Józef Ignaczak

Thermoelasticity with finite wave speeds

Preface Introduction 1. Fundamentals of linear thermoelasticity with finite wave speeds 2. Formulations of initial-boundary value problems 3. Existence and uniqueness theorems 4. Domain of influence theorems 5. Convolutional variational principles 6. Central equation of thermoelasticity with finite wave speeds 7. Exact aperiodic-in-time solutions of Green-Lindsay theory 8. Kirchhoff type formulas and integral equations in Green- Lindsay theory 9. Thermoelastic polynomials 10. Moving discontinuity surfaces 11. Time-periodic solutions 12. Physical aspects and applications of hyperbolic thermoelasticity 13. Nonlinear hyperbolic rigid heat conductor of the Coleman type References Index

A NOTE ON UNIQUENESS IN THERMOELASTICITY WITH ONE RELAXATION TIME

Abstract Uniqueness for a general initial boundary-value problem of linear dynamic thermoelasticity with one relaxation time is established using the associated conservation law involving higher-order time derivatives.

UNIQUENESS IN GENERALIZED THERMOELASTICITY

A generalized theory of linear thermoelasticity with a single relaxation time is used to formulate a stress-flux-initial-boundary-value problem and prove its uniqueness.

DECOMPOSITION THEOREM FOR THERMOELASTICITY WITH FINITE WAVE SPEEDS

A decomposition theorem of the Boggio type is established for the linear thermoelasticity proposed by Green and Lindsay.

Soliton-like waves in a low temperature nonlinear thermoelastic solid

Abstract An analysis of soliton-like waves propagating in a low-temperature nonlinear thermoelastic solid given in J. Ignaczak, J. Thermal Stresses 13, 73–98 (1990) is extended by introducing a low-temperature parameter ω ϵ (0, 1 ] into the basic equations. The two ω-parametrized one-dimensional soliton-like thermoelastic waves that propagate with finite speeds in a given direction are obtained in an implicit form for a large range of other parameters involved, by using the method similar to that employed by Ignaczak. If ω = 1 the thermoelastic soliton-like waves from Ignaczaks article are recovered. In addition, on the basis of an approximate system of nonlinear governing equations valid for a small ω, two fast-moving soliton-like thermoelastic waves are examined in detail. Each of these two waves reveals a fountain effect in a neighborhood of a moving front, and is close to a thermodynamical equilibrium far from the front. Schematic graphs illustrate both the exact and the approximate soliton-like waves.

A STRONG DISCONTINUITY WAVE IN THERMOELASTICITY WITH RELAXATION TIMES

It is shown that in linear homogeneous isotropic thermoelasticity with two relaxation times the disturbances produced by an instantaneous concentrated source of heat in an infinite region represent a wave of order n = −1 with respect to the displacement u and temperature θ, i.e., 1 * u and 1 * θ, where * denotes convolution with respect to time, suffer jump discontinuities across a propagating spherical surface. Proof of the assertion is based on analysis of an exact series solution to a central initial boundary value problem of the theory. Also, closed forms of the decaying with lime displacement and temperature “amplitudes” of order 1.0 and +1 and hence closed forms of the radial and hoop-stress jumps are obtained. A way to weaken the strong discontinuity thermoelastic wave by improving the smoothness in time of the concentrated source of heat is discussed.

Domain of influence theorem in linear thermoelasticity

Abstract Green-Lindsays characterization of thermoelasticity is used to prove a domain of influence theorem for a homogeneous isotropic linear thermoelastic solid [1], The proof is based on an analogue of the generalized energy identity established by Zaremba [2] for the classical wave equation.

On a refined heat conduction theory for microperiodic layered solids

Abstract A refined averaged theory of a rigid heat conductor with a microperiodic structure is used to solve a one-dimensional initial boundary value problem of heat conduction in a periodically layered plate with a large number of homogeneous isotropic layers. In such a theory, the temperature θ = θ(x,t) (0 ≤ x ≤ L, t ≥ 0)is approximated by θ(x,t) = θ0(x,t) + η(x)θ1(x,t) where θ0(x,t) is a temperature-corrector and η = η(x) is a prescribed microshape function; and the functions θ0 = θ0(x,t) and θ1 = theta;1(x,t) are to be found by solving an initial-boundary value problem described by a system of linear partial differential equations with averaged coefficients subject to suitable initial and boundary conditions. A uniqueness theorem for the averaged problem is proved and two closed-form solutions for a periodically layered semispace are obtained. One of the two solutions represents the temperature field in the layered semispace due to a sudden heating of the boundary plane, while the other stands for the...

A SPATIAL DECAY ESTIMATE FOR TRANSIENT THERMOELASTIC PROCESS IN A COMPOSITE SEMISPACE

A Saint-Venants principle associated with a one-dimensional dynamic coupled ther moelastic effective modulus theory for a microperiodic layered semispace is presented. In such a theory, the displacement u=u(x,t) and the temperature theta=theta(x,t) (x>=0,t>=0) are approximated by u(x,t)=U(x,t)+h(x)V(x,t) and theta(x,t)=THETA(x,t)+ h(x)PHI(x,t), where U(x,t) and PHI(x,t) represent a macrodisplacement and a macrotemperature, respectively; V(x,t) and PHI(x,t)denote a displacement corrector and a temperature corrector, respectively; h=h(x) is a prescribed periodic microshape function; and the pairs (U,THETA) and (V,PHI) are found by solving an initial boundary value problem described by a system of linear partial differential equations with effective thermoelastic moduli subject to suitable initial and boundary conditions. It is shown that the thermoelastic energy associated with a solution to the problem and stored in the semi-space lying beyond a distance x from the loaded boundary x=0 over the time interval [0,t] decays exponentially as x to infinity and its decay length L depends on the time t, an effective velocity of thermoelastic wave (c*l), an effective time (T*), and an effective thermoelastic coupling parameter (epsilon*). In particular, it is shown that for small (large) times the function L reveals behavior of the decay length for a pure thermal (elastic) energy of a semispace.

SOLITON-LIKE SOLUTIONS IN A NONLINEAR DYNAMIC COUPLED THERMOELASTICITY

This paper examines propagation of soliton-like waves in a nonlinear homogeneous isotropic thermoelastic solid in which both the free energy and the heat flux vector depend not only on the absolute temperature and strain tensor but also on an “elastic” heat flow that satisfies an evolution equation. This equation, together with the equation of motion and the energy conservation law, leads to a nonlinear coupled system of partial differential equations from which the temperature, strain, and heat flux fields are to be found. For a one-dimensional case the system admits two closed-form solutions of the soliton type corresponding to two one-way waves propagating with velocities V1, and V2, (0 < V1, < V2) the velocities satisfy a biquadratic equation similar to that for the longitudinal waves of linear thermoelasticity with finite wave speeds. Between the ith temperature soliton Ti, = Ti, (si = x — Vi-,t; |x | < +∞, t ⁢ = 0, i = 1, 2) and the ith strain soliton [udot]i(Si ), a simple relation is established. ...