Two-Dimensional Theory of Elasticity

To provide this basic appreciation, a brief review of two-dimensional theory of elasticity is given below, followed by a summary of the basic formulation of the crack problem. More complete treatments of the theory of elasticity may be found in standard textbooks and other treatises (e.g., Mushkilishevili [5] Sokolnikoff [6] Timoshenko [7]). 

Stress, in its simplest term, is defined as the force per unit area over a surface as the surface area is allowed to be reduced, in the limit, to zero. Mathematically, stress is expressed as follows  

In general, the stresses at a point are resolved into nine components. In Carte- 

The first letter in the subscript designates the plane on which the stress is acting, and the second designates the direction of the stress. 

For two-dimensional problems, two special cases are considered namely, p/anc stress and plane strain. For the case of plane stress, only the in-plane (e.g., the xy-plane) components of the stresses are nonzero and for plane strain, only the inplane components of strains are nonzero. In reality, however, only the average values of the z-component stresses are zero in the plane stress cases. As such, this class of problems is designated by the term generalized plane stress. The conditions for each case will be discussed later. It is to be recognized that, in actual crack problems, these limiting conditions are never achieved. References to plane stress and plane strain, therefore, always connote approximations to these well-defined conditions. 

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Fig. 12.5. A crack from an indent in glass (a) z = 0 (b) z = — 3.8 pm (c) z = —5.2 pm ELSAM, 1.5 GHz. The experimental line-scans superimposed on the images can be compared with the plots calculated using two-dimensional theory (eqns (12.2), (12.13), and (12.14)) with elastic constants from Table 6.3 and values of defocus (a) z = 0 (b) z = —4.2 pm (c) z = —6.8 pm. The values of z in the calculations were chosen for best fit the reason for the discrepancy is not known, though no doubt there are the usual uncertainties associated with thermal drift, the measurement of z, and the frequency and pupil function used (Briggs etal. 1990).

Dislocation Screening at a Crack Tip From the standpoint of the linear theory of elasticity, the problem of crack tip shielding may be seen as a question of solving the boundary value problem of a crack in the presence of a dislocation in its vicinity. As indicated schematically in fig. 11.15, the perspective we will adopt here is that of the geometrically sterilized two-dimensional problem in which both the crack front and the dislocation line are infinite in extent and perfectly straight. As we will see, even this case places mathematical demands of some sophistication. The basic question we... 

For two-dimensional randomly oriented fibers in a composite, approximating theory of elasticity equations with experimental results yielded this equation for the planar isotropic composite stiffness and shear modulus in terms of the longitudinal and transverse moduli of an identical but aligned composite system with fibers of the same aspect ratio ... 

Critical masses of cylindrical cores were determined using one-dimensional transport theory, S4 approximation, In finite cylindrical geometry. Checks on the validity of this procedure were made with a two-dimensional transport code. Elastic scattering for light elements in all cores were determined from ultra-line-group normalmode calculations. 

Fig. 3.6. Comparison of Gaussian theory of elasticity with experimental data, (i) Compression and low extension— the solid line indicates the theoretical result (ii) two-dimensional equi-biaxial extension. In the upper curve, the origin is displaced and indicates hysteresis when the sample stretches to A2=3 (iii) simple extension to break (curve (a)). Curves (b) and (c) are hysteresis curves with displaced origins. (From Treloar, 1944.)...

Classical lamination theory consists of a coiiection of mechanics-of-materials type of stress and deformation hypotheses that are described in this section. By use of this theory, we can consistentiy proceed directiy from the basic building block, the lamina, to the end result, a structural laminate. The whole process is one of finding effective and reasonably accurate simplifying assumptions that enable us to reduce our attention from a complicated three-dimensional elasticity problem to a SQlvable two-dimensinnal merbanics of deformable bodies problem. 

Phase transitions in two-dimensional (adsorbed) layers have been reviewed. For the multicomponent Widom-Rowlinson model the minimum number of components was found that is necessary to stabilize the non-trivial crystal phase. The effect of elastic interaction on the structures of an alloy during the process of spinodal decomposition is analyzed and results in configurations similar to those found in experiments. Fluids and molecules adsorbed on substrate surfaces often have phase transitions at low temperatures where quantum effects have to be considered. Examples are layers of H2, D2, N2, and CO molecules on graphite substrates. We review the PIMC approach, to such phenomena, clarify certain experimentally observed anomahes in H2 and D2 layers and give predictions for the order of the N2 herringbone transition. Dynamical quantum phenomena in fluids are also analyzed via PIMC. Comparisons with the results of approximate analytical theories demonstrate the importance of the PIMC approach to phase transitions, where quantum effects play a role. 

In textbooks, plastic deformation is often described as a two-dimensional process. However, it is intrinsically three-dimensional, and cannot be adequately described in terms of two-dimensions. Hardness indentation is a case in point. For many years this process was described in terms of two-dimensional slip-line fields (Tabor, 1951). This approach, developed by Hill (1950) and others, indicated that the hardness number should be about three times the yield stress. Various shortcomings of this theory were discussed by Shaw (1973). He showed that the experimental flow pattern under a spherical indenter bears little resemblance to the prediction of slip-line theory. He attributes this discrepancy to the neglect of elastic strains in slip-line theory. However, the cause of the discrepancy has a different source as will be discussed here. Slip-lines arise from deformation-softening which is related to the principal mechanism of dislocation multiplication a three-dimensional process. The plastic zone determined by Shaw, and his colleagues is determined by strain-hardening. This is a good example of the confusion that results from inadequate understanding of the physics of a process such as plasticity. 

