Numerical homogenization of the Eshelby tensor at small strains

Numerical homogenization methods, such as the FE2 approach, are widely used to compute the effective physical properties of microstructured materials. Thereby, the macroscopic material law is replaced by the solution of a microscopic boundary value problem on a representative volume element in conjunction with appropriate averaging techniques. This concept can be extended to configurational or material quantities, like the Eshelby stress tensor, which are associated with configurational changes of continuum bodies. In this work, the focus is on the computation of the macroscopic Eshelby stress tensor within a small-strain setting. The macroscopic Eshelby stress tensor is defined as the volume average of its microscopic counterpart. On the microscale, the Eshelby stress tensor can be computed from quantities known from the solution of the physical microscopic boundary value problem. However, in contrast to the physical quantities of interest, i.e. stress and strain, the Eshelby stress tensor is sensitive to rigid body rotations of the representative volume element. In this work, it is demonstrated how this must be taken into account in the computation of the macroscopic Eshelby stress tensor. The theoretical findings are illustrated by a benchmark simulation and further simulation results indicate the microstructural influence on the macroscopic configurational forces.

Eshelbian mechanics of novel materials: Quasicrystals

In this work, the so-called Eshelbian or configurational mechanics of quasicrystals is presented. Quasicrystals are considered as a prototype of novel materials. Material balance laws for quasicrystalline materials with dislocations are derived in the framework of generalized incompatible elasticity theory of quasicrystals. Translations, scaling transformations as well as rotations are examined; the latter presents particular interest due to the quasicrystalline structure. This derivation provides important quantities of the Eshelbian mechanics, as the Eshelby stress tensor, the scaling flux vector, the angular momentum tensor, the configurational forces (Peach–Koehler force, Cherepanov force, inhomogeneity force or Eshelby force), the configurational work, and the configurational vector moments for dislocations in quasicrystals. The corresponding [Formula: see text]-, [Formula: see text]-, and [Formula: see text]-integrals for dislocation loops and straight dislocations in quasicrystals are derived and discussed. Moreover, the explicit formulas of the [Formula: see text]-, [Formula: see text]-, and [Formula: see text]-integrals for parallel screw dislocations in one-dimensional hexagonal quasicrystals are obtained. Through this derivation, the physical interpretation of the [Formula: see text]-, [Formula: see text]-, and [Formula: see text]-integrals for dislocations in quasicrystals is revealed and their connection to the Peach–Koehler force, the interaction energy and the rotational vector moment (torque) of dislocations in quasicrystals is established.

Crystal plasticity: The Hamilton-Eshelby stress in terms of the metric in the intermediate configuration

The Hamilton-Eshelby stress is a basic ingredient in the description of the evolution of point, lines and bulk defects in solids. The link between the Hamilton-Eshelby stress and the derivative of the free energy with respect to the material metric in the plasticized intermediate configuration, in large strain regime, is shown here. The result is a modified version of Rosenfeld-Belinfante theorem in classical field theories. The origin of the appearance of the Hamilton-Eshelby stress (the non-inertial part of the energy-momentum tensor) in dissipative setting is also discussed by means of the concept of relative power.