Abstract
The paper presents a versatile framework for solids which undergo nonisothermal processes with irreversibly changing microstructure at large strains. It outlines rate-type and incremental variational principles for the full thermomechanical coupling in gradient-extended dissipative materials. It is shown that these principles yield as Euler equations essentially the macro- and micro-balances as well as the energy equation. Starting point is the incorporation of the entropy and entropy rate as canonical arguments into constitutive energy and dissipation functions, which additionally depend on the gradient-extended mechanical state and its rate, respectively. By means of (generalized) Legendre transformations, extended variational principles with thermal as well as mechanical driving forces can be constructed. On the thermal side, a rigorous distinction between the quantity conjugate to the entropy and the quantity conjugate to the entropy rate is essential here. Formulations with mechanical driving forces are especially suitable when considering possibly temperature-dependent threshold mechanisms. With regard to variationally consistent incrementations, we suggest an update scheme which renders the exact form of the intrinsic dissipation and is highly suitable when considering adiabatic processes. It is shown that this proposed numerical algorithm has the structure of an operator split. To underline the broad applicability of the proposed framework, we set up three model problems as applications: Cahn-Hilliard diffusion coupled with temperature evolution, where we propose a new variational principle in terms of the species flux vector, as well as thermomechanics of gradient damage and gradient plasticity. In a numerical example we study the formation of a cross shear band.
Highlights
In order to model dissipative size effects in solid materials that are, for example, related to the width of shear bands or the grain size in polycrystals, nonstandard continuum theories have to be elaborated that are based on characteristic length scale parameters
We present a unified framework for the fully coupled thermomechanics of gradientextended dissipative solids that is applicable to a wide range of model problems such as diffusion, plasticity and damage
The paper is organized as follows: In Sect. 2 we consider as motivating example a very simple adiabatic thermomechanical material element and derive rate-type as well as incremental variational principles based on the entropy and entropy rate as canonical variables in the energy and dissipation potential functions
Summary
In order to model dissipative size effects in solid materials that are, for example, related to the width of shear bands or the grain size in polycrystals, nonstandard continuum theories have to be elaborated that are based on characteristic length scale parameters. We present a unified framework for the fully coupled thermomechanics of gradientextended dissipative solids that is applicable to a wide range of model problems such as diffusion, (crystal) plasticity and damage. The strongly coupled multifield problem will exhibit an incremental variational structure which is an extension of the framework of gradient-extended dissipative solids presented in Miehe [49, 50] towards nonisothermal processes. The construction of rate-type and incremental mixed variational principles via (generalized) Legendre transformations, where we propose a new semi-explicit numerical update scheme that has the structure of an operator split and is highly suitable when considering adiabatic processes,. 2 we consider as motivating example a very simple adiabatic thermomechanical material element and derive rate-type as well as incremental variational principles based on the entropy and entropy rate as canonical variables in the energy and dissipation potential functions. To demonstrate the capability of the newly proposed semi-explicit variational update scheme, we show a numerical example that is concerned with adiabatic shear band localization in softening plasticity
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have