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M.Chen, L.L.Zhang, D. Weichert
GAMM 2009, Gdansk, 09.02.-13.02. 2009
Shakedown and limit analysis
of periodic composites
Institute of General MechanicsRWTH Aachen University
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PROBLEM
A Mechanical structure or structural element made of composite material operates beyond the elastic limit. Find the optimal design variables for the composite under Limit and Shakedown conditions. METHOD Use of Direct Methods (LA and SDA) combined with homogenisation techniques
INTRODUCTION
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Basic concepts
Elements of homogenisation theory
Direct Methods applied to composites Design methods
Local design Global design
Conclusions
OUTLINE
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A composite is loaded up to or beyond the elastic limit by monotonous or variable loads
The composite is regarded as an inhomogeneous mechanical structure Global elastic properties to be determined by using homogenisation theory Two types of loading histories:
Monotonic loading Limit Analysis (LA)
Variable loading Shakedown Analysis (SDA)
BASIC CONCEPTS
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ELEMENTS OF HOMOGENISATION THEORY
♦ ASSUMPTIONS
periodic composite (fibre-reinforced)
at least one ductile phase
perfect bonding
theory of elasto-plasticity (ductile phase)
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Concept of representative volume element (RVE)
Heterogeneous Material RVE Homogeneous Material
Localisation Globalisation
/ ; : a small parameter θ θ= xξ
Average field quantities* 1( ) ( ) d ( )
1( ) ( ) d ( )
VV
VV
= < >
= < >
∫
∫V
V
Σxσσ
E xεε
=
=
ξ ξ
ξ ξ
ELEMENTS OF HOMOGENISATION THEORY
* P.Suquet. Université Pierre et Marie Curie. PhD. Thesis (1982)
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In static shakedown theory for composite materials with periodic
microstructure*
1 1 1( )d d de eΣσρσρ + = +∫ ∫ ∫=V V V
V V VV V V
α α
1 d 0V
VV
=∫ ρ
♦ Stress Approach** ♦ Strain Approach**
stress
div =0 in anti-periodic on 0
VVP
⋅ ∂< > =
< > =
e
eσ
σn σΣρ
strain
div = 0 in periodic on
anti-periodic on
VVP
V
∂
⋅ ∂< > =
e
e
e
σuσnεE
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.Débordes. Int. J. Plasticity. 20, 1655-1675 (2004)
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The application of static shakedown theorems leads
genuinely to an optimisation problem
Objective function : load multiplier α
Variables : generalized residual stresses ρ
Subsidiary conditions : yield condition
Here: LA is treated as particular case of SDA
DIRECT METHODS APPLIED TO COMPOSITES
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General strategy: A purely elastic reference solution σ e is calculated for each loading vertex by means of conventional FE-analysis (ANSYS, SAMCEF, ABAQUS). Equilibrium conditions for ρ are satisfied by principle virtual work*:
{ ( )} { ( )}d 0T
VVδρε ξ ξ [ ]{ }={0}Cρ
σ e and [C] are input data for the subsequent SD/LA-module. Then, the maximum loading factor is determined.
DIRECT METHODS APPLIED TO COMPOSITES
* D.Weichert, A. Hachemi and F.Schwabe. Mech. Res. Comm. 26, 309-318 (1999)
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max[ ]{ } {0}
[ ( ) , ] 0
[1, ], [1, 2 ]
ei k i Y
n
F P
i NG k
α
α σ
= + ≤ ∈ ∈
C ρ
σ ρ
Final Mathematical Problem
DIRECT METHODS APPLIED TO COMPOSITES
Optimisation Problem
* A.R.Conn, N.I.M.Gould and Ph.L.Toint. Berlin Heidelberg, Springer-Verlag (1992)
** Pham Dinh Tao; Le Thi Hoai An. SIAM J. Opt. 8, 476-505 (1998 )
LANCELOT* IPDCA** SQP(Augmented Lagrangian method) (Interior Point Method) (Sequential quadratic)
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DESIGN METHODS
Local design: exclusively RVE-level
Global design: interaction between structure and RVEHeterogeneous Material
RVE
LA in global design
Localisation
Micro results
Macro results
Homogenised Parameters
Strain Method
Homogenisation
Globalisation
Limit domain
Yield surface
LM
Approximationof von Mises Yield criterion
Elastic Properties
Plastic Properties
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DESIGN METHODS : Validation test
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Comparation
0
0.2
0.4
0.6
0.8
0 0.2 0.4 0.6 0.8
U1/U0
U2/U0
SD_3D
LM_3D
EL_3D
SD_2D*
LM_2D*
EL_2D*
Limit Load
0
20
40
60
80
100
0 20 40 60 80 100
PX/MPa
PY/MPa
Inc.Meth.
