Well-posedness of the equations of generalized thermoelasticity

Chrzeszczyk, A.

doi:10.1007/bf00041771

Cited by 2 publications

(1 citation statement)

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“…Chrzeszczyk [17] proved the local existence, uniqueness, and continuous dependency for the solution of the initial-value problem for the generalized thermoelastic materials. In the analysis, he wrote the field equations as a quasi-linear hyperbolic system.…”

Section: Introductionmentioning

confidence: 99%

Coupled thermopiezoelectric behaviour of a one-dimensional functionally graded piezoelectric medium based on C–T theory

Akbarzadeh

Babaei

Chen

2011

Proceedings of the Institution of Mechanical Engineers, Part C:

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In this article, the transient thermopiezoelectric behaviour of a one-dimensional (1D) functionally graded piezoelectric medium subjected to a moving heat source is investigated. The formulation is given based on the Chandrasekhariah and Tzou (C–T) generalized thermoelasticity theory to consider the details of energy transport in the material in comparison with the Lord–Shulman (L–S) generalized theory. All material properties are taken to vary exponentially along the length of the medium except for phase lags, the relaxation time, and the specific heat, which are taken to be constant. The governing partial differential equations are given in the three coupled fields of displacement, temperature, and electric potential based on the C–T theory. Using Laplace transform to eliminate the time dependency of the problem, an analytical method is presented to obtain the coupled fields in the Laplace domain. The solutions are then derived in time domain by employing the fast Laplace inversion technique. Numerical results are shown to display the effects of discontinuities on the temperature and stress distribution, non-homogeneity index and the phase-lag constants of heat flux and temperature gradient on the wave propagation of temperature and stress fields based on the dual-phase-lag model of the C–T. Furthermore, the results are compared between the C–T and L–S thermoelasticity theories. Finally, the results are validated with those reported in the literature.

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Section: Introductionmentioning

confidence: 99%

Coupled thermopiezoelectric behaviour of a one-dimensional functionally graded piezoelectric medium based on C–T theory

Akbarzadeh

Babaei

Chen

2011

Proceedings of the Institution of Mechanical Engineers, Part C:

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Non-smooth hyperbolic mixed problems of the second order

Chrzeszczyk

1997

Jpn. j. math

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IntroductionIt is well known that any mathematical model of arbitrary dynamical the ory should satisfy the requirement of the existence of solutions to the correspond ing initial and initial-boundary value problems. Unfortunately the known existence theorems are not strong enough to describe global solutions in elasticity or ther moelasticity, in the general, multidimensional and nonliner case. In some special cases, for example for special materials and very smooth and small initial data one can prove the global in time existence (cf. for example [10,25,32]) but generally, the global solvability in the class of smooth functions is not expected. Furthermore the experience with formation of singularities (cf. [9,13,26,27]) suggests that a sufficiently general, non-linear theory of evolution problems should admit the non classical solutions, i.e. the solutions with discontinuities in some derivatives occur ing in the corresponding equations. Unfortunately almost all papers concerning the existence of solutions for nonlinear, multidimensional hyperbolic and hyperbolic parabolic problems of the second order are restricted to the classical solutions, cf. for example [1, 3-4, 6-8, 10-12, 14-15, 17-18, 22-23, 25, 28-32]. An exceptional character is found in the papers [16,21] but it seems that the results contained there do not cover the systems of elasticity or thermoelasticity.The main purpose of this article is to present some methods of proving the local in time existence results for initial-boundary value problems for second order quasilinear hyperbolic systems modeled on the system of elasticity. The solutions we consider are relatively regular but they may possess singularities in derivatives of the second order and the coefficients of the investigated quasi-linear systems are not necessarily Lipschitz-continuous. As a consequence the methods developed in the papers quoted above are not aplicable and we are forced to apply a new technique. In the case of the initial problem (cf.[5]) it is possible to give relatively short proof of the existence of non-classical solutions for a class of systems generalizing the systems of elasticity and thermoelasticity, using the methods from the book [34].The case of the mixed, initial-boundary value problem is more complicated 188 ANDRZEJ CHRZESZCZYK and therefore in the present paper we restrict ourselves to the hyperbolic system of the type of elasticity. The proof of the existence of solutions to the non-linear mixed problem (cf. Section 4) is based on some quite delicate results concerning the corresponding linear problem. This is the reason for which the largest part of this article is devoted to the analysis of the linear, hyperbolic, mixed problems with non smooth coefficients (cf.

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Well-posedness of the equations of generalized thermoelasticity

Cited by 2 publications

References 8 publications

Coupled thermopiezoelectric behaviour of a one-dimensional functionally graded piezoelectric medium based on C–T theory

Coupled thermopiezoelectric behaviour of a one-dimensional functionally graded piezoelectric medium based on C–T theory

Non-smooth hyperbolic mixed problems of the second order

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