limit loads for structural elements made of … · limit loads for structural elements made of...
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Institute of General Mechanics RWTH Aachen University, Germany
M. Chen, A. Hachemi, D. Weichert
PVP 2012, July 15-19, 2012, Toronto, Canada
LIMIT LOADS FOR STRUCTURAL ELEMENTS MADE OF HETEROGENEOUS MATERIALS
METHOD
INTRODUCTION
PROBLEM
A mechanical structure or structural element made of composite materials operates beyond the elastic limit :
Determination of material properties
Variable loads with unknown evolution in time
Direct methods combined with homogenization technique
Direct methods give information on serviceability without calculating the evolution of mechanical field quantities.
Instantaneous collapse Limit Analysis Failure under variable loads Shakedown Analysis
Prediction of the global material properties by using homogenization theory
CONTENTS
Basic concepts • Static shakedown theorem
Direct methods applied to composites • Elements of homogenization theory • Boundary conditions • Consideration of kinematic hardening
Failure criterion of composites • Loci of yield strength fitting
Numerical examples • Periodic fiber reinforced metal matrix composites
• Porous material
Conclusions
BASIC CONCEPTS
ε
σ
σmax
ο ο
σ
ε ο
σ
ε ο
σ
ε
σmax σmax
σmax
σmin σmin
σmin σmin
Purely elastic Shakedown Low-cycle fatigue Ratcheting
𝜀𝜀𝑝𝑝(𝒙𝒙, 𝑡𝑡) = 0 lim𝑡𝑡→∞
𝜀𝜀�̇�𝑖𝑖𝑖𝑝𝑝 = 0 ∆𝜀𝜀𝑝𝑝(𝒙𝒙) = � �̇�𝜀𝑖𝑖𝑖𝑖
𝑝𝑝 (𝒙𝒙, 𝑡𝑡) d𝑡𝑡 = 0T
0 ∆𝜀𝜀𝑝𝑝(𝒙𝒙) = � �̇�𝜀𝑖𝑖𝑖𝑖
𝑝𝑝 (𝒙𝒙, 𝑡𝑡) d𝑡𝑡 ≠ 0T
0
Schematic illustration of different material behaviors
Definition of load domain*
BASIC CONCEPTS
* König, J.A. : Elsevier, Amsterdam (1987).
A finite number 𝑛 of types of loads:
𝑃 𝒙𝒙, 𝑡𝑡 = 𝑃 𝛽𝑠 𝑡𝑡 ,𝒙𝒙 𝑥 ∈ 𝑉 or SP; 𝑠 = 1,⋯ , 𝑟; 𝛽s− ≤ 𝛽s 𝑡𝑡 ≤ 𝛽𝑠+ 𝑠 = 1,⋯ ,𝑛
Loading domain 𝑃 can be described by a 𝑛-dimensional polyhedron:
𝑃(𝑥, 𝑡𝑡) = 𝑃|𝑃 = �𝜇𝑠 𝑡𝑡𝑛
𝑠=1
𝑃𝑠0 𝑥 , 𝜇𝑠(𝑡𝑡) ∈ 𝜇𝑖−,𝜇𝑖+
Two dimensional loading domain 𝓛 and α𝓛
Static shakedown theorem by Melan*
BASIC CONCEPTS
* Melan, E. : Sitzber. Akad. Wiss., Abt. IIA 145, 195-218 (1936 ).
If there exist a loading factor 𝛼 > 1 and a time-independent residual stress field 𝝆�(𝑥) whose superposition with elastic stresses 𝝈𝑬 does not exceed the yield condition 𝑭 ≤ 0 at any time 𝑡𝑡 > 0 and at all points 𝑥 ∈ 𝑉 in volume 𝑉 of the considered structure,
𝐹 𝛼𝝈𝑬 𝑥, 𝑡𝑡 + 𝝆� 𝑥 , 𝜎𝑌 𝑥 ≤ 0
where: 𝜎𝑌 𝑥 is yield stress.
𝝈𝑬 𝑥, 𝑡𝑡 is purely elastic stress reference.
then the system will shakedown under arbitrary load paths contained within given load domain 𝓛.
Numerical implementation
BASIC CONCEPTS
* Weichert, D., Hachemi, A. and Schwabe, F. : Mech. Res. Comm. 26, 309-318 (1999)
A purely elastic reference stress 𝝈𝑬 is calculated for each loading vertex by means of conventional FE-analysis
𝝈𝑬 and [C] are input data for the subsequent SD/LA-module.
