**22**, 4, pp. 545-563, 1974

### Some Methods of Solving Problems of Non-Linear Thermo-Viscoelasticity

The formulation of the problems of non-linear thermo-viscoelasticity, including also coupled problems is given. For that purpose application of thermodynamics of irreversible processes to the visco-elastic media is analysed. Certain methods of subsequent approximations are presented for the solution of problems of non-linear theory of thermo-viscoelasticity, among other "rapidly convergent" method. The rate of convergence of these methods is analysed. Numerical methods are also considered, mainly the method of nets and the method of finite elements.

**Full Text:**PDF

#### References

[On coupled problems of mechanics of a continuum, Selected works: «Elasticity and non-elasticity»],

[Plasticity (Foundations of the general theory)],

P. LÉVY, Problèmes corcrets d'analyse fonctionnelle, Deuxième edition, Paris 1951.

M. B. GURTIN, Eli STERNBERG, On the linear theory of viscoelasticity, Arch. Ration Mech. and Analys, 11, 4, 291 - 356, 1962.

C. TRUESDELL, W. NOLL, The non-linear field theories of mechanics, Handbuch der Physik, Band III/3, 1965.

[Equa-tions of state in the non-linear theory of viscoelasticity],

[Mathematical theory of non-linear viscoelasticity. Selected works: «Elasticity and non-elasticity»],

[Plasticity],

[Fundamental equations and theorems for a certain model of a physically non-linear medium],

[Quasi-linear theory of viscoelasticity and the perturbation method],

[A contribution to the problem of non-linear theory of viscoelästicity],

H. LEADERMAN, F. MC CRAKIN, O. NAKADA, Large longitudinal retarted elastic deformation of rubberlike network polimers, Trans. Soc. Rheology, N 7, 1963.

[The uniqueness theorem of coupled thermo-viscoelasticity],

[On the method of elastic solutions],

[The convergence of the method of elastic approximations in non-linear viscoelastincity],

[Some problems of existence and approximate solution for quasi-linear elliptic equations and sets of equations in S. L. Sobolev's spaces],

I. BABUSKA, I. HLVACEK, On the existencense and uniqueness of solution in the theory of viscoelasticity, Arch. Mech. Stos., 1, N 18, 1966.

[On a new method for solution of some quasi-static problems in non-linear continuum mechanics],

[Equations of mathematic physics],

[Equations of mathematic physics],

[Introduction into the theory of difference schemes],

[Introduction into the theory. of difference schemes],

[Method of small steps for solving multi-dimensional problems of theoretical physics],

[Difference methods for solving boundary-value problems],

[Fundamental notions of numerical analysis],

[Numerical solutions of problems of the theory of elasticity],

[Solution of boundary value problems of the plane theory of elasticity by means of digital and analogue computers

[Numerical methods in the theory of viscoelasticity],

[Difference methods for solving certain nonstationary systems. Selected works: «Applied mathematics and programming>],

[Numerical application of variational methods],

O.C. ZIENKIEWICZ, Y. K. CHEUNG, The finite element method in structural and continuum mechanics, Mc Graw-Hill Publ. Comp. Lim., NY, 1967.

J. T. ODEN. The finite element method in non-lineare continuum mechanics, Mc Graw-Hill Company, N.Y., 1972.

[Dependence of a solution in the theory of elasticity on the elastic constants],

[Computation of viscoelastic systems according to the numerical elastic method],

[Problem of coupling in the mechanics of polymers],

R. C. PETROF, S. GRATCH, Wawe propagation in a viscoelastic material with temperature-dependent properties and thermomechanical coupling, J. Appl. Mech., Series E, 31, 3, 1964.

[Numerical method for solving coupled problems of thermo-viscoelasticity],

[Algorithms in the theory of elasticity and the strain theory of plasticity],

[Algorithm system for solving, by means of an electronic computer, problems of the theory of elasticity and plasticity. Selected works: «Problems of numerical and applied ma thematics»,]