Elastic solutions

Fig. 2. Elastic solution for the crack tip stress field with. small scale yielding [24].

Study of transverse shearing stress effects is divided in two parts. First, some exact elasticity solutions for composite laminates in cylindrical bending are examined. These solutions are limited in their applicability to practical problems but are extremely useful as checl oints for more broadly applicable approximate theories. Second, various approximations for treatment of transverse shearing stresses in plate theory are discussed. 

The preceding subsection was devoted to a comparison of a special exact elasticity solution with classical lamination theory results. The importance of transverse shear effects was clearly demonstrated. However, that demonstration was for a special problem of rather narrow interest. The objective of this subsection is to display approaches and results for the approximate consideration of transverse shear effects for general laminated plates. 

Completion of the problems will often require thoughtful analysis of the conditions and search for the correct solution. Thus, the problems are often not trivial or straightforward. The required mathematics are senior level except for the elasticity solutions in the micromechanics chapter where obviously the level must be higher (but the elasticity sections can be skipped in lower-level classes). 

Failure can be considered as an actual rupture (stress-rupture) or an excessive creep deformation. Correlation of stress relaxation and creep data has been covered as well as a brief treatment of the equivalent elastic problem. The method of the equivalent elastic problem is of major assistance to designers of plastic products since, by knowing the elastic solution to a problem, the viscoelastic solution can be readily deduced by simply replacing elastic physical constants with viscoelastic constants. 

The physical processes that occur during indentation are schematically illustrated in Fig. 31. As the indenter is driven into the material, both elastic and plastic deformation occurs, which results in the formation of a hardness impression conforming to the shape of the indenter to some contact depth, h. During indenter withdrawal, only the elastic portion of the displacement is recovered, which facilitates the use of elastic solutions in modeling the contact process. 

During the processing of composite materials in a hot press or an autoclave, the laminate is usually kept flat until cure is complete. If the platen surfaces are assumed frictionless, the effect of the constraints is to require that the curvatures K] and k2 be zero throughout cure. To develop the elastic solution under these constrained conditions, the laminated plate theory may be used with conditions of N = 0 and jc, = 0. The resulting midplane strains are given by... 

In Tsai [7], an elasticity solution for stresses in a pressurized thick cylindrical vessel is presented. In this analysis, the longitudinal bending deformation due to end closures is neglected, the formulation of the elasticity problem then reduces to a generalized plane strain analysis. The effects of material selection, layup sequence, and winding angles on the burst strength of thick multilayered cylinders are also addressed. 

The temperature, fiber tension, stresses, and strains vary only in the radial directions. An elasticity solution is employed to calculate the six components of the stresses and strains. The solution procedure follows the established techniques of elasticity solutions. A displacement field is assumed that satisfies the equilibrium equations and the compatibility conditions. The latter requires that at each interface the displacements and the normal stresses in adjacent... 

We shall discuss separately the effect of non-linearity, inhomogeneity and anisotropy and consider how they affect individual tests. The problem of the translation of elastic solutions to the viscoelastic case has been mentioned in the previous chapter and will not be treated further. 

In Fig. 3 the dependence of the displacements on the distance from the crack tip is shown using a normalized scale and assuming ct = 0.10. At a distance behind the crack tip the elastic-plastic boundary has an almost parabolic form, whilst directly at the crack tip — in contrast to the ideal elastic solution (Eq. (4)) — there is a certain displacement which is also the maximum width of the plastic zone. Here, and in the following text, the width of the plastic zone at the crack tip will be denoted by 2v without any coordinate parameter. In fracture mechanics terms it can be expressed as... 

The generalized stress-strain relationships in linear viscoelasticity can be obtained directly from the generalized Hooke s law, described by Eqs. (4.85) and (4.118), by using the so-called correspondence principle. This principle establishes that if an elastic solution to a stress analysis is known, the corresponding viscoelastic (complex plane) solution can be obtained by substituting for the elastic quantities the -multiplied Laplace transforms (8 p. 509). The appUcation of this principle to Eq. (4.85) gives... 

To solve this problem, we shall refer to the elastic solution given in Ref. 15 (p. 25). According to this approach, the displacement of a point x, j) of the half-plane caused by the load applied at (x, j ) is given 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. 

One of the key features that emerges from this solution, and a crucial hint for the type of analyses we will aim for in the future, is the fact that the stresses implied by the Peierls-Nabarro solution do not suffer from the same singularities that plague the linear elastic solution. A detailed examination of this point may be found in Hirth and Lothe (1992) and is illustrated in their eqn (8-13). By introducing an element of constitutive realism in the form of a nonlinear (and in fact nonconvex) interplanar potential, the solution is seen to be well behaved. 

Thermodynamics and Kinetics of Slip by U. F. Kocks, A. S. Argon and M. F. Ashby, Pergamon Press, Oxford England. This book is volume 19 of the series Progress in Materials Science, edited by B. Chalmers, J. W. Christian and T. B. Massalski. This book reflects the considerable knowledge of three leaders in the field. One of the key features that makes the book so useful is that rather than concentrating on elastic solutions it considers how dislocations give rise to observed macroscopic behavior. 

The elastic solution upon which many approaches to the consideration of microstructure are built is that of the so-called Eshelby inclusion in which one considers a single ellipsoidal inclusion in an otherwise unperturbed material. From the standpoint of the linear theory of elasticity, this problem is analytically tractable... 

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