Generalized stress-strain Hooke

Generalized Stress-Strain Hooke s Law for Isotropic Solids 162... 

GENERALIZED STRESS-STRAIN HOOKE S LAW FOR ISOTROPIC SOLIDS... 

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... 

In Section 2.2, the stress-strain relations (generalized Hooke s law) for anisotropic and orthotropic as well as isotropic materials are discussed. These relations have two commonly accepted manners of expression compliances and stiffnesses as coefficients (elastic constants) of the stress-strain relations. The most attractive form of the stress-strain relations for orthotropic materials involves the engineering constants described in Section 2.3. The engineering constants are particularly helpful in describing composite material behavior because they are defined by the use of very obvious and simple physical measurements. Restrictions in the form of bounds are derived for the elastic constants in Section 2.4. These restrictions are useful in understanding the unusual behavior of composite materials relative to conventional isotropic materials. Attention is focused in Section 2.5 on stress-strain relations for an orthotropic material under plane stress conditions, the most common use of a composite lamina. These stress-strain relations are transformed in Section 2.6 to coordinate systems that are not aligned with the principal material... 

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 ... 

It is the slope of the stress-strain curve. If a material is elastic, it recovers to its original length if the applied stress is removed. In general, E depends on the strain of the material. However, for many substances, if the stress is not too large, E is independent of strain. Over this range, the substance is said to obey Hook s law ... 

The radial and transverse stresses can be determined from the stress-strain relationships. Owing to the orthogonality of the spherical coordinates, the formal structure of the generalized Hooke s law, given by Eq. (P4.11), is preserved, so that the nonzero components of the stress tensor are expressed in terms of the strain tensors as... 

As a measure of stiffness, the Young s modulus is important in the predictive behavior of the material being used. For linear analysis, E = stress/strain. For automotive applications, some common materials are steel (E 200 GPa), aluminum (E 70 GPa), and nylon (E 8.5 GPa). As stated earlier, Hook s law is force = spring constant spring displacement (F = KU). The generalized Hook s... 

The stress strain relation (generalized linear Hooke s law) incorporating the thermal expansion of the solid is... 

The moduli of elasticity determined by stress / strain measurements are generally much lower than the lattice moduli of the same polymers (Table 11-3). The difference is to be found in the effects of entropy elasticity and viscoelasticity. Since the majority of the polymer chains in such polymer samples do not lie in the stress direction, deformation can also occur by conformational changes. In addition, polymer chains may irreversibly slide past each other. Consequently, E moduli obtained from stress/strain measurements do not provide a measure of the energy elasticity. Such E moduli are no more than proportionality constants in the Hooke s law equation. The proportionality limit for polymers is about 0.l%-0.2% of the... 

For a unidirectional laminate the elastic stress-strain relations define an orthotropic material for which the generalized form of Hooke s Law, relating the stress o to the strain e,... 

The most prevalent and widely developed constitutive connections of polymers between strain and stress are dealt with in linear elasticity by applying the generalized form of Hooke s law which is presented in Chapter 4 for anisotropic solids of different symmetry classes starting with orthotropic solids and progressing up to isotropic solids. Here and in the following chapters we shall develop only the connection for isotropic solids, which is the most useful one and most often is quite sufficient in development of concepts. 

The rest of this hook consists of property data on many plastics and elastomers. Chapters 2—10 contain multipoint data in the form of plots. The plastics are grouped hy the basic chemical structures of the plastics. Each of these chapters contains a short introduction that descrihes the basic chemical structures of the plastics in that chapter. The figures that follow contain the multipoint data. They are grouped by the t5rpe of data. Generally stress vs. strain curves start the chapter, followed by various modulus measurements, strength measures, other physical properties, and electrical properties. These properties are measured by the appropriate ISO or ASTM standards. 

