theoretical and experimental investigation of magnetostrictive tagged composite beams

Theoretical and Experimental Investigation of Magnetostrictive Tagged Composite Beams

by

Florin G. Jichi

Bachelor of Science Technical University of Timisoara, Romania, 1995

__________________________________________________

Submitted in Partial Fulfillment of the

Requirements for the Degree of Master of Science in the

Department of Mechanical Engineering

College of Engineering & Information Technology

University of South Carolina

2000

_____________________________________ _____________________________________ Department of Mechanical Engineering Department of Mechanical Engineering Director of Thesis Second Reader

________________________ Dean of the Graduate School

ACKNOWLEDGEMENTS

As any research endeavor cannot be done alone, but more likely is a team effort,

there are many members involved. Therefore, I would like to express my gratitude and

deeply thanks to Dr. Victor Giurgiutiu for all his guidance and help throughout my entire

time spent here at USC. Without his support, none of what I accomplished would not

have been possible.

In addition, I would like to sincerely thank Dr Abdel-Moez E. Bayoumi for his time

from his busy schedule. My gratitude also goes to, Adrian Cuc, Paulette Goodman,

Jingjing “Jack” Bao, Greg Nall, Radu Pomirleanu, and Andrei Zagrai, my colleagues,

who made my time spent here so pleasant and fun.

I would like to express my forever gratitude and thanks to my mother, father, sister

and brother-in-law for their trust, incommensurable support, love and encouragement.

ABSTRACT

Among novel non-destructive evaluation techniques for structural health

monitoring, the magnetostrictive-tagged fiber-reinforced composites stand out as

especially suitable due to: (a) distributed sensory properties; (b) non-contact damage

detection; and (c) straight forward manufacturing implementation. Experimental data and

mathematical modeling of a magnetostrictive-tagged fiber reinforced composite specimen

under bending (flexural) loading are presented. A brief review of the state of the art

identifies previous work on axially loaded magnetostrictive composites, but finds no

previous work on bending. Description of bending specimen design and fabrication is

followed by the theoretical analysis and by the description of the experimental set-up and

equipment used. Several analysis models were used. Test data, with and without

magnetic annealing between loading cycles, is presented and results are discussed.

Numerical values for the stress and strain versus magnetic flux density coefficients are

given for both annealed and non-annealed cases. Piezomagnetic coefficients for the

magnetostrictive composite are calculated. The correlation between the results developed

in the presented paper for bending and previously published results for axial loading is

found to be within 10% after correction factors depending on the quantity of the

magnetostrictive material are applied. In conclusion, the usefulness of this method for

structural health monitoring and further work are discussed.

TABLE OF CONTENTS

ACKNOWLEDGEMENTS .........................................................................................................................II

ABSTRACT ................................................................................................................................................ III

TABLE OF CONTENTS ........................................................................................................................... IV

LIST OF FIGURES.................................................................................................................................... VI

LIST OF TABLES.....................................................................................................................................XII

1 INTRODUCTION................................................................................................................................1

1.1 PREVIOUS WORK ON MAGNETOSTRICTIVE TAGGED COMPOSITES FOR STRUCTURAL HEALTH

MONITORING (TENSILE EXPERIMENTS) .............................................................................................................1

1.2 CHARACTERISTICS OF ETREMA’S TERFENOL-D MAGNETOSTRICTIVE MATERIAL.................................5

1.3 PRESENT INVESTIGATION ...................................................................................................................7

2 ANALYSIS OF A MAGNETOSTRICTIVE COMPOSITE BEAM ...............................................8

2.1 STATIC ANALYSIS OF SIMPLY SUPPORTED COMPOSITE BEAM UNDER CENTRAL LOAD........................9

2.2 ANALYSIS OF A MAGNETOSTRICTIVE LAMINATED COMPOSITE ........................................................10

2.2.1 Micromechanics analysis of the MS tagged composite specimen ...........................10

2.2.2 Lamination Analysis ................................................................................................13

2.3 7-PLY BALANCED-ORTHOTROPIC MODEL..........................................................................................17

2.4 14-PLY CROSS-PLY ANGLE MODEL ...................................................................................................20



2.7 CONVERGENCE ANALYSIS ...............................................................................................................25

2.7.1 Convergence of the strain........................................................................................26

2.7.2 Convergence of stress..............................................................................................30

3 MAGNETOSTRICTIVE COMPOSITE BEAM EXPERIMENT.................................................39

3.1 SPECIMEN DIMENSIONAL DESIGN .....................................................................................................39

3.2 DESCRIPTION OF THE MS TAGGED COMPOSITE SPECIMEN .................................................................43

3.3 SPECIMEN PREPARATION FOR THE EXPERIMENT...............................................................................44

3.4 EXPERIMENTAL DESIGN ...................................................................................................................45

3.4.1 Description of the experiment .................................................................................45

3.4.1.1 Clamping fixture design ......................................................................................................48

3.4.1.2 Protective wood fixture design for the strengthen the gaussmeter probe.............................48

3.4.2 List of the equipment used .......................................................................................49

3.4.3 Equipment Calibration ............................................................................................50

3.4.4 LabView Virtual Instrument Environment...............................................................51

4 EXPERIMENTAL PROCEDURE AND RESULTS ......................................................................54

4.1 MAGNETIC ANNEALING....................................................................................................................54

4.2 TESTING PROCEDURE FOR MS TAGGED COMPOSITE BENDING EXPERIMENT.......................................54

4.3 RESULTS WITHOUT MAGNETIC ANNEALING......................................................................................55

4.3.1 Displacement and strain without magnetic annealing ............................................55

4.3.2 Magnetic flux density without magnetic annealing .................................................59

4.4 RESULTS WITH MAGNETIC ANNEALING BETWEEN LOADING-UNLOADING CYCLES............................63

4.4.1 Displacement and strain with magnetic annealing between loading-unloading

cycles .................................................................................................................................63

4.4.2 Magnetic flux density with magnetic annealing between loading-unloading cycles66

4.5 MAGNETIC FLUX DENSITY ANALYSIS ...............................................................................................70

4.6 PIEZOMAGNETIC STRESS AND STRAIN COEFFICIENTS ........................................................................72

4.6.1 Piezomagnetic stress coefficient, e31 without magnetic annealing ..........................73

4.6.2 Piezomagnetic stress coefficient, e31, with magnetic annealing...............................74

4.6.3 Piezomagnetic stress coefficient ,e31 ,using experimental magnetic flux density

without magnetic annealing and the 56-ply strain results model.........................................................75

4.6.4 Piezomagnetic stress coefficient, e31 using experimental magnetic flux density with

magnetic annealing between loading-unloading cycles and the 56-ply strain results model ..............77

4.6.5 Piezomagnetic strain coefficient d31 using experimental magnetic flux density

without magnetic annealing and 56-ply stress model ..........................................................................78

4.6.6 Piezomagnetic strain coefficient ,d31 ,for annealed specimen experiment and 56-ply

stress model .................................................................................................................................79

5 ANALYSIS AND DISCUSSION ......................................................................................................80

5.1 COMPARISON OF PIEZOMAGNETIC COEFFICIENTS FOR MAGNETOSTRICTIVE TAGGED COMPOSITES

WITH PREVIOUSLY PUBLISHED DATA ..............................................................................................................80

5.2 COMPARISON BETWEEN EXPERIMENTAL, DESIGN AND MODEL STRAIN FOR WITH AND WITHOUT

MAGNETIC ANNEALING...................................................................................................................................83

5.3 COMPARISON BETWEEN EXPERIMENTAL AND MODEL PIEZOMAGNETIC STRESS COEFFICIENT

WITHOUT MAGNETIC ANNEALING ...................................................................................................................84

5.4 COMPARISON BETWEEN EXPERIMENTAL AND MODEL PIEZOMAGNETIC STRESS COEFFICIENT WITH

MAGNETIC ANNEALING...................................................................................................................................84

6 CONCLUSIONS ................................................................................................................................86

7 BIBLIOGRAPHY ..............................................................................................................................89

8 APPENDIX.........................................................................................................................................91

LIST OF FIGURES

Figure 1 Magnetic flux in axial and thickness direction during axial loading of

magnetostrictive tagged composite specimens (White, 1999).................................... 3

Figure 2 Transverse (x-axis) magnetic flux density versus load: (a) neat resin;........... 4

Figure 3 Stress/magnetic flux density cyclic testing results: (a) without annealing

between cycles; (b) with annealing between cycles (Quattrone, Berman, and White,

1998) ......................................................................................................................... 5

Figure 4 Strain versus magnetic strength (Butler, 1988)............................................... 6

Figure 5 Relationships between a) magnetic flux density versus magnetic strength, b)

stress versus magnetic flux density, c) stress versus magnetic strength ..................... 6

Figure 6 Schematic of the static system ........................................................................ 9

Figure 7 Schematic of a generic lay-up....................................................................... 14

Figure 8 Balanced-orthotropic lay-up model used in the analysis .............................. 17

Figure 9 Cross-ply model used in the analysis............................................................ 20

Figure 10 Strain distribution of the 7, 14, 28 model with 0°/90° degree ply angle in

longitudinal direction................................................................................................ 26

Figure 11 Convergence of the strain result of the models to the experimental data ..... 28

Figure 12 Strain distribution for the 7, 14, 28 mathematical models with 0°/90° degree

ply angle in transversal direction. ............................................................................. 29

Figure 13 Stress – thickness distribution of the 7-ply balanced-orthotropic model with 0

degree fiber angle...................................................................................................... 30

Figure 14 Stress – thickness distribution in longitudinal direction for the 14-ply 0°/90°

degree fiber angle model........................................................................................... 31

Figure 15 Thickness distributions of longitudinal stress for 28-ply 0°/90° degree

fiber angle model. ..................................................................................................... 32

Figure 16 Thickness distributions of longitudinal stress for 56-ply 0°/90° degree

fiber angle model. ..................................................................................................... 32

Figure 17 Stress-thickness distribution for the 7, 14, 28-ply models with 0°/90°

degree fiber angle...................................................................................................... 33

Figure 18 Convergence of results on the specimen surface under maximum load

condition (Fmax = 58.9 N, Mmax = -88.29 Nm/m) as predicted by various models and

measured by experiment for stress results ................................................................ 35

Figure 19 Stress – thickness distribution in longitudinal direction for 14-ply 90°/0°


Figure 20 Stress - thickness distribution in longitudinal direction for the 28-ply

90°/0° degree fiber angle model ............................................................................... 36



Figure 22 Stress-Thickness Distribution for the 7 ply 0 degree angle model and 14, 28-

ply models with 90°/0° degree fiber angle................................................................ 38

Figure 23 General view of the experiment set-up ......................................................... 45

Figure 24 Schematic of the experiment and data flow.................................................. 46

Figure 25 Position of the gaussmeter probe, strain gage, and displacement transducer47

Figure 26 Concept design of the protective wood fixture ............................................. 48

Figure 27 Overview of the assembly of magnetic sensor and carved wood ................. 49

Figure 28 Front panel of the program created for the MS composite beam

experiment using the LabView virtual instrument.................................................... 52

Figure 29 The logic flow of the program created for the MS composite beam

experiment using the LabView virtual instrument.................................................... 53

Figure 30 Mechanical data resulting from the experiments without magnetic

annealing between loading cycles: displacement vs. load ........................................ 56


annealing between loading cycles: strain vs. load .................................................... 58


annealing: strain vs. displacement ............................................................................ 58

Figure 33 Magnetic flux density responses, without magnetic annealing, to bending

strain in top surface of MS-tagged composite specimens: superposed 10 consecutive

cycles ................................................................................................................... 60

Figure 34 Mean value and standard deviation of the magnetic field ............................ 61

Figure 35 The resulted magnetic field as function of measured strain ......................... 61

Figure 36 Mean values of magnetic flux density and standard deviation versus strain

for the specimen without annealing .......................................................................... 63

Figure 37 Displacement results versus applied force for annealed specimen............... 65

Figure 38 Strain results for the MS composite specimen with magnetic annealing

between cycles .......................................................................................................... 66

