The Stress-Strain Relationships

Using the above definitions of finite strain and stress, we wish to construct stress-strain relationships for finite strains that are analogous to the generalised Hooke s law for small strain elasticity. Each component of stress can be a function of every component of strain, and vice versa. For a linear relation, we would expect equations such as 

We could use this as a starting point for finite elasticity. It would be desirable to reduce the number of elastic constants a, b, and so on, by considerations such as material symmetry. Rather than developing a general theory of finite elasticity, however, we will introduce appropriate restrictions at an early stage, as appropriate for a representation of the behaviour of rubbers. The principal restrictions are driven by the simplifications that  

Volume changes associated with deformation are very small and may be neglected, i.e. a rubber is incompressible. 

First consider the impact of these assumptions on small strain elasticity. 

We now apply the incompressibility en + 22 + 33 = 0. Adding the above equations then gives 

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By combining random flight statistics from Chap. 1 with the statistical definition of entropy from the last section, we shall be able to develop a molecular model for the stress-strain relationship in a cross-linked network. It turns out to be more convenient to work with the ratio of stretched to unstretched lengths L/Lq than with y itself. Note the relationship between these variables ... 

The stress—strain relationship is used in conjunction with the rules for determining the stress and strain components with respect to some angle 9 relative to the fiber direction to obtain the stress—strain relationship for a lamina loaded under plane strain conditions where the fibers are at an angle 9 to the loading axis. When the material axes and loading axes are not coincident, then coupling between shear and extension occurs and... 

Proportional limit the point on the stress-strain curve at which will commence the deviation in the stress-strain relationship from a straight line to a parabolic curve (Figure 30.1). 

Equation (13) is valid for r/Nlp < 0.25 (Fig. 3). At much higher extension ratios, the force must increase indefinitely since the molecule is almost straightened out. The thermodynamic approach to the problem of coil stretching for a freely-jointed chain was considered by Treloar [32], who obtained the following expression for the stress-strain relationship when the two chain ends are kept a distance r apart ... 

In general, during the initial stages of deformation, a material is deformed elastically. That is to say, any change in shape caused by the applied stress is completely reversible, and the specimen will return to its original shape upon release of the applied stress. During elastic deformation, the stress-strain relationship for a specimen is described by Hooke s law ... 

Response of a material under static or dynamic load is governed by the stress-strain relationship. A typical stress-strain diagram for concrete is shown in Figure 5.3. As the fibers of a material are deformed, stress in the material is changed in accordance with its stress-strain diagram. In the elastic region, stress increases linearly with increasing strain for most steels. This relation is quantified by the modulus of elasticity of the material. 

Concret does not have well defined elastic and plastic regions due to its brittle nature. A maximum compressive stress value is reached at relatively low strains and is maintained for small deformations until crushing occurs. The stress-strain relationship for concrete is a nonlinear curve. Thus, the elastic modulus varies continuously with strain. The secant modulus at service load is normally used to define a single value for the modulus of elasticity. This procedure is given in most concrete texts. Masonry lias a stress-strain diagram similar to concrete but is typically of lower compressive strength and modulus of elasticity. 

Stress-strain relationships for soil are difficult to model due to their complexity. In normal practice, response of soil consists of analyzing compression and shear stresses produced by the structure, applied as static loads. Change in soil strength with deformation is usually disregarded. Clay soils will exhibit some elastic response and are capable of absorbing blast-energy however, there may be insufficient test data to define this response quantitatively. Soil has a very low tensile capacity thus the stress-strain relationship is radically different in the tension region than in compression. 

The properties of a material must dictate the applications in which it will best perform its intended use. All materials made to date with polymerized sulphur show time-dependent stress-strain behaviour. The reversion to the brittle behaviour of orthorhombic sulphur is inevitable as the sulphur transforms from the metastable polymeric forms to the thermodynamically stable crystalline structure. The time-span involved of at most 15 months (to date) would indicate that no such materials should be used in applications dependent on the strain softening behaviour. Design should not be based on the stress-strain relationships observed at an age of a few days. Since the strength of these materials is maintained, however, uses based on strength as the only mechanical criterion would be reasonable. 

Our own experience, as well as that of other authors, has shown that very precise measurement for the stress-strain relationship under general biaxial deformation is required to investigate the behavior of the strain energy density function of rubber vulcanizates. Unfortunately, available biaxial extension data are still too meager to deduce the general form of the strain energy density function of rubber-like substances. We wish to take this opportunity to summarize the principal results from our recent efforts, in the hope that they may serve to illustrate the interesting and complex nature of the derivatives 31V/9/,- of such substances. 

