The relationship between stress and strain

When the strain produced in a simple solid is small it is found experimentally that 

This is Hooke s law, first stated in connection with the extension of a loaded spring. The constant in the equation as it is written is called a modulus of elasticity. For an isotropic material the modulus depends on temperature only, but for an anisotropic material it also depends on the direction in the sample in which the measurement is made. The graph of stress against strain for such small strains is a straight line passing through the origin it has the same slope in tension and in compression and is followed 

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By analogy with Eq. (3.1), we seek a description for the relationship between stress and strain. The former is the shearing force per unit area, which we symbolize as as in Chap. 2. For shear strain we use the symbol y it is the rate of change of 7 that is involved in the definition of viscosity in Eq. (2.2). As in the analysis of tensile deformation, we write the strain AL/L, but this time AL is in the direction of the force, while L is at right angles to it. These quantities are shown in Fig. 3.6. It is convenient to describe the sample deformation in terms of the angle 6, also shown in Fig. 3.6. For distortion which is independent of time we continue to consider only the equilibrium behavior-stress and strain are proportional with proportionality constant G ... 

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... 

In the region where the relationship between stress and strain is linear, the material is said to be elastic, and the constant of proportionality is E, Young s modulus, or the elastic modulus. 

The tensile modulus can be determined from the slope of the linear portion of this stress-strain curve. If the relationship between stress and strain is linear to the yield point, where deformation continues without an increased load, the modulus of elasticity can be calculated by dividing the yield strength (pascals) by the elongation to yield ... 

Generalized Hooke s Law. The discussion in the previous section was a simplified one insofar as the relationship between stress and strain was considered in only one direction along the applied stress. In reality, a stress applied to a volnme will have not only the normal forces, or forces perpendicular to the surface to which the force is applied, but also shear stresses in the plane of the surface. Thus there are a total of nine components to the applied stress, one normal and two shear along each of three directions (see Eigure 5.4). Recall from the beginning of Chapter 4 that for shear stresses, the first subscript indicates the direction of the applied force (ontward normal to the surface), and the second subscript indicates the direction of the resnlting stress. Thus, % is the shear stress of x-directed force in the y direction. Since this notation for normal forces is somewhat redundant—that is, the x component of an... 

In general, Hooke s law is the basic constitutive equation giving the relationship between stress and strain. Generalized Hooke s law is often expressed in the following form [20,108] ... 

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. 

It should be noted that the stresses usually used are engineering stresses calculated from the ratio of force and original cross section area whereas true stress is the ratio of the force and the actual cross sectional area at that deformation. Clearly, the relationship between stress and strain depends on the definition of stress used and taking the case of tensile strain, for example, the true stress is equal to the engineering stress multiplied by the extension ratio. 

The relationship between stress and strain in a test piece with bonded end pieces is very dependent on the shape factor of the test piece. This is usually defined as the ratio of the loaded cross-sectional area to the total force-free area (Figure 8.15). The larger the shape factor the more stiff the rubber appears and this property is much exploited in the design of rubber springs and mountings. 

The relationship between stress and strain in this dynamic case can be defined by writing... 

In this case the relationship between stress and strain is given by Equation (6.2) for a material that exhibits a linear relationship between stress o and strain e. 

This property of viscoelasticity is possessed by all plastics to some degree, and dictates that while plastics have solid-like characteristics, they also have liquid-like characteristics (Figure 1.2). This mechanical behavior is important to understand. It is basically the mechanical behavior in which the relationships between stress and strain are time dependent for plastic, as opposed to the classical elastic behavior of steel in which deformation and recovery both occur instantaneously on application and removal of stress.1... 

For certain solid bodies, the relationship between stress and strain is represented by a straight line through the origin (Figure 8-8)... 

Look at this carefully it has an in-phase component (the term in sinot) and an out-of-phase component (the term in coscot). This can be used to define the relationship between stress and strain in terms of two moduli. First writing Equation 13-78 ... 

The relationships between stress and strain, and the influence of time on them are generally described by constitutive equations or rheological equations of state (Ferry, 1980). When the strains are relatively small, that is, in the linear range, the constitutive... 

Using extension and compression tests to develop the relationship between stress and strain and then to conclude that orientation of segments of kamaboko s network was negligible, and similar to the behavior of the vulcanized rubber in small deformation. 