Figure 39 Magnetic flux density results for specimen with magnetic annealing

between cycles .......................................................................................................... 68

Figure 40 Mean value and standard deviation of magnetic flux density results vs. load

for the specimen with magnetic annealing between cycles ...................................... 68

Figure 41 Experimental magnetic flux density results versus strain with magnetic

annealing of the specimen between cycles ............................................................... 69

Figure 42 Mean value and standard deviation of magnetic flux density results vs. strain

for the specimen with magnetic annealing between cycles ...................................... 69

Figure 43 Magnetic flux density in relationship with model longitudinal stresses on the

surface of the composite specimen model ................................................................ 70

Figure 44 Standard deviation of the stress from the mean value versus magnetic flux

density ................................................................................................................... 71

Figure 45 Magnetic flux density versus strain on the top surface of the model

material considered and 0°/90° degree fiber angle without magnetic annealing...... 71

Figure 46 Plot of the average magnetic flux density vs. strain showing the trend lines in

determination of the piezomagnetic stress coefficient e31 for non -annealed specimen

case 74

Figure 47 Plot of the average magnetic flux density vs. strain showing the trend lines in

determination of the piezomagnetic stress coefficient e31 of magnetic annealing of

the specimen.............................................................................................................. 75

Figure 48 Plot of the average magnetic flux density with no magnetic annealing

specimen vs. 56-ply model strain showing the trend lines in determination of the

piezomagnetic stress coefficient e31.......................................................................... 76

Figure 49 Plot of the average magnetic flux density with magnetic annealing vs. 56-ply

model strain showing the trend lines in determination of the piezomagnetic stress

coefficient e31 ............................................................................................................ 77

Figure 50 Trend line for calculating the piezomagnetic strain coefficient.................... 78

Figure 51 Trend line for computing the piezomagnetic strain coefficient d31 for the case

of magnetic annealing of the specimen between cycles ........................................... 79

LIST OF TABLES

Table 1 Dimensions of the MS tagged composite specimen ....................................... 9

Table 2 Mechanical properties considered for the magnetostrictive-tagged woven

composite material specimen.................................................................................... 11

Table 3 Lay-up of the composite models for 14, 28 and 56-ply models ................... 21

Table 4 Comparison of the longitudinal strain on the specimen surface under

maximum load condition (Fmax = 58.9 N, Mmax = - 88.29 Nm/m) as predicted by

various models and measured by experiment ........................................................... 27

Table 5 Stress results on the surface for 7 balanced-orthotropic, 14, 28-ply cross-ply

models 34

Table 6 Stress results on the surface for 7, 14, and 28 ply 0 and 90°/0° cross-ply

models 38

Table 7 List of the equipment used in the magnetostrictive composite beam

experiment................................................................................................................. 50

Table 8 The displacement data without magnetic annealing ..................................... 56

Table 9 Strain data without magnetic annealing........................................................ 57

Table 10 Magnetic flux density results without magnetic annealing........................... 59

Table 11 Displacement results with annealing between loading-unloading cycles..... 64

Table 12 Strain results with annealing between loading-unloading cycles ................. 65

Table 13 Experimental magnetic flux density with magnetic annealing between

loading-unloading cycles .......................................................................................... 67

Table 14 Summary of magnetostrictive coefficient for MS tagged composites, as

determined in the present work and by previous investigators................................. 81

Table 15 Comparison of the design, model and measured strain ................................ 83

1 INTRODUCTION

In recent years, numerous applications in civil engineering construction used

composite materials. The increase of the usage of the composite materials in civil

construction imposed problems of evaluating the in-service composite civil engineering

structures. The evaluation of the composite is a wide area in which engineers and

researchers have proposed several technologies, for in-service Non-destructive evaluation

(NDE). Conventional NDE methods, initially developed for metallic structures, have

been shown to be less effective in monitoring composite structures due to the

micromechanical complexity of the composite material. New NDE technologies are

required. The NDE technology analyzed in this paper is proposed for the inspection of

advanced composite by using the magnetostrictive particle tagging technique.

Magnetostrictive-tagged composites permit: (a) distributed sensory properties; (b) non-

contact damage detection; and (c) straight forward manufacturing implementation.

1.1 PREVIOUS WORK ON MAGNETOSTRICTIVE TAGGED COMPOSITES FOR

STRUCTURAL HEALTH MONITORING (TENSILE EXPERIMENTS)

Terfenol-D is a magnetic anisotropy-compensated alloy TbxDy1-xFe2 that

shows a strong magnetostrictive behavior. The name Terfenol-D represents the

composition of the material and the original name of the Navy Laboratory at which the

work begun. Ter represents Terbium, Fe from iron, nol Naval Ordnance Laboratory and

D Dysprosium. A magnetostrictive (MS) material, such as Terfenol-D, produces

magnetic field when subjected to mechanical strain. This phenomenon is known as the

“converse magnetostrictive effect”. White and collaborators (White, Albers and

Quattrone, 1996; White and Brouwers, 1998) did extensive work on magnetostrictive-

tagged composites under axial loading. White (1999) reviewed the magnetostrictive

tagging methodology of composites for structural health monitoring and gave an update

on recent results. An experiment in which MS-tagged composite specimens were

subjected to uniaxial tension in a testing machine was presented. Neat resin specimens

tagged 2.24% by volume with magnetostrictive Terfenol-D powder were used. The

resulting magnetic field was measured in both the axial and thickness directions (Figure

1). Trovillion et al. (1999) studied the magnetic characteristics of neat resin and glass-

fiber-reinforced magnetostrictive composites subjected to axial load. The fiber reinforced

polymer composite (FRP) specimens consisted of 4 layers of continuous strand glass mat

fibers embedded in a polyester resin. Trovillion et al. (1999) studied the magnetic

characteristics of neat resin and glass-fiber reinforced magnetostrictive composites

subjected to axial load. The fiber reinforced polymer composite (FRP) specimens

consisted of 4 layers of continuous strand glass mat fibers embedded in a polyester resin.

Figure 1 Magnetic flux in axial and thickness direction during axial loading of

magnetostrictive tagged composite specimens (White, 1999)

The top lamina of the composite was impregnated with Terfenol-D powder at a

volume fraction of 2.24% for that lamina. The specimens were subjected to uniaxial

loading under load control at a rate of 0.02 kN/s.

Hall-effect device were used to measure the magnetic field response to subsequent

loading and unloading. As seen in Figure 2, both measuring devices gave similar results.

Magnetic annealing (i.e. rearranging the magnetic dipoles chains of the magnetostrictive

molecules through application of strong magnetic field) was performed by applying a

magnetic field through the thickness of the specimen using a pair of 800 Gauss

permanent magnets.

Nersessian N. and Carman G.P. presented at ASME conference in Orlando results

of tests for five different volume fraction composites namely 10 %, 20%, 30 %, 40 % and

50 %. The composite were tested under constant magnetic field with varying the

mechanical load and constant mechanical load with varying magnetic field conditions.

Results for the constant magnetic filed test indicated that modulus generally increases

with volume fraction and increasing H/Hmax. For low fields, initial dip is noticed in

modulus attributed to domains becoming more mobile at lower magnetic field levels.

Results presented for the constant load test show a strong dependence of strain output on

applied pre-stress.

(a)

0.0 0.2 0.4 0.6 0.8 1.00.0

0.5

1.0

1.5

2.0

2.5 Gauss Probe Hall Effect Chip

Mag

netic

Flu

x D

ensi

ty (

gaus

s)

Load (kN)

(b)

0.0 0.5 1.0 1.5 2.00.00

0.25

0.50

0.75

Ma

gn

etic

Flu

x D

en

sity

(g

auss

)

Load (kN)

Gauss Probe Hall Effect Chip

Figure 2 Transverse (x-axis) magnetic flux density versus load: (a) neat resin;

(b) composite sample (Trovillion et al, 1999)

Quattrone, Berman, and White (1998) studied the magnetic response repeatability

of MS tagged composites under cyclic loading. Terfenol-D active-tagged composites

were subjected to uniaxial tension and the magnetic response in the axial direction under

repeated loading and unloading was measured. Two types of experiments were

performed: without magnetic annealing between loading cycles (Figure 3a) and with

annealing (Figure 3b).

(a)

Slope 45/10 mG/MPa

(b)

Slope 140/10 mG/MPa

Figure 3 Stress/magnetic flux density cyclic testing results: (a) without annealing

between cycles; (b) with annealing between cycles (Quattrone, Berman, and

White, 1998)

Krishnamurthy, Anjanappa, and Wang (1999) consider health-monitoring detection

of delaminations in composite materials using an excitation coil and a sensing coil. The

open-circuit voltage induced in the sensing coil is proportional to the stress generated in

the magnetostrictive layer by the presence of the delamination.

1.2 CHARACTERISTICS OF ETREMA’S TERFENOL-D MAGNETOSTRICTIVE

MATERIAL

An application manual for designing the magnetostrictive transducers using Etrema

Terfenol-D was prepared by J. Butler (1988). A physical description of the

magnetostrictive material, with emphasis on Etrema Terfenol-D, was given. The nominal

Etrema Terfenol-D strain versus magnetic strength, H, as illustrated in this manual, is

shown in Figure 4.

Figure 4 Strain versus magnetic strength (Butler, 1988)

The theoretical stress versus magnetic flux density, stress versus magnetic strength

and magnetic flux density versus magnetic strength curves are presented in Figure 5.

Butler (1988) also provides an analysis of the properties of Etrema Terfenol-D with

emphasis on strain, stress and magnetic field relationships. Some design considerations

and fundamental concepts for designing magnetostrictive transducers are also presented.

Figure 5 Relationships between a) magnetic flux density versus magnetic strength, b)

stress versus magnetic flux density, c) stress versus magnetic strength

1.3 PRESENT INVESTIGATION

In the present investigation, studies of the theoretical and experimental of stress-

strain and magnetic flux density response of a woven tagged composite subjected to

bending loading were performed. Several two-dimensional simulation models for strain

and stress prediction in the composite were investigated. The analysis was performed

under two separate assumptions regarding woven composites modeling: (a) balanced-

orthotropic equivalent and (b) cross-ply equivalent. The experimental investigation

considers the bending moment created by the loading and unloading of a simply

supported MS-tagged composite beam using several weights. At each loading and

unloading step, the magnetic flux density induced by the converse magnetostrictive effect

was measured. Comparison between the experimental and analysis results are then

presented. Based on the comparison of experimental and analysis data, piezomagnetic

stress and strain coefficients under bending are calculated. Comparison of these

coefficients with the results obtained by other investigators for axial loading is

performed. It is shown that, although exact comparison is not fully possible since bending

and axial stress distributions are essentially different, the results are similar. The

piezomagnetic stress and strain coefficients determined in this paper for MS tagged

composite under bending can be used in industry as reference data for design of novel

NDE devices based in the magnetostrictive effect in MS tagged composites.

2 ANALYSIS OF A MAGNETOSTRICTIVE COMPOSITE BEAM

The analytical work performed on the MS tagged woven composite specimen

consisted of: (a) loading analysis; (b) micromechanical analysis; (c) lamination analysis.

One balanced-orthotropic and several cross-ply lamination models were used. Converge

analysis of the strain and stress predictions versus number of plies in the lamination

model were also performed.

A magnetostrictive composite stress-strain model has been developed for

simulating the behavior of the woven magnetostrictive composite. Two dimensional

mathematical model study was performed for the 7 ply balanced-orthotropic model, and

14, 28 and 56-ply multidimensional composite model with 0°/90° and 90°/0° degree.

Influence of the Terfenol-D was considered for the first and last ply of the 7-ply model,

in first 2 plies and last 2 plies for the 14-ply model, for the first 4 plies and last 4 plies of

the 28-ply model and the 8 plies for the 56-ply model.

