAS in Equation (40) and the ANN output in Equation (16). A generally accepted correlation for Nux is

1132Nu 0.332Re Prx x (42)

whose error can be as high as 25% depending on flow conditions [17].

The coupled system in Equations (36), (37), and (38) is traditionally solved by recognizing self-similarity and defining

**

Luy

x

(43)

with the free stream velocity u and kinematic viscosity . The coupled system reduces to the ordinary DE 0f ff (44) where f() is a function alone, and dimensionless temperature is expressed as

Pr

* 0Pr

0

1f d

Tf d

(45)

Equation (44) can be approximated with an iterative technique [18] and Nux in Equation (41) rewritten

12

* * * *

* **

* *, 0 , 0

Nu Rex

x y x y

T Tx

y

(46)

using the chain rule.

The TASs in Equation (40) are optimized with ANNs for a 10 cm long plate, free stream velocity of 1 cm/s, and property data for air at room temperature, i.e. Re=62.9 and Pr=0.707. TASs for u*, v*, and T* appear in Figures 4, 5, and 6 respectively. Figure 7 compares the resulting local Nusselt number in Equation (41) with the accepted

Figure 4: Dimensionless temperature T* in the

Blasius boundary layer as approximated by the ANN.

*x

*y

*T


Figure 5: Dimensionless horizontal speed u* in the


Figure 6: Dimensionless vertical speed v* in the


Figure 7: Percentage error in local Nusselt number for the ANN method and the self-similar method.

correlation in Equation (42) as a percent error, including error in the self-similar approximation using Equation (46) as a benchmark. Error in the ANN method is smaller over most of the plate’s length than the self-similar method, and easily within the 25% error margin. Results are worst near the leading edge of the plate due to the discontinuous BCs in both u* and T* at the origin. The ANN has difficulty in this region as it is updated based on derivatives required to approach infinity at the origin. Indeed the point at the origin in the 40×40 training grid must be removed in order for the method to operate at all. Also note that

*

*

0lim

xv

(47)

in the analytical solution is impossible for the ANN to model correctly. Despite discontinuities, an analytical solution with infinite values, and incompletely specified BCs for u*, the local Nusselt number predicted by the ANN is in general more accurate than the traditional self-similar technique for solving this problem.

MODELLING ENTRANCE FLOW

The normalized Navier-Stokes equations for steady two-dimensional flow ignoring body forces

2 2

2 2

* * * 2 * 2 ** *

* * * * *

* * * 2 * 2 ** *

* * * * *

1 0 andRe

1 0Re

p u u u uu v

x x y x y

p v v v vu v

y x y x y

(48)

along with the continuity equation

* *

* * 0u v

x y

(49)

govern the entry length problem for flow between two parallel plates with a steady and uniform horizontal inlet velocity. The BCs on dimensionless x and y velocity components and dimensionless pressure

*

* * * * *12* 1

2

0 : 1, 0, 0 0

: 0x

u v y p xy

(50)

are satisfied with the TASs

2

2

2 2

2

*11* * *4 1

142* *1

4

2* *124

3* *3

and

t

t

t

yu x y N

x y

v y N

p x N

(51)

The approximations of u*, v*, and p* for laminar flow

with Re=15 appear in Figures 8, 9, and 10 respectively. The ANNs producing these approximations were trained using a 120×40 grid reflecting the domain’s 3:1 aspect ratio. Again, the analytical solution at the entry corners will have infinite derivatives and so those two points are removed from S.

Fully-developed flow can be predicted to occur at a dimensionless entrance length [19]

* 0.06Reel (52)

*x

% e

rror

in N

u x

*x

*y

*v

*x

*y

*u


with a corresponding velocity profile of

2*

* *1 12 4*Re p

u yx

(53)

The fully-developed region is characterized by a velocity uniform in the flow direction. The velocity gradient u*/x* for the u* in Figure 8 averaged over the flow cross-section appears in Figure 11. The gradient drops sharply just after the predicted entrance length of le

*=0.9 to remain below 10-2, representing changes in speed of less than a 1% of the free stream velocity over a distance equal to the hydraulic diameter.

The volumetric flow rate

14

14

* * * * *u dA w u dy

(54)

for a cross-section with dimensionless width w* is equal to ½w* at the inlet with u*=1. Numerically approximating Equation (54) for u* in Figure 8 in the fully-developed region results in 0.500w* for the flow rate. Combining Equations (53) and (54) require that the dimensionless pressure gradient p*/x* in the fully-developed region be 48/Re, or 3.2 in the case explored here. The pressure in Figure 10 has a gradient of 3.19 in the fully-developed region. Equation (53) and the fully-developed velocity profile approximated by the ANN are essentially coincident, as illustrated in Figure 12. CONCLUSION

The length factor artificial neural network (ANN) method has been expanded to solve coupled systems of partial differential equations (DEs). This method uses approximate solutions to the DEs which depend on the output of an ANN but exactly satisfy Dirichlet boundary conditions (BCs) independent of the ANN. The approximate solution is then optimized by updating the ANN via gradient descent.

The main strength of this method is that the same solution method is followed regardless of the form or complexity of the DEs involved, including nonlinear equations like the Navier-Stokes equations. Every problem requires developing a length factor L dependent on the shape of the problem domain, and a function AD based on the value of the solution on the boundary. Most irregularly-shaped domains preclude obvious ad hoc functions for L and AD, but their generation can be automated in a straight-forward process using thin plate splines. Other benefits of the method include an explicit, differentiable approximate solution, the absence of meshing concerns, and success despite omission of some boundary conditions (BCs).

Figure 8: Dimensionless horizontal speed u* for the

entrance flow problem as approximated by the ANN.

Figure 9: Dimensionless vertical speed v* for the


Figure 10: Dimensionless pressure p* for the


Figure 11: Velocity gradient u*/x* averaged over

the flow cross-section as approximated by the ANN, with the location for fully-developed flow as predicted

by Equation (52) indicated with the vertical line.

Figure 12: Dimensionless velocity profile in the fully-

developed region as determined analytically and approximated by the ANN.