From the viewpoint of the mechanics of continua, the stress-strain relationship of a perfectly elastic material is fully described in terms of the strain energy density function W. In fact, this relationship is expressed as a linear combination erf the partial derivatives of W with respect to the three invariants of deformation tensor, /j, /2, and /3. It is the fundamental task for a phenomenologic study of elastic material to determine W as a function of these three independent variables either from molecular theory or by experiment. The present paper has reviewed approaches to this task from biaxial extension experiment and the related data. The results obtained so far demonstrate that the kinetic theory of polymer network does not describe actual behavior of rubber vulcanizates. In particular, contrary to the kinetic theory, the observed derivative bW/bI2 does not vanish. 

The ISO standard clearly differentiates between bonded and unbonded test pieces and in an appendix gives the stress strain relationships, taking account of shape factor. In the scope it is pointed out that comparable results will only be obtained for bonded test pieces if they are of the same shape, and that lubricated and bonded test pieces do not give the same results. There is, however, a very curious little introduction that gives a very narrow view of when compression data is needed and makes a dubious claim about use on thin samples when hardness measurement would be difficult - so is an accurate compression test on thicknesses below 2 mm. 

It is customary when evaluating plasticizers in polyvinyl chloride to compare them at concentrations which produce a standard apparent modulus in tension, as measured at room temperature. Since the stress-strain relationship is generally nonlinear it is necessary to specify a given point on the stress-strain curve as well as the rate of loading or straining. The efficiency may be expressed as the concentration of a given plasticizer necessary to produce this standaid modulus. Other properties, e g., indentation hardness, may take the place of tensile modulus. 

Missing from the literature are standardized and comparative physical strength data for adhesives in neat form or as applied to the skin. The most comprehensive examples of the stress-strain relationships of excised human skin are appended (Appendix 2.6.1.) three axes of any material are demonstrated below, where X- and T-axes are lateral (perpendicular to each other), but in the same plane, whereas the Z direction is not in the same plane, but perpendicular to X- and T-axes as demonstrated in Fig. 3.10. 

The criteria for sutureless adhesives are still being developed. Although the criteria are not well defined, the following Table 3.2 embodies certain essential properties that will continue to apply until in vivo testing is complete. These criteria are offered as a template for collecting data after application of adhesives in animals, and the data listed in Table 3.3 are indications of the stress-strain relationships to be expected in humans. 

In a rheomety experiment the two plates or cylinders are moved back and forth relative to one another in an oscillating fashion. The elastic storage modulus (G - The contribution of elastic, i.e. solid-like behaviour to the complex dynamic modulus) and elastic loss modulus (G" - The contribution of viscous, i.e. liquid-like behaviour to the complex modulus) which have units of Pascals are measured as a function of applied stress or oscillation frequency. For purely elastic materials the stress and strain are in phase and hence there is an immediate stress response to the applied strain. In contrast, for purely viscous materials, the strain follows stress by a 90 degree phase lag. For viscoelastic materials the behaviour is somewhere in between and the strain lag is not zero but less than 90 degrees. The complex dynamic modulus ( ) is used to describe the stress-strain relationship (equation 14.1 i is the imaginary number square root of-1). 

Force versus displacement data are directly useful in comparing some simple mechanical properties of particles from different samples, for example, the force required to break the particle and the deformation at breakage. However, these properties are not intrinsic, that is, they might depend on the particular method of measurement. Determination of intrinsic mechanical properties of the particles requires mathematical models to derive the stress-strain relationships of the material. 

M.F. Florstemeyer et al A large deformation atomistic study examining crystal orientation effects on the stress-strain relationship. Int. J. Plasticity 18, 203-209 (2002)... 

Toughness—The property that describes the stress-strain relationship of a poly-... 

Najafbadi and Yip (18) have investigated the stress-strain relationship in iron under uniaxial loading by means of a MC simulation in the isostress isothermal ensemble. At finite temperatures, a reversible b.c.c. to f.c.c. transformation occurs with hysteresis. They found that the transformation takes place by the Bain mechanism and is accompanied by sudden and uniform changes in local strain. The critical values of stress required to transform from the b.c.c. to the f.c.c. structure or vice versa are lower than those obtained from static calculations. Parrinello and Rahman (14) investigated the behavior of a single crystal of Ni under uniform uniaxial compressive and tensile loads and found that for uniaxial tensile loads less than a critical value, the f.c.c. Ni crystal expanded along the axis of stress reversibly. 

Stress can be evaluated by using strain gages60 located in that part of the equipment or structure which is suspected to be critical. Using the measured values of strain in the component, together with the known mechanical properties the stress of the component or structure of interest can be evaluated using the stress-strain relationships. The locations where the strains are to be measured in complex structures are obtained by techniques involving photoelasticity or brittle coating techniques. The preliminary identification of the... 

By analyzing the compositional dependent relaxation time, the stress-strain relationships of polymer composites are determined as a function of the filler concentration and strain rate. As the volume fraction of filler increases, both the effective elastic modulus and yield stress increases. However, the system becomes more brittle at the same time. 

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