LANCELOT
IPDCA
Matrix(A1) Fiber(A12O3)
E(GPa) υ
σy(MPa)
700.380
3700.3
2000
Comparison between 2D and 3D elements
* F. Schwabe. RWTH-Aachen. Phd. Thesis (2000)
Comparison with incremental method
Matrix Fiber
E(GPa) υ
σy(MPa)
210 0.3280
2.1 0.2 140
DESIGN METHODS : Validation test
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11 0 22 0
M / M /
a K b KE E K E E K
= = ⋅= ⋅ =
tan
M = constant value
K
a b
ϕ=
⋅ =
, ,
Normalized Shakedown Domain
0
0.2
0.4
0.6
0.8
0 0.2 0.4 0.6 0.8
(U1/U0)/K
(U2/
U0)*
K
45
35
30
20
Normalized Elastic Domain
0
0.1
0.2
0.3
0.4
0 0.1 0.2 0.3 0.4
(U1/U0)/K
(U1/U0)*K 45
35
30
20
Normalized Limit Domain
0
0.2
0.4
0.6
0.8
0 0.2 0.4 0.6 0.8 1
(U1/U0)/K
(U2/U0)*K 45
35
30
20
22E
11E
Influence of geometrical parameters
DESIGN METHODS : Local design
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DESIGN METHODS : Local design
Square pattern Rotated pattern Hexagonal pattern
Radius of fiber R=15 R=15 R=15
A 50 70.7 93.0
B 50 70.7 53.7
Volume fraction 7.0686Square pattern
Rotated pattern
Hexagonal pattern
Shakedown Domain
0
1
2
3
4
5
0 1 2 3 4 5 6U1/U0
U2/U0
Square
Rotated Square
Hexagonal
Influence of pattern
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DESIGN METHODS : Global design
20.0635 - 0.8389 72.3360R RE +=
0 5 10 15 20 25 30 35 400.24
0.26
0.28
0.3
0.32
0.34
Radius of Fiber/mm
Poi
sson
Rat
io
homogenized Poisson Ratio
20.000047 0.000871 0.298097R Ru += − +
Homogenisation Theory
Material Al Al2O3
E (MPa) 70000 370000
υ 0.3 0.3
σy (MPa) 80 2000
0 5 10 15 20 25 30 35 4060
80
100
120
140
160
Radius of Fiber/mm
You
ngs
Mod
ulus
/Mpa
homogenized Youngs Modulus
Elastic property
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Limit Domain
0
20
40
60
80
100
0 20 40 60 80 100
PX/MPa
PY/MPa
Limit_R25
Limit_R15
DESIGN METHODS: Global design
Py
Px
State 1: Onset of plasticity
State 2: Debonding of the interface
State 3: Overall plastic flow
Plastic property
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Limit Domain
0
50
100
150
200
250
0 50 100 150 200 250
PX/MPa
PY/MPa
R5
R10
R15
R20
R25
R30
R35
R40
> ca.30%η
Limit domain increased quickly
DESIGN METHOD: Global design
Plastic property
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CONCLUSIONS
Direct Methods (SDA and LA) appear to be adequate tools
for the assessment of limit states of composites.
Perspectives:
♦ Design of composites
♦ Combination with global optimisation strategies
♦ Multi-physical problems
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Thanks for
your attention!