*u

*y

*x

* *u x

*x

*y

*p

*x

*y

*v

*x

*y

*u


This manuscript offers the first time the length factor method has been shown to solve coupled systems of DEs. Also for the first time, the method has been shown to accurately solve reasonably complicated problems of interest in science and engineering, including finding the local Nusselt number in the Blasius boundary layer and solving the Navier-Stokes equations for laminar flow between parallel plates. Accurate solutions for these problems are obtainable despite the fact that the analytical solutions involve discontinuities impossible for the continuous trial approximate solution (TAS) to model exactly. Optimization of the TAS is minimally affected by such discontinuities since they arise from BCs which are automatically satisfied independent of the ANN due to the unique design of the length factor method. Since the value of the TAS is correctly specified near the discontinuity, the DEs need not be satisfied accurately there. However, points in the domain too close to discontinuities should not be used for training since optimization will concentrate on regions with infinite derivatives at the expense of the remainder of the domain. While using a square training grid does avoid complicated meshing schemes, problems with discontinuous solutions do require modest attention to remove training points with infinite derivatives.

Additionally, the method offers a continuous TAS so that quantities such as temperature, velocity, and pressure gradients are available analytically everywhere in the domain rather than requiring numerical approximation. Also of interest is the ability of the method to accurately solve problems with incomplete boundary conditions.

While not as recognized as traditional numerical techniques such as the finite element method (FEM), ANN methods, and the length factor method in particular, offer alternatives eliminating some vexing issues with the FEM. The length factor method shows promise as a tool for solving difficult problems in science and engineering. REFERENCES [1] G. Smith, Numerical Solution of Partial Differential Equations: Finite Difference Methods, Oxford, UK: Clarendon, 1978. [2] S. Rao, The Finite Element Method in Engineering, Boston: Butterworth-Heinemann, 1999. [3] C. Brebbia, J. Telles, and L. Wrobel, Boundary Element Techniques: Theory and Applications in Engineering, Berlin: Springer-Verlag, 1984. [4] L. Aarts, P. Van Der Veer, and van der Veer, “Neural Network Method for Solving Partial Differential Equations,” Neural Processing Letters, vol. 14, 2001, pp. 261-271. [5] S. He, K. Reif, and R. Unbehauen, “Multilayer Neural Networks for Solving a Class of Partial Differential Equations,” Neural Networks, vol. 13, 2000, pp. 385-396.

[6] I. Lagaris, A. Likas, and D. Papageorgiou, “Neural-network Methods for Boundary Value Problems with Irregular Boundaries,” Neural Networks, IEEE Transactions on, vol. 11, 2000, pp. 1041-1049. [7] N. Mai-Duy and T. Tran-Cong, “An Efficient Indirect RBFN-based Method for Numerical Solution of PDEs,” Numerical Methods for Partial Differential Equations, vol. 21, 2005, pp. 770-790. [8] K. McFall and J. Mahan, “Artificial Neural Network Method for Solution of Boundary Value Problems With Exact Satisfaction of Arbitrary Boundary Conditions,” IEEE Transactions on Neural Networks, vol. 20, 2009, pp. 1221-1233. [9] A. Meade and A. Fernandez, “The Numerical Solution of Linear Ordinary Differential Equations by Feedforward Neural Networks,” Mathematical and Computer Modelling, vol. 19, 1994, pp. 1-25. [10] D. Parisi, M. Mariani, and M. Laborde, “Solving Differential Equations with Unsupervised Neural Networks,” Chemical Engineering and Processing, vol. 42, 2003, pp. 715-721. [11] I.G. Tsoulos, D. Gavrilis, and E. Glavas, “Solving Differential Equations with Constructed Neural Networks,” Neurocomputing, vol. 72, 2009, pp. 2385-2391. [12] R. Yentis and M. Zaghloul, “VLSI Implementation of Locally Connected Neural Network for Solving Partial Differential Equations,” Circuits and Systems I: Fundamental Theory and Applications, IEEE Transactions on, vol. 43, 1996, pp. 687-690. [13] L. Zagorchev and A. Goshtasby, “A Comparative Study of Transformation Functions for Nonrigid Image Registration,” Image Processing, IEEE Transactions on, vol. 15, 2006, pp. 529-538. [14] K. Hornik, M. Stinchcombe, and H. White, “Multilayer Feedforward Networks are Universal Approximators,” Neural Networks, vol. 2, 1989, pp. 359-366. [15] J. Jang, C. Sun, and E. Mitzutani, Neuro-fuzzy and Soft Computing, Englewood Cliffs, NJ: Prentice-Hall, 1997. [16] M. Albaali and R. Fletcher, “An Efficient Line Search for Nonlinear Least-squares,” Journal of Optimization Theory and Applications, vol. 48, 1986, pp. 359-377. [17] F.P. Incropera and D.P. DeWitt, Fundamentals of Heat and Mass Transfer, John Wiley, 2007. [18] A. Mills, Heat and Mass Transfer, CRC Press, 1994. [19] B.R. Munson, D.F. Young, and T.H. Okiishi, Fundamentals of Fluid Mechanics, Wiley, 2006.



October 1-2, Atlanta, Georgia USA

EFFECTS OF ENVIRONMENTAL AGING ON THE THERMAL AND MECHANICAL PROPERTIES OF VINYL ESTER NANOCOMPOSITES

Ahmad Almagableh, P. Raju Mantena Dept. of Mechanical Eng., Univ. of Mississippi

University, MS 38677 USA

ABSTRACT In this paper, the Dynamic Mechanical Analyzer is used to study the effects of hygrothermal exposure on material properties of graphite platelet and nanoclay reinforced vinyl ester nanocomposites. An attempt is made to empirically model the change in glass transition temperature under hygrothermal effects. Aged specimens were observed to have drop in glass transition temperature as well as a reduction in the damping peak magnitude, indicating possible formation of microcavities (irreversible morphological changes) mainly in the vinyl ester with nanoclay reinforcements. A dominant trend of drop in the room temperature storage modulus in all nanocomposites was also observed, though the drop is more pronounced in nanoclay specimens and increases with duration of aging exposure. Empirical Predictions of degraded modulus as a function of temperature did agree quiet well with experimental data from DMA, and error between modeled data and the experimental is becoming more pronounced in the rubbery region for nano-reinforced specimens. INTRODUCTION Accurate prediction of the durability or life of composite structures in service environments is a major concern. Service environments are temperature, moisture, mechanical loads (static or cyclic) or a combination of these, known as hygrothermomechanical environments. Plasticization of polymer based composites by water absorption leads to a drop in the glass transition temperature, Tg, and further degradation of composite mechanical properties. Warm, moist environments can considerably change the performance of a material as demonstrated in several polymeric systems [1]. The available free volume (pores) in a resin alters the water equilibrium concentration and can, in addition, occupy micro-voids and other morphological defects. Water absorbed in the polymer is generally grouped into free water and bound water [1].Water molecules, which are gathered in the free volume of the polymer and are relatively free to travel through the microvoids and pores, are identified as free water, while water molecules that are