Finite element discretization
Principle of virtual work
� 𝛿𝜺 𝑇 𝛼𝝈𝐸 + 𝝆�𝑉
d𝑉 = � 𝛿𝒖 𝑇 𝐩∗𝜕𝑉
d𝑆 + � 𝛿𝒖 𝑇 𝐟∗𝑉
d𝑉
Equilibrium conditions for 𝝆� are satisfied by principle of virtual work * :
� 𝛿𝜺 𝑇
𝑉𝝆� d𝑉 = 𝛿𝒖 � 𝑩𝑇 𝝆�
𝑉d𝑉 = 0
⇒ � 𝑩𝑇 𝝆� d𝑉𝑉𝑒
= � 𝑩𝑇 𝝆� 𝐽 d𝑟 d𝑠 d𝑡𝑡1
−1= � 𝐽 𝑩𝑇 𝝆�
𝑁𝑁𝐸
𝑗=1
= 0
⇒
� � 𝐽 𝑩𝑇 𝝆�𝑁𝑁𝐸
𝑗=1
𝑁𝐸
𝑘=1
= 𝑪 𝝆� = 0
Numerical implementation
BASIC CONCEPTS
Mathematic formulation of shakedown problem
Objective function Load factor 𝛼 Variables 𝛼 and residual stress field 𝝆� Linear equality constraints Self-equilibrated condition Nonlinear non-equality constraints Yield condition
• Algorithm Augmented Lagrangian method Sequential quadratic programming Interior Point Methed ……
• Software Packages LANCELOT *, … SNOPT, NPSOL,NLPQL IPDCA**, IPOPT*** … ……
Large-scale optimization
* Conn, A.R., Gould, N.I.M. and Toint, Ph.L. : Berlin Heidelberg, Springer-Verlag (1992).
** Pham Dinh Tao; Le Thi Hoai An. : SIAM J. Opt. 8, 476-505 (1998 ).
*** Wächter, A. and Biegler, L. T. : Math. Program.106(1), 25-57 (2006).
with NGS is the total number of Gauss Points; 𝑛 is the number of independent loads; 𝑃𝑘 is the load vertex. 𝑘 =1 corresponds to limit analysis; 𝑘 = 2𝑛 corresponds to shakedown analysis.
max𝛼
� 𝑪 𝝆� = 0𝐹 𝛼𝝈𝑖𝐸 𝑃𝑘 + 𝝆�𝑖 ,𝜎𝑌𝑖 ≤ 0𝑖𝑖 ∈ 1,𝑁𝑁𝑆 , 𝑘 ∈ 1, 2𝑛
Objective
Micro-/Mesoscopic Level --------------------------------------- Numerical Model of limit and shakedown analysis of RVE
Numerical Solution of limit and shakedown problem (FEM & Optimization)
Macroscopic Level ---------------------------------------- Comparison of loading carrying capacity of composites with different fiber distributions
Prediction of elastic and plastic material properties
Homogenization
DIRECT METHODS APPLIED TO COMPOSITES
Assumptions
• Periodic composites
• At least one ductile phase
DIRECT METHODS APPLIED TO COMPOSITES
* Suquet, P. : Doctor Thesis, Pierre & Marie Curie, (1982)
Heterogeneous Material RVE Homogenized Material
Localisation Globalisation
𝜉𝜉 = 𝑥 𝜃𝜃⁄ , θ: a small parameter
𝜮𝜮(𝑥) =1𝑉� 𝝈(𝜉𝜉) d𝑉 = ⟨𝝈(𝜉𝜉)⟩𝑉
𝑬(𝑥) =1𝑉� 𝜺(𝜉𝜉) d𝑉 = ⟨𝜺(𝜉𝜉)⟩𝑉
Concept of representative volume element (RVE)
Average field quantities *
Homogenization theory
DIRECT METHODS APPLIED TO COMPOSITES
* Weichert, D., Hachemi, A. and Schwabe, F. : Mech. Res. Comm. 26(3), 309-318 (1999). ** Suquet, P. : Doctor Thesis, Pierre & Marie Curie, (1982).
