Elasticity Thermoelasticity

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. 

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. 

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). 

Thermoelasticity Rubberlike elasticity that a rigid plastic displays ... 

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-... 

These deductions from basic facts of observation interpreted according to the rigorous laws of thermodynamics do not alone offer an insight into the structural mechanism of rubber elasticity. Supplemented by cautious exercise of intuition in regard to the molecular nature of rubberlike materials, however, they provide a sound basis from which to proceed toward the elucidation of the elasticity mechanism. The gap between the cold logic of thermodynamics applied to the thermoelastic behavior of rubber and the implications of its... 

Force-temperature ("thermoelastic") relations lead to a quantitative assessment of the relative amounts of entropic and energetic components of the elasticity of the network. 

Thus, this consideration shows that the thermoelasticity of the majority of the new models is considerably more complex than that of the phantom networks. However, the new models contain temperature-dependent parameters which are difficult to relate to molecular characteristics of a real rubber-elastic body. It is necessary to note that recent analysis by Gottlieb and Gaylord 63> has demonstrated that only the Gaylord tube model and the Flory constrained junction fluctuation model agree well with the experimental data on the uniaxial stress-strain response. On the other hand, their analysis has shown that all of the existing molecular theories cannot satisfactorily describe swelling behaviour with a physically reasonable set of parameters. The thermoelastic behaviour of the new models has not yet been analysed. 

In the swollen state the situation is somewhat better. In many swollen networks Eq. (III-26) is reasonably well obeyed and the application of the thermo-elasticity equation yields a 3 In 0/3 T value which is reasonably independent of the deformation at least in good diluents (46). It should be pointed out that the experimental error in measuring the thermoelasticity of open, swollen systems is quite appreciable. This derives from the fact that the conversion term for an open system is more complicated because of the change in diluent content with temperature (86,87) ... 

Hoeve, C. A. J.., and A. Ciferri Limitations of the application to semi-crystalline fibers of thermoelastic relations for high elastic materials A reply to W. Prins. J. Polymer Sci. 60, 68 (1962). 

From the dynamic mechanical investigations we have derived a discontinuous jump of G and G" at the phase transformation isotropic to l.c. Additional information about the mechanical properties of the elastomers can be obtained by measurements of the retractive force of a strained sample. In Fig. 40 the retractive force divided by the cross-sectional area of the unstrained sample at the corresponding temperature, a° is measured at constant length of the sample as function of temperature. In the upper temperature range, T > T0 (Tc is indicated by the dashed line), the typical behavior of rubbers is observed, where the (nominal) stress depends linearly on temperature. Because of the small elongation of the sample, however, a decrease of ct° with increasing temperature is observed for X < 1.1. This indicates that the thermal expansion of the material predominates the retractive force due to entropy elasticity. Fork = 1.1 the nominal stress o° is independent on T, which is the so-called thermoelastic inversion point. In contrast to this normal behavior of the l.c. elastomer... 

Abstract In this contribution, the coupled flow of liquids and gases in capillary thermoelastic porous materials is investigated by using a continuum mechanical model based on the Theory of Porous Media. The movement of the phases is influenced by the capillarity forces, the relative permeability, the temperature and the given boundary conditions. In the examined porous body, the capillary effect is caused by the intermolecular forces of cohesion and adhesion of the constituents involved. The treatment of the capillary problem, based on thermomechanical investigations, yields the result that the capillarity force is a volume interaction force. Moreover, the friction interaction forces caused by the motion of the constituents are included in the mechanical model. The relative permeability depends on the saturation of the porous body which is considered in the mechanical model. In order to describe the thermo-elastic behaviour, the balance equation of energy for the mixture must be taken into account. The aim of this investigation is to provide with a numerical simulation of the behavior of liquid and gas phases in a thermo-elastic porous body. 

For most particulate composites the mismatch between the particles and the matrix is more important than the anisotropy of either component (though alumina/aluminium titanate composites provide a notable exception and are described below). The main features of the stresses can therefore be understood in terms of a simple elastic model assuming thermoelastic isotropy and consisting of a spherical particle in a concentric spherical shell of matrix with dimensions chosen to give the appropriate volume fractions. The particles are predicted to be under a uniform hydrostatic stress, ap after cooling. If the particles have a larger thermal expansion coefficient than the matrix, this stress is tensile, and vice versa. For small particle volume fractions the stress... 

