Application of Shakedown Theory and Numerical Methods
In this lecture, the methodological framework for the numerical application of shakedown analysis is developed. Examples of applications such as the analysis and optimisation of composite materials and the analysis of various plates and shells problems ar
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Abstract. In this lecture, the methodological framework for the numerical application of shakedown analysis is developed. Examples of applications such as the analysis and optimisation of composite materials and the analysis of various plates and shells problems are presented.
3.1 Application to Composite Materials The advantage of composite materials compared to conventional materials stemming just from the possibility to design materials for specific technological purposes where in some sense controversial material properties are required turns out to complicate the assessment of the material's performance. The development of such materials demands an optimal design of the microstructure and a fundamental understanding of the role of the microstructure on the overall properties. The microstructural parameters controlling the macroscopic properties are on one hand the morphology of the microstructure and on the other hand the constitutive behaviour of each individual component. The correlation between the microstructure and the macroscopic properties is addressed by homogenisation technique (Suquet, 1983). 3.1.1 Considered Composite We consider a material which is composed of inclusions, embedded according to a regular pattern as shown in Figure 17 in a homogeneous elastic-plastic metal matrix. In a first step, it is assumed that the material can be regarded as two-dimensional. In this case, these inclusions represent fibres with constant cross sections in the case of plane strains and disk-shaped inclusions in the case of plane stresses. The macroscopic behaviour of this heterogeneous material is observed on the scale x and the microscopic behaviour on the scale y. For reasons of simplicity, only infinitesimal geometrical transformations are considered. The generalised stresses l:(x) and strains E(x) on the macro-level are linked to generalised stresses s(y) and strains e(y) on the micro-level through (Hill, 1963)
l:(x) =
= ~ J(JI) f s(y) dV
(3.1)
E(x) =
=~ !JI) e(y) dV
(3.2)
D. Weichert et al. (eds.), Inelastic Behaviour of Structures under Variable Repeated Loads © Springer-Verlag Wien 2002
240
D. Weichert, A. Hachemi
where V denotes the volume of the (periodic) representative volume element (RVE) as shown in Figure 17. Composite material
r· .-
.
' 1 -.-
I -.
·r· ' r '
-:~ ·~ ··-~~ .p------. ~:-: :~ :·-...-.; . ---;-
-
..
Matrix
Inclusion
I, a time-independent field of generalised stresses s(y) = [p(y), 1t(y)]T and a Sanctuary of Elasticity (see §1.8.7) (3.4) with
(3.5)
Chapter 4: Application of Shakedown Theory and Numerical Methods
241
then the periodic composite material shakes down (Weichert et al., 1999a, 1999b). Here, the safe state of generalised stresses sCsl is defined as usual (see, e.g., Mandel, 1976) sCsl = asCcl +
s
(3.6)
where sCcJ = [aCe), 0] is the generalised stress field which would occur in the RVE(c) under the same boundary conditions as the actual RVE such that the following relations hold for given macroscopic strains E Div aCe) uCcJ £(c) aCcl The field of
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