Stress and strain rate

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. 

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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... 

Viscoelasticity A combination of viscous and elastic properties in a plastic with the relative contribution of each being dependent on time, temperature, stress, and strain rate. It relates to the mechanical behavior of plastics in which there is a time and temperature dependent relationship between stress and strain. A material having this property is considered to combine the features of a perfectly elastic solid and a perfect fluid. 

Unfortunately the high Peclet number regime is where many rheological measurements are most easily made. High stresses and strain rates allow the development of simpler instrumental designs and lower sensitivities are required. It is also important to be aware of the fact that many applications require very high deformation regimes and it is... 

Viscous Forces In the momentum equation (Navier-Stokes equation), forces F acting on the system result from viscous stresses. It is necessary to relate these stresses to the velocity field and the fluid s viscosity. This relationship follows from the stress and strain-rate tensors, using Stokes postulates. 

Stress and Strain Rate The stress and strain-rate state of a fluid at a point are represented by tensors T and E. These tensors are composed of nine (six independent) quantities that depend on the velocity field. The strain rate describes how a fluid element deforms (i.e., dilates and shears) as a function of the local velocity field. The stress and strain-rate tensors are usually represented in some coordinate system, although the stress and strain-rate states are invariant to the coordinate-system representation. 

We have discussed stresses and strain rates. A critical objective is to relate the two, leading to equations of motion governing how fluid packets are accelerated by the forces acting on them. Generally, we are working toward a differential-equation description of a momentum balance, F = ma. The approach is to represent both the forces and the accelerations as functions of the velocity field. The result will be a system of differential equations in which velocities are the dependent variables and the spatial coordinates and time are the independent variables (i.e., the Navier-Stokes equations). 

The basis for connecting the stress and strain-rate tensors was postulated first by G. G. Stokes in 1845 for Newtonian fluids. He presumed that a fluid is a continuous medium and that its properties are independent of direction, meaning they are isotropic. His insightful observations, itemized below, have survived without alteration, and are an essential underpinning of the Navier-Stokes equations ... 

Consider first the trivial case of a static fluid. Here there can only be normal forces on a fluid element and they must be in equilibrium. If this were not the case, then the fluid would move and deform. Certainly any valid relationship between stress and strain rate must accommodate the behavior of a static fluid. Hence, for a static fluid the strain-rate tensor must be exactly zero e(/- = 0 and the stress tensor must reduce to... 

The principal coordinates provide an extraordinarily useful conceptual framework within which to develop the fundamental relationships between stress and strain rate. For practical application, however, it is essential that a common coordinate system be used for all points in the flow. The coordinate system is usually chosen to align as closely as possible with the natural boundaries of a particular problem. Thus it is essential that the stress-strain-rate relationships can be translated from the principal-coordinate setting (which, in general, is oriented differently at all points in the flow) to the particular coordinate system or control-volume orientation of interest. Accomplishing this objective requires developing a general transformation for the rotation between the principal axes and any other set of axes. 

In the principal coordinates, of course, there are only three nonzero components of the stress and strain-rate tensors. Upon rotation, all nine (six independent) tensor components must be determined. The nine tensor components are comprised of three vector components on each of three orthogonal planes that pass through a common point. Consider that the element represented by Fig. 2.16 has been shrunk to infinitesimal dimensions and that the stress state is to be represented in some arbitrary orientation (z, r, 6), rather than one aligned with the principal-coordinate direction (Z, R, 0). We seek to find the tensor components, resolved into the (z, r, 6) coordinate directions. 

There was a sigmoidal relationship between the logarithms of stress and strain rate. 

Strictly speaking, the viscosity rj, measured with shear deformation viscometers, should not be used to represent the elongational terms located on the diagonal of the stress and strain rate tensors. Elongational flows are briefly discussed later in this chapter. A rheologist s... 

Real materials are neither truly Hookean nor truly Newtonian, though some exhibit Hookean or Newtonian behavior under certain conditions (Barnes et al., 1989). Real materials may exhibit nonlinearity, which is a lack of direct proportionality between stress and strain, or between stress and strain rate. Real materials may exhibit either predominantly elastic behavior or predominantly viscous behavior, or a measurable combination of the two, depending on the stress or strain and the duration of its application (Barnes et al., 1989). Such materials are termed viscoelastic. Barnes et al. (1989) pointed out that it is better to classify rheological behavior than to classify materials, a given material can then be included in more than one rheological class depending on experimental conditions. 

