Aris Tsakmakis

Conventional continuum thermodynamics is characterized by a classical form of the energy law and the second law of thermodynamics in the form of the Clausius–Duhem inequality. This thermodynamic framework fails to capture certain features in material response as, e.g., length scale effects, temperature waves in rigid heat conductors and diffusion phenomena. Typically, such material characteristics are associated with pronounced non-localities in time and space. To tackle these issues, non-conventional thermodynamic approaches might be appropriate. The employment of non-conventional thermodynamics is very attractive, as it enables the extension of the applicability of conventional thermodynamics in a simple way. A specific non-conventional thermodynamic framework has previously been proposed as a generalization of irreversible thermodynamics. Energy supply effects were neglected in this work. However, energy supply terms may become important when discussing thermodynamical consistency of many physical models. The present paper extends the applicability of the proposed non-conventional thermodynamic framework by accounting for energy supply densities and demonstrates its capabilities by addressing Cattaneo’s heat conduction law—known for predicting temperature waves—and thermo-diffusion coupling theories. It is shown that, within the adopted thermodynamics, the considered physical models are thermodynamically consistent and that the resulting field theories admit formulations within a variational framework for rate problems; in this sense, the models are properly formulated.

The paper addresses the numerical calculation of multi‐axial fatigue crack propagation in ductile materials under proportional and non‐proportional cyclic loading conditions. The analysis is based on a phase field fracture theory proposed in earlier work, which describes crack propagation on the basis of the evolution of plastic strain. A plane stress formulation of the theory is developed and used to analyze both length scale effects and comparison with experimental results. The considered loading histories comprise proportional and non‐proportional cyclic tension/compression and shear loadings. Generally, the predicted responses are in good agreement with the experimental results for proportional loading histories. Qualitative differences seem to exist, however, in what concerns the number of predicted cracks for non‐proportional cyclic loading conditions.

Phase field models have been successfully applied in recent years to a variety of fracture mechanics problems, such as quasi-brittle materials, dynamic fracture mechanics, fatigue cracks in brittle materials, as well as ductile materials. The basic idea of the method is to introduce an additional term in the energy functional describing the state of material bodies. A new state variable is included in this term, the so-called phase field, and enables to determine the surface energy of the crack. This approach allows to model phenomena such as crack initiation, crack branching and buckling of cracks, as well as the modelling of the crack front in three-dimensional geometries, without further assumptions. There is yet no systematic investigation of the influence of strain hardening on crack development within the phase field method. Thus, the aim of the paper is to provide an analysis of the effect of kinematic and isotropic hardening on the evolution of the phase field variable.

The fundamental idea in phase field theories is to assume the presence of an additional state variable, the so-called phase field, and its gradient in the general functional used for the description of the behaviour of materials. In linear elastic fracture mechanics the phase field is employed to capture the surface energy of the crack, while in damage mechanics it represents the variable of isotropic damage. The present paper is concerned, in the context of plasticity and ductile fracture, with a commonly used phase field model in fracture mechanics. On the one hand, an appropriate framework for thermodynamical consistency is outlined. On the other hand, an analysis of the model responses for cyclic loading conditions and pure kinematic or pure isotropic hardening are shown.

The configurational force concept is known to describe adequately the crack driving force in linear fracture mechanics. It seems to represent the crack driving force also for the case of elastic-plastic material properties. The latter has been recognized on the basis of thermodynamical considerations. In metal plasticity, real materials exhibit hardening effects when sufficiently large loads are applied. Von Mises yield function with isotropic and kinematic hardening is a common assumption in many models. Kinematic and isotropic hardening turn out to be very important whenever cyclic loading histories are applied. This holds equally regardless of whether the induced deformations are homogeneous or non-homogeneous. The aim of the present paper is to discuss the effect of nonlinear isotropic and kinematic hardening on the response of the configurational forces and related parameters in elastic-plastic fracture problems.