2.1 STATIC ANALYSIS OF SIMPLY SUPPORTED COMPOSITE BEAM UNDER

CENTRAL LOAD

To determine the bending moment generated by the centrally-placed load, a static

analysis of simply supported composite beam was performed. In this analysis, the

dimensions and mechanical properties of the woven composite presented in Table 1 were

used. A schematic of the system and how the load is applied is shown in Figure 6. The

loading and unloading of the specimen was assumed to be performed using 2-kg weights.

Table 1 Dimensions of the MS tagged composite specimen

Characteristics Dimension [mm]

Span, L 600

Width, w 100

Thickness, t 6.5

Thickness of the lamina 0.92857

Composite Material Plate

Strain Gages

Force given by the bricks

2-Kg bricks

L

w

Figure 6 Schematic of the static system

The force exerted by the one brick is calculated considering the weight of 2 kg and

gravitational acceleration of 9.81 m/s2. Thus the force of the brick is

F = 2.09.81 = 19.62 N (1)

The moment per unit width exerted by the weights is:

F LM

4 w

(2)

Where L is the actual length of the beam, w- the width of the beam, F is the forced

applied by one weight. During the loading cycle, up to three weights were sequentially

put on the specimen and then removed. The maximum force value, corresponding to

three weights, was Fmax = 58.9 N. The corresponding moment per unit width was Mmax =

88.29 Nm/m.

2.2 ANALYSIS OF A MAGNETOSTRICTIVE LAMINATED COMPOSITE

2.2.1 Micromechanics analysis of the MS tagged composite specimen

Since a woven layer has fibers in two directions, warp and fill, the analysis of a

woven composite layer cannot be directly performed through the Classical Lamination

Theory (CLT), which assumes fibers aligned with just one direction. An equivalence

principle needs to be applied. Tsai (1992) suggested that the predictions of elastic

constants and strength of woven composite could be made using classical micro and

macro mechanics with appropriate empirical correction factors (Tsai, 1992, page 7-14).

One approach is to replace a woven composite layer with two equivalent conventional

layers representing the weave and the warp of the original fabric. The micro-mechanics

stiffness and strength formulas are then applied to the equivalent plies. One shortcoming

of this approach is that the order of 0 and 90 plies may strongly affect the results.

Another approach is the replace the balanced woven composite layer by an orthotropic

layer with averaged properties. For a woven composite having fibers aligned with the

loading axes, this hypothesis yields balanced-orthotropic behavior. In our analysis, both

approaches were taken. In either case, a conventional micromechanics analysis (Jones,

1999) was first applied to determine the basic MS composite properties from the

properties of its constituents, given in Table 2.

Table 2 Mechanical properties considered for the magnetostrictive-tagged woven

composite material specimen

Mechanical properties Dimension Young’s modulus of the fiber, Ef

1 72.4 GPa

Young’s modulus of the matrix, Em1 3.25 GPa

Young’s modulus for the Terfenol-D, EMS2 30 GPa

Poisson ratio of the fiber, f1 0.2

Poisson ratio of the matrix, m1 0.3

Fiber weight fraction, wf 0.3

Fiber density, f1 2.54 g/ml

Matrix density, m1 1.18 g/ml

Terfenol-D density, MS3 9.25·103 kg/m3

Thermal expansion coefficient for the fiber, αf1 5·10-6 m/m per ºC

Thermal expansion coefficient for the matrix, αm1 3.0·10-6 m/m per ºC

Terfenol-D weight fraction 15

Terfenol-D particle size4 38-40 μm

Note: 1Malik, 1992; 2Butler, 1988; 3De Laicheisserie, 1993, Trovillion et al., 1999.

The fiber volume fraction vf is calculated as (Malik, 1992, page 81, equation 2.5):

f

ff

f f

f m

w

vw 1 w

(3)

For the values given in Table 2, the fiber volume fraction values yield vf = 0.16.

Since the sum of the volume fractions is equal to 1, the volume fraction for the

matrix is vm= 1-vf for the plies which does not include the Terfenol-D, and vm = 1- vf -

vMS for the plies that contain MS material. The basic moduli of elasticity in the material

axes were calculated following (Jones) 1999 as:

1 f f m m MS MS

1 1 1 12 f f m m MS MS

E E v E v E v

E [(E ) v (E ) v (E ) v ]

(4)

The ply Poisson’s ratio was calculated as:

12 f f m m MS MS

221 12

1

v v v

E

E

(5)

Shear modulus for the fiber, matrix and Terfenol-D for each ply were defined as:

f

ff

EG

2 1

,

mm

m

EG

2 1

,

MSMS

MS

EG

2 1

(6)

Following Jones (1999, page 134), we calculate the shear modulus for the plies as:

f m MS12

f m MS m _ MS f MS MS f m

G G GG

v G G v G G v G G

(7)

Thermal expansion coefficients for the ply were:

f1 f f m1 m m MS MS MS1

f f m m MS MS

E v E v E v

E v E v E v

(8)

2 f f1 f m m1 m MS MS MS 1 121 v 1 v 1 v (9)

and

1

2

[ ]

(10)

2.2.2 Lamination Analysis

The elastic behavior of multidirectional plies can be described in terms of the

stiffness matrix, the compliance matrix, (Jones, 1975). The goal is to determine the

strain-stress distribution in the material. The plane stress stiffness matrix [Q] of a

orthotropic composite ply is computed as:

1 21 1

12 21 12 21

12 2 2

12 21 12 21

12

E E0

1 1

E E[Q] 0

1 1

0 0 G

(11)

Where E1 and E2 are given by Equation (4). The composite is assumed to consist of a lay-

up of multiple plies. The lay-up schematic for generic composite is presented in Figure 7

where x-y are the loading axes. In our case, the angle of the plies is either 0 or 90-degree.

y

x

z

θ 1

θ k

θ n

Figure 7 Schematic of a generic lay-up

Considering the transformation matrix:

2 2

2 2

2 2

cos sin 2 cos sin

[T] sin cos 2 cos sin

cos sin cos sin cos sin

(12)

The stiffness matrix in loading axes [Q]is:

T1 1[Q] [T] [Q] [T] (13)

Knowing the stiffness matrix, one calculates the extensional stiffness matrix [A],

coupling stiffness matrix [B], bending stiffness matrix [D] for the composite laminate:

k 1 k kk

2 2k k 1k 1

k

3 3k k 1k 1

k

[A] [Q] (z z )

1[B] [ [Q] (z z )]

2

1[D] [ [Q] (z z )]

3

(14)

The variable zk represents the distance from the midplane to the bottom of the kth

lamina; zk-1 represents the distance from the midplane to the top of the kth lamina. These

distances were calculate using the formula:

layerk

Nz h (k )

2

(15)

In matrix form the general equation is:

0N A B

M B D

(16)

Where {N} is the load vector, {M} is the moment vector, {ε0} is the mid-surface strain

vector, and {κ} is the curvature vector. For the design purposes, knowing the loading, the

state of deformation general solution is:

10 A B N

B D M

(17)

Next, the effect of thermal conditions is analyzed by considering only the thermal effect,

i.e. without loading F=0. The loads derived from the above-mentioned conditions are:

layerN

k 1T k 1 k k 1 k 1k 1

[N] [Q] (z z ) T

,

layerN 2 2k 1

k 1T k 1k 1

z z[M] [Q] ( ) T

2

(18)

Here, the term k 1 represents the coefficient of thermal expansion in the loading axes,

which is calculated using the transformation matrix and the coefficients of thermal

expansion in the material axes:

T

k 1k 1 k 1{ } [T] { } (19)

The loads and moments due to thermal effect:

layerN

k 1T k 1 k k 1 k 1k 1

{N} [Q] { } (z z ) T

(20)

layerN 2 2k k 1

k 1T k 1 k 1k 1

z z{M} [Q] { } ( ) T

2

(21)

The load and moment vector are composed as:

T

T

N N N

M M M

(22)

A general loading vector it is defined as:

0

1

2

0

1

2

N

N

N{NM}

M

M

M

(23)

Solving the state of deformation Equation (17) yields the loading axes strain on the top

and bottom of each ply is calculated.

topk 1 0 k 1

0 kbotk 1

{ } { } z { }

{ } { } z { }

(24)

Using the strains in loading axes, the strains in material axes are calculated for the top

and the bottom ply:

T1

k 1{ } [T] { } (25)

Consequently, the stresses in the longitudinal (L) and transversal (T) directions at the top

and the bottom of each ply are:

k 1top k 1 topk 1{ } [Q] { }

(26)

k 1 k 1bot k 1 bot{ } [Q] { }

(27)

2.3 7-PLY BALANCED-ORTHOTROPIC MODEL

X

Y

Z

Ply 1

Ply n

Ply k

Figure 8 Balanced-orthotropic lay-up model used in the analysis

The balanced-orthotropic model, presented in Figure 8, uses the assumption that a

balanced woven composite can be represented by a composite having averaged

properties. Using the longitudinal and transverse moduli for a uni-directional fiber

composite layer we calculated the averaged properties:

1 2 L T

1E E (E E )

2

(28)

The MS Terfenol-D volume fraction and Poisson’s ratio were

vMS = 2.24% and νMS = 0.3 for the plies with Terfenol-D, and zero for the other plies.

Two indexes were defined in order to make a distinction between the laminas with

and without Terfenol-D, Nlayer=7, Nlayers=6.. Performing the CLT analysis yields the

balanced-orthotropic (BO) matrices:

7 6

6 7BO

6

6.596 10 7.291 10 0

[A] 7.291 10 6.596 10 0

0 0 9.668 10

(N/m) (29)

12 13

13 13BO

3.638 10 2.27 10 0

[B] 2.274 10 9.09 10 0

0 0 0

(N) (30)

BO

244.47 25.897 0

[D] 25.897 190.99 0

0 0 34.346

(N·m) (31)

The thermal effect was included for complete analysis procedure but since the cure

temperature was the same with environmental temperature, k 1T 0 , then all the stresses

due to thermal effects are zero.

Considering only the bending moment, the load vector is zero and the moment

vector is:

vector

M

{M } 0

0

(Nm/m) (32)

The values for this loading vector are:

0

0

0 (N / m)NM

88.29 (N)

0

0

(33)

The strain and curvature vector is determined then using the general state of

deformation equation mentioned above. Thus the strain and curvatures are:

0

0

0

0

and

0.366

0.050

0

(1/m) (34)

The values for the stresses in longitudinal direction for the top and bottom layer

are:

L _ top

12.72

8.72

5.23

1.74

1.74

5.23

9.08

(MPa)

L _ bot

9.08

5.23

1.74

1.74

5.23

8.72

12.72

(MPa) (35)

The strains in the longitudinal direction for the top and bottom layer are:

top

1347

962

577

192

192

577

962

(με)

bot

962

577

192

192

577

962

1347

(με) (36)

2.4 14-PLY CROSS-PLY ANGLE MODEL

X

Y

Z

Ply 1

Ply n

Ply k

Figure 9 Cross-ply model used in the analysis

In the cross-ply analysis (Figure 9), each woven composite layer is approximated

by a couple of unidirectional cross-ply layers. The cross-ply layer can be either 0/90or

90/0. Since the choice between 0/90and 90/0 directly affects the accuracy of the

results on the specimen surfaces, we initially considered both cases in our analysis.

