A Variational Framework for the Thermomechanics of Gradient-Extended Dissipative Solids – with Applications to Diffusion, Damage and Plasticity
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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 additional arguments into constitutive energy and dissipation functions which 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. To underline the broad applicability of the proposed framework, we exemplarily set up three model problems, namely Cahn-Hilliard diffusion coupled with temperature evolution as well as thermomechanics of gradient damage and gradient plasticity. In a numerical example we study the formation of a cross shear band.
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