Concepts of Stress and Strain

The output of such a tensile test is recorded (usually on a computer) as load or force versus elongation. These load-deformation characteristics depend on the specimen size. For example, it requires twice the load to produce the same elongation if the cross-sectional area of the specimen is doubled. To minimize these geometrical factors, load and elongation are normalized to the respective parameters of engineering stress and engineering strain. Engineering stress a- is defined by the relationship 

Compression stress-strain tests may be conducted if in-service forces are of this type. A compression test is conducted in a manner similar to the tensile test, except that the force is compressive and the specimen contracts along the direction of the stress. Equations 6.1 and 6.2 are utilized to compute compressive stress and strain, respectively. By convention, a compressive force is taken to be negative, which yields a negative stress. Furthermore, because /q is greater than /, compressive strains computed from Equation 6.2 are necessarily also negative. Tensile tests are more common because they are easier to perform also, for most materials used in structural applications, very little additional information is obtained from compressive tests. Compressive tests are used when a material s behavior under large and permanent (i.e., plastic) strains is desired, as in manufacturing applications, or when the material is brittle in tension. 

These same mechanics principles allow the transformation of stress components from one coordinate system to another coordinate system with a different orientation. Such treatments are beyond the scope of the present discussion. 

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In this idealization, determinate stmctures are defined by geometry, and concepts of stress and strain are not needed. 

This brings us to the concepts of stress and strain and their ratio, which we call Young s modulus. Like Hooke, Young (Figure 13-7) was a brilliant scientist who made major contributions to many fields (e.g., the wavelike nature of light, color perception), but his communication skills were somewhere between abysmal and incomprehensible. His definition of the modulus (>- 1800) given his name reads ... 

It is desirable at this juncture to outline very briefly the concepts of stress and strain. For a more comprehensive discussion, the reader is referred to standard textbooks on the theory... 

Chemical reactions in boundary lubrication are different from static reactions even if the reactive substances involved are the same. The temperature to activate a chemical reaction on rubbing surfaces is usually lower than that required in the static chemical process. Some believe this is because of the naked surfaces and structural defects created by the friction/wear process, which are chemically more active. Kajdas proposed a new concept that accumulations of stress and strain in friction contacts could cause emission of low-... 

Employing the concepts of stress and conjugate strain, and their proper mathematical formulation as. second-rank tensors, now enables us to deal with mechanical work in a general anisotropic piece of matter. One realization of sudi a system are fluids in confinement to whidi this book is devoted. However, at the core of our subsequent treatment are thermal properties of confined fluids. In other words, we need to understand the relation between medianical work represented by stress-strain relationships and other forms of energy such as heat or chemical work. This relation will be formally... 

On the other hand, Fukada et al.[5] found piezoelectricity properties in bone which was stressed. There are several reports [11,20,21] which are based on evidence that bone demonstrates a piezoelectric effect. This is used to explain the concept of stress- or strain-induced bone remodelling which is often refered to as Wolfs law[3]. Thus, bone converts mechanical stress to an electrical potential that influences the activity of osteoclasts and osteoblasts[l]. It is also known that the interior structure of bone(trabecular architecture) is arranged in compressive and tensile systems corresponding to the principal stress directions[4]. The role of the voltage signals induced in bamboo and palm we found may also be similar to the piezoelectric effect in bone. 

The definitions of stress and strain developed earlier were for uniform stress states but, often, one has to deal with situations in which the stress and strain are non-uniform and vary from point to point in a body. In these cases, one must consider stresses and strains in a more general way. When a body is loaded in a complex way, different particles of the body will be displaced relative to one another. It is important, therefore, to define both the coordinate of a point and its displacement. The position of a particle P can be defined by its coordinates x, X2, Xj in a set of cartesian axes V, X, Xy which are fixed and independent of the body. As a shorthand, the coordinates can be written as x., where /= 1,2 and 3. Suppose, as shown in Fig. 2.13, that the deformation and movement of the body displaces the particle at P to P, such that the new coordinates are X +M, X2+U2 and X3+M3. The vector u. (same subscript notation) is then the displacement of P. There must, however, be a relationship between the vectors u. and x. because if u.v/as a constant for all the particles in the body, this would only represent a rigid translation of the body and not a deformation. It is the relationship between the two vectors that leads to the concept and precise definition of strain. An essential part of this definition is that u. varies from one particle to another in a body, that is u.=f x.). 