dispersed in the polymer matrix and linked to the polar groups of the polymer are designated as bound water. A loosely bound water within the polymer network is recognized as the one which can be released easily upon heating [2], and strongly bound water (frozen) is difficult to remove from the polymer network. Moisture uptake in a polymeric composite can lead to chemical degradation effects, both reversible and irreversible, including weakening the intermolecular bonds among the functional groups of the chains (plasticization) [3], debonding at filler-matrix interfaces [4], leaching of unreacted functional groups [5], micro-structural damage such as cavities or pores in the matrix [6], increased crosslinking [7], and degradation of the matrix properties due to hydrolysis and oxidation during long-term exposure to water [5]. Mechanical and thermal behavior could also be affected due to moisture absorption, the thermal properties such as the glass transition temperature and damping properties, and mechanical properties including tensile strength, modulus, failure strain [8] and fracture toughness [9] can be significantly altered. The kinetics of water diffusion is assumed to follow the one-dimensional Fick’s second law, which considers that the driving force of diffusion is the water concentration gradient. Some other models have also been applied to describe more complex diffusion processes. A two-phase model has been utilized to interpret the Langmuir diffusion process [10], in which the absorbed water is divided into a free phase and a strongly bound phase. Alvarez and Vazquez [11] have studied the effect of water absorption on the behavior of vinylester (VE) and polyester–glass fiber composites at two different temperatures. A relation was found between the glass transition temperature (Tg), the bath temperature and the plasticization of the network-structure. Poor fiber–matrix interfaces were observed by a scanning electron microscope (SEM) due to water absorption. Extraction of silane coupling in composite materials was observed at a temperature of 80 C. In addition, unsaturated polyester matrix absorbs more water than VE due to its higher


hydrophilic character and void content. It was observed that mechanical properties (flexural modulus and interfacial shear strength) decrease remarkably with immersion time at high temperatures.

Plasticization of polymeric based composites under water absorption leads to a drop in the glass transition temperature, Tg, and further degradation of composite properties [12]. In this study, the effect of hygrothermal exposure on glass transition temperature and damping of nanoclay and graphite platelet reinforced vinyl ester nanocomposites are characterized and modeled empirically.

MATERIALS AND ENVIRONMENTAL AGING The polymeric matrix used was a vinyl ester resin (Ashland specialty chemical, Division of Ashland INC, Columbus, OH). DERAKANE 510A-40 vinyl ester resin is a brominated bisphenol-A based vinyl ester consisting of 38 wt. % styrene, and modified to produce the maximum degree of fire retardancy combined with enhanced chemical resistance and toughness. These additives are Butanone peroxide, N,N-Dimethylaniline, Cobalt Naphthenate, and 2-4-Pentanedione, all supplied from Sigma Aldrich (St. Louis, Mo). Exfoliated graphite nanoplatelets (xGnP) were produced according to the method described in Ref [13]. Figure 1 (a) shows a TEM morphology for edge view of xGnP inside the matrix. These xGnP nanocomposites have exfoliated and dispersed graphite platelets with 1 nm thickness and several hundred nanometers widths. The distance between layers is in the range of 10~30 Å and the size of the layered graphite extends from several hundred nanometers to several microns. The nanoclay was Cloisite 30B from Southern Clay Products, Inc as shown in Figure1 (b). Nanocomposites were prepared by dispersing about 3000 gm of epoxy vinyl ester resin solution with different percentages of nanoclay or nanographite in a 1 gal container for 4 hours, followed by 4 passes through a flow cell connected to a 100 W sonicator. 1% Butanone peroxide, 0.2% of 2-4 Pentanedione, 0.1% N,N-Dimethylaniline, and 0.2% Cobalt Naphthenate were added to the mixed vinyl ester resin solution in order and mixed for 10 min. The above mixed resin solution was mixed for 2 min with FlackTek speed mixer at 3000 RPM. The well-mixed vinyl ester resin solution with nanoclay or nanographite was poured into a 13”13”0.4” mold, let stand for 30 minutes at room temperature and then was

post cured at 80 oC for 3 hours. Prismatic samples with

nominal dimension of 35 x 10 x 1.6 mm size were prepared from these plates and tested before and after environmental aging in a DMA using the single-cantilever clamp fixture. The hygrothermal aging effect on both mechanical and thermal properties was studied through conducting a freeze-thaw cycling, in which a specimen is submerged in

a water bath, subjected to a temperature cycling from 15 F to 45 F each over a 6 hour period. Aged Specimens were divided into two sets based on the immersion time. The first set was taken out after 60 days and tested and the other is left inside the environmental chamber for 60 more days. Specimens subjected to freeze-thaw cycling were tested using the TA Instruments model Q800 Dynamic Mechanical Analyzer to obtain viscoelastic properties.

(a)

(b)

Figure 1. TEM images of nanoparticle dispersion for (a) xGnP, and (b) Cloisite® nanoclay [14].

MOISTURE UPTAKE It should be noted that nano-specimens were weighed before and after aging exposure, using digital meter with a 1% gm resolution, thus moisture measurements using this technique might be not sufficient accurate with relative to the weight of water molecules absorbed. Water absorption (percent) as a function of immersion time was determined using the following Equation:


(1) 100%

o

ot

M

MMM

where Mt is the weight at a time, t, and Mo is the weight at dry conditions. Table 1 lists moisture absorbed by weight percentage versus immersion time.

