Richard B. Hetnarski

STATE SPACE AFPROACH TO THERMOELASTICITY

The method of the matrix exponential, which constitutes the basis of the state space approach of modern control theory, is applied to the nondimensional equations of coupled thermoelasticity. The disadvantages of using the thermoelastic potential are thus avoided. The results obtained can be used to generate solutions in the Laplace-transform domain to a broad class of problems in thermoelasticity. Applications to problems pertaining to a half-space and a layer are presented.

Generalized thermoelasticity: response of semi-space to a short laser pulse

An analysis of laser-induced waves propagating in an absorbing thermoelastic semi-space of the Green-Lindsay type is presented. A volume model of absorption in which a laser-induced heat has the form of a product of an exponentially decreasing function of the semi-space depth and a “skewed” Gaussian temporal profile is assumed. A closed-form solution to the associated one-dimensional initial-boundary value problem is obtained. A qualitative analysis of the solution shows that for a fixed cross-section of the semi-space the stress-temperature response of the body is represented by a pair of smooth transiental functions of time.

GENERALIZED THERMOELASTICITY: CLOSED-FORM SOLUTIONS

A study of the one-dimensional thermoelastic waves produced by an instantaneous plane source of heat in homogeneous isotropic infinite and semi-infinite bodies of the Green-Lindsay (G-L) type is presented. Closed-form Greens functions corresponding to the plane heat source are obtained using the decomposition theorem for a potential-temperature wave of the G-L theory. Qualitative analysis of the results is included.

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.

CONNECTION BETWEEN THE THERMOELASTIC POTENTIAL AND THE STATE SPACE FORMULATION OF THERMOELASTICITY

The connection between the thermoelastic potential and the state space approach to thermoelasticity is established. It is shown that a formulation based on a state vector that consists of the Laplace transforms of the thermoelastic potential and its three spatial derivatives has advantages owing to the simpler nature of the matrix involved.

COUPLED THERMOELASTICITY OF A LAYERED MEDIUM

The transfer matrix method recently developed by the authors is applied to a layered medium. The components of the state vector are taken as the temperature, the displacement, the heat flux, and the strain. Interface conditions are automatically satisfied by multiplying the initial state vector by a layer matrix, and then by a point matrix that accounts for the finite discontinuity of the strain across the interface. The missing components of the initial state vector are determined from the corresponding boundary conditions. Global and local response is then determined by continued matrix multiplication.

Thermal Stresses in Materials Due to Laser Heating

Publisher Summary This chapter discusses various aspects of thermal stresses in materials because of laser heating. Stimulated emission requires that a substantial quantity of energy be injected into a system of atoms in order to excite the majority of atoms to higher energy states. As the atoms begin to drop to lower energy states, they spontaneously emit photons, which can stimulate other atoms to emit photons. The stimulated photons have exactly the same phase, frequency, polarization, and direction as the stimulating photons. Thermal stresses due to laser heating of a material play an important role in materials processing applications, and in laser component performance. Excessive thermal stress can lead to surface and subsurface cracking of a material and, hence, degrade its performance in a future application. The thermoelastic stress field is the sum of a particular stress field and a homogeneous stress field. The imposition of the zero shear stress boundary condition allows one to investigate only the radial and circumferential stress distributions in the surface plane. The chapter also discusses the problem of a half space heated with a laser beam having a mode structure that displays an azimuthal variation of energy.

TRANSIENT THERMAL STRESSES DUE TO A LOCAL SOURCE OF HEAT MOVING OVER THE SURFACE OF AN INFINITE ELASTIC SLAB

Abstract A solution is given for the transient thermal stresses due to a local source of heat that moves at constant speed over the surface of an infinite elastic slab. The transient temperature distribution is obtained by means of the Fourier and Laplace transforms, and the associated thermal stresses are obtained by making use of the thermoelastic displacement function and the Galerkin function.

Second Sound in a Cracked Layer Based on Lord–Shulman Theory

In this study, a cracked layer is considered under thermal shock and is analyzed once by the Lord–Shulman theory and once by classical theory. In this way the effect of second sound of Lord–Shulman theory on a cracked layer is investigated. The Galerkin method is invoked to obtain finite element modeling of the cracked layer. The eight node rectangular element is used and the nodes near the crack tip are replaced to introduce the crack tip singularity (Barsoum element). The discretized form of Navier and energy equations are solved simultaneously in time domain by Newmark integration algorithm. The J integral formulation in dynamical thermal form is implemented to obtain stress intensity factors from the finite element solution of the problem.

DIRECT APPROACH TO THERMOELASTICITY

The Laplace transforms with respect to time of the governing equations of one-dimensional thermoelasticity are obtained, with the displacement and temperature fields coupled. Elimination of the Laplace transform of either field variable between the resulting linear, simultaneous, and coupled ordinary differential equations results in ordinary differential equations in the other field variable. The characteristic equations are identical for both variables. Depending on the sequence in which the elimination is carried out, the solutions are obtained in two different forms. The equivalence of these forms is established by using the properties of the characteristic roots. (This process provides an alternative to the state approach recently developed by the authors.) Applications to problems pertaining to a half-space and a layer of finite thickness are presented.