Static direct methods for periodic composites *
Localization problem – Boundary conditions **
Strain method
Stress method
Periodicity
Uniform strain is imposed on 𝜕𝑉 : 𝒖 = 𝑬 ∙ 𝝃 on 𝜕𝑉
Uniform stress is imposed on 𝜕𝑉 : 𝝈 ∙ 𝒏 = 𝜮𝜮 ∙ 𝒏 on 𝜕𝑉
Anti-periodicity of stress : 𝝈 ∙ 𝒏 anti-periodic on 𝜕𝑉 Decomposition of local strain : 𝜀𝜀 𝒖 = 𝑬 + 𝜀𝜀 𝒖per = 𝑬 + 𝜺per Average of 𝜺∗ over the RVE : 𝜺per = 0
Under thermal loading
𝜮𝜮 =1𝑉� 𝛼𝝈𝑬 + 𝝆� d𝑉𝑉
=1𝑉� 𝛼𝝈𝑬 d𝑉𝑉
+1𝑉� 𝝆� d𝑉𝑉
with 1𝑉� 𝝆� d𝑉𝑉
= 0
DIRECT METHODS APPLIED TO COMPOSITES
* Magoariec, H., Bourgeois, S. and Débordes, O. : Int. J. Plasticity. 20, 1655-1675 (2004). ** Chen, M., Hachemi, A. and Weichert, D. : Proc. Appl. Math. Mech. 106, 405-406 (2010).
Before deformation after deformation symmetrical part
Periodic composites – strain method *
Finite element discretization using non-conforming element ** • Material Model: elastic-perfectly plastic • von Mises yield criterion
max𝛼
� 𝑪 𝝆� = 0𝐹 𝛼𝜎𝑖𝐸 𝑃�𝑘 + 𝝆�𝑖 ,𝜎𝑌𝑖 ≤ 0𝑖𝑖 ∈ 1,𝑁𝑁𝑆 , 𝑘 = 1,⋯ , 2𝑛
Nr. Vriables: 6NGS+1 Nr. Equality Constraint: 3NK+9NE Nr. Inequality Constraint: NL*NGS NL=1, limit analysis; NL=2𝑛, shakedown analysis
𝓟𝐬𝐬𝐬𝐬𝐬𝐬
div 𝝈𝐸 = 0 in 𝑉 𝝈𝐸 = 𝒅: 𝑬 + 𝜺per in 𝑉 𝝈𝐸 ∙ 𝒏 anti − periodic on 𝜕𝑉 𝒖per periodic on 𝜕𝑉 𝜺 = 𝜠
𝓟𝐬𝐬𝐬𝐬𝐬𝐬𝐬𝐫𝐬 �div 𝝆� = 0 in 𝑉
𝝆� ∙ 𝒏 anti − periodic on 𝜕𝑉
Unlimited kinematic hardening
DIRECT METHODS APPLIED TO COMPOSITES
Consideration of kinematic hardening
Sup { 𝛼 } s.t. 𝐹 𝛼𝝈𝐸 + 𝝆� − 𝝅,𝝈𝒀 ≤ 0
Limited kinematic hardening
* Weichert D., Gross-Weege J. Int. J. Mech. Sci., 30, 757-767 (1988). ** Stein E., Zhang G. and König J.A. Int.J.Plast., 8,1-31 (1992).
Sup { 𝛼 } s.t. 𝐹 𝛼𝝈𝐸 + 𝝆� − 𝝅,𝝈𝒀 ≤ 0 𝐹 𝛼𝝈𝐸 + 𝝆�,𝝈𝑼 ≤ 0*
OR 𝐹 𝝅,𝝈𝑼 − 𝝈𝒀 ≤ 0**
where : 𝝅 is back stress.
where : 𝝈𝑼 is ultimate stress.
FAILURE CRITERION OF COMPOSITES
Failure : Every material has certain strength, expressed in terms of stress or strain, beyond which the structures fracture or fail to carry the load.
For heterogeneous material, consisted of two or more than two phases materials, how to determine the Failure Criterion?
Why Need Failure Criterion for Composites? • To determine weak and strong directions • To guide local design of composites • To guide global design of composites-structures
Macromechanical Failure Theories in Composites ?? • Maximum stress theory • Maximum strain theory • Tsai-Hill theory (Deviatoric strain energy theory) • Tsai-Wu theory (Interactive tensor polynomial theory) • …… • Loci of yield strength fitting based on limit macroscopic stress domain
* Hosford, W.F. : Oxford University (1993)
𝜎12 + 𝜎2
2 − � 2𝑅𝑅𝑅𝑅+1
�𝜎1𝜎2 − 𝑋𝑋2 = 0
where 𝑅𝑅 = 2 �𝑍𝑍𝑋𝑋�
2− 1
with 𝑋𝑋 = 1√𝐹+𝐻𝐻
and 𝑍𝑍 = 1√2𝐹
R : is a measure of the plastic
anisotropy of a rolled metal
Hill’s yield criterion in plane stress*
FAILURE CRITERION OF COMPOSITES
-1.5 -1 -0.5 0 0.5 1 1.5-1.5
-1
-0.5
0
0.5
1
1.5R=1R>1R<1
The plastic strain ratio R is a parameter that indicates the ability of a sheet metal to resist thinning or thickening when subjected to either tensile or compressive forces in the plane of the sheet.