Table 1. Moisture absorption (wt. %) for vinyl ester nanocomposites as a function of immersion time.

Material M (%) after 60 days

M (%) after 120 days

Vinyl ester matrix (VE)

3.0 4.5

VE+1.25wt. percent nanoclay

1.4 2.8

VE+2.5 wt. percent nanoclay

1.4 5.5

VE+1.25wt. percent xGnP

1.2 2.5

VE+2.5 wt. percent xGnP

1.4 2.8

DYNAMIC MECHANICAL ANALYZER To obtain the thermal transition characteristics of the vinyl ester nanocomposites, both prior to and after environmental aging, dynamic measurements were carried out using the TA Instrument model Q800 DMA on prismatic specimens with 0, 1.25 and 2.5 wt. percent graphite nanoplatelet (xGnP) and nanoclay reinforcements. Specimens were deformed at constant amplitude (25 )m over a 1 Hz single frequency, with

temperature ramped at 3 Co per minute starting from 30 Co (RT) to 150 Co . EXPERIMENTAL RESULTS Moisture uptake by resin causes hydrolysis and chain scission over the long term and plasticization in the short term due to moisture absorption, which generally results in a decrease in storage modulus (E’) as well as drop in the glass-transition temperatures, linked to the degree of molecular mobility [15]. Vinyl ester nanocomposites exhibited a reduction in Tg as a function of immersion time except for the case of 2.5 wt. percent xGnP (Table 2). A drop in the room temperature storage modulus in the nanocomposites was also observed which is more pronounced in nanoclay specimens and increases with duration of aging exposure, whereas those reinforced with xGnP experienced steady modulus variation insensitive to aging time. For example, pure vinyl ester resin had a maximum drop of about 400 MPa in room temperature modulus compared to 150 MPa for those reinforced with 1.25 wt. % xGnP based upon 120 days of immersion time. Intermolecular hydrogen bonding by water molecules (plasticization) can be manifested by a decrease in the room temperature modulus which is found in nanoclay

composites. However, it is clear that stress relaxation allows increased molecular movement, which in turn allows further cross-linking to occur. In fact, the water uptake process at the outset is quite complex due to the competing effects of plasticization and crosslinking and the mechanism is further complicated due to a possible cure progression under immersion in aqueous solution which cause higher modulus and increased brittleness [15]. Data scatter tends to be greater between aged specimens than those unaged, with a tendency towards greater scatter with longer aging exposure. The reason may be because of uneven plasticization taking place or specimen shrinkage during temperature ramp in DMA tests.

Table 2. Viscoelastic properties for vinyl ester

nanocomposites before and after aging exposure.

Material Conditions Tg

(oC)

Peak of loss factor

Initial modulus (MPa)

Vinyl ester VE

Dry 125 1.23 2950 60 days 124 1.08 2950 120 days 117.5 0.93 2600

1.25 % nanoclay/VE

Dry 122 1.2 2900 60 days 122 1 3400 120 days 119 1 2700

2.5 % nanoclay/VE

Dry 124 1.25 3100 60 days 121 0.96 3000 120 days 121 0.9 2700

1.25 % xGnP/VE

Dry 126 1 3500 60 days 121 1 3300 120 days 122 1 3350

2.5 % xGnP/VE Dry 129 0.94 3800

60 days 123 0.7 3700 120 days 128 0.83 3500

Unlike the trend observed for nanoclay specimens, Tg was initially reduced in xGnP specimens and then rose again with further moisture absorption, in some cases it becomes close to the original dry Tg of the material. Comyn [16] also reports similar behavior for epoxy where, Tg initially reduced and then it went up again with further moisture absorption. These xGnP reinforced specimens show a more considerable increase in Tg (120 days compared to 60 days), which could be evidence of embrittlement or antiplasticization phenomena. Antiplasticization is basically leaching of free (non-crosslinked) styrene monomer residues that act as a


plasticizer in the material after longer immersion time in water. In addition to the above effects, it is believed that major effect occurs due to rapid heating induced by the DMA test, which has also been observed by Akay et al. [17]. MODELING GLASS TRANSITION TEMPERATURE The hygrothermal degradation based on plasticization of vinyl ester nanocomposites is modeled based on the following empirical Equation 2. Equation 2 is based on the experimental observation that degradation in property (stiffness) is gradual until the temperature T approaches Tgw [12]. The value of Tgw is obtained from DMA experiments as the peak of loss-factor (Tan δ peak).

(2) )(

2/1

OgO

gw

o TT

TT

E

TE

Where E= matrix stiffness after hygrothermal degradation. Eo= matrix stiffness before degradation (baseline). T= temperature at which E is to be predicted (F). Tgo= glass transition temperature for reference dry condition. Tgw=glass transition temperature at fully saturated conditions (120 aging days). To= test temperature at which Eo was measured (room temperature).

(a)

(b)

(c)

(d)

(e)

Figure 2. Emperical model predictions and experimental results of modulus vs temperature for (a) brominated vinyl ester (VE), (b)1.25 wt.% nanoclay/ VE, (c) 2.5 wt. % nanoclay/ VE, (d) 1.25 wt. % xGnP/ VE and (e) 2.5 wt. % xGnP/ VE. The degraded modulus as a function of temperature obtained from DMA along with that predicted at approximately fully saturated conditions (Equation 2) is plotted in Figures 2 (a-e). The predicted degradation of modulus as a function of temperature did agree quite well


with experimental data from DMA; however, the degraded modulus based on model prediction was generally less than the experimental modulus for all specimens tested. Moreover, error between modeled and experimental data in the rubbery region is more pronounced for nano-reinforced specimens especially with 2.5 wt. %. Again, this can be an indication of irreversible damage taking place in the form of fiber–matrix debonding (cross-linking) which can be directly assessed from the rubbery region [18].

