Thermoelasticity with Finite Wave Speeds (Oxford Mathematical Monographs)

Generalized dynamic thermoelasticity is a vital area of research in continuum mechanics, free of the classical paradox of infinite propagation speeds of thermal signals in Fourier-type heat conduction. Besides that paradox, the classical dynamic thermoelasticity theory offers either unsatisfactory or poor descriptions of a solid's response at low temperatures or to a fast transient loading (say, due to short laser pulses). Several models have been developed and intensively studied over the past four decades, yet this book, which aims to provide a point of reference in the field, is the first monograph on the subject since the 1970s. Thermoelasticity with Finite Wave Speeds focuses on dynamic thermoelasticity governed by hyperbolic equations, and, in particular, on the two leading theories: that of Lord-Shulman (with one relaxation time), and that of Green-Lindsay (with two relaxation times). While the resulting field equations are linear partial differential ones, the complexity of the theories is due to the coupling of mechanical with thermal fields. The mathematical aspects of both theories - existence and uniqueness theorems, domain of influence theorems, convolutional variational principles - as well as with the methods for various initial/boundary value problems are explained and illustrated in detail and several applications of generalized thermoelasticity are reviewed. CONTENTS......Page 6 PREFACE......Page 11 INTRODUCTION......Page 13 1.1.1 Basic considerations......Page 20 1.1.2 Global balance law in terms of (u[sub(i)], υ)......Page 26 1.1.3 Global balance law in terms of (S[sub(ij)], q[sub(i)])......Page 28 1.2.1 Basic considerations......Page 30 1.2.2 Global balance law in terms of (u[sub(i)], υ)......Page 33 1.2.3 Global balance law in terms of (S[sub(ij)], q[sub(i)])......Page 34 1.3.1 Basic considerations......Page 37 1.3.2 Global balance law in terms of (u[sub(i)], υ)......Page 44 1.3.3 Global balance law in terms of (S[sub(ij)], υ)......Page 45 2.1 Conventional and non-conventional characterization of a thermoelastic process......Page 49 2.1.1 Two mixed initial-boundary value problems in the L–S theory......Page 50 2.1.2 Two mixed initial-boundary value problems in the G–L theory......Page 52 2.2 Relations among descriptions of a thermoelastic process in terms of various pairs of thermomechanical variables......Page 53 3.1 Uniqueness theorems for conventional and non-conventional thermoelastic processes......Page 56 3.2 Existence theorem for a non-conventional thermoelastic process......Page 62 4.1 The potential–temperature problem in the Lord–Shulman theory......Page 70 4.2 The potential–temperature problem in the Green–Lindsay theory......Page 78 4.3 The natural stress–heat-flux problem in the Lord–Shulman theory......Page 84 4.4 The natural stress–temperature problem in the Green–Lindsay theory......Page 90 4.5.1 A thermoelastic wave propagating in an inhomogeneous anisotropic L–S model......Page 99 4.5.2 A thermoelastic wave propagating in an inhomogeneous anisotropic G–L model......Page 102 5.1 Alternative descriptions of a conventional thermoelastic process in the Green–Lindsay theory......Page 105 5.2 Variational principles for a conventional thermoelastic process in the Green–Lindsay theory......Page 112 5.3 Variational principle for a non-conventional thermoelastic process in the Lord–Shulman theory......Page 122 5.4 Variational principle for a non-conventional thermoelastic process in the Green–Lindsay theory......Page 125 6.1 Central equation in the Lord–Shulman and Green–Lindsay theories......Page 130 6.2 Decomposition theorem for a central equation of Green–Lindsay theory. Wave-like equations with a convolution......Page 133 6.3 Speed of a fundamental thermoelastic disturbance in the space of constitutive variables......Page 146 6.4 Attenuation of a fundamental thermoelastic disturbance in the space of constitutive variables......Page 158 6.4.1 Behavior of functions k[sub(1.2)] for a fixed relaxation time t[sub(0)]......Page 159 6.4.2 Behavior of functions k[sub(1.2)] for a fixed ∊......Page 160 6.5.1 Analysis of λ at fixed t[sub(0)]......Page 162 6.5.2 Analysis of λ at fixed ∊......Page 163 6.5.3 Analysis of the convolution kernel......Page 165 7.1 Fundamental solutions for a 3D bounded domain......Page 171 7.2 Solution of a potential–temperature problem for a 3D bounded domain......Page 183 7.3 Solution for a thermoelastic layer......Page 189 7.4 Solution of Nowacki type; spherical wave of a negative order......Page 194 7.5 Solution of Danilovskaya type; plane wave of a negative order......Page 211 7.6 Thermoelastic response of a half-space to laser irradiation......Page 216 8.1 Integral representations of fundamental solutions......Page 236 8.2 Integral equations for fundamental solutions......Page 240 8.3 Integral representation of a solution to a central system of equations......Page 241 8.4 Integral equations for a potential–temperature problem......Page 251 9.1 Recurrence relations......Page 260 9.2 Differential equation......Page 268 9.3 Integral relation......Page 271 9.4 Associated thermoelastic polynomials......Page 273 10.1 Singular surfaces propagating in a thermoelastic medium; thermoelastic wave of order n (≶0)......Page 276 10.2 Propagation of a plane shock wave in a thermoelastic half-space with one relaxation time......Page 280 10.3 Propagation of a plane acceleration wave in a thermoelastic half-space with two relaxation times......Page 289 11.1 Plane waves in an infinite thermoelastic body with two relaxation times......Page 299 11.2 Spherical waves produced by a concentrated source of heat in an infinite thermoelastic body with two relaxation times......Page 313 11.3 Cylindrical waves produced by a line heat source in an infinite thermoelastic body with two relaxation times......Page 321 11.4.1 Integral representations and radiation conditions for the fundamental solution in the Green–Lindsay theory......Page 329 11.4.2 Integral representations and radiation conditions for the potential–temperature solution in the Green–Lindsay theory......Page 333 12.1.1 Physics viewpoint and other theories......Page 340 12.1.2 Consequence of Galilean invariance......Page 342 12.1.3 Consequence of continuum thermodynamics......Page 344 12.2.1 Homogeneous case......Page 348 12.2.2 Heterogeneous case and homogenization......Page 351 12.2.3 Plane waves in non-centrosymmetric micropolar thermoelasticity......Page 352 12.3 Surface waves......Page 355 12.4.1 Flexural vibrations of a thermoelastic Bernoulli–Euler beam......Page 358 12.4.2 Numerical results and discussion......Page 361 12.5.1 Anomalous heat conduction......Page 362 12.5.2 Fractal media......Page 365 13.1 Basic field equations for a 1D case......Page 371 13.2.1 Closed-form solution to a time-dependent heat-conduction Cauchy problem......Page 374 13.2.2 Travelling-wave solutions......Page 377 13.3 Asymptotic method of weakly non-linear geometric optics applied to the Coleman heat conductor......Page 385 REFERENCES......Page 402 ADDITIONAL REFERENCES......Page 411 E......Page 423 L......Page 424 S......Page 425 Z......Page 426 D......Page 427 H......Page 428 N......Page 429 S......Page 430 W......Page 431 A Unique Monograph In A Fast Developing Field Of Generalized Thermoelasticity - An Area Of Active Research In Continuum Mechanics - This Text Focuses On Thermoelasticity Governed By Hyperbolic Equations, Rather Than On A Wide Range Of Continuum Theories. Józef Ignaczak, Martin Ostoja-starzewski. Includes Bibliographical References And Indexes.