Additionally, the number of the layers was gradually increased from 14 to 28 and 56, and

a convergence study was performed. During this convergence study, as the number of

layers were doubled, the thickness of the layer was correspondingly halved, such that the

overall thickness of the specimen was maintained. The difference in 14, 28 and 56-ply

mathematical models procedure from the balanced-orthotropic model is that the Young’s

modulus of elasticity in longitudinal and transversal direction were not averaged The 28-

ply and 56-ply models were obtained by subsequent subdivision of the 0/90 layers and

application of symmetry principles. The ply angle distributions for the 14, 28 and 56-ply

models are presented in Table 3:

Table 3 Lay-up of the composite models for 14, 28 and 56-ply models

Number of Layers Lay-up

14 [(0/90)3/0]S

28 [(0/90)7/90]S

56 [(0/90)14]S

Terfenol-D is assumed present in the first two layers and last two layers and the for

the Young’s modulus values in the longitudinal and transversal directions are:

E1 = 14.96 GPa, E2 = 3.94 GPa. The Young’s modulus of elasticity values for the

layers that does not contain magnetostrictive material are: E1 = 14.31 GPa, E2 = 3.84

GPa. The A, B and D matrices for the 14-ply model were:

7 6 11

6 7 914

11 9 6

6.6 10 7.29 10 2.51 10

[A] 7.29 10 5.60 10 1.83 10

2.51 10 1.83 10 9.67 10

(N/m) (37)

12 13

13 1314

3.64 10 2.27 10 0

[B] 2.27 10 9.09 10 0

0 0 0

(N) (38)

14

244.47 25.9 0

[D] 25.9 190.99 0 (N m)

0 0 34.35

(39)

Stress and strain results in longitudinal direction for the 14-ply 0°/90° cross-ply

laminate, at the top and bottom of the plies, are:

L _ top

18.01

3.95

12.32

2.57

7.39

1.28

2.46

0.00

- 0.64

- 4.93

- 1.92

- 9.85

- 3.29

-15.44

(MPa),

L _ bot

15.44

3.29

9.85

1.92

4.93

0.64

0.00

- 2.46

- 1.28

- 7.39

- 2.57

-12.32

- 3.95

-18.01

(MPa),

top

1191

1021

851

680

510

340

170

0

-170

-340

-510

-680

-851

-1021

(με),

bot

1021

851

680

510

340

170

0

- 170

- 340

- 510

- 680

- 851

-1021

-1191

(με) (40)

For the 90°/0° cross-ply laminate the stress and strain results are:

L _ top

5.94

19.81

4.14

12.64

2.48

6.32

0.83

0.00

- 3.16

- 1.66

- 9.48

- 3.31

-16.51

- 5.10

(MPa),

L _ bot

5.10

16.51

3.31

9.48

1.66

3.16

0.00

- 0.83

- 6.32

- 2.48

-12.64

- 4.14

-19.81

- 5.94

(MPa),

top

1524

1307

1089

871

653

436

218

0

- 218

- 436

- 653

- 871

-1089

-1307

(με),

bot

1307

1089

871

653

436

218

0

- 218

- 436

- 653

- 871

-1089

-1307

-1524

(με) (41)


The 28-ply model is computed in a similar manner with the 14-ply model. The changes

occur in the indices, the ply angle, and the thickness of the lamina. The A, B and D

matrices were:

7 6 11

6 7 914

11 9 6

6.1 10 7.29 10 1.38 10

[A] 7.29 10 6.1 10 3.18 10

1.38 10 2.18 10 8.97 10

(N/m) (42)

12 13 13

13 13 1314

13 13

3.64 10 1.25 10 1.05 10

[B] 2.27 10 9.09 10 1.05 10

1.05 10 1.05 10 101.23

(N) (43)

14

231.19 25.9 0

[D] 25.9 204.27 0 (N m)

0 0 30.59

(44)

The stress and strain results are presented next:

L _ top

19.06

4.53

16.33

3.83

13.03

3.06

10.42

2.38

7.82

1.70

5.21

1.02

2.61

0.34

0.00

- 1.30

- 0.68

- 3.91

- 1.36

- 6.52

- 2.04

- 9.12

- 2.72

-11.73

- 3.49

-14.97

- 4.18

-17.69

L _ bot

17.69

4.18

14.97

3.49

11.73

2.72

9.12

2.04

6.52

1.36

3.91

0.68

1.30

0.00

- 0.34

- 2.61

- 1.02

- 5.21

- 1.70

- 7.82

- 2.38

-10.42

- 3.06

-13.03

- 3.83

-16.33

- 4.53

-19.06

top

1259

1169

1079

989

899

809

719

629

539

449

259

269

179

89

0

-89

-179

-269

-359

-449

-539

-629

-719

-809

-899

-989

-1079

-1169

bot

1169

1079

980

899

809

719

629

539

449

359

269

179

89

0

-89

-179

-269

-359

-449

-539

-629

-719

-809

-899

-989

-1079

-1169

-1259

(45)


The 56-ply model was obtained by subsequent subdivision of the 0 and 90 layers

and application of symmetry principles. The changes appear in setting the indexes and the

thickness of the lamina. The A, B and D matrices for the 56-ply model were:

7 6 11

6 7 956

11 9 6

6.22 10 7.29 10 2.82 10

[A] = 7.29 10 5.97 10 2.06 10 (N/m)

2.82 10 2.06 10 9.67 10

(46)

-13

-1356

13

-72.14 -3.98 10 0

[B] = -3.98 10 72.14 0 (N)

0 0 9.09 10

(47)

56

224.47 25.90 0

[D] = 25.90 211 0 (N m)

0 0 34.35

(48)

The maximum strain on the surface of the specimen model was found to be:

= 1296 (49)

The corresponding maximum stress was:

= 19.6 MPa (50)

2.7 CONVERGENCE ANALYSIS

The convergence of the strain and stress distribution for various models was

studied. Attention was focused on the longitudinal stress and strain under maximum load

conditions (Fmax = 58.9 N, Mmax = - 88.29 Nm/m). The strain values predicted on the top

and bottom surfaces were also compared with the experimental value measured during

the tests described in Chapter 4.

2.7.1 Convergence of the strain

Figure 10 shows the distribution of the longitudinal strain for the 7-ply balanced-

orthotropic model and the 14-ply and 28-ply 0/90 cross-ply models. (The 56-ply results

were not plotted on Figure 10 to avoid cluttering the drawing). The sign convention used

in Figure 10 is negative to the left and positive to the right.

14 plies 0/90 deg 28 plies 0/90 deg

7-ply balanced-orthotropic model

Figure 10 Strain distribution of the 7, 14, 28 model with 0°/90° degree ply angle in

longitudinal direction

The study shows that the largest strain value was predicted by the 7-ply balanced-

orthotropic model. This value is also very close to the experimental value (1347 με vs.

1333 με). The convergence of the strain distribution for the cross-ply models is apparent

as the models are becoming more refined i.e. strain distribution is getting closer to the

experimental data as the number of plies employed in the model is increased. The strain

values predicted by each model, at the surface of the composite, and comparison with

experimental data, are presented in Table 4.

Table 4 Comparison of the longitudinal strain on the specimen surface under

maximum load condition (Fmax = 58.9 N, Mmax = - 88.29 Nm/m) as predicted

by various models and measured by experiment

Strain [με]

7-ply balanced-orthotropic

model

14-ply 0/90 model

28-ply 0/90 model

56-ply 0/90 model

Experimental

1347 1191 1259 1296 1333

The reason for the difference between the balanced-orthotropic model and the other

models is that the balanced-orthotropic model considers each of the plies at the same

angle. Thus, the strain that could be carried is larger than the more realistic models in

which the strains are lower because the angle of ply is alternating from 0 to 90 degree.

The results show that the plies with transverse fiber to the longitudinal direction of

measuring (90 degree fiber angle) shrink and oppose the longitudinal stress direction. The

convergence of the strain data of the models is presented in Figure 11.

Str

ain

[m

icro

str

ain

]

1000

1055

1110

1165

1220

1275

1330

1385

Number of the plies

Strain response of the models for surface Experimental Baseline Strain

Strain response for quasi-isotropic model

7 28 5614

Figure 11 Convergence of the strain result of the models to the experimental data

The figure shows that the maximum strain given by the 7-ply balanced-orthotropic

model, is approximately the same as the experimentally obtained strain. The graph also

shows that the strain prediction by the cross-ply models improves with increasing number

of layers.

Balanced –orthotropic model

14-ply 0/90 deg model

28-ply 0/90 deg model

Figure 12 Strain distribution for the 7, 14, 28 mathematical models with 0°/90° degree

ply angle in transversal direction.

The strain distribution for each model in transversal direction is presented in Figure

12.The strains on the left side of the symmetry line represent positive strains and on the

right side represent negative strains. This case shows the strain distribution for the 14 and

28 ply models have close results. It is also observed that for the 7-ply model the strains

are larger.

2.7.2 Convergence of stress

For the 7-ply balanced-orthotropic model, the longitudinal stress distribution is

presented in the Figure 11. It can be noticed the stress distribution is rather linear, in

accordance with the balanced-orthotropic assumption. The slight discontinuity between

the outer first and last layers and the other layers is due to the outer layers being stiffer

due to presence of MS tagging in the modified layers, layer 0 and 6.

Stresses -Thickness Distribution in Longitudinal Direction for 7 plies

Figure 13 Stress – thickness distribution of the 7-ply balanced-orthotropic model with 0

degree fiber angle.

The stress distribution for the 14-ply 0°/90° cross-ply model is presented in Figure

15.

Stress - Thickness Distribution for 14 plies 0/90 degree fiber angle model

Figure 14 Stress – thickness distribution in longitudinal direction for the 14-ply 0°/90°

degree fiber angle model

In this model, the stress distribution is different then the stress distribution for the

7-ply balanced-orthotropic model.It can be noticed the stress in the layers with 90-degree

fiber angle is lower than the stress in the layers with 0-degree fiber angle. For the layers

that contain the magnetostrictive material an increased stress is noticed. The cause of this

is the increased stiffness due to Terfenol-D. Similar model behavior is observed for 28

ply and 56-ply 0°/90° cross-ply models as can be shown in Figure 15 and 16.

Stress - Thickness Distribution for 28 plies 0/90 degree fiber angle model

Figure 15 Thickness distributions of longitudinal stress for 28-ply 0°/90° degree fiber

angle model.

Stress - Thickness Distribution for 7, 14, 28, 56 plies models

Figure 16 Thickness distributions of longitudinal stress for 56-ply 0°/90° degree fiber

angle model.

The longitudinal stress distribution for the 14, 28-ply 0/90 cross-ply model is

presented in Figure 17. In the cross-ply models, the stress distribution is substantially

different then the stress distribution from the 7-ply balanced-orthotropic model. The

stresses in the layers with 90-degree fiber angle are substantially smaller. This produce

alternating changes from high stress to low stress as the layer stiffness change from the

high to low in accordance with the 0 and 90 orientations of the cross-ply laminate.

7 p lie s b a lan ce d -o rth o tro p ic m o d e l

1 4 p lie s 0 /9 0 d eg m o d e l

2 8 p lie s 0 /9 0 d eg m o d e l

Figure 17 Stress-thickness distribution for the 7, 14, 28-ply models with 0°/90° degree

fiber angle

The 56 ply 0°/90° degree fiber angle model is presented separately since the graph

would have been hard to visualize and the stress distribution for each model would have

been too hard to follow if they were superimposed on Figure 17. The results reveal that

the layers that contain Terfenol-D are stiffer than the other layers.

The 7 ply balanced-orthotropic model stress distribution shows that the model does

not accurately simulate the real woven composite material because larger stress values

were observed than in the cross-ply. This statement is made from a stress distribution

point of view. Depending on the fiber orientation and the stress amplitude, the stress

carried is consequently proportional. For the 0 degree fiber orientation the stress is

larger. For the fiber oriented to a 90-degree angle to the longitudinal direction, the stress

amplitude is small. However, this shows that even the transversal fiber carry load. For

analysis purposes, Table 4 presents the largest stresses on the surface for each model.

Table 5 Stress results on the surface for 7 balanced-orthotropic, 14, 28-ply cross-

ply models

7 ply balanced-orthotropic model

[MPa]

14 ply 0°/90° model [MPa]



12.72 18.01 19.06 19.63

Table 5 shows that the longitudinal stresses obtained by the 14, 28, and 56-ply

models are larger than the stresses obtained with the balanced-orthotropic model. These

results are due to increased stiffness in the longitudinal direction and less stiffness in

transverse direction typical of cross-ply models. In the case of the balanced-orthotropic

model the stiffness in both direction are the same, but of lower value than the longitudinal

cross-ply stiffness..

Figure 18 presents the results of Table 5 in graphical form. It can be appreciated

that the balanced-orthotropic model grossly underestimates the stresses, whereas the

cross-ply models shown a definite convergence. It can be estimated that, by further

increasing the number of layers in the model, convergence towards an asymptotic value

of around 20 MPa would be obtained.

