Basic shear response

The mechanics of stability and localization of deforma tion for plastic rigid materials has been considered by Hill (1957), who stated that, in general, the conditions for loss of load-carrying capacity under controlled boundary displacements or large distortions under constant boundary loads are different from the conditions for localization by inhomogeneous deformation. The conditions for the latter are more akin to those of uniqueness. For the plastic rigid bar the two conditions coincide. As Hart (1967) has shown, and as was developed in more detail by Hutchinson and Neale (1977), the two conditions are always separate for strain-rate-sensitive materials, where much stable extension can follow a load maximum. 

When uniqueness conditions can no longer be satisfied and localization is possible, the rate of development of such localization strongly depends on the magnitude of geometrical and physical inhomogeneities. 

We consider first the conditions for impending localization in strain-hardening and strain-rate-sensitive plastic solids where the plastic resistance can also be pressure-dependent (Argon 1973). 

In all plastic deformation, plastic strain results from the accumulation of stress-driven and thermally assisted unit events that were discussed in Chapters 7-9 for amorphous metals, glassy polymers, and semi-crystalline polymers. In 3D, under a general applied stress tensor, plastic deformation can be initiated at 0 K when the deviatoric shear component of the applied stress, which we designate as r, reaches a threshold plastic-shear resistance r that is, in general, dependent on a previous history of plastic deformation, which is characterized by a deviatoric plastic shear strain -f (Chapter 3). 

We develop first the considerations related to shear response in a ID context of plastic-shear flow to state the basic kinetic response of the solid, where s stands for an applied shear stress, t stands for a threshold plastic-shear resistance, and y is taken to be the plastic-shear strain yP. As a useful simplification, we first consider the material to be rigid on the basis that the plastic-shear increments are large, in comparison with the elastic-strain increments. At temperatures T OK, for which the elastic moduli of the solid are significantly lower than at 0 K, we expect that the rate-independent plastic-shear resistance z z and will have the same temperature dependence as the elastic-shear modulus (Chapter 4). Then, where the plastic response in a rate-independent manner is initiated when s = z(T), under conditions of s z T), a plastic response is still possible by thermal assistanee and occurs at a (plastic) shear rate of (Argon 1973) 

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For clarity we deal with the plastic response mechanisms and their kinetics in simple shear in the absence of a mean normal stress (pressure or negative pressure) in a ID framework. However, we also discuss the important effects of a mean normal stress on shear separately in order to elucidate its modulating effects on shear flow. Thus, our goal is to present a skeletal mechanistic framework of large-strain plastic response in glasses of the simplest structure and bonding. There are excellent operational generalizations of the basic shear response into 3D forms of deformation, which we point out in a final section. 

The elastic and viscoelastic properties of materials are less familiar in chemistry than many other physical properties hence it is necessary to spend a fair amount of time describing the experiments and the observed response of the polymer. There are a large number of possible modes of deformation that might be considered We shall consider only elongation and shear. For each of these we consider the stress associated with a unit strain and the strain associated with a unit stress the former is called the modulus, the latter the compliance. Experiments can be time independent (equilibrium), time dependent (transient), or periodic (dynamic). Just to define and describe these basic combinations takes us into a fair amount of detail and affords some possibilities for confusion. Pay close attention to the definitions of terms and symbols. 

It is a flow characteristic where a material has basically abnormal flow response when force is applied. That is, their viscosity is dependent on the rate of shear. They do not have a straight proportional behavior with application of force and rate of flow (Figure 3.1). When proportional, the behavior has a Newtonian flow. 

In this book, we review the most basic distinctions and similarities among the rheological (or flow) properties of various complex fluids. We focus especially on their linear viscoelastic behavior, as measured by the frequency-dependent storage and loss moduli G and G" (see Section 1.3.1.4), and on the flow curve— that is, the relationship between the "shear viscosity q and the shear rate y. The storage and loss moduli reveal the mechanical properties of the material at rest, while the flow curve shows how the material changes in response to continuous deformation. A measurement of G and G" is often the most useful way of mechanically characterizing a complex material, while the flow curve q(y ) shows how readily the material can be processed, or shaped into a useful product. The... 

The basic equations of motion contain coefficients of shear viscosity for the fluids. This gives rise to a dependency of porosity diffusion on the rate of fluid transmission through the pore space. In low viscosity cases such as crustal fluids in deep interconnected fracture networks, transmission is relatively rapid in reservoirs largely saturated with viscous oil, transmission of pressure effects can be quite slow. For a 10,(XX) cP oil filling 88% of the pore space of a 30% porosity 2-3 Darcy sand, it took about 5 weeks to see a substantial response at distances of 300 m from the excitation well which was being aggressively pulsed at the right frequency (Spanos et al., 2003). 

Loads on a fabricated product can produce different t3q>es of stresses within the material. There are basically static loads (tensile, modulus, flexural, compression, shear, etc.) and dynamic loads (creep, fatigue torsion, rapid loading, etc.). The magnitude of these stresses depends on many factors such as applied forces/loads, angle of loads, rate and point of application of each load, geometry of the structure, manner in which the structure is supported, and time at temperature. The behavior of the material in response to these induced stresses determines the performance of the structure. 

These transverse entanglements, separated by a typical length ta govern the elastic response of solutions, in a way first outlined by Isambert and Maggs. A more complete discussion of the rheology of such solutions can be found in Morse and Hiimer et al.The basic result for the mbber-like plateau shear modulus for such solutions can be obtained by noting that the mrmber density of entropic constraints (entanglements) is thus where n = fl[a ) is... 

The influence of temperature on basic mechanical properties of the hardened adhesive such as bulk flexural modulus and shear strength is illustrated in Figs. 2.20 and 2.21, respectively (26). From these figures it is evident that the response of all five adhesives to temperature variations within the range 15 " -65 °C is similar. The most noticeable feature of the curves is the rapid deterioration in both stiffness and strength at a temperature close to the measured HDT of the adhesive. 

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