- Computational Configurational Mechanics (2006)
- In contrast to the spatial motion setting, the material motion setting of continuum mechanics is concerned with the response to variations of material placements of particles with respect to the ambient material. The material motion point of view is thus extremely prominent when dealing with defect mechanics to which it has originally been introduced by Eshelby more than half a century ago. Its primary unknown, the material deformation map is governed by the material motion balance of momentum, i.e. the balance of material forces on the material manifold in the sense of Eshelby. Material (configurational) forces are concerned with the response to variations of material placements of 'physical particles' with respect to the ambient material. Opposed to that, the common spatial (mechanical) forces in the sense of Newton are considered as the response to variations of spatial placements of 'physical particles' with respect to the ambient space. Material forces as advocated by Maugin are especially suited for the assessment of general defects as inhomogeneities, interfaces, dislocations and cracks, where the material forces are directly related to the classical J-Integral in fracture mechanics, see also Gross & Seelig. Another classical example of a material - or rather configurational - force is emblematized by the celebrated Peach-Koehler force, see e.g. the discussion in Steinmann. The present work is mainly divided in four parts. In the first part we will introduce the basic notions of the mechanics and numerics of material forces for a quasi-static conservative mechanical system. In this case the internal potential energy density per unit volume characterizes a hyperelastic material behaviour. In the first numerical example we discuss the reliability of the material force method to calculate the vectorial J-integral of a crack in a Ramberg-Osgood type material under mode I loading and superimposed T-stresses. Secondly, we study the direction of the single material force acting as the driving force of a kinked crack in a geometrically nonlinear hyperelastic Neo-Hooke material. In the second part we focus on material forces in the case of geometrically nonlinear thermo-hyperelastic material behaviour. Therefore we adapt the theory and numerics to a transient coupled problem, and elaborate the format of the Eshelby stress tensor as well as the internal material volume forces induced by the gradient of the temperature field. We study numerically the material forces in a bimaterial bar under tension load and the time dependent evolution of material forces in a cracked specimen. The third part discusses the material force method in the case of geometrically nonlinear isotropic continuum damage. The basic equations are similar to those of the thermo-hyperelastic problem but we introduce an alternative numerical scheme, namely an active set search algorithm, to calculate the damage field as an additional degree of freedom. With this at hand, it is an easy task to obtain the gradient of the damage field which induces the internal material volume forces. Numeric examples in this part are a specimen with an elliptic hole with different semi-axis, a center cracked specimen and a cracked disc under pure mode I loading. In the fourth part of this work we elaborate the format of the Eshelby stress tensor and the internal material volume forces for geometrically nonlinear multiplicative elasto-plasticity. Concerning the numerical implementation we restrict ourselves to the case of geometrically linear single slip crystal plasticity and compare here two different numerical methods to calculate the gradient of the internal variable which enters the format of the internal material volume forces. The two numerical methods are firstly, a node point based approach, where the internal variable is addressed as an additional degree of freedom, and secondly, a standard approach where the internal variable is only available at the integration points level. Here a least square projection scheme is enforced to calculate the necessary gradients of this internal variable. As numerical examples we discuss a specimen with an elliptic inclusion and an elliptic hole respectively and, in addition, a crack under pure mode I loading in a material with different slip angles. Here we focus on the comparison of the two different methods to calculate the gradient of the internal variable. As a second class of numerical problems we elaborate and implement a geometrically linear von Mises plasticity with isotropic hardening. Here the necessary gradients of the internal variables are calculated by the already mentioned projection scheme. The results of a crack in a material with different hardening behaviour under various additional T-stresses are given.