Strain and stress

In the following chapters, Chapters 4-13, a variety of forms of mechanical response of polymers are presented that are overwhelmingly concentrated on elucidation of molecular mechanisms of strain production under stress in elastic and inelastic deformation as well as in fracture and in nurturing of toughness. In this, the basic tools are stresses and strains considered both under idealized conditions of rate- and temperature-independent form, and, most often, under more realistic conditions over a wide range of temperatures and imposed strain rates. 

In dealing with the wide-ranging forms of mechanical response of polymers, a basic understanding of the forms of deformation and of the constitutive connections among stress, strain, and strain rate is essential. This chapter is devoted to a brief review of linear elasticity and of plasticity that recur over the entire set of phenomena dealt with in the book. 

In the following chapters we use a very familiar form of representation of stresses by a double-subscript notation in which the first subscript represents the direction of the normal of the area across which the stress is acting and the second is the direction of action of the stress. In the Cartesian rectangular system of coordinates, the subscripts i and j stand for x, y, z or xi,X2,X. In the cylindrical, polar coordinate system, the subscripts stand for r, 6, z. 

In rare cases stresses could refer to the initial undeformed shape of the body, for which they are labeled as nominal stresses. 

Stresses transform from one coordinate-axis system to another according to well-defined transformation laws that utilize direction cosines of the angles of rotation between the final and initial coordinate-axis systems. Matrixes that obey such transformation laws are referred to as tensors (McClintock and Argon 1966). There are three sets of stress relations that are scalar and invariant in coordinate-axis transformations. The first such stress invariant of particular interest is the mean normal stress o- , defined as. 

Let us consider the forces acting on a volume element of a solid body in the form of a parallelepiped (Fig. 4.6). 

A stress is a force per unit area. It acts on two opposite parallel faces (couple). In particular, (j is the stress parallel to acting on the faces perpendicular to e . The stresses r normal to the faces are positive in the case of an extension of the solid. 

This is an axial vector. If the volume element is in equilibrium, M = 0. It thus follows that 

The stress tensor is symmetric. For inhomogeneous stresses where is a function of position, consult J. F. Nye, Physical Properties of Crystals. 

A uniaxial stress r parallel to the direction I = (/i, I2,13) is given by the tensor 

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One of the simplest ways to model polymers is as a continuum with various properties. These types of calculations are usually done by engineers for determining the stress and strain on an object made of that material. This is usually a numerical finite element or finite difference calculation, a subject that will not be discussed further in this book. 

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 relaxation and creep experiments that were described in the preceding sections are known as transient experiments. They begin, run their course, and end. A different experimental approach, called a dynamic experiment, involves stresses and strains that vary periodically. Our concern will be with sinusoidal oscillations of frequency v in cycles per second (Hz) or co in radians per second. Remember that there are 2ir radians in a full cycle, so co = 2nv. The reciprocal of CO gives the period of the oscillation and defines the time scale of the experiment. In connection with the relaxation and creep experiments, we observed that the maximum viscoelastic effect was observed when the time scale of the experiment is close to r. At a fixed temperature and for a specific sample, r or the spectrum of r values is fixed. If it does not correspond to the time scale of a transient experiment, we will lose a considerable amount of information about the viscoelastic response of the system. In a dynamic experiment it may... 

We commented above that the elastic and viscous effects are out of phase with each other by some angle 5 in a viscoelastic material. Since both vary periodically with the same frequency, stress and strain oscillate with t, as shown in Fig. 3.14a. The phase angle 5 measures the lag between the two waves. Another representation of this situation is shown in Fig. 3.14b, where stress and strain are represented by arrows of different lengths separated by an angle 5. Projections of either one onto the other can be expressed in terms of the sine and cosine of the phase angle. The bold arrows in Fig. 3.14b are the components of 7 parallel and perpendicular to a. Thus we can say that 7 cos 5 is the strain component in phase with the stress and 7 sin 6 is the component out of phase with the stress. We have previously observed that the elastic response is in phase with the stress and the viscous response is out of phase. Hence the ratio of... 

Figure 3.14 General representations of stress and strain out of phase by amount 5 (a) represented by oscillating functions and (b) represented by vectors.

Let C i be a bounded domain with a smooth boundary L, and n = (ni,n2,n3) be a unit outward normal vector to L. Introduce the stress and strain tensors of linear elasticity (see Section 1.1.1),... 

The stress and strain tensors aij u),Sij u) are defined by the Hooke and Cauchy laws... 

The elasticity of a fiber describes its abiUty to return to original dimensions upon release of a deforming stress, and is quantitatively described by the stress or tenacity at the yield point. The final fiber quaUty factor is its toughness, which describes its abiUty to absorb work. Toughness may be quantitatively designated by the work required to mpture the fiber, which may be evaluated from the area under the total stress-strain curve. The usual textile unit for this property is mass pet unit linear density. The toughness index, defined as one-half the product of the stress and strain at break also in units of mass pet unit linear density, is frequentiy used as an approximation of the work required to mpture a fiber. The stress-strain curves of some typical textile fibers ate shown in Figure 5. 

One simple rheological model that is often used to describe the behavior of foams is that of a Bingham plastic. This appHes for flows over length scales sufficiently large that the foam can be reasonably considered as a continuous medium. The Bingham plastic model combines the properties of a yield stress like that of a soHd with the viscous flow of a Hquid. In simple Newtonian fluids, the shear stress T is proportional to the strain rate y, with the constant of proportionaHty being the fluid viscosity. In Bingham plastics, by contrast, the relation between stress and strain rate is r = where is... 

