Three-Dimensional Stress and Strain

The stress existing at any point in a material may always be resolved into components acting on the faces of a diifferential element in three arbitrary coordinate directions (Fig. 17.1). The stress components acting on the faces of the element are of two types normal stresses (forces acting normal to the surface per unit surface area) and shear stresses (forces acting parallel to the surface per unit surface area). The first subscript conventionally identifies the direction perpendicular to the surface in question, and the second identifies the direction of the force itself. In general, there are three normal stresses, Th, T22, and T33, and six shear stresses, ti2, Ti3, T21, 1 23, T32. These nine quantities, 

The stress tensor is usually broken into an isotropic or hydrostatic pressure and 

Caution 1 The negative signs in (17.2) and (17.3) arise from the historical conventions of treating both hydrostatic pressure (an inward force on the element) and tensile stress (an outward force on the element) as positive. These conventions are not adhered to universally, however. 

Caution 2 There are about as many different forms of notation as there have been books and papers written in this area. Be sure to understand the notation before plowing through any of the literature— That s half the battle. 

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While the model can be formulated for three-dimensional stresses and strain states, here only the imiaxial deformation is dealt with. It is important to add that the model has both a creep formalism and a relaxation formalism. The relevant equations are presented in the next paragraphs. In addition, although this is a nonlinear theory, it is a small strain theory... 

Rather than a plane-stress state, a three-dimensional stress state is considered in the elasticity approach of Pipes and Pagano [4-12] to the problem of Section 4.6.1. The stress-strain relations for each orthotropic layer in principal material directions are... 

The macromechanical behavior of a lamina was quantitatively described in Chapter 2. The basic three-dimensional stress-strain relations for elastic anisotropic and orthotropic materials were examined. Subsequently, those relations were specialized for the plane-stress state normally found in a lamina. The plane-stress relations were then transformed in the plane of the lamina to enable treatment of composite laminates with different laminae at various angles. The various fundamental strengths of a lamina were identified, discussed, and subsequently used in biaxial strength criteria to predict the off-axis strength of a lamina. 

In summary, polarizing microscopy provides a vast amount of information about the composition and three-dimensional structure of a variety of samples. The technique can reveal information about thermal history and the stresses and strains to which a specimen was subjected during formation. Polarizing microscopy is a relatively inexpensive and well accessible investigative and quality control tool. 

Viscoelasticity deals with the dynamic or time-dependent mechanical properties of materials such as polymer solutions. The viscoelasticity of a material in general is described by stresses corresponding to all possible time-dependent strains. Stress and strain are tensorial quantities the problem is of a three dimensional nature (8), but we shall be concerned only with deformations in simple shear. Then the relation between the shear strain y and the stress a is simple for isotropic materials if y is very small so that a may be expressed as a linear function of y,... 

In the last chapter we discussed the relation between stress and strain (or instead rate-of-strain) in one dimension by treating the viscoelastic quantities as scalars. When the applied strain or rate-of-strain is large, the nonlinear response of the polymeric liquid involves more than one dimension. In addition, a rheological process always involves a three-dimensional deformation. In this chapter, we discuss how to express stress and strain in three-dimensional space. This is not only important in the study of polymer rheological properties in terms of continuum mechanics " but is also essential in the polymer viscoelastic theories and simulations studied in the later chapters, into which the chain dynamic models are incorporated. 

Thus, in the three-dimensional stress space, the von Mises criterion can be represented by a circular cylinder of radius / 3, with its axis equally inclined to the three principal stress axes. Stress points plotted within the cylinder are wholly elastic and when the stress point reaches the surface of the cylinder, yielding occurs. Strain hardening will be represented by a radial expansion of the cylinder. 

Osteocytes are mature cells derived from the osteoblasts implanted in mineralized bone matrix. They have a minor contribution to new bone formation compared to osteoblasts. Osteocytes are arranged around the central lumen of an osteon and between lamellae (Fig. 1). Osteocytes have an interconnecting three-dimensional (3D) network. They are linked with adjacent osteocytes through small channels called canaliculi. Osteocytes are responsive to physiological stress and strain signals in bone tissue and also help to balance osteoblastic and osteoclastic activity to deposit new bone and to dissolve old bone, respectively. Additionally, they act as transporting agents of minerals between bone and blood. 

See Fig. 2-4 for the geometrical relationships in a two-dimensional analog of the three-dimensional stress-strain condition. So far the discussion of stress-strain relationships has been on static stress conditions. Cyclical stresses such as are applied in the operation of machine parts, in vibrating structures, and in the rapid flexing of structures that occurs in vehicles produce a different set of stress response problems. 

The simplest theories of plasticity exclude time as a variable and ignore any feature of the behaviour that takes place below the yield point. In other words, we assume a rigid-plastic material whose stress-strain relationship in tension is shown in Figure 11.9. For stresses below the yield stress there is no deformation. Yield can be produced by a wide range of stress states and not just by simple tension. In general, therefore, it must be assumed that the yield condition depends on a function of the three-dimensional stress field. In a Cartesian axis set, this is defined by the six components of stress cr , ayy, a z, Oxy, Oy and However, the numerical values of these components depend on the orientation of the axis set, and it is crucial that the 3ueld criterion be independent of the observer s chosen viewpoint. It is therefore more straightforward to make use of the principal stresses, whose directions and values are determined uniquely by the nature of the stress field. If the material itself is such that its tendency to yield is independent of... 

