[Paper Review] Thermo-visco-elaticity for models with growth conditions in Orlicz spaces

Filip Z. Klawe|arXiv (Cornell University)|Sep 9, 2014

Geometric Analysis and Curvature Flows26 references3 citations

TL;DR

This paper establishes the existence of weak solutions for a quasi-static thermo-visco-elastic model with temperature-dependent visco-elastic behavior in generalized Orlicz spaces, incorporating non-homogeneous material properties through $x$-dependent $N$-functions. The analysis extends previous results in Lebesgue spaces by using modular convergence, monotonicity methods, and Young measure theory to handle non-standard growth conditions.

ABSTRACT

We study a quasi-static evolution of thermo-visco-elastic model. We act with external forces on non-homogeneous material body, which is a subject of our research. Such action may cause deformation of this body and may change its temperature. Mechanical part of the model contains two kinds of deformation: elastic and visco-elastic. Mechanical deformation is coupled with the temperature and they may influence each other. Since constitutive function on evolution of visco-elastic deformation depends on temperature, the visco-elastic properties of material also depend on temperature. We consider the thermodynamically complete model related to hardening rule with growth condition in generalized Orlicz spaces. We provide the proof of existence of solutions for such class of models.

Motivation & Objective

• To extend the existence theory of thermo-visco-elastic models beyond Lebesgue spaces to generalized Orlicz spaces with $x$-dependent $N$-functions.
• To model non-homogeneous materials where the visco-elastic response depends on spatially varying growth conditions.
• To incorporate temperature-dependent constitutive laws for visco-elastic strain evolution in a thermodynamically consistent framework.
• To establish existence of weak solutions under general growth conditions without assuming upper or lower bounds on the $N$-function's growth.
• To generalize the Norton-Hoff model by allowing faster growth rates than in $L^p$ spaces, approaching Prandtl-Reuss behavior.

Proposed method

• Formulate a quasi-static thermo-visco-elastic system coupling momentum balance, energy balance, and visco-elastic strain evolution with temperature-dependent constitutive laws.
• Use generalized Orlicz spaces $L_M$ with $x$-dependent $N$-functions to model materials with spatially varying mechanical and thermal response.
• Apply the Galerkin method to construct approximate solutions in finite-dimensional subspaces.
• Employ monotonicity and renormalization techniques to pass to the limit in the nonlinear terms involving the stress and strain tensors.
• Use modular convergence and weak compactness in $L_M$ and $L_{M^*}$ spaces to handle non-standard growth conditions.
• Utilize Young measure theory to characterize the limit of products of weakly convergent sequences in the heat equation.

Experimental results

Research questions

• RQ1 Can the existence of weak solutions be established for thermo-visco-elastic models with visco-elastic growth conditions in generalized Orlicz spaces?
• RQ2 How can non-homogeneous material behavior with $x$-dependent $N$-functions be rigorously incorporated into a thermodynamically consistent model?
• RQ3 What techniques are required to handle the lack of uniform upper and lower bounds on the $N$-function's growth in the context of nonlinear PDEs?
• RQ4 How can the coupling between temperature and visco-elastic strain evolution be preserved in the limit process under general growth conditions?
• RQ5 Can the convergence of nonlinear terms in the energy equation be characterized using Young measures when standard $L^p$ tools fail?

Key findings

• The paper proves the existence of weak solutions to a quasi-static thermo-visco-elastic system with temperature-dependent visco-elastic strain evolution in generalized Orlicz spaces.
• The existence result holds under general growth conditions on the $N$-function without requiring upper or lower bounds, extending previous results in Lebesgue spaces.
• Modular convergence in $L_M$ and $L_{M^*}$ spaces is used to control the nonlinearities arising from the $x$-dependent $N$-function.
• The use of Young measures ensures the correct limit of the product term $\boldsymbol{T}^d : \boldsymbol{G}(\theta, \boldsymbol{T}^d)$ in the heat equation.
• The Galerkin approximation combined with renormalization and monotonicity methods allows passage to the limit in the nonlinear system.
• The framework accommodates non-homogeneous materials by allowing the $N$-function and the stress operator $\boldsymbol{D}$ to depend on the spatial variable $x$.