0

5

10

15

20

25

0 7 14 28 56

Number of the layers

Lo

ng

itu

din

al s

tres

s [M

Pa

]

balanced-orthotropic model cross-ply models

Figure 18 Convergence of results on the specimen surface under maximum load

condition (Fmax = 58.9 N, Mmax = -88.29 Nm/m) as predicted by various

models and measured by experiment for stress results

Stress analysis for 90/0cross-ply models was also performed. As before, the

maximum load case (Fmax = 58.9 N, Mmax = - 88.29 Nm/m) was considered. The next

graphs refer to the mathematical models with 90°/0° degree fiber angle. In Figure 19, 20

and 21 the 14, 28 and 56-ply models with 90°/0° degree fiber angle are presented.



Figure 20 Stress - thickness distribution in longitudinal direction for the 28-ply 90°/0°


Stress - Thickness Distribution in Longitudinal Direction for 56 plies 90/0 Degree Fiber Angle Model



In Figure 22 the stress distributions for the 14 and 28 cross-ply models with 90°/0°

degree fiber angle are presented. 56-ply model is not included to not complicate the graph

and make it hard to visualize the stress – thickness distribution. In this graph, the 7-ply

balanced-orthotropic model is also represented. The thickness distribution for the

balanced-orthotropic model is similar to 0°/90° models stress distribution. Here is visible,

in addition, the influence of the Terfenol-D in increasing the stiffness of the material with

results in larger stress

7-ply balanced-orthotropicmodel

14-ply 90/0 deg model28-0ply90/0 deg model

Figure 22 Stress-Thickness Distribution for the 7 ply 0 degree angle model and 14,

28-ply models with 90°/0° degree fiber angle

For analysis purposes Table 6 presents the largest stresses on the surface for each

model. In this case, the comparison shows that the stress carried in the transverse

direction, when the top ply has 90 degree ply angle, is smaller than the stress given by

the balanced-orthotropic model.

Table 6 Stress results on the surface for 7, 14, and 28 ply 0 and 90°/0° cross-ply

models

7 ply balanced-orthotropic model [MPa]




12.72 5.94 5.55 5.4

3 MAGNETOSTRICTIVE COMPOSITE BEAM EXPERIMENT

The experimental results alluded to the previous section were obtained during a

carefully conducted experiments, as described next.

3.1 SPECIMEN DIMENSIONAL DESIGN

For pre-design purpose, the mechanical properties of the specimen were evaluated

with a very simplified theory that took into account the following data:

1 Volume fraction of the fiber: vf = 0.5

2 Volume fraction of the matrix vm = 1- vf = 0.5.

3 Young’s modulus of the E-glass fiber the: 9

fE 72.4 10 Pa

4 Young’s modulus of the resin 9

mE 3.25 10 Pa .

We assumed an average Young’s modulus for random fiber composite calculated

with (Malik, 1992, equation 3.42, page 130):

11 22

3 5E E E

8 8

(51)

E11 and E22 are calculated using the rule of mixture for longitudinal modulus, E11,

and transverse modulus E22 (Malik, 1992, equations 3.26,3.27, page 123).

11 f f m mE E v E v (52)

f m22

f m m f

E EE

E v E v

(53)

The result is 9E 17.808 10 Pa .

The shear modulus for the composite (Malik, 1992, equation 3.43, page 130)

material is assumed to be:

11 22

1 1G E E

8 4

(54)

The result is 9G 6.061 10 Pa .

Simple statics was used to determine the bending moment of a simply supported

composite beam centrally loaded by a concentrated force. To dimension the specimen for

further computations, some mechanical and dimensional quantities were assumed. The

equivalent Young’s modulus, E, calculated above was considered. The width was

measured on the composite specimen, w = 100 mm. The maximum deflection was

considered not to exceed the value of qmax = 36 mm based work previously done in the

area of materials testing.

To determine the thickness of the composite, the length, L to thickness, t, ratio was

calculated:

1

3max

max

2 w E qk

F

,

Lk

t

(55)

The maximum force was calculated based on the force of one weight and the

number of weights to be used in the experiment. The maximum force to be used in the

experiment was considered to be the force exerted by ten weights, Fmax = 196.2 N. The

thickness is then

max

a

3 F kt

E w

(56)

t = 5.051 mm.

The length of the specimen was calculated as:

L t k (57)

with L = 449.9 mm, thus t = 6.5 mm.

Furthermore the length of the specimen is adjusted to L = 579.6 mm. Rounding up,

the final length for following calculations is L = 600 mm.

The design dimensions of the specimen we choose to be:

Length – 1000 (mm) (58)

Width – 100 (mm) (59)

Thickness – 6.75 (mm) (60)

The next step was to calculate the moment of inertia for determining the deflection.

The moment of inertia was calculated using the well-known formula for a rectangle:

3w tI

12

(61)

The result is: 3 4I 2.563 10 mm .

Then the deflection is

3

F

LFq

E I 24

(62)

The result is

F

4.104

8.208

12.312

16.417

q [mm]20.521

24.625

28.729

32.833

36.937

for

19.62

39.24

58.86

78.48

F (N)98.1

117.72

137.34

156.96

176.58

(63)

At this point, the maximum stress and strain were calculated. Because of symmetry, the

stress and strain have same absolute value when equally distanced from mid-plane. The

values for the strain are:

83.6

167.5

250.8

334.4

(MPa)417.9

501.5

585.1

668.7

752.3

445

889

1334

1778

( )2223

2668

3112

3557

4002

(64)

The maximum stress was max

tM

2I

and the maximum strain was max

max E

.

Assuming the flexural strength to be 6

FS 150 10 Pa and a safety factor of SF=1.5,

admissible tensile stress is calculated as

F

a

S

SF

(65)

The result is 6

a 100 10 Pa . The admissible strain value is

a

a E

(66)

εa = 5617 με. (67)

Comparison of the result with the strain predicted above shows that adequate safety

is built into the specimen. These simple calculations were used to ensure that the

specimen was adequate for experimentation.

3.2 DESCRIPTION OF THE MS TAGGED COMPOSITE SPECIMEN

An MS tagged composite specimen was fabricated at Reichhold Chemicals

(Raleigh, N.C.) by binding 7 layers of fiberglass woven roving 36oz./sq.yd. and Atlac

580-05 Urethane-modified Vinyl Ester Resins. The specimen was 1000 mm long, and

had a 100 mm by 6.5 mm cross section. MS Terfenol-D tagging powder was used in the

two outside layers in the middle 500 mm of the span. Out of 1000 grams of resin, 250

grams had the MS powder. The resin was cured with 1 % MEKP (methyl ethyl ketone

peroxide) at room temperature for approximately 90 minutes. The target weight fraction

of the glass fibers in the composite was wf = 30 %. The weight fraction of MS Terfenol-D

tagging in the resin was 15 %.

3.3 SPECIMEN PREPARATION FOR THE EXPERIMENT

The specimen was instrumented with strain gages in the mid-span section on the

upper and lower surfaces. The specimen surface was prepared for bonding the strain

gages using the procedures given by Measurements Group (Raleigh, N.C.). The bonding

area was made planar using abrasive sandpaper. Next, the surface was cleaned using

Measurements Group surface conditioner M-Prep Conditioner A (a water based acidic

surface cleaner). Measurements Group strain gages types CEA-06-125UT-120 were

applied with M-Bond AE-10 Kit gage adhesive. To protect the strain gages and reduce

their exposure to the environment, the strain gages were covered with one sided Scotch

tape.

3.4 EXPERIMENTAL DESIGN

3.4.1 Description of the experiment

LakeshoreGaussmeter

StrainIndicator

MagnetostrictiveComposite

National InstrumentAmplifier

SolartronDisplacementTransducer

Figure 23 General view of the experiment set-up

The overall configuration of the experiment can be seen in Figure 23. The

experimental setup permitted the simultaneous measurement of beam deflection,

mechanical strains, and magnetic response of the magnetostrictive (MS) tagged beam.

Data flow through the data acquisition and processing modules is presented schematically

in Figure 24. The MS-tagged composite beam was supported on concrete blocks (500

mm equivalent span) and loaded gradually with an incremental number of clay weights (2

kgf = 19.6 N each). Strain gages were placed on both the upper and lower surfaces of the

specimen at the mid-span and were connected in a half bridge configuration to the strain

indicator. The magnetic flux density produced from the magnetostrictive particles was

measured by the gaussmeter.

Magnetic Sensor

Strain Gages

Strain Indicator

Gaussmeter

SCXI Unit DAQ PCLVDT

Displacement Transducer

Figure 24 Schematic of the experiment and data flow

For correlation purposes, the mid-span displacement was also measured. An LVDT

displacement transducer and a non-magnetic (aluminum and brass) clamping fixture were

used. Details of the mid-span instrumentation are given in Figure 25. Initial trials showed

that the strain gauge and LVDT electromagnetic fields did not influence the Gaussmeter

reading of the magnetostrictively induced magnetic field.

The MS tagged composite beam, supported on concrete blocks, was loaded

gradually with 2-kg weights. In order to avoid excessive strain in the composite material,

the number of loading weights was limited to three. The test procedure was as follow:

load the specimen, acquire data and repeat until all the three weights were on the

specimen. Next, reverse the procedure and perform unloading. The strain gages were

connected to a strain indicator, which provided the actual strain value at top and bottom

surfaces. At the same time, the displacements were measured with an LVDT

displacement transducer. Hence, the relationship between loads, displacements and

strains could be established. Simultaneously, the magnetic field developed by the MS

active tagging particles was detected using the gaussmeter. The gaussmeter provided the

value of the magnetic flux density read by the probe on the surface of the MS tagged

composite material. The data was collected using National Instruments LabView

software and associate hardware consisting of a SCXI signal-conditioning module, and

the Gateway computer. The information was processed using National Instruments

LabView software. The data collection was developed for simultaneous on-line operation

with the strain gages, displacement transducer and the gaussmeter. At each loading step,

complete data collection was performed. The configuration of the strain gages,

displacement transducer and position of the gaussmeter probe are presented in Figure 25.

Strain Gage

Gaussmeter Probe

The rod of the Displacement Transducer

Clamping Fixture

Figure 25 Position of the gaussmeter probe, strain gage, and displacement

transducer

3.4.1.1 Clamping fixture design

A clamping device was designed in order to achieve a better position of the

displacement transducer on the magnetostrictive composite material .The fixture is

composed of 2 aluminum parts (see Figure 25), which are kept attached to the composite

material with a brass bolt and nut. A brass bolt was chosen for use in order to diminish

the magnetic influence of all the components to the gaussmeter probe.

3.4.1.2 Protective wood fixture design for the strengthen the gaussmeter probe

A special fixture had to be constructed to ensure proper and consistent alignment of

the gaussmeter probe with respect to the composite surface. Its design is presented in

Figure 26:

Wood

Sensor Tip2 mm Sensor Rod

Figure 26 Concept design of the protective wood fixture

Considering the fragility of the gaussmeter probe and the advice from Lakeshore

Cryotonics, a wood fixture for the protection of the probe tip was designed and

fabricated. The distance from the tip of the sensor to the upper surface of the specimen

was 2 mm. The open area was covered with clear tape. The sensor rod was attached to the

wood fixture using Pro Seal Blue RTV Silicone made by Pacer Technology (Rancho

Cucamonga, CA.). A view of this apparatus is given in the Figure 27.

Sensor Tip

Machine Carved Wood

Sensor Rod

Figure 27 Overview of the assembly of magnetic sensor and carved wood

3.4.2 List of the equipment used

In this experiment, many different pieces of equipment were used to achieve the

simultaneous measurements of beam deflection, mechanical strains, and magnetic

response of the MS tagging particles. The equipment list is presented in Table 7.

Table 7 List of the equipment used in the magnetostrictive composite beam

experiment

Name Model Manufacturer

Strain Gages CEA-06-125UT-120 Measurement Group, Inc.

Strain Indicator P-3500 Measurement Group, Inc.

