progress in plastic design of composites - rwth aachen university
TRANSCRIPT
Institute of General Mechanics RWTH Aachen University
M. Chen, A. Hachemi, D. Weichert
3nd International Workshop on Direct Methods, October 20-21, 2011, Athens
Progress in plastic design
of composites
BACKGROUND
*http://www.boeing.com
Composites: Increasing application Multi-scale approach (Meso-/Micromechanics) Local and global design
Direct Methods: Unknown loading history Economic design ( permission of plastic deformation)
Composite Materials at Boeing 787 *
CONTENTS
Direct methods applied to composites โข Elements of homogenization theory โข Finite element discretization
Kinematic hardening โข Unlimited kinematic hardening โข Limited kinematic hardening
Failure criterion of composites โข Plane stress โข Plane strain ( 3D general stress state)
Numerical illustration โข Effective material properties โข Yield criterition fitting โข Kiematic hardening
Conclusions
Research Goals
Microscopic 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 different types of composites Prediction of elastic and plastic properties
Homogenization
DIRECT METHODS APPLIED TO COMPOSITES
Assumptions โข Fieber-reinforced periodic composites (RVE)
โข Perfect bonding
โข Elasto-plastic material behaviour
ELEMENTS OF HOMOGENIZATION THEORY
* Suquet, P.: Doctor Thesis, Pierre & Marie Curie, (1982)
Concept of representative volume element (RVE)
Heterogeneous Material RVE Homogenized Material
Localisation Globalisation
๐๐ = ๐ฅ๐ฅ ๐๐โ , ฮธ: a small parameter
Average field quantities*
๐ฎ๐ฎ(๐ฅ๐ฅ) =1๐๐ ๏ฟฝ ๐๐(๐๐) d๐๐ = โจ๐๐(๐๐)โฉ๐๐
๐ฌ๐ฌ(๐ฅ๐ฅ) =1๐๐ ๏ฟฝ ๐บ๐บ(๐๐) d๐๐ = โจ๐บ๐บ(๐๐)โฉ๐๐
DIRECT METHODS APPLIED TO COMPOSITES
* D.Weichert, A.Hachemi and F.Schwabe. Mech. Res. Comm. 26(3), 309-318 (1999)
** H.Magoariec, S.Bourgeois and O.Dรฉbordes. Int. J. Plasticity. 20, 1655-1675 (2004)
๐ฎ๐ฎ =1๐๐๏ฟฝ(๐ผ๐ผ๐๐๐๐ + ๐๐๏ฟฝ) d๐๐๐๐
=1๐๐๏ฟฝ ๐ผ๐ผ๐๐๐๐ d๐๐๐๐
+1๐๐๏ฟฝ ๐๐๏ฟฝ d๐๐๐๐
๐ค๐ค๐ค๐ค๐ค๐คโ 1๐๐๏ฟฝ ๐๐๏ฟฝ d๐๐๐๐
= 0
Static direct methods for periodic composites*
๐๐๐ฌ๐ฌ๐ฌ๐ฌ๐ฌ๐ฌ๐ฌ๐ฌ๐ฌ๐ฌ๐ฌ๐ฌ
โฉโชโจ
โชโง div ๐๐๐ธ๐ธ = 0 in ๐๐
๐๐๐ธ๐ธ = ๐ ๐ : (๐ฌ๐ฌ + ๐บ๐บ๐ฉ๐ฉ๐ฉ๐ฉ๐ฌ๐ฌ) in ๐๐ ๐๐๐ธ๐ธ โ ๐๐ anti โ periodic on ๐๐๐๐ ๐๐per periodic on ๐๐๐๐ โจ๐บ๐บโฉ = ๐ ๐
๐๐๐ฌ๐ฌ๐ฌ๐ฌ๐ฌ๐ฌ๐ฌ๐ฌ๐ฌ๐ฌ๐ฌ๐ฌ๐ฌ๐ฌ๐ฉ๐ฉ๐ฌ๐ฌ ๏ฟฝdiv ๐๐๏ฟฝ = 0 in ๐๐
๐๐๏ฟฝ โ ๐๐ anti โ periodic on ๐๐๐๐
Localization problem โ strain method**
Before deformation after deformation symmetrical part
FINITE ELEMENT DISCRETISATION
Choose proper finite element Restriction of plane element Scale of optimization problem