Odemark s equivalent thickness concept consists of the transformation of a two or more layer system with different characteristic properties, E and p, into an equivalent one-layer system with equivalent thickness but one elastic modulus, that of the bottom (last) layer. Thus, one elastic, isotropic and homogeneous layer results and calculations of stresses and strains are easier. The transformation a two-layer system into an equivalent one-layer system is explained in Figure 11.13. 

The finite element (FE) technique is an approximate numerical method for solving differential equations. Within the field of adhesive technology, it is most commonly used to determine the state of stress and strain within a bonded joint. It can also be used to determine moisture diffusion, natural frequencies of vibration and other field problems. Although this article will concentrate on the stress analysis, the same concepts can be applied to these other applications of finite element analysis. The basis of any finite element method is the discretization of the (irregular) region of interest into a number of... 

The DCB test, the blister test, and several other geometries are somewhat amenable to the analytical analyses needed for fracture mechanics. As a consequence, most early fracture mechanics analyses focused on such geometries. Modern computational methods, particularly finite element methods (FEM), have lifted this restriction. A brief outline of how FEM might be used for this purpose may be helpful. Inherent in fracture mechanies is the concept that natural cracks or other crack-like discontinuities exist in materials, and that failure of an object generally initiates at such points [13,16,17,23-25]. Assuming that a crack (or a debonded region) is situated in an adhesive bond line, modern computation techniques can be used to facilitate the computation of stresses and strains throughout a body, even where analytical solutions may not be convenient or even possible. 

The Stress-Rang e Concept. The solution of the problem of the rigid system is based on the linear relationship between stress and strain. This relationship allows the superposition of the effects of many iadividual forces and moments. If the relationship between stress and strain is nonlinear, an elementary problem, such as a siagle-plane two-member system, can be solved but only with considerable difficulty. Most practical piping systems do, ia fact, have stresses that are initially ia the nonlinear range. Using linear analysis ia an apparendy nonlinear problem is justified by the stress-range concept... 

The mechanical concepts of stress are outlined in Fig. 1, with the axes reversed from that employed by mechanical engineers. The three salient features of a stress-strain response curve are shown in Fig. la. Initial increases in stress cause small strains but beyond a threshold, the yield stress, increasing stress causes ever increasing strains until the ultimate stress, at which point fracture occurs. The concept of the yield stress is more clearly realised when material is subjected to a stress and then relaxed to zero stress (Fig. Ih). In this case a strain is developed but is reversed perfectly - elastically - to zero strain at zero stress. In contrast, when the applied stress exceeds the yield stress (Fig. Ic) and the stress relaxes to zero, the strain does not return to zero. The material has irreversibly -plastically - extended. The extent of this plastic strain defines the residual strain. 

Define stress and strain. Explain how these two concepts are interrelated and provide three every day examples of stresses and their corresponding strains. 

To characterize the mechanical properties of a material, one first needs a basic understanding of the concepts of stress, strain, and deformation, as they provide the tools necessary. The deformation properties of a material can be determined by applying a stress, either in compression or tension, and determining dimensional changes in the specimen. The applied stress will result in an elongation of the specimen, e = Al/l0. The elongation is called the strain, while the stress is defined as the applied load divided by the area over which it is applied. 

