Global large solutions to initial boundary value problems in one- dimensional nonlinear thermoviscoelasticity. (English) Zbl 0809.35135

The existence of global smooth solutions to the initial boundary value problems of one dimensional nonlinear thermoviscoelasticity is analyzed. The expressions of the constitute relations are as follows \[ e= \widehat{e} (u,\theta), \qquad \sigma=- \widehat{p}(u,\theta)+ \widehat{\mu} (u)v_ x, \qquad q=- \widehat{\kappa} (u,\theta) \theta_ x \] \((v_ x= \partial v/\partial x\), \(\theta_ x= \partial\theta/ \partial x)\) where \(e\) is the internal energy, \(u\) is the deformation gradient, \(\theta\) is the temperature, \(\sigma\) is the stress, \(v\) is the velocity, \(x\) describes the position of the material elements in a chosen reference configuration. It is assumed that \(\widehat{e} (u,\theta)\), \(\widehat{p} (u,\theta)\), \(\widehat{\mu}(u)\) and \(\widehat {\kappa} (u,\theta)\) are twice continuously differentiable for \(0\leq u<\infty\), \(0\leq \theta <\infty\) and the relations \[ \widehat{\mu}(u)u> \mu_ 0>0 \text{ for some constant }\mu_ 0, \qquad {{\partial\widehat {e}} \over {\partial u}}=- \widehat{p} (u,\theta)+ \theta {{\partial \widehat{p}} \over {\partial\theta}} \] must be satisfied. The main theorem is proved with the help of the Leray-Schauder fixed point theorem.


35Q72 Other PDE from mechanics (MSC2000)
35M10 PDEs of mixed type
74B99 Elastic materials
74H99 Dynamical problems in solid mechanics
74A15 Thermodynamics in solid mechanics
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