Elastic Formulation

Figure 4-51 Free-Body Diagram for an Angle-Ply Laminate 4.6.2 Elasticity Formulation...

The main problem with closed form analyses is in accounting realistically for adhesive and adherend non-linearity. Also, even the elastic formulations produce complex equations which are difficult to solve. Fortunately, it is now relatively easy to set up these equations on desk top computers and to get almost instantaneous solutions which give a good indication of the stresses acting in a lap joint under tensile loading. 

Using the equilibrium equations of the elasticity theory enables one to determine the stress tensor component (Tjj normal to the plane of translumination. The other stress components can be determined using additional measurements or additional information. We assume that there exists a temperature field T, the so-called fictitious temperature, which causes a stress field, equal to the residual stress pattern. In this paper we formulate the boundary-value problem for determining all components of the residual stresses from the results of the translumination of the specimen in a system of parallel planes. Theory of the fictitious temperature has been successfully used in the case of plane strain [2]. The aim of this paper is to show how this method can be applied in the general case. 

In integrated photoelasticity it is impossible to achieve a complete reconstruction of stresses in samples by only illuminating a system of parallel planes and using equilibrium equations of the elasticity theory. Theory of the fictitious temperature field allows one to formulate a boundary-value problem which permits to determine all components of the stress tensor field in some cases. If the stress gradient in the axial direction is smooth enough, then perturbation method can be used for the solution of the inverse problem. As an example, distribution of stresses in a bow tie type fiber preforms is shown in Fig. 2 [2]. 

Submitting the main topic, we deal with models of solids with cracks. These models of mechanics and geophysics describe the stationary and quasi-stationary deformation of elastic and inelastic solid bodies having cracks and cuts. The corresponding mathematical models are reduced to boundary value problems for domains with singular boundaries. We shall use, if it is possible, a variational formulation of the problems to apply methods of convex analysis. It is of importance to note the significance of restrictions stated a priori at the crack surfaces. We assume that nonpenetration conditions of inequality type at the crack surfaces are fulfilled, which improves the accuracy of these models for contact problems. We also include the modelling of problems with friction between the crack surfaces. 

The equilibrium problem for an elastic body occupying the domain fig can be formulated as follows. We want to find a function W = u,v,w)... 

We have to stress that the analysed problems prove to be free boundary problems. Mathematically, the existence of free boundaries for the models concerned, as a rule, is due to the available inequality restrictions imposed on a solution. As to all contact problems, this is a nonpenetration condition of two bodies. The given condition is of a geometric nature and should be met for any constitutive law. The second class of restrictions is defined by the constitutive law and has a physical nature. Such restrictions are typical for elastoplastic models. Some problems of the elasticity theory discussed in the book have generally allowable variational formulation... 

Criteria of Elastic Failure. Of the criteria of elastic failure which have been formulated, the two most important for ductile materials are the maximum shear stress criterion and the shear strain energy criterion. According to the former criterion, from equation 7... 

Viscoelastic Measurement. A number of methods measure the various quantities that describe viscoelastic behavior. Some requite expensive commercial rheometers, others depend on custom-made research instmments, and a few requite only simple devices. Even quaHtative observations can be useful in the case of polymer melts, paints, and resins, where elasticity may indicate an inferior batch or unusable formulation. Eor example, the extmsion sweU of a material from a syringe can be observed with a microscope. The Weissenberg effect is seen in the separation of a cone and plate during viscosity measurements or the climbing of a resin up the stirrer shaft during polymerization or mixing. 

The polysulfide impression materials can be formulated to have a wide range of physical and chemical characteristics by modifying the base (polysulfide portion), and/or the initiator system. Further changes may be obtained by varying the proportion of the base to the catalyst in the final mix. Characteristics varied by these mechanisms include viscosity control from thin fluid mixes to heavy thixotropic mixes, setting-time control, and control of the set-mbber hardness from a Shore A Durometer scale of 20 to 60. Variations in strength, toughness, and elasticity can also be achieved. 

The theory is initially presented in the context of small deformations in Section 5.2. A set of internal state variables are introduced as primitive quantities, collectively represented by the symbol k. Qualitative concepts of inelastic deformation are rendered into precise mathematical statements regarding an elastic range bounded by an elastic limit surface, a stress-strain relation, and an evolution equation for the internal state variables. While these qualitative ideas lead in a natural way to the formulation of an elastic limit surface in strain space, an elastic limit surface in stress space arises as a consequence. An assumption that the external work done in small closed cycles of deformation should be nonnegative leads to the existence of an elastic potential and a normality condition. 

