Klaus Hackl

Non–convex potentials and microstructures in finite–strain plasticity

A mathematical model for a finite–strain elastoplastic evolution problem is proposed in which one time–step of an implicit time–discretization leads to generally non–convex minimization problems. The elimination of all internal variables enables a mathematical and numerical analysis of a reduced problem within the general framework of calculus of variations and nonlinear partial differential equations. The results for a single slip–system and von Mises plasticity illustrate that finite–strain elastoplasticity generates reduced problems with non–quasiconvex energy densities and so allows for non–attainment of energy minimizers and microstructures.

On the relation between the principle of maximum dissipation and inelastic evolution given by dissipation potentials

We study the evolution of systems described by internal variables. After the introduction of thermodynamic forces and fluxes, both the dissipation and dissipation potential are defined. Then, the principle of maximum dissipation (PMD) and a minimum principle for the dissipation potential are developed in a variational formulation. Both principles are related to each other. Several cases are shown where both principles lead to the same evolution equations for the internal variables. However, also counterexamples are reported where such an equivalence is not valid. In this case, an extended PMD can be formulated.

Generalized standard media and variational principles in classical and finite strain elastoplasticity

This paper uses the concept of generalized standard media in order to develop a rational theory of small and finite strain elastoplasticity. Invariance requirements for yield function and flow rule are derived especially for the finite strain case. It is shown that isotropic and kinematic hardening can be incorporated into the general concept in a natural way. Finally the thermodynamic background is discussed and used to derive a novel unified variational principle.

Application of the multiscale FEM to the modeling of cancellous bone

This paper considers the application of multiscale finite element method (FEM) to the modeling of cancellous bone as an alternative for Biot’s model, the main intention of which is to decrease the extent of the necessary laboratory tests. At the beginning, the paper gives a brief explanation of the multiscale concept and thereafter focuses on the modeling of the representative volume element and on the calculation of the effective material parameters, including an analysis of their change with respect to increasing porosity. The latter part of the paper concentrates on the macroscopic calculations, which is illustrated by the simulation of ultrasonic testing and a study of the attenuation dependency on material parameters and excitation frequency. The results endorse conclusions drawn from the experiments: increasing excitation frequency and material density cause increasing attenuation.

On the Calculation of Microstructures for Inelastic Materials using Relaxed Energies

The convexity of energy functionals for inelastic materials is analyzed on the basis of an incremental variational principle. Non-quasiconvex problems give rise to microstructures and often exhibit mesh-dependent results when being solved by standard solution methods, e.g., FEM. A partial rank-one convexification enables a reduction of the mesh-dependency and allows to predict the occurrence and distribution of microstructures independent of the numerical realization.

Rate theory of nonlocal gradient damage-gradient viscoinelasticity

Abstract A general concept for the analysis of damage evolution in heterogeneous media is proposed. Since macroscopic failure is governed by physical mechanisms on two different length-scale levels, the macro- and mesolevel, we introduce a 6-dimensional kinematical model with manifold structure accounting for discontinuous fields of microcracks, microvoids and microshear bands. As point of departure a variational functional is introduced with a Lagrangian density depending on macro- and microdeformation gradients and of a damage variable representing scalar-, vector- and/or tensor-type quantities. To derive the equations of motion for viscoinelastic damage evolution on macro- and mesolevel, we introduce into the Lagrangian the macro- and microdeformation gradients, damage variable and also their gradients and time rates. The equations of motion on macro- and mesolevel are derived for non-equilibrium states. We assume that the Lagrangian can be split into two contributions, a time-independent and a time-dependent one which can be identified with the Helmholtz free energy and a dissipation potential. This split of the Lagrangian can be used to decompose the stresses and forces into reversible and irreversible ones. The latter can be considered as dissipative driving stresses and driving forces, respectively, on defects. The model presented in this paper can be considered as a framework, which enables to derive various nonlocal and gradient, respectively, damage theories by introducing simplifying assumptions. As special cases a scalar damage and a solid-void model are considered.

Dissipation distances in multiplicative elastoplasticity

We study finite-strain elastoplasticity in a new formulation proposed in [8,1,7]. This theory does not need smoothness and is based on energy minimization techniques. In particular, it gives rise to robust algorithms. It is based on two scalar constitutive functions: an elastic potential and a dissipation potential which give rise to an energy functional and a dissipation distance.

Micromechanical concept for the analysis of damage evolution in thermo-viscoelastic and quasi-brittle materials

Abstract The aim of this paper is to develop a thermodynamically consistent micromechanical concept for the damage analysis of viscoelastic and quasi-brittle materials. As kinematical damage variables a set of scalar-, vector-, and tensor-valued functions is chosen to describe isotropic and anisotropic damage. Since the process of material degradation is governed by physical mechanisms on levels with different length scale, the macro- and mesolevel, where on the mesolevel microdefects evolve due to microforces, we formulate in this paper the dynamical balance laws for macro- and microforces and the first and second law of thermodynamics for macro- and mesolevel. Assuming a general form of the constitutive equations for thermo-viscoelastic and quasi-brittle materials, it is shown that according to the restrictions imposed by the Clausius–Duhem inequality macro- and microforces consist of two parts, a non-dissipative and a dissipative part, where on the mesolevel the latter can be regarded as driving forces on moving microdefects. It is shown that the non-dissipative forces can be derived from a free energy potential and the dissipative forces from a dissipation pseudo-potential, if its existence can be assured. The micromechanical damage theory presented in this paper can be considered as a framework which enables the formulation of various weakly nonlocal and gradient, respectively, damage models. This is outlined in detail for isotropic and anisotropic damage.

A study on the principle of maximum dissipation for coupled and non-coupled non-isothermal processes in materials

The expression for entropy production owing to diffusion used in Hackl et al. (2010 Proc. R. Soc. A 467, 1186–1196. (doi:10.1098/rspa.2010.0179)) has led to some discussion concerning its definition and comparability with terms found in the literature. Therefore, in this addendum, we introduce a slight modification which leads to a more consistent result that can be better interpreted in the light of existing literature.

Relaxed Potentials and Evolution Equations for Inelastic Microstructures

We consider microstructures which are not inherent to the material but occur as a result of deformation or other physical processes. Examples are martensitic twin-structures or dislocation walls in single crystals and microcrack-fields in solids. An interesting feature of all those microstructures is, that they tend to form similar spatial patterns, which hints at a universal underlying mechanism. For purely elastic materials this mechanism has been identified as minimisation of global energy. For non-quasiconvex potentials the minimisers are not anymore continuous deformation fields, but small-scale fluctuations related to probability distributions of deformation gradients, so-called Young measures. These small scale fluctuations correspond exactly to the observed microstructures of the material. The particular features of those, like orientation or volume fractions, can now be calculated via so-called relaxed potentials. We develop a variational framework which allows to extend these concepts to inelastic materials. Central to this framework will be a Lagrange functional consisting of the sum of elastic power and dissipation due to change of the internal state of the material. We will obtain time-evolution equations for the probability-distributions mentioned above. In order to demonstrate the capabilities of the formalism we will show an application to crystal plasticity.