Secret and joy of configurational mechanics: From foundations in continuum mechanics to applications in computational mechanics

Abstract

Key words Configurational mechanics, computational mechanics. This manuscript contains the material presented in a plenary lecture given at the GAMM annual meeting 2007 in ZurichOver the recent years configurational mechanics has developed into a very active and successful topic both in continuum mechanics as well as in computational mechanics. On the continuum mechanics side the basic idea is to consider energy variations that go along with changes of the material configuration. Configurational forces are then energetically dual to these configurational changes. Configurational forces take the interpretation as being the driving forces in the kinetics of defects; like e.g., cracks, inclusions, phase boundaries, dislocations and the like. On the computational side it turns out that a discretisation scheme brings in artificial, discrete configurational forces that indicate in a certain sense the quality, e.g., of a finite-element mesh. This information can then be used to optimize the nodal material positions. Surprisingly, even driven by energetical arguments, it turns out that a finite element mesh optimized with respect to discrete configurational forces also renders superior results in terms of classical error measures. The manuscript will span the field from the underlying theoretical foundations over the algorithmic challenges to various computational applications.

Effects of the smoothness of partitions of unity on the quality of representation of singular enrichments for GFEM/XFEM stress approximations around brittle cracks

“…This fact may be better observed by assessing local measures as stress intensity factors. This investigation, considering the configurational forces method [74], has been conducted by the present authors and will be the topic of a forthcoming paper. Thus, the higher accuracy of the C k -GFEM in the singular enrichment zone is particularly significant and important for LEFM problems.…”

“…Additional investigations of the local quality of stress fields around singularities, through the assessment of stress intensity factors, using configurational mechanics (see [74], for instance), will be shown in a forthcoming paper. The behavior for different C 0 discretizations, considering C 0 FEM-based PoU's, in terms of strain energy is shown in Fig.…”

Inverse‐motion‐based form finding for quasi‐incompressible finite electroelasticity

“…Based on the pressure contribution specified in Equation (3.13) and # D Â 1 , one obtains the relations ıq D J ıp and ı# D Â 2 ı Â. This leads to the following mixed formulation in terms of the total Cauchy stress tensor, as proposed for the purely elastic problem by Govindjee and Mihalic [8,9] and Steinmann et al [12] by different lines of argumentation,…”

“…Based on the pressure contribution specified in Equation and ϑ = θ − 1 , one obtains the relations δq = − J δp and δϑ = − θ − 2 δ θ . This leads to the following mixed formulation in terms of the total Cauchy stress tensor, as proposed for the purely elastic problem by Govindjee and Mihalic and Steinmann et al by different lines of argumentation, (see also appendix B). The total Cauchy stress tensor is introduced as …”

“…A similar work was published by Germain et al , where anisotropic hyperelasticity was formulated in the logarithmic strain space. The original idea of Govindjee and Mihalic was interpreted in the more general framework of configurational mechanics by Steinmann et al . Recently, Germain and Steinmann presented initial results for inverse form finding methods for elastoplastic solids.…”