CONCLUSIONS Hygrothermal aging effect on graphite platelet and nanoclay reinforced vinyl ester nanocomposites was studied by conducting a freeze-thaw cycling, in which a specimen is submerged in a water bath, subjected to a temperature cycling from 15 F to 45 F for 60 and 120 days. Aged specimens were observed to have drop in Tg as well as a reduction in the damping peak magnitude, indicating possible formation of microcavities (irreversible morphological changes) mainly in the vinyl ester with nanoclay reinforcements. A dominant trend of drop in the room temperature storage modulus in all nanocomposites was also observed, though, the drop is more pronounced in nanoclay specimens and increases with duration of aging exposure. For xGnP specimens, Tg, was initially reduced and then rose again with further moisture absorption, which could be an evidence of embrittlement or antiplasticization phenomena. Furthermore, stable variation of initial storage modulus as a function of aging time because of possible cross-linking (antiplasticization) in xGnP specimens, and there is some recovery in properties of these aged specimens compared to ambient as they are heated during the DMA test. The hygrothermal degraded stiffness (storage modulus) as a function of temperature was modeled empirically. Prediction of degraded modulus as a function of temperature did agree quiet well with experimental data from DMA, however, degraded modulus based on the model prediction was generally less than the experimental modulus for all specimens tested and error between modeled data and the experimental is becoming more pronounced in the rubbery region for nano-reinforced specimens. Further work involves empirical modeling of mechanical properties (strength and stiffness) for brominated vinyl ester nanocomposites under hot-wet conditions. ACKNOWLEDGMENTS

This investigation was supported by ONR Grant N00014-07-1-1010, Office of Naval Research, Solid Mechanics Program (Dr. Yapa D.S. Rajapakse, Program Manager). The nanoclay and graphite platelet vinyl ester

composite plates were manufactured by Dr. Larry Drzal’s group at Michigan State University.

REFERENCES [1] Diamant, Y., Marom, G.., and Broutman, L. J., 1981, “The Effect of Network Structure on Moisture Absorption of Epoxy Resins,’’ J. Appl. Polymer Science., 26, pp. 3015-3025. [2] Maggana, C., and Pissis, P., 1999, “Water Sorption and Diffusion Studies in an Epoxy Resin System,’’ J. Polymer Science: Part B: Polymer Physics., 37, pp. 1165-1182. [3] Ivanova, K. I., Pethrick, R. A., and Affrossman, S., 2000, “Investigation of Hydrothermal Ageing of a Filled Rubber Toughened Epoxy Resin Using Dynamic Mechanical Thermal Analysis and Dielectric Spectroscopy,’’ J. Polymer, 41, pp. 6787-6796. [4] Bowditch, M. R., 1996, “The Durability of Adhesive Joints in the Presence of Water,” International Journal of Adhesion and Adhesives., 16 (2), pp. 73-79. [5] Antoon, M .K., and Koenig, J. L., 1981, “ Irreversible Effects of Moisture on the Epoxy Matrix in Glass-Reinforced Composites,” Journal of Polymer Science: Polymer Physics Edition, 19, pp.197-212. [6] Apicella, A., Nicolais, L., Astarita, G.., and Drioli, E., 1979, “Effect of Thermal History on Water Sorption, Elastic Properties and the Glass Transition of Epoxy Resins,” J. Polymer, 20, pp. 1143-1148. [7] Liu, J., Lai, Z., Kristiansen, H., and Khoo, C., 1998, “Overview of Conductive Adhesive Joining Technology in Electronics Packaging Applications,” Proc. 3rd International Conference on Adhesive Joining and Coating Technology in Electronics Manufacturing, IEEE, pp.1-18. [8] Kasturiarachchi, K. A., and Pritchard, G.., 1985, “Scanning Electron Microscopy of Epoxy-glass Laminates Exposed to Humid Conditions,” Journal of Materials Science., 20, pp. 2038-2044. [9] Butkus, L. M., Mathern, P. D., and Johnson, W. S., 1998, “Tensile Properties and Plane-Stress Fracture Toughness of Thin Film Aerospace Adhesives,” Journal of Adhesion., 66, pp. 251- 273. [10] Bonniau, P., and Bunsell, A. R., 1981, “A Comparative Study of Water Absorption Theories Applied to Glass Epoxy Composites,” Journal of Composite Materials., 15 (3), pp. 272-293. [11] Alvarez, V., and Vazquez, A., 2007, “Cyclic Water Absorption Behavior of Glass–Vinylester and Glass–Epoxy Composites,” Journal of Composite Materials., 41, pp. 1275-1289. [12] Chamis, C., and Sinclair, J., 1982, “Durability/ Life of Fiber Composites in Hygrothermomechanical Environments,” Journal of Composite material, testing and design, ASTM STP 787, pp. 498-512. [13] Drzal, L.T., Fukushima H., “Expanded Graphite Products Produced Therefrom,’’ US Patent No. 7, 550-529.


[14] “Laboratory of Polymer Technology-Polymer Blends”, http://polymeeri.tkk.fi.

[15] Karbhari, V., 2006, “Dynamic Mechanical Analysis of the Effect of Water on E-glass–Vinyl ester Composites,” Journal of Reinforced Plastics and Composites., 25, pp. 631-644. [16] Comyn, J., 1989, “Interaction of water with epoxy resins, Polymers in a marine environment,” 2nd Conference, (IMarE), pp. 153-162. [17] Akay, M., Kongahmun, S., and Stanley, A., 1997, “Influence of moisture on the thermal and mechanical properties of autoclaved and oven-cured Kevlar 49/epoxy laminates,” J. Composite Science Technology., 57, pp. 565- 571. [18] Almagableh, A., Mantena, P. R., Alostaz, A., Liu, W., and Drzal, L.T., 2009 “Effects of Bromination on the Viscoelastic Response of Vinyl ester Nanocomposites,”, J. eXPRESS Polymer Letters., 3 (11), pp. 724–732.