When the inertial forces can be neglected and the deformations are infinitesimal, the relationships between stress and strain can be assimilated into the relationships between force and displacement through a coefficient directly related to the geometry of the system, which, somewhat inadequately, is called a form factor... 

Though the preceding equations represent a convenient way to express the relationship between stress and strain, it is necessary that they be consistent with the integral formulation of Section 16.2 (6). Consequently, taking the Laplace transform of both formulations and identifying them, we obtain... 

As a consequence of the stress applied, the body will change its shape, and as a result, there will be a change in length of the body, i.e., strain. The relationships between stress and strain are illustrated in Fig. 2. 

In the ideal case of a Hookean body, the relationship between stress and strain is fully linear, and the body returns to its original shape and size, after the stress applied has been relieved. The proportionality between stress and strain is quantified by the modulus of elasticity (unit Pa). The proportionality factor under conditions of normal stress is called modulus of elasticity in tension or Young s modulus E), whereas that in pure shear is called modulus of elasticity in shear or modulus of rigidity (G). The relationships between E, G, shear stress, and strain are defined by ... 

Stress (ct) is the resistance to strain. Figure 2.5 shows a typical stress-strain curve for a material. For a material whose stress-strain curve is similar to that of figure 2.5, the relationship between stress and strain is linear for small values of strain. Hooke s law applies to this linear region and is expressed as... 

Young s Modnlns of Elasticity - In the elastic region, the relationship between stress and strain of a polymer, undergoing tensile or compres-sional strain, is linear (Hooke s Law). In this relationship, stress is proportional to strain. The coefficient of proportionality in this stress-strain relationship is called Young s Modulus of Elasticity. 

A uniaxial stress is usually designated by the symbol o, and a shearing stress by X. For certain lipids that behave like solid food systems, the relationship between stress and strain is represented by a straight line through the origin, up to the so-called limit of elasticity. The proportionality factor E for uniaxial stress is called Young s modulus, or the modulus of elasticity. For a shear stress, the modulus is called Coulomb modulus, or the tensile modulus G. 

Linear viscoelasticity is the simplest type of viscoelastic behavior observed in polymeric liquids and solids. This behavior is observed when the deformation is very small or at the initial stage of a large deformation. The relationship between stress and strain may be defined in terms of the relaxation modulus, a scalar quantity. This is defined in Equation 22.7 for a sudden shear deformation ... 

The relationship between stress and strain of a 3D rock mass is given by ... 

Stiffness properties of RPs are used (as with other materials) for the usual purpose of estimating stresses and strains in a structural design, and to predict buckling capacity under compressive loads. Also, stiffhess properties of individual plies of a layered flat plate approach may be used for the calculation of overall stiffiiess and strength properties. The relationship between stress and strain of unreinforced or RPs varies firom viscous to elastic. Most RPs, particularly RTSs are intermediate between viscous and elastic. The type of plastic, stress, strain, time, temperature, and environment all influence the degree of their viscoelasticity. 

In Chapter 2, stress and strain were defined, the compatibility and equilibrium equations were introduced and the relationship between stress and strain was defined. Thus, any solution that satisfies all these equations and the appropriate boundary conditions will be the solution that gives the stress and strain distribution for a particular loading geometry. For the most general problems, the scientific process can be difficult but for plane stress and plane strain problems in elastically isotropic bodies the solution involves a single differential equation. 

Strain produced. Above this so-called proportionality limit, the relationship between stress and strain can be quite different (Section 11.5). For this reason, the modulus of elasticity of polymers is usually measured over a strain of 0.2% and over a time of 100 s. Moduli measured over higher strains or longer times are lower. 

FIGURE 12.7 Schematic indication of the relationship between stress and strain for wood in different anatomical directions. 

Structural Analysis Sometimes called Theory of Structures, the Structural Analysis offers further analytical tools for computing the member forces and the deflections of beams, trusses, and frames. The law of materials used in the development of the analytical tools is limited to that of linearly elastic, which means the relationship between stress and strain is proportional, and once the force acting on a structure is removed, the structure members return to its original position. The linearly elastic material law is applicable to most situations when the deflection of a structure is much smaller relative to the dimension of the... 

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