The constitutive model makes use of the decomposition of the rate of deformation D into an elastic, De, and a plastic part, Dp, as D = De + Dp. Prior to yielding, no plasticity takes place and Dp = 0. In this regime, most amorphous polymers exhibit viscoelastic effects, but these are neglected here since we are primarily interested in those of the bulk plasticity. Assuming the elastic strains and the temperature differences (relative to a reference temperature T0) remain small, the thermoelastic part of the response is expressed by the hypoelastic law... 

Kemer, E.H. (1956). The elastic and thermoelastic properties of composite media. Proc. Phys. 

In this relation, a and a, are the thermal expansion coefficients of the substrate and film. These depend on the temperature T. If the film is homogeneous and elastically isotropic, the in plane thermoelastic stress is expressed by ... 

For a sinusoidal steady excitation and small deflections, the elastic and viscoelastic solutions are formally similar, as the separation of variables methodology outlined above suggests. Thus, in this case, the viscoelastic response is dependent on only the specific material properties of the sample under study. Moreover, on the basis of one of the hypotheses mentioned above, the thermoviscoelastic problem can be reduced to a thermoelastic one. Therefore, in the present context only the elastic solution of the problem will be discussed. 

Most micromechanical theories treat composites where the thermoelastic properties of the matrix and of each filler particle are assumed to be homogeneous and isotropic within each phase domain. Under this simplifying assumption, the elastic properties of the matrix phase and of the filler particles are each described by two independent quantities, usually the Young s modulus E and Poisson s ratio v. The thermal expansion behavior of each constituent of the composite is described by its linear thermal expansion coefficient (3. It is far more complicated to treat composites where the properties of some of the individual components (such as high-modulus aromatic polyamide fibers) are themselves inhomogeneous and/or anisotropic within the individual phase domains, at a level of theory that accounts for the internal inhomogeneities and/or anisotropies of these phase domains. Consequently, there are very few analytical models that can treat such very complicated but not uncommon systems truly adequately. 

The unique cellular morphologies of foams play a key role in determining their deformation mechanisms [51. They also allow the development of very simple alternative equations based on the mechanical models of beam theory (a branch of civil engineering) combined with scaling concepts, to estimate both the thermoelastic properties and the strengths of foams. Such simple relationships have been developed for foams manifesting elastomeric, elastic-plastic and elastic-brittle responses to mechanical defonnation. While much of this work has focused on the responses of foams to compressive defonnation because of the special importance of this deformation mode in many applications of foams, the responses of foams to tensile and shear deformation have also been considered within this theoretical framework. 

As was noted above, local nonuniform heating of the crystal in the pore region should give rise to stresses around the pore. In the framework of the theory of thermoelasticity, an expression relating the stresses to the temperature distribution can be obtained. For this purpose, using the methods described in Gatewood [36] a relationship can be derived for the radial component of the elastic stress tensor at the pore boundary ... 

The thermoelastic effect is the temperature change that results from stretching an elastic material or fiber. The work done on the material is aiven bv... 

Pig. 12. Changes on shortening in the thermoelastic force (curves 1 and la), the potential-elastic force (curves 2 and 2a) and the mechanical force (curves 3 and 3a). Left, model fiber (A. Weber and H. H. Weber, 1951) right, actomyosin thread (Portzehl, 1950b). 

Terzaghi for saturated elastic medium with a single porosity. The Duhamel-Neumann extension to Hooke s law gives the thermoelastic constitutive relationship as,... 

As far as the thermoelasticity performance of a polymer network k concerned, it is generally true that the hi er the concentration of crosslinks or the lower the dimensions of loopholes in the network, the more significant are the changes in the polymer I operties. The elasticity of a polymer is also enhanced by i iyskally entangled chains, whose number increases with the number of crosslinks, the character of the crosslinks being interrdated with the character of the physical aitan ements [27]. This may be illustrated by experiments with two crosslinked samples prepared from low density polyethylene. The first sample was crosslinked via the silane pathway, the second by... 

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