Gaussmeter Model 450 Lakeshore Cryotonics, Inc.

LVDT displacement transducer

B-50 Solartron Company

SCXI amplifier unit SCXI-1000 National Instruments Company

LabView NI Professional Measurement Suite

National Instruments Company

PCMCIA card DAQcard-AI-16E-4 16 channel

National Instruments Company

Composite Material Specimen

Reichold Chemicals

Permanent Magnets 6”x4”x1” Ceramic 8 Magnet Adams Magnetic Products

3.4.3 Equipment Calibration

In order to obtain valid data the equipment had to be calibrated. The calibrated

equipment was Solartron Displacement Transducer, Lakeshore Gaussmeter, National

Instruments Data Acquisition Card, and Measurement Group Strain Indicator.

Calibration of the displacement transducer implied establishing the relationship

between the measured displacements and the output voltage. The equation that defines

the displacement is:

outV

Displ (mm)2

(68)

For the Lakeshore gaussmeter, the calibration was achieved by setting the range

field of the magnetic field and corresponding output voltage. The range of the magnetic

field was established to be G06.0 . The corresponding output voltage for the maximum

and minimum values of the range field was V3 . The National Instruments data

acquisition card was calibrated using National Instrument software. For the Measurement

Group strain indicator the calibration required adjusting the reading of the output voltage

with a calibration coefficient for the real strain. This calibration factor was 0.45.

3.4.4 LabView Virtual Instrument Environment

The LabView visual programming language software used in the experiments was

purchased from National Instruments Company. For processing and visualizing the data,

we created a virtual instrument visual interface using the LabView programming

language. The front panel and the logic flow are shown in the Figure 28 and 29.

The front panel is shown in the Figure 28.

Figure 28 Front panel of the program created for the MS composite beam experiment

using the LabView virtual instrument

The front panel shows the virtual instrument for recording the increase and

decrease of the loading force, and the reading channels for displacement transducer,

gaussmeter, and strain gages. Figure 28 shows three graphs: (a) applied force versus

measured displacement; (b) milistrains given by the strain indicator versus applied force;

and (c) magnetic field read by the gaussmeter versus applied force. Another value

presented on the front panel is the brick weight, in case different types of weights are

used. Total force applied represents the force applied by the weights present at the time of

measurement. The logic flow of the software is presented in the Figure 29.

Figure 29 The logic flow of the program created for the MS composite beam

experiment using the LabView virtual instrument.

The data flows from the left to right. There are two “while loops” which control all

the flowing data in the program.The principal while loop is required by the language

programming in LabView. The inside loop contains the elements of creating , formatting

and saving of the data for each type of reading: displacements, magnetic

fields,milistrains. On the left are the input elements for the data. These elements are the

inputs from the data acquisition card. On the right, inside the while loop, are the elements

which read the data from the saved files in order to display, in the form of the graph, the

information on the front panel. There are some logic loops which control the logic of

taking and displaying the measurements.

4 EXPERIMENTAL PROCEDURE AND RESULTS

4.1 MAGNETIC ANNEALING

Before detailing the experimental procedure, we need to define the term “magnetic

annealing”. Magnetic annealing consists of applying a strong magnetic field that aligns

the magnetic domains of the MS tagging particles. Thus, the initial response under load is

enhanced. We used a pair of permanent magnets (Adams Magnetic Products, Garland,

TX) generating 740 Gauss magnetic flux density. The magnetic annealing was applied

with the specimen in the no-load position. During the set of experiments with magnetic-

annealing, we applied the magnetic annealing between every loading cycle.

4.2 TESTING PROCEDURE FOR MS TAGGED COMPOSITE BENDING EXPERIMENT

The MS-tagged composite beam, supported on concrete blocks (500 mm equivalent

span), was loaded and unloaded gradually with 2-kg weights. In order to avoid excessive

strain in the composite material, the number of loading weights was limited to three. The

test procedure was as follow: gradually load the specimen, acquire data and repeat until

all the weights were on the specimen. Next, reverse the procedure and perform gradual

unloading. The strain gages were connected to a strain indicator, which provided the

actual strain value at top and bottom surfaces. At the same time, the displacements were

measured with an LVDT displacement transducer. Hence, the relationship between loads,

displacements and strains could be established. Simultaneously, the magnetic field

developed by the MS active tagging particles was detected using the gaussmeter. The

gaussmeter provided the value of the magnetic flux density read by the probe on the

surface of the MS tagged composite material. The data was collected using National

Instruments LabView software and associated hardware consisting of a SCXI signal-

conditioning module, and the Gateway computer. The information was processed using

National Instruments LabView software. The data collection was developed for

simultaneous on-line operation with the strain gages, displacement transducer and the

gaussmeter. At each loading step, complete data collection was performed.

4.3 RESULTS WITHOUT MAGNETIC ANNEALING

In the first set of experiments the acquisition of the data was performed without

annealing the specimen between cycles. Displacement, strain and magnetic field levels

were also analyzed.

4.3.1 Displacement and strain without magnetic annealing

Table 7 shows the displacement results obtained for the experiment without

magnetic annealing and the force generated by the weights in loading and unloading

procedure.

Table 8 The displacement data without magnetic annealing

Displacement [mm]

Cycle 1

Cycle 2

Cycle 3

Cycle 4

Cycle 5

Cycle 6

Cycle 7

Cycle 8

Cycle 9

Cycle10

0 0.221 0.211 0.229 0.235 0.236 0.243 0.242 0.248 0.251 0.269

19.62 3.097 3.122 3.082 3.057 3.105 3.104 3.116 3.122 3.128 3.128

39.24 5.436 5.567 5.431 5.363 5.571 5.529 5.549 5.581 5.481 5.448

58.86 8.296 8.492 8.232 8.168 8.466 8.356 8.484 8.521 8.231 8.187

39.24 5.645 5.693 5.53 5.04 5.676 5.637 5.695 53579 3.201 3.213

19.62 3.202 3.225 3.161 3.134 3.2 3.205 3.204 3.203 3.201 3.213

0 0.257 0.256 0.256 0.262 0.249 0.269 0.268 0.272 0.291 0.283

Next, the dependency of the displacement with the applied force is presented in

Figure 28. It can be seen that there is a linear relationship between displacement and

force. It must be emphasis that this data depends on the accuracy of the data acqusition

card and National Instrument amplifier.


annealing between loading cycles: displacement vs. load

Good repeatability of the readings during loading and unloading was observed.

Both of these observations validate the experimental set-up. Table 8 presents the strain

data obtained. The precision of the voltage output of the Measurement Group Strain

Indicator was lower than National Instruments devices. For a very small strain displayed

on the strain indicator for instance 2-3 microstrains for zero loads, the voltage output had

the same value of noise. Thus a perfect zero output couldn’t be obtained. The error of the

output voltage versus real strains is lower than 2%. The linear relationship between strain

and force is presented in the Figure 29.

Table 9 Strain data without magnetic annealing

Strain ε [milistrains]

Cycle 1 Cycle 2 Cycle 3 Cycle 4 Cycle 5 Cycle 6 Cycle 7 Cycle 8 Cycle 9 Cycle10 Average strain

Standard deviation

%

0 0 0 0 0 0 0 0 0 0 0 0 0

19.62 443.9 461.8 466.6 454.6 469.7 468.7 456.6 463.5 454.6 467.7 460.7 1.78

39.24 858.1 883.6 872.2 853.2 888.1 879.1 870.8 871.2 877.7 886.3 874.0 1.30

58.86 1342.2 1347.1 1345.0 1309.5 1343.7 1318.1 1330.2 1339.9 1326.8 1332.3 1333.4 0.93

39.24 875.3 906.4 888.8 877.4 900.8 895.0 888.1 895.3 888.1 897.7 891.2 1.10

19.62 459.4 473.9 468.7 463.2 477.0 463.9 467.7 473.2 470.8 475.2 469.2 1.21

0 0 0 0 0 0 0 0 0 0 0 0 0


annealing between loading cycles: strain vs. load

By analyzing the strain data, linearity and repeatability of the strain versus force

data can be seen. The strain and displacement linear relationship is presented in the

Figure 30, which shows a linear dependency of the strain with displacement for a

specimen subjected to loading.


annealing: strain vs. displacement

4.3.2 Magnetic flux density without magnetic annealing

The magnetic flux density response of the specimen is presented in Table 10. The

specimen at the time of data acquisition had not been annealed for more than two days.

Thus, the results are indicative of the response of a field-deployed MS-tagged composite

material. The small difference between readings in different cycles illustrates the

deviation expected from the use of general-purpose magnetic flux density measurement.

The data acquired by equipment depends on the position of the gaussmeter probe, the

distance from the sensor tip to the surface of the specimen, and the environment. These

observations indicate that non-annealed MS-tagged composites present satisfactory

repeatability and small hysteresis in bending. White and Brouwers (1998) made similar

observations for axially loaded specimens and axially measured magnetic fields.

Table 10 Magnetic flux density results without magnetic annealing

Magnetic Flux Density B [mG]

Cycle 1 Cycle 2 Cycle 3 Cycle 4 Cycle 5 Cycle 6 Cycle 7 Cycle 8 Cycle 9 Cycle10

Average value

Standard deviation %

0 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0 0

19.62 4.73 4.79 5.67 5.57 5.39 5.25 5.41 3.36 3.88 3.92 4.79 0.16

39.24 9.48 9.96 10.68 10.64 9.92 10.39 8.96 8.96 7.98 7.73 9.56 3.97

58.86 16.60 16.53 16.63 16.05 16.74 16.37 16.85 15.15 15.02 14.43 16.03 5.05

39.24 12.27 11.08 10.96 10.65 11.41 10.23 11.11 9.26 8.15 8.86 10.39 11.54

19.62 6.62 5.85 6.31 5.94 6.79 5.3 6.14 4.07 3.49 2.69 5.31 25.23

0 0.87 1.6 0.62 0.81 0.58 0.29 1.09 0.81 1.06 0.89 0.86 38.37

The zero load reading was substracted from the other values in order to eliminate

the influence of the environmental magnetic field. Ten cycles were considered in order to

see the repeatability of the magnetic flux results. Some difference in magnetic flux

density readings resulted between cycles were noticed. Though, the strain and

displacement reading were almost identical. This difference between cycles in magnetic

flux density readings arises from the fact that the magnitude of the magnetic flux density

is very small, of order of miligauss, and hence the influence of the environmental

magnetism has an impact on the stress-induced magnetic flux density readings. This

influence made the gaussmeter display to constantly fluctuate during readings. Graphical

representation of the magnetic flux density versus applied force is given in Figure 33.

The standard deviation of the magnetic flux density is presented in the Figure 34.

0

2

4

6

8

10

12

14

16

18

0 20 40 60 80

Force [N]

cycle 1cycle 2cycle 3cycle 4cycle 5cycle 6cycle 7cycle 8cycle 9cycle 10M

agn

etic

Flu

x D

ensi

ty B

[mG

]

Figure 33 Magnetic flux density responses, without magnetic annealing, to bending

strain in top surface of MS-tagged composite specimens: superposed 10

consecutive cycles

Figure 34 Mean value and standard deviation of the magnetic field

One can also see in Figure 33 that the magnetic field has a small hysteresis. This

phenomenon is present because of changes in the microscopic magnetic status of the

tagging particles (White and Brouwers, 97-98, pp 8).

Figure 35 The resulted magnetic field as function of measured strain

Figure 35 shows the raw data of magnetic flux density vs. strain obtained during the

ten cycles. Figure 35 shows that strain can be predicted when the magnitude of the

magnetic flux density B is known. At the time the experiment was performed and the data

was acquired, the specimen was not annealed for more than two days. Thus, the results

are indicative of the response of a field-deployed MS-tagged composite material. The

small difference between readings in different cycles illustrates the deviation expected

from the use of general-purpose magnetic flux density measurement equipment.

Figure 36 shows the average-trend correlation between the magnetic flux density

and strain for the specimen without annealing. The data presented are the mean values

and standard deviation of the experimental results. These observations indicate that non-

annealed MS-tagged composites present satisfactory repeatability and small hysteresis in

bending. White and Brouwers (1998) made similar observations for axially loaded

specimens and axially measured magnetic fields.

