Thermoelastic

Theory of the fictitious temperature field allows us to analyze the problems of residual stresses in glass using the mathematical apparatus of thermoelasticity. In this part we formulate the boundary-value problem for determining the internal stresses. We will Lheretore start from the Duhamel-Neuinan relations... 

Puro, A. The inverse problem of thermoelasticity of optical tomography. J. Appl. Maths. Mechs. 1993, 57(1) I4I-I45. 

The thermoelastic law, valid only within the elastic range of isotropic and homogeneus materials, relates the peak to peak temperature changes to the peak to peak amplitude of the periodic change in the sum of principal stresses. 

The SPATE technique is based on measurement of the thermoelastic effect. Within the elastic range, a body subjected to tensile or compressive stresses experiences a reversible conversion between mechanical and thermal energy. Provided adiabatic conditions are maintained, the relationship between the reversible temperature change and the corresponding change in the sum of the principal stresses is linear and indipendent of the load frequency. 

Here we have assumed that the temperature 9 does not depend on Let the plate be isotropic and aij = Pij = (5 %, where 5 is a constant. Then (1.33) gives the quasi-static model of a thermoelastic plate ... 

This system is called a model of the thermoelastic plate which is analysed in Sections 3.3 and 3.4. For more precise and, therefore, more cumbersome relations for thermoelastic plates see (Nowacki, 1962). 

In this section we consider the boundary value problem for model equations of a thermoelastic plate with a vertical crack (see Khludnev, 1996d). The unknown functions in the mathematical model under consideration are such quantities as the temperature 9 and the horizontal and vertical displacements W = (w, w ), w of the mid-surface points of the plate. We use the so-called coupled model of thermoelasticity, which implies in particular that we need to solve simultaneously the equations that describe heat conduction and the deformation of the plate. The presence of the crack leads to the fact that the domain of a solution has a nonsmooth boundary. As before, the main feature of the problem as a whole is the existence of a constraint in the form of an inequality imposed on the crack faces. This constraint provides a mutual nonpenetration of the crack faces ... 

Here [ ] is the jump of a function across the crack faces and v is the normal to the surface describing the shape of the crack. Thus, we have to find a solution to the model equations of a thermoelastic plate in a domain with nonsmooth boundary and boundary conditions of the inequality type. 

The system of equations (3.92)-(3.94) is a model one. More precise (and more bulky) equations for a thermoelastic plate can be found, for instance, in (Nowacki, 1962). 

We can now give an exact statement of the equilibrium problem for a plate. Suppose that / G L Q ). An element (0, x) G 17 is said to be a solution to the equilibrium problem for a thermoelastic plate with a crack if it satisfies the variational inequality... 

In this section cracks of minimal opening are considered for thermoelastic plates. It is proved that the cracks of minimal opening provide an equilibrium state of the plate, which corresponds to the state without the crack. This means that such cracks do not introduce any singularity for the solution, and actually we have to solve a boundary value problem without the crack. 

Khludnev A.M. (1996d) The equilibrium problem for a thermoelastic plate with a crack. Siberian Math. J. 37 (2), 394-404. 

Nowacki W. (1962) Problems of thermoelasticity. USSR Acad. Sci., Moscow (in Russian). 

Zuazua E. (1995) Controllability of the linear system of thermoelasticity. J. Math. Pures Appl. 74 (4), 291-315. 

In the book, two- and three-dimensional bodies, plates and shells with cracks are considered. Properties of solutions are established existence of solutions, regularity up to the crack faces, convergence of solutions as parameters of a system are varying and so on. We analyse different constitutive laws elastic, thermoelastic, elastoplastic. The book gives a new outlook on the crack problem, displays new methods of studying the problems and proposes new models for cracks in elastic and nonelastic bodies satisfying physically suitable nonpenetration conditions between crack faces. 

Most, if not all, microwave biological effects and potential medical appHcations are beheved to be the result of heating, ie, thermal effects. The phenomenon of microwave hearing, ie, the hearing of clicking sounds when exposed to an intense radar-like pulse, is generally beheved to be a thermoelastic effect (161). Excellent reviews of the field of microwave bioeffects are available (162,163). 

Another anomalous property of some nickel—iron aHoys, which are caHed constant-modulus aHoys, is a positive thermoelastic coefficient which occurs in aHoys having 27—43 wt % nickel. The elastic moduH in these aHoys increase with temperature. UsuaHy, and with additions of chromium, molybdenum, titanium, or aluminum, the constant-modulus aHoys are used in precision weighing machines, measuring devices, and osciHating mechanisms (see Weighing AND proportioning). 

In the following development we consider a plane wave of infinite lateral extent traveling in the positive Xj direction (the wave front itself lies in the Xj, Xj plane). When discussing anisotropic materials we restrict discussion to those propagation directions which produce longitudinal particle motion only i.e., if u is the particle velocity, then Uj = Uj = 0. The <100>, <110>, and <111 > direction in cubic crystals have this property, for example. The derivations presented here are heuristic with emphasis on the essential qualitative features of plastic flow. References are provided for those interested in proper quantitative features of crystal anisotropy and nonlinear thermoelasticity. 

D.C. Wallace, Thermoelastic-Plastic Flow in Solids, Los Alamos Report LA-10119, 1985. 

Wallace, D.C., Thermoelastic-Plastic Flow in Solids, Los Alamos National Laboratory, University of California Report No. X-4 84-114U, Los Alamos, NM, 82 pp., February 1984. 

Thermoelasticity Rubberlike elasticity that a rigid plastic displays ... 

C. C. Chamis and G. P. Sendeckyj, Critique on Theories Predicting Thermoelastic Properties of Fibrous Composites, Journal of Composite Materials, July 1968, pp. 332-358. 

The three-dimensional thermoelastic anisotropic strain-stress relations are... 

Derive the thermoelastic stress-strain relations for an orthotropic lamina under plane stress, Equation (4.102), from the anisotropic thermoelastic stress-strain relations in three dimensions. Equation (4.101) [or from Equation (4.100)]. 

Lord Kelvin s close associate, the expert experimentalist J. P. Joule, set about to test the former s theoretical relationship and in 1859 published an extensive paper on the thermoelastic properties of various solids—metals, woods of different kinds, and, most prominent of all, natural rubber. In the half century between Gough and Joule not only was a suitable theoretical formula made available through establishment of the second law of thermodynamics, but as a result of the discovery of vulcanization (Goodyear, 1839) Joule had at his disposal a more perfectly elastic substance, vulcanized rubber, and most of his experiments were carried out on samples which had been vulcanized. He confirmed Gough s first two observations but contested the third. On stretching vulcanized rubber to twice its initial length. Joule ob-... 

Table XXXIII.—Thermoelastic Measurements of Joule on Vulcanized Rubber...

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