October 1-2, 2010, Atlanta, Georgia, USA

MOLECULAR DYNAMIC SIMULATION TO DETERMINE TEMPERATURE DEPENDENT YOUNG’S MODULUS OF MONOLAYER

AND BILAYER GRAPHENE SHEETS

Roy Downs B.S. Student*

Dakota Scrivener B.S. Student*

Sachin S. Terdalkar PostDoc**

Joseph J. Rencis Professor*

*Department of Mechanical Engineering, University of Arkansas Fayetteville, AR, USA

**Department of Mechanical Engineering, Indiana University-Purdue University Indianapolis Indianapolis, IN, USA

ABSTRACT Large-scale molecular dynamics simulations for

the nano-indentation of monolayer and bilayer

graphene sheets are performed to calculate Young’s

modulus. The simulations are compared with

experimental values and found to be consistent. Effects

from temperature variations on Young’s modulus were

studied using molecular dynamics simulations. Very

little change in the Young’s modulus was observed with

temperature variation. The effect of simulation size on

measured value of Young’s modulus was also studied,

and it was found that the Young’s modulus approached

1 TPa (closer to experimental value) as the sheet

diameter increased.

INTRODUCTION Graphene is one of the most popular materials to

study in the nano-materials area due to its excellent

material properties. Graphene is a monolayer

hexagonal lattice with sp2 hybridized carbon atoms. It

is the “building block of all graphitic materials” [1]

such as bucky-balls, nanotubes, and graphite.

Mechanically, graphene is the stiffest and strongest

material known to mankind. It has a Young’s modulus

of 1 TPa and an intrinsic strength of 130 GPa [2],

approximately 200 times stronger than structural steel

[3]. Some potential applications for the single layer

atom structure are nano-electro-mechanical system

(NEMS), nano-sensors, nano-electronics, super

capacitors, solar cells, etc. [4]. Young’s modulus is a material property that

measures the stiffness of a material. The Young’s

moduli for CNTs and graphene sheets have been

determined experimentally with transmission electron

microscope (TEM) [5] and numerically with Molecular

Dynamics (MD) simulation [6,7]. A majority of the

research has been conducted with CNTs at room

temperature and temperatures ranging from 0 K to

2000 K [7]. TEM and MD simulation were used to

observe CNTs, and results show that the Young’s

modulus is insensitive to tube temperatures less than

1100 K, but decreases at temperatures greater than

1100 K [7].

Monolayer graphene was studied using atomic

force microscopy (AFM) and MD simulation.

Researchers used the AFM to observe graphene at

static temperature and measured a mean Young’s

modulus of 1TPa [2]. Using MD simulations it was

found that at temperatures below 500 K the Young’s

modulus increases with higher temperature.

Temperatures above 500 K tend to cause a decrease in

Young’s modulus [6]. Both CNTs and graphene tend to

exhibit high stiffness even at high temperatures. The

results calculated from the MD simulations presented

in this paper are different than those calculated by

Jiang et al. [6].

Researchers from Columbia University, in a recent

paper by Lee et al. [2], performed nano-indentation

experiments of monolayer graphene on a silicon

substrate. They were able to calculate the Young’s

modulus of graphene experimentally. The force-

indentation data was numerically analyzed using

membrane theory to calculate Young’s modulus and the

mean value was 1 TPa [2] with excellent repeatability

in the measured value. This is the first time Young’s

modulus of graphene was measured experimentally.

Graphene lacks a natural band gap, which limits its

use in semiconductor devices such as laser diodes and

transistors [8]. A recent breakthrough discovered that a

band gap can be introduced into bilayer graphene [9].

However, band gaps are only tunable in a bilayer

graphene sheet. This makes the bilayer graphene sheet

more ideal for semiconductor devices than monolayer

graphene.


In this paper, nano-indentation of a bilayer

graphene sheet was performed. The focus of this paper

is to determine the Young’s modulus of bilayer

graphene and compare it to monolayer graphene.

Researchers in a recent paper have found using MD

Simulations that the Young’s modulus of bilayer

graphene is 0.8 TPa [10]. These results are different

than the MD simulation results presented in this paper.

MOLECULAR DYNAMICS MODEL

In this paper, MD simulation model is used to

perform nano-indentation on monolayer and bilayer

graphene sheets. The MD simulations are carried out

using the Large-scale Atomic/Molecular Massively

Parallel Simulator (LAMMPS) [11]. Simulations for

constant temperature were carried out on a 96 CPU

Linux cluster Trillion in the Department of Mechanical

Engineering at the University of Arkansas, Fayetteville.

Trillion was decommissioned in June 2010. Further

simulations for varying temperature were carried out

on the Star of Arkansas at the University of Arkansas,

Fayetteville [12]. Post-processing was performed

using the engineering visualization software Ensight

[13].

Figure 1a shows the nanoindentation experimental

setup by Lee et al. [1] that is employed as a base for the

MD simulation model used in this paper. Due to

length-scale limitations of MD simulations a smaller

diameter graphene sheet model is used as compared to

the experiment by Lee et al. [2]. Figures 1b and 1c

show a MD simulation model of the monolayer

graphene sheet containing 52873 atoms. This model is

used to determine the force-indentation curve of a

simply supported circular monolayer graphene sheet.

Atoms in the outer diametric thickness (shown in red)

of 15Å are fixed representing a rigid substrate. The

remaining atoms (shown in green), with a diameter of

390 Å, are free and coupled to the external bath. The

atoms in the outer diameter are fixed so the sheet does

not translate in the z-direction. In the xy-plane

additional roller constraints are used in the x- and y-

directions to restrain the sheet from translating in those

directions and rotating about the z-axis. Figure 1b

shows four regions where these constraints are applied

and two regions are enlarged for clarity. The thickness

of a monolayer graphene sheet is assumed to be 0.335

nm [2] based on the interlayer spacing of graphite.

This method is commonly used to determine the

graphene sheet thickness since there is no defined

thickness for a monolayer graphene sheet. Another

method that is used to define the thickness is based on

the Brenner Potential, assuming the Young’s modulus

of graphene to be the same as graphite [10]. A rigid

spherical indenter with a diameter of 100 Å is used to

apply a load to the graphene sheet as shown in Figure

1c.

Figure 1d shows a MD model for a bilayer

graphene sheet. The constraints and parameters are the

same as those used in the monolayer model except that

the bilayer has 105769 atoms, twice the number found

in the monolayer. The thickness of bilayer graphene

sheet is assumed to be 0.67 nm, twice the thickness of

monolayer graphene sheet. This is the most common

method employed to determine sheet thickness;

however, there are other methods that can be used [10].