(a)8-node first order element (b) 20-node second order
element (c) 8-node non-conforming
element *
โฉโชโชโชโจ
โชโชโชโง๐ข๐ข = ๏ฟฝ๐๐๐ค๐ค๐ข๐ข๐ค๐ค +
8
๐ค๐ค=1
๐ผ๐ผ1(1 โ ๐๐2) + ๐ผ๐ผ2(1 โ ๐ ๐ 2) + ๐ผ๐ผ3(1 โ ๐ค๐ค2)
๐ฃ๐ฃ = ๏ฟฝ๐๐๐ค๐ค๐ฃ๐ฃ๐ค๐ค +8
๐ค๐ค=1
๐ผ๐ผ4(1 โ ๐๐2) + ๐ผ๐ผ5(1 โ ๐ ๐ 2) + ๐ผ๐ผ6(1 โ ๐ค๐ค2)
๐ค๐ค = ๏ฟฝ๐๐๐ค๐ค๐ค๐ค๐ค๐ค +8
๐ค๐ค=1
๐ผ๐ผ7(1 โ ๐๐2) + ๐ผ๐ผ8(1 โ ๐ ๐ 2) + ๐ผ๐ผ9(1 โ ๐ค๐ค2)
(๐๐1,๐๐2,๐๐3) (๐๐4,๐๐5, ๐๐6)
(๐๐7,๐๐8,๐๐9)
* Wilson,E.L. Imcompatible displacement models. Numerical and Computer Methods in Structural Mechanics.(1973)
FINITE ELEMENT DISCRETISATION
max๐ผ๐ผ
๏ฟฝ [๐ช๐ช]{๐๐๏ฟฝ} = 0๐น๐น๏ฟฝ๐ผ๐ผ๐๐๐ค๐ค๐ธ๐ธ๏ฟฝ๐๐๏ฟฝ๐๐๏ฟฝ + ๐๐๏ฟฝ๐ค๐ค ,๐๐๐๐๐ค๐ค๏ฟฝ โค 0๐ค๐ค โ [1,๐๐๐๐๐๐],๐๐ = 1,โฏ , 2๐๐
Nr. Vr: 6NGS+1 Nr. Equality Constr: 3NK+9NE Nr. Inequality Constr: NL*NGS NL=1, limit analysis; NL=2๐๐ , shakedown analysis
Finite element discretization using non-conforming element: โข Material Model: elastic-perfectly plastic โข Von Mises yield criterion
Large-scale optimization
โข Algorithm Augmented Lagrangian method Sequential quadratic programming Interior Point Methed โฆโฆ
โข Software Packages LANCELOT, โฆ SNOPT, NPSOL,NLPQL IPOPT, IPDCA, IPSAโฆ โฆโฆ
HARDENING MATERIAL MODEL
Unlimited kinematic hardening Limited kinematic hardening
Isotropic hardening & Kinematic hardening
HARDENING MATERIAL MODEL
โข Material model: Matrix -- Unlimited kinematic hardening Fiber -- elastic-perfectly plastic
max๐ผ๐ผ
โฉโชโจ
โชโง [๐ช๐ช]{๐๐๏ฟฝ} = 0 ๐น๐น๏ฟฝ๐ผ๐ผ๐๐๐๐๐ธ๐ธ๏ฟฝ๐๐๏ฟฝ๐๐๏ฟฝ + ๏ฟฝฬ ๏ฟฝ๐๐๐ ,๐๐๐๐๐๐f ๏ฟฝ โค 0 ๐๐ โ [1,๐๐๐๐๐๐๐น๐น]๐น๐น๏ฟฝ๐ผ๐ผ๐๐๐ค๐ค๐ธ๐ธ๏ฟฝ๐๐๏ฟฝ๐๐๏ฟฝ + ๏ฟฝฬ ๏ฟฝ๐๐ค๐ค โ ๐ฝ๐ฝ๐ค๐ค ,๐๐๐๐๐ค๐คm ๏ฟฝ โค 0 ๐ค๐ค โ [1,๐๐๐๐๐๐๐๐]๐๐ = 1,โฏ , 2๐๐
๐ท๐ท:back stress field ๐๐๐๐:yield strength Nr. Vr: 6NGSF+6NGSM+1 Nr. Equality Constr: 3NK+9NE Nr. Inequality Constr: NL*NGS NGS=NGSF+NGSM
โข Material model: Matrix -- limited kinematic hardening Fiber -- elastic-perfectly plastic
max๐ผ๐ผ
โฉโชโจ
โชโง
[๐ช๐ช]{๐๐๏ฟฝ} = 0 ๐น๐น๏ฟฝ๐ผ๐ผ๐๐๐๐๐ธ๐ธ๏ฟฝ๐๐๏ฟฝ๐๐๏ฟฝ + ๏ฟฝฬ ๏ฟฝ๐๐๐ ,๐๐๐๐๐๐f ๏ฟฝ โค 0 ๐๐ โ [1,๐๐๐๐๐๐๐น๐น] ๐น๐น๏ฟฝ๐ผ๐ผ๐๐๐ค๐ค๐ธ๐ธ๏ฟฝ๐๐๏ฟฝ๐๐๏ฟฝ + ๏ฟฝฬ ๏ฟฝ๐๐ค๐ค โ ๐ฝ๐ฝ๐ค๐ค ,๐๐๐๐๐ค๐คm ๏ฟฝ โค 0 ๐น๐น๏ฟฝ๐ผ๐ผ๐๐๐ค๐ค๐ธ๐ธ๏ฟฝ๐๐๏ฟฝ๐๐๏ฟฝ + ๏ฟฝฬ ๏ฟฝ๐๐ค๐ค ,๐๐๐๐๐ค๐คm ๏ฟฝ โค 0 ๐ค๐ค โ [1,๐๐๐๐๐๐๐๐]๐๐ = 1,โฏ , 2๐๐
๐ท๐ท:back stress field ๐๐๐๐:yield strength ๐๐๐๐: ultimate strength Nr. Vr: 6NGS+6NGSM+1 Nr. Equality Constr: 3NK+9NE Nr. Inequality Constr: NL*(NGS+NGSM)
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 material, how to determine the Failure Criterion?