In this introduction, the viscoelastic properties of polymers are represented as the summation of mechanical analog responses to applied stress. This discussion is thus only intended to be very introductory. Any in-depth discussion of polymer viscoelasticity involves the use of tensors, and this high-level mathematics topic is beyond the scope of what will be presented in this book. Earlier in the chapter the concept of elastic and viscous properties of polymers was briefly introduced. A purely viscous response can be represented by a mechanical dash pot, as shown in Fig. 3.10(a). This purely viscous response is normally the response of interest in routine extruder calculations. For those familiar with the suspension of an automobile, this would represent the shock absorber in the front suspension. If a stress is applied to this element it will continue to elongate as long as the stress is applied. When the stress is removed there will be no recovery in the strain that has occurred. The next mechanical element is the spring (Fig. 3.10[b]), and it represents a purely elastic response of the polymer. If a stress is applied to this element, the element will elongate until the strain and the force are in equilibrium with the stress, and then the element will remain at that strain until the stress is removed. The strain is inversely proportional to the spring modulus. The initial strain and the total strain recovery upon removal of the stress are considered to be instantaneous. 

We seek to nnderstand the response of a material to an applied stress. In Chapter 4, we saw how a flnid responds to a shearing stress through the application of Newton s Law of Viscosity [Eq. (4.3)]. In this chapter, we examine other types of stresses, snch as tensile and compressive, and describe the response of solids (primarily) to these stresses. That response usually takes on one of several forms elastic, inelastic, viscoelastic, plastic (ductile), fracture, or time-dependent creep. We will see that Newton s Law will be useful in describing some of these responses and that the concepts of stress (applied force per unit area) and strain (change in dimensions) are universal to these topics. 

Other coordinate systems may be used for failure surface representations in addition to stress space. Blatz and Ko (11) indicate that either stress (Stress space is most commonly used because the failure surface concept was originally applied to metals, for which stress and strain are more simply related. Viscoelastic materials, on the other hand, may show a multitude of strain values at a given stress level, depending on test conditions. 

The notion of fluid strains and stresses and how they relate to the velocity field is one of the fundamental underpinnings of the fluid equations of motion—the Navier-Stokes equations. While there is some overlap with solid mechanics, the fact that fluids deform continuously under even the smallest stress also leads to some fundamental differences. Unlike solid mechanics, where strain (displacement per unit length) is a fundamental concept, strain itself makes little practical sense in fluid mechanics. This is because fluids can strain indefinitely under the smallest of stresses—they do not come to a finite-strain equilibrium under the influence of a particular stress. However, there can be an equilibrium relationship between stress and strain rate. Therefore, in fluid mechanics, it is appropriate to use the concept of strain rate rather than strain. It is the relationship between stress and strain rate that serves as the backbone principle in viscous fluid mechanics. 

Hooke s law relates stress (or strain) at a point to strain (or stress) at the same point and the structure of classical elasticity (see e.g. Love, Sokolnikoff) is built upon this linear relation. There are other relationships possible. One, as outlined above (see e.g. Green and Adkins) involves the large strain tensor Cjj which does not bear a simple relationship to the stress tensor, another involves the newer concepts of micropolar and micromorphic elasticity in which not only the stress but also the couple at a point must be related to the local variations of displacement and rotation. A third, which may prove to be very relevant to polymers, derives from non-local field theories in which not only the strain (or displacement) at a point but also that in the neighbourhood of the point needs to be taken into account. In polymers, where the chain is so much stiffer along its axis than any interchain stiffness (consequent upon the vastly different forces along and between chains) the displacement at any point is quite likely to be influenced by forces on chains some distance away. 

We will use readily available plastic films to demonstrate stress-strain behavior. Students should be able to relate the physical behavior of thin films to the concepts of orientation and crystallinity. They should be able to explain terms such as cold drawing, yielding, and machine and transverse directions. 

The concept of stress-induced dilatation affecting the relaxation time or rate has been suggested by others (5, 6, 7, 8). The density of most solids decreases under uniaxial stress because the lateral contraction of the solid body does not quite compensate for the longitudinal extension in the direction of the stress, and the body expands. The Poisson ratio, the ratio of such contraction to the extension, is about 0.35 for many polymeric solids it would be 0.5 if no change in density occurred, as in an ideal rubber. The volume increase, AV, accompanying the tensile strain of c, can be described by the following equation ... 

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