Nesse artigo será apresentada uma formulação de termodinâmica estendida para materiais viscoelásticos com condução de calor. O requerimento da invariância Galileana sobre as equações de balanço e o princípio de entropia conduzem à introdução de multiplicadores de Lagrange, que estabelecem equações constitutivas para os fluxos. Uma condição de hiperbolicidade do sistema de equações é obtido por meio da concavidade da densidade de entropia.

In this paper an extended thermodynamic formulation for viscoelastic materials with heat conduction is presented. The requirement of Galilean invariance on the balance equations and the entropy principle lead to the introduction of Lagrange multipliers, which provide constitutive equations for the flux. A hyperbolic condition of the equations system is obtained through the concavity of the entropy density.

A method of successive Lagrangian formulation of linear approximation for solving boundary value problems of large deformation in finite elasticity is proposed. Instead of solving the nonlinear problem, by assuming time steps small enough and the reference configuration updated at every step, we can linearize the constitutive equation and reduce it to linear boundary value problems to be solved successively with incremental boundary data. Moreover, nearly incompressible elastic body is considered as an approximation to account for the condition of incompressibility. For the proposed method, numerical computations of pure shear of a square block for Mooney-Rivlin material are considered and the results are compared with the exact solutions. Mathematical subject classification: Primary: 65C20; Secondary: 74B20.