Figure 36 Mean values of magnetic flux density and standard deviation versus strain

for the specimen without annealing

4.4 RESULTS WITH MAGNETIC ANNEALING BETWEEN LOADING-UNLOADING

CYCLES

4.4.1 Displacement and strain with magnetic annealing between loading-unloading

cycles

A new set of experimental data was obtained by applying magnetic annealing

between cycles. Ten loading-unloading cycles were conducted, with magnetic annealing

between each cycle. The annealing procedure was performed with 2 magnets with

average magnetic field magnitude of 740 G. Adams Magnetic Products (Garland, TX)

supplied the magnets for this experiment. The procedure followed was analogous to that

previously use, with exception that annealing between cycles (load and unload the

specimen with all the weights) was performed. The annealing time was on average 1.5

minutes. Displacement, strain and magnetic field levels were collected and analyzed. The

strain and displacement results showed a very good repeatability with less than 1.7%

standard deviation. The displacement results are presented in Table 11.

Table 11 Displacement results with annealing between loading-unloading cycles

Displacement [mm]


0 0.215 0.308 0.418 0.395 0.368 0.277 0.357 0.343 0.278 0.356

19.62 2.927 3.004 3.049 3.109 2.898 2.964 3.027 3.007 2.895 3.052

39.24 5.479 5.611 5.589 5.64 5.433 5.557 5.604 5.56 5.405 5.637

58.86 8.03 8.467 8.318 8.487 8.259 8.411 8.475 7.925 8.135 8.442

39.24 5.588 5.765 5.739 5.738 5.585 5.691 5.75 5.68 5.527 5.753

19.62 3.029 3.146 3.16 3.212 3.078 3.101 3.131 3.111 3.03 3.168

0 0.35 0.414 0.515 0.517 0.405 0.363 0.426 0.419 0.406 0.471

The displacement results are presented graphically in Figure 35. The graph shows

the linearity of the displacement with loading and unloading. The strain results are

presented in the Table 11.

Figure 37 Displacement results versus applied force for annealed specimen

Table 12 Strain results with annealing between loading-unloading cycles

Strains ε [milistrains]


Average value

Standard deviation,

%

0 0 0 0 0 0 0 0 0 0 0 0 0

19.62 440.4 432.1 432.5 443.5 434.9 437.0 437.3 436.6 439.0 434.9 436.8 0.80

39.24 886.7 882.6 879.1 882.6 877.4 875.3 874.3 869.4 881.9 880.1 878.9 0.55

58.86 1310.2 1359.5 1323.0 1365.4 1344.0 1340.9 1336.4 1296.4 1245.4 1359.9 1337.1 1.6

39.24 895.3 896.0 895.3 894.3 878.4 890.1 876.7 880.1 889.5 898.8 889.4 0.91

19.62 447.0 452.8 447.0 457.0 444.6 448.0 447.3 450.1 445.9 452.5 449.2 0.84

0 0 0 0 0 0 0 0 0 0 0 0 0

It should be noted that the strains in Table 12 are very similar to those in Table 9,

indicating that magnetic annealing does not influence the mechanical response. The

linearity of the strain with applied force is presented in Figure 38.

0

200

400

600 800

1000

1200 1400

1600

0 20 40 60 80

Force [N]

Str

ain

[m

icro

stra

ins]

cycle 1 cycle 2 cycle 3 cycle 4 cycle 5 cycle 6 cycle 7 cycle 8 cycle 9 cycle 10

Figure 38 Strain results for the MS composite specimen with magnetic annealing

between cycles

4.4.2 Magnetic flux density with magnetic annealing between loading-unloading cycles

The experimental magnetic field data are given in Table 13. The 10 cycles show

that the measurements are repeatable and consistent. As in the experiment without

magnetic annealing, the small difference between readings in different cycles illustrates

the deviation to be expected from the use of general-purpose magnetic flux density

measurement equipment.

Table 13 Experimental magnetic flux density with magnetic annealing between

loading-unloading cycles

Magnetic Flux Density B [mG]

Cycle 1

Cycle 2

Cycle 3

Cycle 4

Cycle 5

Cycle 6

Cycle 7

Cycle 8

Cycle 9

Cycle10 Average value

Standard deviation

%

0 0.05 0.00 1.6 0.70 -0.25 -0.05 -0.05 0.47 0.33 0.64 0.3 0

19.62 11.03 8.18 9.98 8.42 11.92 16.18 14.99 17.10 14.88 15.29 12.80 24.37

39.24 23.88 20.63 50.45 16.45 20.43 24.14 22.89 25.5 22.64 23.02 21.99 11.09

58.86 32.59 28.19 29.18 26.75 29.05 32..5 31.23 33.91 30.65 30.37 31.43 7.85

39.24 33.79 28.69 29.38 26.37 28.81 32.93 31.54 34.94 30.56 30.13 30.71 8.04

19.62 32.84 27.57 27.63 25.83 30.02 33.88 33.42 35.11 30.87 30.36 30.75 9.46

0 36.34 30.27 29.95 20.33 24.03 29.49 29.72 30.06 26.20 25.29 28.17 14.73

The plot of magnetic flux density versus force is given in Figure 39. It can be seen

that, on the increasing branch of the curve, the magnetic field response is almost linear.

The maximum magnetic flux values are about double the values obtained without

annealing (32 mG vs. 16 mG, respectively). This proves that magnetic annealing

increases the initial magnetostrictive response. The increasing branch also shows a slight

negative curvature, i.e., the marginal response per unit load becomes less pronounced as

the load increases. On the decreasing branch of the curve, the marginal response is very

small, and significant hysteresis is present. These observations indicate that annealed

MS-tagged composites possess a higher initial response, but also a very pronounced

hysteresis. Mean average and standard deviation of the magnetic flux density are

presented in Figure 38.

Figure 39 Magnetic flux density results for specimen with magnetic annealing

between cycles

Figure 40 Mean value and standard deviation of magnetic flux density results vs. load

for the specimen with magnetic annealing between cycles

Figure 41 shows the correlation between the magnetic flux density and strain for

the specimen with annealing between cycles. Mean average and standard deviation of the

magnetic flux density is presented in Figure 42.

Figure 41 Experimental magnetic flux density results versus strain with magnetic

annealing of the specimen between cycles

Figure 42 Mean value and standard deviation of magnetic flux density results vs.

strain for the specimen with magnetic annealing between cycles

4.5 MAGNETIC FLUX DENSITY ANALYSIS

Based on the stress results of the model and experimental magnetic flux density

data Figure 43 shows the relationship between two variables. The graph shows

convergence of stress results for the models without averaging the Young’s Modulus.

Figure 43 Magnetic flux density in relationship with model longitudinal stresses on the

surface of the composite specimen model

The results show linearity between longitudinal stress obtained using bending

moment of the weights involved in experiment and magnetic flux density (B) measured

during the experiment. For purpose of calculating the magnetostrictive strain, coefficient

standard deviation of the stress from average has been produced. This is presented in

Figure 44.

Figure 44 Standard deviation of the stress from the mean value versus magnetic flux

density

Similar behavior is observed in the magnetic flux density versus calculated strain of

the models. This is presented in following figure, Figure 45.

Figure 45 Magnetic flux density versus strain on the top surface of the model material

considered and 0°/90° degree fiber angle without magnetic annealing.

4.6 PIEZOMAGNETIC STRESS AND STRAIN COEFFICIENTS

For monolithic magnetostrictive Terfenol-D material, the piezomagnetic

coefficients, e31, and d31 are well known and available from standard references (Butler,

1988; De Lacheisserie, 1993). These coefficients correlate the magnetic flux density, B3,

with the strain, ε1, and stress, σ1. For magnetostrictively tagged composites, such simple

relationships cannot be directly established, since the composite contains three separate

phases (fiber, resin, and magnetostrictive tagging powder) of which only one is

magnetostrictive. However, effective piezomagnetic coefficients can be established to

allow a macroscale relationship between the recorded magnetic flux density and the strain

and stress in the tagged composite. To this purpose, we have calculated both the overall

e31 and the d31 coefficients, as shown in the following sections. The e31 coefficient was

calculated because it can be determined directly, from the measured strain and magnetic

flux. The d31 coefficient was calculated in order to facilitate comparison with the d31 data

reported in the literature for axially loaded MS composite specimens. It should be noted

that the d31 coefficient cannot be determined directly from experimental data alone, since

no simple device exist to directly measure the stress inside an composite beam in

bending. We determined the d31 coefficient using the stress predicted by the lamination

analysis, and the magnetic flux density measured during the experiments. Details of this

process are given next.

4.6.1 Piezomagnetic stress coefficient, e31 without magnetic annealing

By using the mean value of magnetic field versus strain for the loading steps,

piezomagnetic strain coefficient was calculated. The trend line helps to determine this

coefficient as a ratio of magnetic flux density B, and strain .

According to De Lacheisserie (1993), the piezomagnetic stress coefficient e31,

defines the relationship between the applied strain, ε1, and the resulting flux density B3, in

the form:

3 31B d (69)

3 31 εB e (70)

The Figure 46 shows the trend line and the point that was chosen to determine the

piezomagnetic stress coefficient. The trend line was considered only for the loading steps.

Thus, the piezomagnetic stress coefficient e31, for the case when the specimen is not

magnetically annealed, is:

-431

N13.7 10

A me

(71)

0

2

4

6

8

10

12

14

16

18

0 500 1000 1500

Experimental Strain [microstrains]

Mag

net

ic F

lux

Den

sity

B [

mG

]

Slope 13.6/1000 mG/με

Figure 46 Plot of the average magnetic flux density vs. strain showing the trend lines

in determination of the piezomagnetic stress coefficient e31 for non -

annealed specimen case

4.6.2 Piezomagnetic stress coefficient, e31, with magnetic annealing

Piezomagnetic stress coefficient e31 was calculated using the average of the

magnetic flux density function of the strain experimentally obtained when annealing the

specimen between cycles.

0

5

10 15

20 25

30

35

40

0 500 1000 1500

Experimental Strain ε [microstrains]

Exp

erim

enta

l M

agn

etic

Fie

ld B

[m

G]

Slope 24.5/1000 mG/με

Figure 47 Plot of the average magnetic flux density vs. strain showing the trend lines

in determination of the piezomagnetic stress coefficient e31 of magnetic

annealing of the specimen

Considering the plot in Figure 47 the piezomagnetic stress coefficient e31 for the

case when the specimen is annealed we obtained:

431

N24.5 10

A me

(72)

4.6.3 Piezomagnetic stress coefficient ,e31 ,using experimental magnetic flux density

without magnetic annealing and the 56-ply strain results model

We calculated the piezomagnetic stress coefficient, e31, using the plot of the

average values of the experimentally determined magnetic flux density, B3, and the

calculated strain ε1. In our investigation, we used several lamination models (7-ply

through 56-ply), and compared their predicted strain with experimentally measured

values. This showed that the 56-ply model was most adequate.

An analysis of the piezomagnetic stress coefficient, when correlating between the

experimental magnetic flux density results of non-annealed specimen case and the strain

results of the 56-ply model, is now shown. For the piezomagnetic stress coefficient e31 we

used:

0 2 4 6 8

10 12 14 16 18

0 200 400 600 800 1000 1200 1400

56-ply model strain [microstrains]

Exp

erim

enta

l M

agn

etic

Flu

x

Den

sity

B [

mG

]

Figure 48 Plot of the average magnetic flux density with no magnetic annealing

specimen vs. 56-ply model strain showing the trend lines in determination

of the piezomagnetic stress coefficient e31.