X10. The next three rows present the viscosity rj, the surface tension, and its tenqterature dependence, in the liquid state. The next properties are the coefficient of linear thermal expansion a and the sound velocity, both in the solid and in the liquid state. A number of quantities are tabulated for the presentation of the elastic properties. For isotropic materials, we list the volume compressihility k = —(l/V)(dV/dP), and in some cases also its reciprocal value, the bulk modulus (or compression modulus) the elastic modulus (or Young s modulus) E the shear modulus G and the Poisson number (or Poisson s ratio) fj,. Hooke s law, which expresses the linear relation between the strain s and the stress a in terms of Young s modulus, reads a = Ee. For monocrystalline materials, the components of the elastic compliance tensor s and the components of the elastic stiffness tensor c are given. The elastic compliance tensor s and the elastic stiffness tensor c are both defined by the generalized forms of Hooke s law, a = ce and e = sa. At the end of the list, the tensile strength, the Vickers hardness, and the Mohs hardness are given for some elements. 

Using these tensors, a generalized form of Hooke s law can be written in which each component of the stress tensor is a linear combination of the strain components of the strain tensor, as shown in Eq. (11). 

In most cases the material cannot be regarded as one-dimensional, and forces and deformations in all three principal directions must be taken into account. The normal and shear stresses are interrelated by the equations of mechanical equilibrium. The relationship between stresses and deformations can be described by the generalized Hooke s law in the case of linear elastic behavior. The stress-strain relationship of materials showing more complicated behavior can often be described by advanced theories based on the generalized Hooke s law. All models contain constants that must be determined experimentally, on materials equilibrated in moisture content and temperature. 

If the material is subjected to a time-dependent strain, the situation becomes more complicated. However, in the case of a linear viscoelastic material (like many food products) the superposition principle can be applied the response of the stress to a strain increment is independent of the already existing strain. The effect of the strain as a function of time can therefore be integrated, and the generalized Hooke s law can be extended to describe the stress-strain behavior of linear viscoelastic materials relatively easily. 

The generalized form of Hooke s law, which proposes a linear relationship between stress and strain at vanishingly small strains, is then given by Eq. (6). 

The latter equation is the uniaxial stress-strain relation for a polymer analogous to Hooke s law for a material that is time independent but is valid only for the case of a constant input of strain. The relaxation test provides the defining equation for the material property identified as the relaxation modulus. More general differential and integral stress-strain relations for an arbitrary loading will be developed in later Chapters. 

Under complex loading and in the elastic domain, the stress-strain relationship is referred to as the generalized Hooke s law ... 

Young s modulus E is the ratio of nominal stress to strain, as shown in equation (70). However, vulcanized rubbers do not obey Hooke s law (as is shown in Figure 5), so E is not a constant. The stress-strain relationship is generally assumed to be linear over small tensile or compressive strains, and Young s modulus is usually defined as the slope of the stress-strain curve in this range of deformation. " Hardness measurement is another way of determining values of this modulus. It is noteworthy that the slope of the stress-strain curve in the tensile and... 

As indicated earlier the different stress-strain curves are not characteristic for particular, chemically defined species of polymers but for the physical state of a polymeric solid. If the environmental parameters are chosen accordingly transitions from one type of behavior (e.g. brittle, curve a) to another (e.g. ductile, curve c) will be observed. These phenomenological aspects of polymer deformation are discussed in detail in [14], [52—53], [55—57], and in the general references of Chapter 1. A decrease of rate of strain or an increase of temperature generally tend to increase the ductility and to shift the type of response from that of curve a) towards that of curves c) and d). At small strains (between zero and about one per cent) the uniaxial stress o and the strain e are linearly related (Hooke s law) ... 

Hooke s law describes the relationship between stress and strain of an elastic material. In a general formulation, the stress-strain relationship is a tensorial equation... 

The generalized Hooke s law relating stresses to strains can be written in contracted notation as... 

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] ... 

Elasticity is the inherent property in bodies by which they recover their former figure or state after the force (stress) of external pressure, tension, or distortion have been removed (as for instance elasticity of gases, rubber, etc). Any force or distribution of forces which acts upon a body and is balanced by equal and opposite forces in the body is, in general, termed as a stress, although.the term is more particularly applied to the force per unit area acting upon the body. The change in size per unit size, or the change in some dimension per its unit, produced by the stress is called a strain. For each substance and for each kind of strain there is some limit beyond which Hooke s Law does not apply. 

The general relations between strains and stresses are represented by Hooke s law as... 

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