Hill’s yield criterion in plane strain (general stress state)
FAILURE CRITERION OF COMPOSITES
“y-convention” ,i.e.(Z,Y’,Z’’):
• Rotate the 𝑥1𝑦𝑦1𝑧𝑧1-system about the 𝑧𝑧1-axis (Z) by angle Ψ;
• Rotate the current system about the new 𝑦𝑦-axis (Y’) by angle Θ;
• Rotate the current system about the new 𝑧𝑧 –axis (Z’’) by angle Φ.
�𝑥2𝑦𝑦2𝑧𝑧2
� = 𝑇𝑇−1 �𝑥1𝑦𝑦1𝑧𝑧1
�
Rotation matrix is:
𝑇𝑇𝑥1𝑦𝑦1𝑧𝑧1 = �cosΨ − sinΨ 0sinΨ cosΨ 0
0 0 1��
cos Θ 0 sin Θ0 1 0
−sin Θ 0 cos Θ��
cosΦ − sinΦ 0sinΦ cosΦ 0
0 0 1�
with
Ψ =𝜋𝜋4
; Θ= tan−1�√2� ; Φ = −𝜋𝜋4
Hill’s yield criterion in plane strain (general stress state)
FAILURE CRITERION OF COMPOSITES
• Calculation of principle stress
• Projection in 𝛑𝛑-plane
𝐹(𝜎2 − 𝜎3)2 + 𝑁(𝜎3 − 𝜎1)2 + 𝐻𝐻(𝜎1 − 𝜎2)2 − 1 = 0
𝐹,𝑁,𝐻𝐻 are defined as:
𝐹 =12 �
1𝑌𝑌2 +
1𝑍𝑍2 −
1𝑋𝑋2� ; 𝑁 =
12 �
1𝑍𝑍2 +
1𝑋𝑋2 −
1𝑌𝑌2� ; 𝐻𝐻 =
12 �
1𝑋𝑋2 +
1𝑌𝑌2 −
1𝑍𝑍2�
Let: �𝜎1𝜎2𝜎3
� = 𝑇𝑇 �𝛾𝛾1𝛾𝛾2𝛾𝛾3
�
(𝐹 + 1.866𝐻𝐻 + 0.134𝑁)𝛾𝛾12 + (𝐹 + 0.134𝐻𝐻 + 1.866𝑁)𝛾𝛾2
2 +(𝑁 − 2𝐹 + 𝐻𝐻)𝛾𝛾1𝛾𝛾2 = 1
Stress Invariants 𝜎𝑛 Independent on coordinate system
𝜎𝑖𝑖𝑖𝑖 Dependent on coordinate system
Hill’s yield criterion in plane strain (general stress state)
FAILURE CRITERION OF COMPOSITES
For homogenous material, 𝑋𝑋 = 𝑌𝑌 = 𝑍𝑍, i.e. 𝐹 = 𝑁 = 𝐻𝐻:
𝛾𝛾12 + 𝛾𝛾2
2 = 𝐶𝐶 ; here: 𝐶𝐶 =1
3𝐹 =23𝜎𝑌𝑌
2
For transversely homogenous material, assume 𝑌𝑌 = 𝑍𝑍 , i.e. 𝑁 = 𝐻𝐻
(𝐹 + 2𝐻𝐻)𝛾𝛾12 + (𝐹 + 2𝐻𝐻)𝛾𝛾2
2 + (2𝐻𝐻 − 2𝐹)𝛾𝛾1𝛾𝛾2 = 1
(𝐹 + 1.866𝐻𝐻 + 0.134𝑁)𝛾𝛾12 + (𝐹 + 0.134𝐻𝐻 + 1.866𝑁)𝛾𝛾2
2 +(𝑁 − 2𝐹 + 𝐻𝐻)𝛾𝛾1𝛾𝛾2 = 1
NUMERICAL EXAMPLES
E (MPa) 𝜐 𝜎𝑌 (MPa) 𝜎𝑈 (MPa)
Matrix (Al) 70e3 0.3 80 120
Fiber (Al2O3) 370e3 0.3 2000 ---
Material properties:
Periodic composites: Perfect bounding Different fiber distribution Different volume fraction (0 ~ 50%)
Periodic fiber reinforced metal matrix composites
Unidirectional fiber reinforced periodic composites
Representative volume element
Finite element model
Square pattern
Rotated pattern
Hexagonal pattern
0.200
0.220
0.240
0.260
0.280
0.300
0.320
0 10 20 30 40 50
Poiss
on R
atio
Fiber Volume Fraction (%)
Homogenized Poisson Ratio
Square-X,Y
R-Square-X,Y
Hexagonal-X
Hexagonal-Y
40
60
80
100
120
140
160
0 10 20 30 40 50
Youn
g's
Mod
ulus
(G
Pa)
Fiber Volume Fraction (%)
Homogenized Young's Modulus
Square-X,Y
R-Square-X,Y
Hexagonal-X
Hexagonal-Y
NUMERICAL EXAMPLES
Homogenized elastic material properties • Elastic-perfectly plastic
Matrix: • Elastic-perfectly plastic (SD) • Unlimited kinematic hardening(SDH-UN) • Limited kinematic hardening(SDH)
Admissible shakedown displacement domain
Admissible limit displacement domain
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4U2/