Why Need Failure Criterion for Composites? โข To determine weak and strong directions โข To guide local design of composites โข To guide global design of composites-structures
Macromechanical Failure Theories in Composites ?? โข Maximum stress theory โข Maximum strain theory โข Tsai-Hill theory (Deviatoric strain energy theory) โข Tsai-Wu theory (Interactive tensor polynomial theory) โข โฆโฆ โข Loci of yield strength fitting based on limit macroscopic stress domain
* Hosford, W.F.: Oxford University (1993)
๐๐12 + ๐๐2
2 โ ๏ฟฝ 2๐ ๐ ๐ ๐ +1
๏ฟฝ๐๐1๐๐2 โ ๐๐2 = 0
where ๐ ๐ = 2 ๏ฟฝ๐๐๐๐๏ฟฝ
2โ 1
with ๐๐ = 1โ๐น๐น+๐ป๐ป
and ๐๐ = 1โ2๐น๐น
R : is a measure of the plastic
anisotropy of a rolled metal
Hillโs 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.
Hillโs yield criterion in plane strain (general stress state)
FAILURE CRITERION OF COMPOSITES
โy-conventionโ ,i.e.(Z,Yโ,Zโโ):
โข Rotate the ๐ฅ๐ฅ1๐ฆ๐ฆ1๐ง๐ง1-system about the ๐ง๐ง1-axis (Z) by angle ฮจ;
โข Rotate the current system about the new ๐ฆ๐ฆ-axis (Yโ) by angle ฮ;
โข Rotate the current system about the new ๐ง๐ง โaxis (Zโโ) by angle ฮฆ.
๏ฟฝ๐ฅ๐ฅ2๐ฆ๐ฆ2๐ง๐ง2
๏ฟฝ = ๐๐โ1 ๏ฟฝ๐ฅ๐ฅ1๐ฆ๐ฆ1๐ง๐ง1
๏ฟฝ
Rotation matrix is:
๐๐๐ฅ๐ฅ1๐ฆ๐ฆ1๐ง๐ง1 = ๏ฟฝcosฮจ โ sinฮจ 0sinฮจ cosฮจ 0
0 0 1๏ฟฝ๏ฟฝ
cos ฮ 0 sin ฮ0 1 0
โsin ฮ 0 cos ฮ๏ฟฝ๏ฟฝ
cosฮฆ โ sinฮฆ 0sinฮฆ cosฮฆ 0
0 0 1๏ฟฝ
with
ฮจ =๐๐4
; ฮ= tanโ1๏ฟฝโ2๏ฟฝ ; ฮฆ = โ๐๐4
Hillโs yield criterion in plane strain (general stress state)
FAILURE CRITERION OF COMPOSITES
โข Calculation of principle stress
โข Projection in ๐๐-plane
๐น๐น(๐๐2 โ ๐๐3)2 + ๐๐(๐๐3 โ ๐๐1)2 + ๐ป๐ป(๐๐1 โ ๐๐2)2 โ 1 = 0
๐น๐น,๐๐,๐ป๐ป are defined as:
๐น๐น =12 ๏ฟฝ
1๐๐2 +
1๐๐2 โ
1๐๐2๏ฟฝ ; ๐๐ =
12 ๏ฟฝ
1๐๐2 +
1๐๐2 โ
1๐๐2๏ฟฝ ; ๐ป๐ป =
12 ๏ฟฝ
1๐๐2 +
1๐๐2 โ
1๐๐2๏ฟฝ
Let: ๏ฟฝ๐๐1๐๐2๐๐3
๏ฟฝ = ๐๐ ๏ฟฝ๐พ๐พ1๐พ๐พ2๐พ๐พ3
๏ฟฝ
(๐น๐น + 1.866๐ป๐ป + 0.134๐๐)๐พ๐พ12 + (๐น๐น + 0.134๐ป๐ป + 1.866๐๐)๐พ๐พ2
2 +(๐๐ โ 2๐น๐น + ๐ป๐ป)๐พ๐พ1๐พ๐พ2 = 1
Stress Invariants ๐๐๐๐ Independent on coordinate system
๐๐๐ค๐ค๐๐ Dependent on coordinate system
Hillโs yield criterion in plane strain (general stress state)
FAILURE CRITERION OF COMPOSITES
For homogenous material, ๐๐ = ๐๐ = ๐๐, i.e. ๐น๐น = ๐๐ = ๐ป๐ป:
๐พ๐พ12 + ๐พ๐พ2
2 = ๐ถ๐ถ ; here: ๐ถ๐ถ =1
3๐น๐น =23๐๐๐๐
2
For transversely homogenous material, assume ๐๐ = ๐๐ , i.e. ๐๐ = ๐ป๐ป
(๐น๐น + 2๐ป๐ป)๐พ๐พ12 + (๐น๐น + 2๐ป๐ป)๐พ๐พ2
2 + (2๐ป๐ป โ 2๐น๐น)๐พ๐พ1๐พ๐พ2 = 1
(๐น๐น + 1.866๐ป๐ป + 0.134๐๐)๐พ๐พ12 + (๐น๐น + 0.134๐ป๐ป + 1.866๐๐)๐พ๐พ2
2 +(๐๐ โ 2๐น๐น + ๐ป๐ป)๐พ๐พ1๐พ๐พ2 = 1
NUMERICAL ILLUSTRATION
Numerical Example
