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

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

Basic concepts

Elements of homogenisation theory

Direct Methods applied to composites Design methods

Local design Global design

Conclusions

OUTLINE

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

ELEMENTS OF HOMOGENISATION THEORY

♦ ASSUMPTIONS

periodic composite (fibre-reinforced)

at least one ductile phase

perfect bonding

theory of elasto-plasticity (ductile phase)

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)

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)

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

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)

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)

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

DESIGN METHODS : Validation test

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

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

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

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

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

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

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

Thanks for

your attention!