Partially Plastic Thick-Walled Cylinders. As the internal pressure is increased above the yield pressure, P, plastic deformation penetrates the wad of the cylinder so that the inner layers are stressed plasticady while the outer ones remain elastic. A rigorous analysis of the stresses and strains in a partiady plastic thick-waded cylinder made of a material which work hardens is very compHcated. However, if it is assumed that the material yields at a constant value of the yield shear stress (Fig. 4a), that the elastic—plastic boundary is cylindrical and concentric with the bore of the cylinder (Fig. 4b), and that the axial stress is the mean of the tangential and radial stresses, then it may be shown (10) that the internal pressure, needed to take the boundary to any radius r such that is given by... 

Little error is introduced using the idealized stress—strain diagram (Eig. 4a) to estimate the stresses and strains in partiady plastic cylinders since many steels used in the constmction of pressure vessels have a flat top to their stress—strain curve in the region where the plastic strain is relatively smad. However, this is not tme for large deformations, particularly if the material work hardens, when the pressure can usuady be increased above that corresponding to the codapse pressure before the cylinder bursts. 

More recently, Raman spectroscopy has been used to investigate the vibrational spectroscopy of polymer Hquid crystals (46) (see Liquid crystalline materials), the kinetics of polymerization (47) (see Kinetic measurements), synthetic polymers and mbbers (48), and stress and strain in fibers and composites (49) (see Composite materials). The relationship between Raman spectra and the stmcture of conjugated and conducting polymers has been reviewed (50,51). In addition, a general review of ft-Raman studies of polymers has been pubUshed (52). 

Physical Properties. Raman spectroscopy is an excellent tool for investigating stress and strain in many different materials (see Materlals reliability). Lattice strain distribution measurements in siUcon are a classic case. More recent examples of this include the characterization of thin films (56), and measurements of stress and relaxation in silicon—germanium layers (57). 

The need to ensure that the stresses ia piping systems meet the appropriate code requirements and the concern that cycHc stresses resulting from events such as periodic heating and cooling of the piping may lead to fatigue failures, make accurate evaluation of the stresses and strains ia piping systems a necessity. 

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... 

Fig. 17. Viscoelastic material stress ( and strain (---------) ampHtudes vs time where 5 is the phase angle that defines the lag of the strain behind the...

This behavior is usually analy2ed by setting up what are known as complex variables to represent stress and strain. These variables, complex stress and complex strain, ie, T and y, respectively, are vectors in complex planes. They can be resolved into real (in phase) and imaginary (90° out of phase) components similar to those for complex modulus shown in Figure 18. 

Measurement of Residual Stress and Strain. The displacement of the 2 -value of a particular line in a diffraction pattern from its nominal, nonstressed position gives a measure of the amount of stress retained in the crystaUites during the crystallization process. Thus metals prepared in certain ways (eg, cold rolling) have stress in their polycrystalline form. Strain is a function of peak width, but the peak shape is different than that due to crystaUite size. Usually the two properties, crystaUite size and strain, are deterrnined together by a computer program. 

Deformation Under Loa.d. The mechanical behavior of coal is strongly affected by the presence of cracks, as shown by the lack of proportionahty between stress and strain in compression tests or between strength and rank. However, tests in triaxial compression indicate that as the confirming pressure is increased different coals tend to exhibit similar values of compressive strength perpendicular to the directions of these confining pressures. Except for anthracites, different coals exhibit small amounts of recoverable and irrecoverable strain underload. 

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... 

A flowing fluid is acted upon by many forces that result in changes in pressure, temperature, stress, and strain. A fluid is said to be isotropic when the relations between the components of stress and those of the rate of strain are the same in all directions. The fluid is said to be Newtonian when this relationship is linear. These pressures and temperatures must be fully understood so that the entire flow picture can be described. 

FIG. 20-70 The influence of moisture as a percentage of sample saturation S on granule deformabihty. Here, deformation strain (AL/L) is measured as a function of applied stress, with the peak stress and strain denoted by tensile strength and critical strain (AL/L) of the material. Dicalcium phosphate with a 15 wt % binding solution of PVP/PVA Kolhdon VAG4. [Holm et al., Powder Tech., 43, 213 (1.9S.5J,] With land permission from Elsevier Science SA, Lausanne, Switzerland. 

Another generalization uses referential (material) symmetric Piola-Kirchhoff stress and Green strain tensors in place of the stress and strain tensors used in the small deformation theory. These tensors have components relative to a fixed reference configuration, and the theory of Section 5.2 carries over intact when small deformation quantities are replaced by their referential counterparts. The referential formulation has the advantage that tensor components do not change with relative rotation between the coordinate frame and the material, and it is relatively easy to construct specific constitutive functions for specific materials, even when they are anisotropic. 

The deformation may be viewed as composed of a pure stretch followed by a rigid rotation. Stress and strain tensors may be defined whose components are referred to an intermediate stretched but unrotated spatial configuration. The referential formulation may be translated into an unrotated spatial description by using the equations relating the unrotated stress and strain tensors to their referential counterparts. Again, the unrotated spatial constitutive equations take a form similar to their referential and current spatial counterparts. The unrotated moduli and elastic limit functions depend on the stretch and exhibit so-called strain-induced hardening and anisotropy, but without the effects of rotation. 

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