AB stacking was used for the bilayer model where one

layer is shifted half a lattice spacing from the other

layer as shown in Figure 2.

The treatment of covalent bonds in LAMMPS is

governed by a potential that regulates intermolecular

interactions. The Adaptive Intermolecular Reactive

Empirical Bond-order (AIREBO) [14] potential is used

in this paper for the carbon-carbon interaction in the

graphene sheet. The AIREBO potential is defined by a

sum over pairwise interactions, composed of covalent

bonding (reactive empirical bond-order (REBO))

interactions, LJ terms, and torsion interactions [14].

The MD model used in Figure 1d for a bilayer

graphene sheet is different from the one used by Neek-

Amal and Peeters [10]. Their MD model used a

different indenter, potential, and constraints. The

indenter was a square-based pyramid and atoms

modeled the indenter. This paper assumed a rigid

spherical indenter. A Lennard-Jones potential was used

for the interaction between the indenter and graphene

sheet as well as the two graphene sheets. The

Brenner’s bond order potential was used for the

carbon-carbon interaction for the bilayer graphene

sheet. This paper used an AIREBO potential for the

carbon-carbon interaction in the graphene sheet.

Constraints were applied to the edge of the sheet in all

directions, i.e., x-, y-, and z-translations are zero. This

represents a rigidly clamped boundary in the absence

of residual stress [10]. This paper assumes a simply

supported boundary.

The following two cases were considered in this

paper using MD simulations:

1. Varying Graphene Sheet Temperature.

2. Young’s Modulus versus Graphene Sheet

Diameter.

The MD model shown in Figures 1b and 1c for the

monolayer and bilayer graphene sheet, respectively, is

used for most of the simulations performed in this

paper except for case 2 above where the graphene sheet

diameter is varied.


(a)

(b) (c)

(d)

Figure1. Experimental and molecular dynamics simulation of the nano-indentation of circular monolayer and bilayer graphene sheets. (a) Experimental setup of circular monolayer graphene sheet (light purple)

on a silicon substrate (dark purple) undergoing nano-indentation [2]. (b) Top view of simply supported graphene sheet with roller constraints applied in the four rectangular regions for molecular dynamics simulation. The red atoms are fixed and the green atoms are free. The constraints are applied to the

right, left, top, and bottom of the graphene sheet to restrain translation in the x-y plane and rotation about the z-axis. (c) Isometric view of simply supported monolayer graphene sheet with rigid spherical indenter

for molecular dynamics simulation. (d) Isometric view of simply supported bilayer graphene sheet with rigid spherical indenter for molecular dynamics simulation. Layer spacing is based on AB stacking in

Figure 2.

z

y

x

Rigid Spherical Indenter

100 Å Diameter Indenter

Monolayer Graphene Sheet

420 Å Diameter Graphene Sheet

390 Å Diameter Free Atoms

Rigid Spherical Indenter

100 Å Diameter Indenter

Bilayer Graphene Sheet

420 Å Diameter Graphene Sheet

390 Å Diameter Free Atoms

AFM Indenter

Tip

Silicon Substrate

(Dark Purple)

Circular Monolayer

Graphene Sheet

(Light Purple)


Figure 2. AB stacking of three layers of

graphene in the case of graphite. AB stacking was used to model the bilayer graphene sheet in

this paper. RESULTS FOR VARYING GRAPHENE SHEET TEMPERATURE

Multiple temperature simulations were applied to

the monolayer and bilayer graphene sheets using MD

simulation to determine the variation in Young’s

modulus with temperature. The graphene sheet and

indenter diameter were assumed constant for the MD

simulation. Simulations were carried out using

constant NVT ensemble [15] which assumes a constant

number of atoms, constant volume, and temperature

throughout the simulation. Temperatures varying from

300 K to 2700 K, at increments of 300 K, were applied

to monolayer and bilayer graphene sheets to study the

variation in Young’s modulus. Indentation of graphene

sheet is performed by using a rigid spherical indenter

with constant velocity. The indentation force on the

graphene sheet is calculated by analyzing the reaction

force on the indenter as shown in Figure 3. The

Young’s modulus can be determined from the force-

indentation data curve. The data calculated from the

MD simulation is fitted based on the non-linear

relationship from membrane theory given by [2]

3

2 2 3

0

D DF a E q a

a a

(1a)

21/ 1.05 0.15 0.16q v v

(1b)

where F is the force applied to the graphene sheet by

the indenter, 2

0

D is the in-plane pretension stress in the

graphene sheet that covers the hole on the rigid

substrate, is the indentation, a is the diameter of

graphene sheet, 2 D

E is the in-plane Young’s modulus,

and q is a dimensionless constant dependent on

Poisson’s ratio v . The Poisson’s ratio used in

Equation 1b is 0.167 and was determined through a

MD simulation of a monolayer graphene sheet [16].

All variables are known, except for 2

0

D and

2 D

E , in

Equation 1a and were determined by least-square

fitting the MD data. Equation 1a is only valid when

indenter is in the center of the graphene sheet, i.e., not

eccentric. The equation is used to determine the in-

plane Young’s modulus of the graphene sheet. The

bulk Young’s modulus is hard to define for monolayer

graphene sheet. A way to calculate the bulk Young’s

modulus is to assume the thickness of graphene as the

interlayer spacing in graphite [2] resulting in the value

of bulk Young’s modulus E given by, nm./E D 33502

where 0.335 nm is the distance between graphene

sheets in graphite. To be consistent with bilayer

graphene sheet, Young’s Modulus is calculated by

nm./E D 67002, where 0.670 is double the thickness

of monolayer graphene sheet. The diameter of the

graphene sheet defined by variable a in Equation 1a is

the inner diameter or the section with free atoms (green

atoms) shown in Figure 1b.