โข Homogenization theory- Effective material properties
โข Failure criterion fitting
โข Kinematic hardening
E (Gpa) ๐ ๐๐๐ (Mpa) ๐๐๐ (Mpa)
Matrix (Al) 70e3 0.3 80 120
Fiber (Al2O3) 370e3 0.3 2000 ---
Material properties:
Periodic composites:
Homogenized elastic material property
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
Square pattern (SQ)
Rotated pattern (TS)
Hexagonal pattern (HEX)
Axial macroscopic limit stress
Homogenized axial strength
Comparison
Failure curve fitting
-3 -2 -1 0 1 2 3-3
-2
-1
0
1
2
3Displacement Domain
ELSDLM
-150 -100 -50 0 50 100 150-150
-100
-50
0
50
100
150Addmissible Stress Domain
ELSDLM
-4 -2 0 2 4-4
-3
-2
-1
0
1
2
3
4Displacement Domain
ELSDLM
-200 -100 0 100 200-250
-200
-150
-100
-50
0
50
100
150
200
250Addmissible Stress Domain
ELSDLM
Plain Strain
Plain Stress Vf: 40%
Vf: 40%
-2 -1 0 1 2-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5Displacement Domain
LM
-1.5 -1 -0.5 0 0.5 1 1.5-1.5
-1
-0.5
0
0.5
1
1.5Addmissible Stress Domain
LM
R=0.9146
Not good at deforming
Failure curve fitting โ plane stress
-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 = 592.4 Mpa = 7.41 ๐๐m Y = Z= 98.5 MPa = 1.23 ๐๐m
Failure curve fitting โ plane strain
Kinematic hardening
โข Elastic-perfectly plastic material (SD) โข Limited kinematic hardening(SDHard) โข Unlimited kinematic hardening(SDHard-UN)
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4U2/
U0
U1/U0
Plain strain V40% - Shakedown Domain
SD
SDHard
SDHard-UN
AP
Matrix:
Numerical results
Limited kinematic Unlimited kinematic El
astic
Stre
ss
Elastic-perfectly Re
sidua
l Stre
ss
Numerical results
Limited kinematic Unlimited kinematic Ba
ck S
tress
Elastic-perfectly
Tota
l Stre
ss
CONCLUSIONS
Summary
Perspectives
โข Application of a 3D nonconforming element to the limit and shakedown analysis of composites.
โข Prediction of transverse elastic material properties of unidirectional continuous fiber reinforced metal matrix composites based on homogenization theory.
โข Definition of yield loci for periodic composites and the prediction of plastic material properties by using yield surface fitting.
โข Consideration of the hardening enlarged the shakedown domain. Different material model, different mechanism.
โข Apply direct methods to other composites types and more complex problems.
โข The general quantitative yield criterion fitting according to limit domain is still not solved yet.
โข 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.