The fundamental rheological characterization of a material requires the experimental determination of a constitutive equation (a rheological equation of state) that mathematically relates stress and strain, or stress and strain rate. The constants in the constitutive equation are the rheological properties of the material. 

Equation (4) can be used to compare the change in creep behavior at zero time and for an infinitely long time. Consider that a composite is initially loaded at an infinitely rapid rate to a constant creep stress strain-rate relation for the composite (crc, ec) becomes... 

To more clearly understand the transient changes in constituent stress and strain rate that occurs between the limits of Eqns. (7) and (8), the transient creep and changes in fiber and matrix stress are plotted in Fig. 5.4 as a function of log(cr) and log(e). For convenience in presenting the interaction of the constituents, a normalized creep equation is used for each constituent ... 

Fig. 5.4 (a, b and c) Initial and final (steady-state) strain rate of a hypothetical composite as a function of normalized stress. The dashed lines represent the creep rate of the constituents, (b) and (c), which detail the two framed regions in Fig. 5.4(a), show the transient paths of the stress and strain rate for the composite and its constituents for the two corresponding stress regimes cr and a<. The dashed lines in (b) and (c) show the creep rate of the constituents (excluding the elastic components), which follow the monolithic creep behavior of each phase the total strain rate of the composite and the constituents must remain equal. 

Although a reference stress and strain rate was used in the above discussion, practical composites may work in a regime far away from this equilibrium point this does not affect the results of the above discussion. As discussed in detail elsewhere,31 plots such as Fig. 5.4 are useful for estimating the constituent creep behavior from experimental studies of the initial and final creep behavior of a composite. 

This dependence is certainly different from the amplitude of the RR stress and strain-rate fields which is Kjlt. This is an illustration of why the amplitude of the RR-field, C(t), is not necessarily the crack driving force parameter. This is in contrast to the ambient temperature situation wherein the strain energy release rate correlates exactly with either G (= K2/E) or /, both of which also govern the amplitude of the appropriate elastic or elastic-plastic stress fields. 

The first quantitative study of deformation mechanisms in ABS polymers was made by Bucknall and Drinkwater, who used accurate exten-someters to make simultaneous measurements of longitudinal and lateral strains during tensile creep tests (4). Volume strains calculated from these data were used to determine the extent of craze formation, and lateral strains were used to follow shear processes. Thus the tensile deformation was analyzed in terms of the two mechanisms, and the kinetics of each mechanism were studied separately. Bucknall and Drinkwater showed that both crazing and shear processes contribute significantly to the creep of Cycolac T—an ABS emulsion polymer—at room temperature and at relatively low stresses and strain rates. 

The effects of very high stresses and strain-rates have been investigated in microhardness experiments. In these experiments, loads of 50-500 g (corresponding to stresses as high as 2 GPa) are exerted by a diamond or sapphire Vickers indenter for about 20 seconds at temperatures up to 1,(X)0°C. Clearly, steady-state flow is never achieved but such experiments have provided important information about the dislocations involved in the deformation of olivine, for example. 

The types of free dislocations show that [100] slip was dominant during the high-temperature deformation, and [001] slip was dominant during the low-temperature deformation. Extrapolation of experimental data (Ave Lallement and Carter 1970) to natural deformation strain-rates suggests a transition temperature of 600 -800 C, which compares with a value of 700°-800°C inferred from the mineral assemblages. The stresses and strain-rates during the two stages of deformation can, in principle, be estimated from microstructural information, as already discussed. 

Figure 8.5 A diagram of relations among stresses and strain rates. The two full-line arcs are based on the definition of a material s viscosity, and the two dotted arcs are based on mechanical equilibrium or conservation of momentum. From these, two relations are derived that resemble Fick s first and second laws, represented by wiggly lines.

II. Viscous behavior (viscous flow) is characterized by proportionality between the shear stress and strain rate, i.e. by linear dependence between t and the rate of shear, y =dy/dt, given by Newton s law ... 

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