The force-indentation results from Equation 1a

show that monolayer graphene at 0 K results in a

calculated value of 1.07 TPa for Young’s modulus as

shown in Figure 3a. The bilayer graphene sheet has a

calculated Young’s Modulus of 1.06 TPa. In Figure 3b,

the temperature is 300 K and the calculated Young’s

modulus is 1.07 TPa for monolayer graphene and 1.05

TPa for bilayer graphene. Figure 3c shows indentation

at 1500 K and the resulting Young’s modulus is 1.11

TPa for monolayer and 1.07 TPa for bilayer. Figure 3a

versus 3c shows a large amount of scatter in the MD

simulation data at a temperature of 1500 K versus 0 K.

The magnitude of the scatter of data is due to atom

vibrations. The vibrations are small at low

temperatures and increase at higher temperatures.

Figures 4a and 4b show the calculated Young’s

modulus versus the temperature from 300 K to 2700 K

for monolayer and bilayer graphene sheets respectively.

During MD simulations, initial velocities of atoms

were generated using a random number generator. In

order to obtain an accurate average of graphene’s

material property calculated from MD simulations,

multiple simulations were carried out by changing the

seed of the random number generator. The results for

monolayer and bilayer graphene are similar to each

other with little variation in the Young’s modulus due

to temperature. In contrast to the MD simulations for

bilayer graphene carried out by Neek-Amal and Peeters

[10], the results acquired in this paper differ

significantly when compared to Neek-Amal and

Peeters [10] simulation results. The primary reason for

the difference might be due to the boundary conditions

used by Neek-Amal and Peeters [10]. Neek-Amal and

Peeters [10] assumed a fixed edge and this paper

assumed a simply supported edge. Secondly, Neek-

Amal and Peeters [10] used the Brenner’s bond order

potential for the carbon-carbon interaction and a

Lennard-Jones potential for the indenter-graphene

interaction and interaction between the two graphene

layers. This paper used the AIREBO potential for the

graphene sheet which has been observed to more

Layer

Shifted Half

a Lattice

Spacing

from Other

Layers


accurately represent carbon-carbon interaction [14].

Thirdly, Neek-Amal and Peeters assumed a square-

based pyramid and atoms modeled the indenter,

whereas, this paper used a rigid spherical indenter. The

spherical indenter is consistent with an AFM tip.

(a)

(b)

(c)

Figure 3. Force-indentation curves for three various temperatures. (a) 0 K has a Young’s modulus of 1.07 TPa for monolayer and 1.06 TPa for bilayer. (b) 300 K has a Young’s modulus of 1.07 TPa for

monolayer and 1.05 TPa for bilayer. (c) 1500 K has a Young’s modulus of 1.11 TPa for monolayer and 1.07 for bilayer.


(a)

(b)

Figure 4. Young’s modulus versus temperature cases for monolayer and bilayer graphene sheets. (a) 300 K – 2700 K values of Young’s modulus compared to temperature are relatively the same values monolayer

graphene sheet. (b) 300 K – 2700 K values of Young’s modulus compared to temperature are relatively the same values bilayer graphene sheet.

RESULTS FOR VARYING MONOLAYER AND BILAYER GRAPHENE SHEET DIAMETER

In order to understand the influence of simulation

size on the measured value of Young’s modulus, MD

simulations were performed varying the diameter of the

graphene sheet while keeping the ratio between the

indenter diameter and graphene sheet diameter free

atoms constant at 0.256. This allowed us to analyze the

effect of monolayer and bilayer graphene sheet

diameter on the value of the Young’s modulus. Four

graphene sheet diameters considered in this paper

included 200 Å, 300 Å, 350 Å, and 370 Å. Four

temperature cases were considered to calculate the

variation in Young’s modulus. The four temperature

cases used were 0 K, 300 K, 900 K, and 1500 K.

Figure 5 shows how the graphene sheet diameter

affected the Young’s modulus. As the diameter of the

graphene sheets increased, the value of the Young’s

modulus decreased. Previously, it has been observed

that with a decrease in the size of a material, there is an

increase in stiffness [17]. A similar phenomenon is

observed in the simulations done in this paper. In bulk

metals an increase in temperature decreases the

Young’s modulus; however, in case of graphene the

Young’s modulus is relatively constant for all

temperature values. In the case of metals the principle

reason for a decrease in the Young’s modulus with

increase in temperature is due to dislocation nucleation.

In graphene, dislocations cause it to be more ductile,

hence reducing the Young’s modulus. With repeated

calculations using graphene we were unable to create

dislocations in the form of Stone-Wales defects using

MD simulation. The primary reason being the bond

order interatomic potential used is unable to map the

Stone-Wales defect nucleation in graphene.


(a) (b)

Figure 5. (a) Young’s modulus versus the diameter of monolayer graphene sheet. (b) Young’s modulus versus the diameter of bilayer graphene sheet. The Young’s modulus increases as the diameter

of the graphene sheet decreases. Temperature has little effect on the value of Young’s modulus in relation to the graphene sheet diameter.

CONCLUSION

Molecular dynamics simulation was used to

simulate the indentation of monolayer and bilayer

graphene sheets subjected to temperature change. The

results show that the Young’s modulus of monolayer

and bilayer graphene sheets is not affected by

temperature change. The Young’s modulus of

monolayer and bilayer graphene sheets remains

constant even at high temperatures. According to MD

calculations, the monolayer graphene sheet has a

slightly higher Young’s modulus than bilayer graphene

sheet. For monolayer graphene, Young’s modulus

versus sheet diameter shows that with increasing

diameter, the Young’s modulus tends toward 1 TPa.

For bilayer graphene, Young’s modulus versus sheet

diameter shows that with increasing diameter, the

Young’s modulus tends toward 1.1 TPa. Smaller sized

monolayer and bilayer graphene sheets are stiffer than

larger ones. ACKNOWLEDGEMENT

Roy Downs was supported by the Student

Undergraduate Research Fellowship (SURF) Program

from the Arkansas Department of Higher Education

(ADHE). Dakota Scrivener was supported through

grant EEC-1005201 from the National Science

Foundation (NSF). Dakota Scrivener carried out this

work during the NSF Research Experiences for

Undergraduates (REU) Program in Nanomaterials and

Nanomechanics at the University of Arkansas,

Fayetteville from May 17th

to July 23rd

, 2010. Roy

Downs served as an undergraduate student mentor for

Dakota Scrivener during the REU program.

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