The concept of homogeneous thermodynamical potentials for non linear structural rigidity optimization

For structural rigidity optimization under the assumption of small strains and small displacements, a numerically efficient approach is to consider the minimization of the compliance with distributed parameters in the form of a double minimization problem over the design parameters and the statically admissible stress field [

1

]. Initially introduced in the framework of linear elasticity, it was used with the homogenization method to relax the illness posed problem of topology optimization (repartition of void and material in a fixed domain).

Using the concept of homogeneous thermodynamical potentials, we present the general form of the simultaneous extension of this optimization algorithm to a class of nonlinear elastic materials and a class of nonlinear structural analysis. The main idea is to formulate the local equations such that there exists a variational equality for the problem.

Class of nonlinear elastic materials: the stress and strain are related by a behavior law deriving from two dual (by the Legendre transform) homogeneous thermodynamical potentials (e.g. piecewise linear elasticity, power law nonlinear elasticity and dissymmetric in tension-compression power law). In this special case, those two potentials are proportional by a factor related to the degree of homogeneity [

2

].

Class of nonlinear structural analysis: the nonlinear phenomenon is modelled by a behavior law deriving from an homogenous potential. This of course limits the number of nonlinear structural phenomenons that can be taken into account. For example, a frictionless unilateral contact can be modelled as an interface with a behavior law relating the normal component of the stress vector to the normal displacement.

The degree of homogeneity is chosen (imposed) identical for every potential considered. The compliance and the complementary energy are proportional by a factor related to the degree of homogeneity. The optimization algorithm is then extended to nonlinear material behaviors and to a structural nonlinearity if we assume that this homogeneity degree is independent of the design parameters. Different numerical examples are given: optimal repartition of composite fibers with different behavior in tensioncompression, topology optimization of a structure with frictionless unilateral contact.