Variational sensitivity analysis of elastoplastic structures applied to optimal shape of specimens

Abstract

The aim of this paper is to improve the shape of specimens for biaxial experiments with respect to optimal stress states, characterized by the stress triaxiality. Gradient-based optimization strategies are used to achieve this goal. Thus, it is crucial to know how the stress state changes if the geometric shape of the specimen is varied. The design sensitivity analysis (DSA) of the stress triaxiality is performed using a variational approach based on an enhanced kinematic concept that offers a rigorous separation of structural and physical quantities. In the present case of elastoplastic material behavior, the deformation history has to be taken into account for the structural analysis as well as for the determination of response sensitivities. The presented method is flexible in terms of the choice of design variables. In a first step, the approach is used to identify material parameters. Thus, material parameters are chosen as design variables. Subsequently, the design parameters are chosen as geometric quantities so as to optimize the specimen shape with the aim to obtain a preferably homogeneous stress triaxiality distribution in the relevant cross section of the specimen.

Shape optimization of the X0-specimen: theory, numerical simulation and experimental verification

“…The experimental data is taken from [24] and the material parameters have been determined by a curve fitting procedure based on gradient information obtained following the same variational principles as described in this paper for shape sensitivities. The procedure is described in [31] in detail. The parameters found are indicated in Table 1.…”

“…Thus, its total variation has to vanish, cf. [2,30,31,44]. We assume that external forces are design independent (δ R ext = 0 → δ R = δ R int ).…”

“…2, a pull-back operation to parameter space B can be performed and all required variations can easily be obtained, see e.g. [2,26,31,33] for details. Finally, a push-forward to the reference configuration leads to…”

“…See e.g. [31] for details on the implementation utilizing the Matlab function fmincon available in the Matlab Optimization Toolbox. Further details on the approach, discretization usingF-finite elements, cf.…”

“…Further details on the approach, discretization usingF-finite elements, cf. [15], and important implementation aspects can be found in [31]. 5 Optimization problems…”