An energy‐entropy‐consistent time stepping scheme for nonlinear thermo‐viscoelastic continua

Abstract

This paper deals with an energy-entropy-consistent time integration of a thermo-viscoelastic continuum in Poissonian variables. The four differential evolution equations of first-order are transformed by a new General Equation for NonEquilibrium Reversible-Irreversible Coupling (GENERIC) format into a matrix-vector notation. Since the entropy is a primary variable, we include thermal constraints to affect the temperatures at the boundary of the body. This enhanced GENERIC format with thermal constraints yields with the related degeneracy conditions structure preservation properties for a system with thermal constraints. The properties of an isolated system are in addition to a constant total linear and angular momentum, the constant total energy, an increasing total entropy and a decreasing Lyapunov function. The last one is a stability criterion for thermo-viscoelastic systems and also for unisolated systems without loads valid. The discretization in time is done with a new TC (Thermodynamically Consistent) integrator. This ETC integrator is constructed such, that the algorithmic structural properties after the space-time discretization reflect the underlying enhanced GENERIC format with thermal constraints. As discretization in space the finite element method is used. A projection of the test function of the thermal evolution equation is necessary for an energy-consistent discretization in space. The enhanced GENERIC format with thermal constraints, which is here given in the strong evolution equations, contains the external loads. The consistency properties are discussed for representative numerical examples with different boundary conditions. The coupled mechanical system is solved with a multi-level Newton-Raphson method based on a consistent linearization.

An energy‐stable mixed formulation for isogeometric analysis of incompressible hyperelastodynamics

“…The inner radius of the disk is 0.5 m, the outer radius is 1.5 m, and the thickness is 1 m. The material of the disk is Neo‐Hookean with density ρ 0 =10 kg/m 3 and shear modulus c 1 =7.5 Pa. Both the geometrical and material settings follow the benchmark example in the work of Krüger et al The initial displacement is zero, and the spinning motion is initiated by an initial angular velocity of 1 rad/s in the x ‐ y plane, that is, We choose the reference scales as L 0 =1 m, M 0 =1 kg, and T 0 =1 s. The geometry of the domain can be exactly parametrized by connecting four thick‐walled cylinders shown in Figure and adjusting the coordinates of the control points. Therefore, we have for the discrete pressure function space.…”

Energy‐momentum‐entropy consistent numerical methods for large‐strain thermoelasticity relying on the GENERIC formalism

“…In particular, the formulation in terms of the internal energy density relies on , whereas the entropy‐based formulation relies on . Originally, the projection has been introduced in the framework of the entropy‐based formulation in Romero (see also the work of Krüger et al).…”

“…Another advantageous feature of the GENERIC framework is that it facilitates the use of different sets of independent state variables (see the work of Öttinger and Mielke. The entropy was initially preferred as a thermodynamic state variable in GENERIC‐based integrators (see the work of Romero) and Krüger et al). The work by Mielke has laid the theoretical foundation for the development of GENERIC‐based integrators relying on the temperature as thermodynamic state variable (see the work of Martín et al, Martín, and Martín and García Orden).…”

Variational integrators of higher order for flexible multibody systems

“…This is applied to the D'ALEMBERT term δW e,nc which is defined by the material law in [5] as well as the potential energy V e and the kinetic energy T e . In this way, we obtain…”

“…The variable c e i (S, t) is the corresponding internal variable to c e . The used material model for the viscoelastic rope is discribed by the free energy function in [5]. The space discretization is based on one-dimensional elements and linear LAGRANGE polynomials G a with GAUSS points ξ h in space as their approximation.…”