Internal Variables in Thermoelasticity

Abstract

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Thermodynamical Extension of a Symplectic Numerical Scheme with Half Space and Time Shifts Demonstrated on Rheological Waves in Solids

“…with its form written via its roots, 8 Namely, the problem of differing multiplicities for |ξ| = 1 can be wiped out by the boundary conditions. reveals, on one side, that, in order to have both |ξ + | ≤ 1 and |ξ + | ≤ 1, both magnitudes have to be 1 (since their product is 1), which, on the other side, also implies…”

Weakly Nonlocal Non-Equilibrium Thermodynamics: the Cahn-Hilliard Equation

“…This idea was used independently by in several works [10,2,37], but as a method was introduced in classical irreversible thermodynamics, where the starting point is the Gibbs relation for the classical extensives [34,35,26]. Gerard Maugin applied the approach in several cases, including internal variables [36,56,54,57]. Let us assume that the evolution equation of the internal variable ξ is written in a general form as…”

“…The relation of the two approaches and a more systematic background with complete continuum mechanics on material manifolds is developed e.g. in [57]. Let us calculate the time derivative of the entropy density and separate the full divergences with Leibnitz rule:…”

“…One may observe that natural boundary conditions emerge assuming a zero entropy flux. These are convenient for numerical solutions and effectively substitute the natural boundary conditions of variational principles [2,57].…”

Morphing continuum theory for turbulence: Theory, computation, and visualization

“…Equations 19 connect the thermodynamic fluxes and the thermodynamic forces, and can be referred to as Onsager's reciprocal relations derived in 1931 [15,16] leading to his Nobel Prize in Chemistry in 1968. It should be noted that the reciprocity is the condition for the existence of the dissipative potential [59,60]. With further algebraic manipulation, the linear constitutive equations for the morphing continuum are…”