We have already seen in Section 10.3.4 that yield can be modelled using the Eyring process. It provides a convincing representation of both the temperature-and strain rate dependence of the yield stress. However, the discussions of Chapter 10 were confined to one-dimensional states of stress, whereas we now appreciate that yield criteria are essentially functions of the three-dimensional stress state. Also, in view of the discussion in the previous section, it is of interest to explore its applicability to pressure dependence. Both pressure dependence and the extension of the Eyring process to general stress states are considered here. 

The proposed blood-myocardium composite model enables the regional analysis to be realistically implemented by application of the finite element method at reasonable computer cost. It permits not only regional stress and strain calculations but more importantly the in vivo quantification of myocardial fiber contraction of the heart during cardiac cycles. The diastolic and systolic fiber strains can be easily computed at any instant of a cardiac cycle by use of Eqs. (8) and (9) if the myocardial layer thickness t and the blood volume fraction/in that layer are known. Presently, the layer thickness t can be calculated using the three-dimensional displacement data of the implanted markers which can be monitored by the biplane or computer-aided tomographic technique, and the blood volume... 

Hadingham PT (1983) The stress state in the human left ventricle. Adv Cardiovascular Phys 5 88-105 Heethaar RM, Pao YC, Ritman EL (1977) Computer aspects of three-dimensional finite-element analysis of stresses and strains in the intact heart. Comp and Biomed Res 10 291-295 Horowitz A, Perl M, Sideman S Minimization of fiber length changes and mechanical work in the heart muscle. Submitted for publication... 

A characteristic property of amorphous polymers is the ability to sustain large strains. For cross-linked three-dimensional networks the strain is usually recoverable and the deformation process reversible. The tendency toward crystallization is greatly enhanced by deformation since chains between points of cross-linkages are distorted from their most probable conformations. A decrease in conformational entropy consequently ensues. Hence, if the deformation is maintained, less entropy is sacrificed in the transformation to the crystalline state. The decrease in the total entropy of fusion allows crystallization, and melting, to occur at a higher temperature than would normally be observed for the same polymer in the absence of any deformation. This enhanced tendency toward crystallization is exemplified by natural rubber and polyisobutylene. These two polymers crystallize very slowly in the absence of an external stress. However, they crystallize extremely rapidly upon stretching. 

Meanwhile, Denis Poisson (1781-1840) escaped poverty through the new education system to become a teacher of physics and mechanics. The ratio named after him enables the three-dimensional effects of strains and vibrations to be considered. Noticing the correspondence between the equations describing heat flows through solid bodies and strain fields, he initiated the idea that stresses can be imagined as flows of force. 

Generalized Hooke s Law As noted previously, Hooke s law for one dimension or for the condition of uniaxial stress and strain for elastic materials is given by a = E e. Using the principle of superposition, the generalized Hooke s law for a three dimensional state of stress and strain in a homogeneous and isotropic material can be shown to be. 

Principle. The quantity, E (s), in transform space is analogous to the usual Young s modulus for a Unear elastic materials. Here, the Unear differential relation between stress and strain for a viscoelastic polymer has been transformed into a linear elastic relation between stress and strain in the transform space. It will be shown in the next chapter that the same result can be obtained from integral expressions of viscoelasticity without recourse to mechanical models, so that the result is general and not limited to use of a particular mechanical model. Therefore, the simple transform operation allows the solution of many viscoelastic boundary value problems using results from elementary solid mechanics and from more advanced elasticity approaches to solids such as two and three dimensional problems as well as plates, shells, etc. See Chapters 8 and 9 for more details on solving problems in the transform domain. 

As many nonlinear approaches are beyond the intended level and scope of this text, the focus will be on single integral mathematical models which are an outgrowth of linear viscoelastic hereditary integrals and lead to an extended superposition principle that can be used to evaluate nonlinear polymers. The emphasis will be on one-dimensional methods but these can be readily extended to three dimensions using deviatoric and dilatational stresses and strains as was the case for linear viscoelastic stress analysis as discussed in Chapters 2 and 9. 

There are other classes of fluids, such as Herschel-Bulkley fluids and Bingham plastics, that follow different stress-strain relationships, which are sometimes useful in different drilling and cementing applications. For a discussion on three-dimensional effects and a rigorous analysis of the stress tensor, the reader should refer to Computational Rheology. For now, we will continue our discussion of mudcake shear stress, but turn our attention to power law fluids. The governing partial differential equations of motion, even for simple relationships of the form given in Equation 17-57, are nonlinear and therefore rarely amenable to simple mathematical solution. For example, the axial velocity v (r) in our cylindrical radial flow satisfies... 

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