The referential formulation is translated into an equivalent current spatial description in terms of the Cauchy stress tensor and Almansi strain tensor, which have components relative to the current spatial configuration. The spatial constitutive equations take a form similar to the referential equations, but the moduli and elastic limit functions depend on the deformation, showing effects that have misleadingly been called strain-induced hardening and anisotropy. Since the components of spatial tensors change with relative rigid rotation between the coordinate frame and the material, it is relatively difficult to construct specific constitutive functions to represent particular materials. 

The deformation may be viewed as composed of a pure stretch followed by a rigid rotation. Stress and strain tensors may be defined whose components are referred to an intermediate stretched but unrotated spatial configuration. The referential formulation may be translated into an unrotated spatial description by using the equations relating the unrotated stress and strain tensors to their referential counterparts. Again, the unrotated spatial constitutive equations take a form similar to their referential and current spatial counterparts. The unrotated moduli and elastic limit functions depend on the stretch and exhibit so-called strain-induced hardening and anisotropy, but without the effects of rotation. 

This is the condition (A.85) that the elastic limit function / be an isotropic tensor functions of its arguments. By analogy with the hypoelastic constitutive equation, the name hypoinelasticity suggests itself for this formulation. 

This is the condition (A.85) for to be isotropic, as in the hypoinelastic formulation (5.116i). It is therefore clear that, when the moduli c and b, as well as the elastic limit function do not include the special dependence on F indicated in (5.155) and (5.160), then objectivity demands that they be isotropic tensor functions of their arguments, and the spatial formulation reduces to the hypoinelastic formulation. 

It is the dependence of the spatial constitutive functions on the changing current configuration through F that renders the spatial constitutive equations objective. It is also this dependence that makes their construction relatively more difficult than that of their referential counterparts. If this dependence is omitted, then the spatial moduli and elastic limit functions must be isotropic to satisfy objectivity, and the spatial constitutive equations reduce to those of hypoinelasticity. Of course, there are other possible formulations for the spatial constitutive functions which are objective without requiring isotropy. One of these will be considered in the next section. 

Irwin [23] developed an expression for the mode I stress intensity factor around an elliptical crack embedded in an infinite elastic solid subjected to uniform tension. The most general formulation is given by ... 

The net effect is that tackifiers raise the 7g of the blend, but because they are very low molecular weight, their only contribution to the modulus is to dilute the elastic network, thereby reducing the modulus. It is worth noting that if the rheological modifier had a 7g less than the elastomer (as for example, an added compatible oil), the blend would be plasticized, i.e. while the modulus would be reduced due to network dilution, the T also would be reduced and a PSA would not result. This general effect of tackification of an elastomer is shown in the modulus-temperature plot in Fig. 4, after the manner of Class and Chu. Chu [10] points out that the first step in formulating a PSA would be to use Eqs. 1 and 2 to formulate to a 7g/modulus window that approximates the desired PSA characteristics. Windows of 7g/modulus for a variety of PSA applications have been put forward by Carper [35]. 

Whereas the quasi-chemical theory has been eminently successful in describing the broad outlines, and even some of the details, of the order-disorder phenomenon in metallic solid solutions, several of its assumptions have been shown to be invalid. The manner of its failure, as well as the failure of the average-potential model to describe metallic solutions, indicates that metal atom interactions change radically in going from the pure state to the solution state. It is clear that little further progress may be expected in the formulation of statistical models for metallic solutions until the electronic interactions between solute and solvent species are better understood. In the area of solvent-solute interactions, the elastic model is unfruitful. Better understanding also is needed of the vibrational characteristics of metallic solutions, with respect to the changes in harmonic force constants and those in the anharmonicity of the vibrations. 

In a recent version of the Tobolsky and Eyring formulation, the rate of mechanochemical degradation was considered as a Thermally Activated Barrier to Scission (TABS) process. The elastic energy function f(v /) was explicitly considered in terms of the frictional hydrodynamic drag force acting over the entire macromolecule [100]. A more detailed account of this model will be presented in Sect. 5.1. 

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