In the asperity filtering regime, the Greenwood and Williamson theory no longer properly models the local contact pressure since the model contains no notion of asperity width. We describe a simple statistical method for incorporating width effects in two-dimensional polishing. A more sophisticated but more complex approach based on elasticity theory that takes into account asperity shape and the interaction of the asperity with trench structures of similar size can be found in Reference 27. The statistical approach assumes that... 

Persson and co-workers [265 267] consider a rough, rigid surface with a height prohle h x ). where x is a two-dimensional vector in the x-y plane. In reaction to /z(x) and its externally imposed motion, the rubber will experience a (time-dependent) normal deformation 8z(x, f). If one assumes the rubber to be an elastic medium, then it is possible to relate 52(q, ), which is the Fourier transform (F.T.) of 8z(x, f), to the F.T. of the stress a(q, ). Within linear-response theory, one can express this in the rubber-hxed frame (indicated by a prime) via... 

Recall from chap. 2 that often in the solution of differential equations, useful strategies are constructed on the basis of the weak form of the governing equation of interest in which a differential equation is replaced by an integral statement of the same governing principle. In the previous chapter, we described the finite element method, with special reference to the theory of linear elasticity, and we showed how a weak statement of the equilibrium equations could be constructed. In the present section, we wish to exploit such thinking within the context of the Schrodinger equation, with special reference to the problem of the particle in a box considered above and its two-dimensional generalization to the problem of a quantum corral. 

A computational design procedure of a thermoelectric power device using Functionally Graded Materials (FGM) is presented. A model of thermoelectric materials is presented for transport properties of heavily doped semiconductors, electron and phonon transport coefficients are calculated using band theory. And, a procedure of an elastic thermal stress analysis is presented on a functionally graded thermoelectric device by two-dimensional finite element technique. First, temperature distributions are calculated by two-dimensional non-linear finite element method based on expressions of thermoelectric phenomenon. Next, using temperature distributions, thermal stress distributions are computed by two-dimensional elastic finite element analysis. 

Most of the crack problems that have been solved are based on two-dimensional, linear elasticity (i.e., the infinitesimal or small strain theory for elasticity). Some three-dimensional problems have also been solved however, they are limited principally to axisymmetric cases. Complex variable techniques have served well in the solution of these problems. To gain a better appreciation of the problems of fracture and crack growth, it is important to understand the basic assumptions and ramifications that underlie the stress analysis of cracks. 

For concentrated emulsions and foams, Prin-cen [1983, 1985] proposed a stress-strain theory based on a two-dimensional cell model. Consider a steady state shearing of such a system. Initially, at small values of strain, the stress increases linearly as in elastic body. As the strain value increases, the stress reaches its yield value, then at stiU higher deformation, it catastrophically drops to the negative values. The reason for the latter behavior is the creation of unstable cell structure that provides the recoil mechanism. The predicted dependencies for modulus and the yield stress were expressed as ... 

The molecular theory of rubberlike elasticity predicts that the first coefficient, Ci, is proportional to the number N of molecular strands that make up the three-dimensional network. The second coefficient, C, appears to reflect physical restraints on molecular strands like those represented in the tube model (Graessley, 2004) and is in principle amenable to calculation. The third parameter,, is not really independent. When the strands are long and flexible, it will be given approximately by 3X, where Xm is the maximum stretch ratio of an average strand. But is inversely proportional to N for strands that are randomly arranged in the unstretched state (Treloar, 1975). Jm is therefore expected to be inversely proportional to Ci. Thus the entire range of elastic behavior arises from only two fundamental molecular parameters. 

The basic principle of SHPB experiment is the stress wave propagation theory in elastic slender bar, which is based on two basic assumptions that one is one-dimensional assumption (also known as plane assumption) and other is uniform stress assumption. The one-dimensional assumption considered that each cross section in elastic bars always keeps the plane state in the propagation process of the stress wave in the slender bar. Uniform stress assumption assumed that the stress... 

The validity of Princen s theory for concentrated water-in-oil emulsions was also investigated by Ponton et al. (2001), using the droplet size distribution determined by laser diffractometry based on the Mie theory model. Comparing the surface-volume diameter and the mass fractions of emulsions depicted an increase in the particle size with the volume fraction reduction. They showed that their experimental data (as obtained by oscillatory measurements and droplet-size distribution) corroborated the expression of the elastic shear modulus for the two-dimensional model proposed by Princen and Kiss (1986). In this model, G is proportional to (a/Rsv) l v ( l v- l c) where a is the interfacial tension, Rsv is the volume-surface radius (as obtained by laser diffractometry), and Oy and Oc are the volume fraction and the critical volume fraction, respectively (Ponton et al. 2001). The latter was found to be 0.714 experimentally, which is close to the value obtained by Princen ( 0.712) (Ponton et al. 2001). 

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