U0
U1/U0
SD
SDH
SDH-UN
AP -6
-4
-2
0
2
4
6
-6 -4 -2 0 2 4 6U2/
U0
U1/U0
LMLMH
NUMERICAL EXAMPLES
Fiber: Square pattern distributed Elastic-perfectly plastic Volume fraction 40%
Displacement domain for kinematic hardening
Admissible macroscopic shakedown stress domain
Admissible macroscopic limit stress domain
NUMERICAL EXAMPLES
Macroscopic stress domain for kinematic hardening
-3
-2
-1
0
1
2
3
-3 -2 -1 0 1 2 3Σ22/σm
Σ11/σm
SD
SDH
SDH-UN -5
-4
-3
-2
-1
0
1
2
3
4
5
-5 -3 -1 1 3 5Σ22/σm
Σ11/σm
LM
LMH
Matrix: • Elastic-perfectly plastic (SD) • Unlimited kinematic hardening(SDH-UN) • Limited kinematic hardening(SDH)
Fiber: Square pattern distributed Elastic-perfectly plastic Volume fraction 40%
-500 0 500
-800
-600
-400
-200
0
200
400
600
800
Admissible Stress Domain
LM
-200 -100 0 100 200-200
-150
-100
-50
0
50
100
150
200Admissible Stress Domain-Projected in Pi-plane
LM
Major axis: a=241.83 Minor axis: b=67.78
X = 296.18 Mpa = 3.70 𝜎m Y = Z= 98.5 MPa = 1.23 𝜎m
NUMERICAL EXAMPLES
Prediction of plastic material properties • Elastic-perfectly plastic
NUMERICAL EXAMPLES
E (MPa) 𝜐 𝜎𝑌 (MPa) Steel (Outside) 200e3 0.30 360
Aluminum (Inside) 72e3 0.33 100
Material properties:
Porous material
Items Dimensions (mm) Width 8 Height 6
Radius of the hole 1.5 Thickness of single layer 2 Thickness of two layers 4
Dimensions of RVE
NUMERICAL EXAMPLES
Porous material
-8
-6
-4
-2
0
2
4
6
8
-8 -6 -4 -2 0 2 4 6 8U2/U
0
U1/U0
SD_SteSD_AluSD_2Mat
-8
-6
-4
-2
0
2
4
6
8
-8 -6 -4 -2 0 2 4 6 8U2/U
0
U1/U0
LM_SteLM_AluLM_2Mat
-400
-200
0
200
400
-400 -200 0 200 400Σ22(
MPa)
Σ11(MPa)
SD_SteSD_AluSD_2Mat
-400
-200
0
200
400
-400 -200 0 200 400
Σ22(
MPa)
Σ11(MPa)
LM_SteLM_AluLM_2Mat
• Displacement domain • Macroscopic stress domain
CONCLUSIONS
Summary
Perspectives
• Application of direct methods to composites. Three boundary conditions are discussed.
• Prediction of transverse elastic material properties of unidirectional continuous fiber reinforced metal matrix composites based on homogenization theory.
• Consideration of the hardening enlarged the shakedown and limit domain.
• Definition of yield loci for periodic composites and the prediction of plastic material properties by using yield surface fitting.
• Apply direct methods to other composites types and more complex problems.
• Consider thermal loads, as well as the temperature dependent yield strength.
• The presented results for the fiber-reinforced composite are only numerical, and a future effort has to be made in order to compare with experimental results quantitatively.
Thanks for
your attention!