Thus, the values for the piezomagnetic stress coefficient e31 was found as:

431

N13.25 10

A me

(73)

4.6.4 Piezomagnetic stress coefficient, e31 using experimental magnetic flux density

with magnetic annealing between loading-unloading cycles and the 56-ply strain

results model

In a similar way with previous coefficient determination, the correlation between

experimental magnetic flux density data for the specimen annealed between cycles and

the strain data of the 56-ply model was determined. The value for the coefficient is then:

431

N25 10

A me

(74)

56-ply Model Strain [microstrains]

0 5

10 15 20 25 30 35

0 200 400 600 800 1000 1200 1400

Exp

erim

enta

l M

agn

etic

Flu

x D

ensi

ty B

[m

G]

Figure 49 Plot of the average magnetic flux density with magnetic annealing vs. 56-

ply model strain showing the trend lines in determination of the

piezomagnetic stress coefficient e31

4.6.5 Piezomagnetic strain coefficient d31 using experimental magnetic flux density

without magnetic annealing and 56-ply stress model

The piezomagnetic strain coefficient d31 was determined using the experimental

magnetic flux density of the not annealed specimen and the 56-ply model stress results

(Figure 50). Based on the trend line in Figure 50 the piezomagnetic strain coefficient, d31

was calculated by considering the ratio of the magnetic flux density to strain. From the

graph, we calculated the piezomagnetic strain coefficient d31 for the not annealed

specimen to be:

1431

m10.2 10

Ad

(75)

0 2

4 6 8

10 12 14 16 18

0 5 10 15 20 25 56-ply model Stress [MPa]

Exp

erim

enta

ly M

agn

etic

Flu

x D

ensi

ty B

[m

G]

Figure 50 Trend line for calculating the piezomagnetic strain coefficient

4.6.6 Piezomagnetic strain coefficient ,d31 ,for annealed specimen experiment and 56-

ply stress model

In a similar way, the piezomagnetic strain coefficient is determined for the case of

magnetic annealing of the specimen between cycles. As stated before, the piezomagnetic

coefficient d31 was considered to be the ratio between experimentally obtained magnetic

flux density B and 56-ply model stress. Considering the graph above for the

piezomagnetic strain coefficient the value found is:

1431

m15.6 10

Ad

(76)

Figure 51 Trend line for computing the piezomagnetic strain coefficient d31 for the

case of magnetic annealing of the specimen between cycles

5 ANALYSIS AND DISCUSSION

5.1 COMPARISON OF PIEZOMAGNETIC COEFFICIENTS FOR MAGNETOSTRICTIVE

TAGGED COMPOSITES WITH PREVIOUSLY PUBLISHED DATA

The present authors used the experimentally-derived curves reported by other

investigators (Quatttrone, Berman, and White, 1996) to identify average values of stress-

magnetic flux density coefficients as depicted in Figure 3. These coefficients express the

magnetic flux density developed when a given axial stress is applied to the MS tagged

neat-resin composite specimen (Please recall that the piezomagnetic stress coefficient

defines magnetic flux density per unit strain, and vice-versa). The values of these

coefficients were entered in the last two columns of Table 14. Table 14 shows that the

“raw” piezomagnetic strain coefficients obtained in our composite experiments are about

4 to 10 times lower then those published by Quattrone, Berman, and White (1998) for

neat-resin tension specimens.

Table 14 Summary of magnetostrictive coefficient for MS tagged composites, as

determined in the present work and by previous investigators

Bending Series Tensile Series

Not annealed Annealed Not annealed Annealed

Composite type Woven Woven Neat resin Neat resin

Chained No No Yes Yes

Measured B Field Direction Transverse Transverse Axial Axial

Piezomagnetic Stress Coefficient (N/Am) e31 =13..710-4 e31 = 24.510-4 X X

Raw d31 = 10.210-14 d31 = 15.610-14

Corrected for MS volume

fraction d31 = 33.710-14 d31 = 51.510-14

Note: The tensile series for neat resin not annealed was published by Trovillion, J et all

(1999). The tensile series data for annealed neat resin was published by Quattrone, Berman, and

White (1998). The bending series refers data were published by Giurgiutiu, et al. (1999). The

piezomagnetic coefficients were calculated by the authors of the present report from the data

published in the references cited above.

The reasons for this difference are:

1. Different type if stress distribution, bending vs. axial. The axial stress

distribution is uniform across section, while the bending distribution is

linearly varying from maximum positive to negative values. Hence, only a

small portion of the composite (outer fibers) is actually loaded to the

maximum stress. Thus, for the same stress values in the outer fibers, a

tensile specimen will produce much more magnetism than bending

specimen. Since the gaussmeter probe was measuring the overall response,

it results that its readings are expected to be lower in the bending specimen

than in the tensile specimen.

2. Previous investigators, performing tensile experiments, measured the

magnetic flux density, B3, parallel to the applied stress, σ3. In our bending

experiments, the applied load generated stress in the σ1, which is

perpendicular to the direction of the measured B3.

3. In the tensile net-resin specimens used by the previous investigators, the

Terfenol-D particles were dispersed throughout the neat-resin body,

whereas in our composite bending experiments an important portion of the

specimen was occupied by the reinforcing fibers. Hence, the Terfenol-D

particles could only be dispersed in the 81% volume allocated to resin.

Additionally, Terfenol-rich resin was applied only in the outer layers of the

composite lay-up. In essence, this means that, in our composite bending

specimens, the Terfenol volume ratio was 0.68%, whereas in neat-resin

tensile specimens it was 2.24%, which is 3.3 times higher.

When the difference in MS tagging volume fraction was compensated for, the

values of piezomangetic strain coefficients measured by us in bending without annealing

between cycles, became comparable with those measured by Quattrone, Berman and

White (1996) under axial loading and similar non-annealing conditions. This shows

convergence of the two investigations and increased confidence in the industrial

application potential of this technology.

5.2 COMPARISON BETWEEN EXPERIMENTAL, DESIGN AND MODEL STRAIN FOR

WITH AND WITHOUT MAGNETIC ANNEALING

Since the 56-ply model 0°/90° degree fiber angle was considered to be the closest

in simulation of the composite material, the results of the model are included in

comparison with classical theoretical prediction and experimental strain obtained.

Comparing the information about the strain, results and theoretical values, we can

conclude that the results meet the predicted behavior of the composite material specimen.

The comparison is presented in the Table 15.

Table 15 Comparison of the design, model and measured strain

Experimental Strain [με] με με

Non Annealed Annealed

0 0 0 0 0

19.62 445 432 460 436

39.24 889 842 874 878

58.86 1334 1296 1333 1337

39.24 889 842 891 889

19.62 445 432 469 449

0 0 0 0 0

As can be seen for both of annealing and not annealing, the results of the predicted

and mathematical model are close to experimental results. These comparisons consider

the loading and unloading of only three weights, and for the mathematical model used, 56

ply model with 0°/90° degree fiber angle.

5.3 COMPARISON BETWEEN EXPERIMENTAL AND MODEL PIEZOMAGNETIC

STRESS COEFFICIENT WITHOUT MAGNETIC ANNEALING

The purpose of comparing the piezomagnetic coefficients is to check how close is

the model to the experimental data. In determining the mathematical piezomagnetic stress

coefficient, the magnetic flux density experimentally obtained has been used with model

strain calculated. The model strain calculated was considered for the 56 ply with 0°/90°

degree ply angle. For the experiment, considering the non-annealed specimen the

piezomagnetic stress coefficient has been obtained (64):

431

N13.7 10

A me

. As

previously shown for the 56-ply model has been obtained (66):

431

N13.25 10

A me

.

By comparison, both coefficients, the 56 ply mathematical model proves to give good

level of accuracy in prediction.

5.4 COMPARISON BETWEEN EXPERIMENTAL AND MODEL PIEZOMAGNETIC

STRESS COEFFICIENT WITH MAGNETIC ANNEALING

Here a comparison between piezomagnetic stress coefficient what was obtained for

the 56-ply model and the one for the material with annealing between loading cycles is

presented. As before, the experimental piezomagnetic stress coefficient was calculated as

the ratio of the value of magnetic flux density and 1000 strains. This fraction was

calculated using the graph presented in Figure 47. Using the value for the piezomagnetic

stress coefficient of the 56-ply model, which was obtained in paragraph 4.5.4,

431

Ne 25 10

A m

to the value for the composite material (on the surface) that was

obtained at paragraph 4.5.2,

431

Ne 24.5 10

A m

, we observe that the results are very

close. In this case the difference between piezomagnetic coefficients is small. The reason

for this is that the magnetic flux density (B) is different from the not annealed case. The

prediction of the mathematical model when is used experimental magnetic field data is

very good.

6 CONCLUSIONS

This thesis has presented data obtained from the experimental study of

magnetostrictive-tagged fiber reinforced composites under bending (flexural) load and

from two-dimensional mathematical models. A brief review of the state of the art

revealed that no previous work on bending of magnetostrictive-tagged composites has

been published yet.

A magnetostrictive-tagged composite bending specimen was designed, fabricated,

and tested. An analysis procedure of laminate woven composites was applied to predict

the state of stress and strain in a simple supported composite beam specimen.

Convergence of results was studied using an increasing number of layers from 7 to 56.

The main equipment and the experimental setup and testing procedure have been

described. Test data was obtained with and without annealing between loading cycles.

The results were processed and presented in terms of magnetic response vs. strain and vs.

predicted stress. For tests without annealing between cycles, the results showed linearity

and low hysteresis. For tests with annealing between cycles, an increase in response and

pronounced hysteresis were both observed.

The piezomagnetic stress coefficient, e31, was calculated using measured magnetic

flux vs. measured strain for both annealed and non-annealed experiments. The

piezomagnetic strain coefficient, d31, was also calculated using the measured magnetic

flux density and the stress predicted by a 56-ply model with 0/90 cross-ply layers.

Comparison between the present bending results and the previously published results for

axial loading was performed.

The values of these coefficients are presented in chapter 5. Piezomagnetic stress

coefficient d31 has been calculated using the mathematical model for future reference.

The strain data used are from the 56-ply model with 0°/90° degree fiber angle.

This paper has proved that the strain vs. magnetic flux density coefficient for

magnetostrictive-tagged composites is portable from neat-resin composite to fiber

reinforced composites and can be used as a “master” design coefficient. This concept, not

previously reported in literature, resolves a major issue related to the design of

magnetostrictive-tagged composites with various fiber - reinforcement architecture,

which share a common design criteria base on strain or deflection constraints. Further

work needs to be performed to identify the variation of strain vs. magnetic field

coefficient with magnetostrictive tagging density and to develop a three dimensional

mathematical model. On this line, one has to verify whether the saturation phenomenon

previously reported in neat-resin magnetostrictive-tagged composites, also appears in

fiber-reinforced composites, and to find tagging density values at which it becomes

present. This investigation has shown that a clear relationship can be established between

the magnetic flux density measured at the surface of a MS tagged composite and the

stress state inside the specimen. Previous investigators had proved this relationship for

axial loading. In this paper, for the first time, MS tagged composite results under bending

loading are presented. Both axial results (presented by previous investigators) and

bending results (present in this paper) indicate a good consistency and repeatability of the

magnetic flux response resulting from loading and unloading of the MS-tagged

composite.

This paper also showed the concept that magnetostrictive tagging can be applied

sparsely, i.e., only in critical places. In our specimen, magnetostrictive tagging was

applied only in the outer layer and only in the mid-span region. This concept allows

tailoring of the magnetostrictive tagging and results in considerable cost saving and

improved application efficiency.

The impact of using MS-tagged composite for in-service NDE of composite

structure could be very significant. Unlike conventional strain gages, which need to be

permanently applied and wired on the structure, the MS-tagging method can be done by

simple surface scanning using a traveling magnetic flux gage. This opens the opportunity

for measuring stress patterns in large structural areas to identify stress distributions and

“hot spot”.

At present, the MS-tagging NDE methodology is still in its infancy. Considerable

work needs still to be done before large-scale industrial implementation is possible.

Further research work needs to cover: (a) development of portable, hand-held equipment

for magnetic flux measurement of large structural areas using surface scanning

techniques; (b) further characterization of magnetic response of MS-tagged composite of

various tagging densities and distributions; (c) evaluation of long-term behavior under

various environmental conditions

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theoretical and experimental investigation of magnetostrictive tagged composite beams

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