Purely straining shear

These expressions are written in the coordinates TZ, 0 obtained from the initial point by revolution by the angle Ad and related to the principal axes of the symmetric tensor E (in the principal axes, the tensor E can be reduced to the diagonal form with diagonal entries E and -E). Purely straining shear corresponds to the value fi = 0, and simple shear is determined by the parameters E = 0 and E2 = — fi. 

General methodology includes preparation of the solid texture. The measurement techniques may be the application of forces such as pure stress (also strain), shear, or their combinations and the measurement of the resistance to this procedure (Table III). Also used are special methods such Table III. Measurable Properties of Texture... 

Fig. 1. Schematic of a family of two-dimensional steady incompressible shear flows showing the streamline patterns at the top and the corresponding velocity components at the bottom. By varying X continuously from — 1 to +1, the flow can be varied from pure rotation (without strain) to pure strain (without rotation).

The expressions (8-44) and (8 45) represent a complete, exact solution of the creeping-flow equations for a completely arbitrary linear flow. Among the linear flows of special interest are axisymmetric pure strain, which was solved by means of the eigenfunction expansion for axisymmetric flows in the previous chapter, and simple shear flow, for which... 

Gki = G-ik - 0 (k = 1, 2, 3). Since the fluid is incompressible, we can represent the shear tensor as the sum of a symmetric and antisymmetric tensor, which correspond to the purely straining and purely rotational components of the fluid motion at infinity ... 

An arbitrary shear Stokes flow past a fixed cylinder is described by the stream function (2.7.9). We restrict our discussion to the case 0 2 fi < 1, in which there are four stagnation points on the surface of the cylinder. Qualitative streamline patterns for a purely straining flow (at CIe 0) and a purely shear flow (at CIe = 1) are shown in Figure 2.10. 

It follows from (4.11.3) that in the region -1 < fl < +1, the mean Sherwood number varies only slightly (the relative increment in the mean Sherwood number as Iflfil varies from 0 to 1 is at most 1.3%). In the special cases of purely straining (CIe = 0) and purely shear (Ifi l = 1) linear Stokes flow past a circular cylinder, formula (4.11.3) turns into those given in [342, 343]. 

The same approach can be used to find the interface structure for different kinds of time-dependent strain. For example, both the shear or the vortex flows described in Sect. 2.7.1 lead to interfaces described by (5.11) with /(2r1/2) — x/4Dt/3 at long times. In these cases the interface region, aligned with the shear flow, widens diffusively in time. We have Ct VDt and the production per unit length p = Crit) + A(t)Cr(t) (D/t)1/2 decreases in time. At any fixed time both the interface width and the production behave as Pe-1/2, as in the pure strain case. 

For finite strain in isotropic media, only states of homogeneous pure strain will be considered, i.e. states of uniform strain in the medium, with all shear components zero. This is not as restrictive as it might first appear to be, because for small strains a shear strain is exactly equivalent to equal compressive and extensional strains applied at 90° to each other and at 45° to the original axes along which the shear was applied (see problem 6.1). Thus a shear is transformed into a state of homogeneous pure strain simply by a rotation of axes by 45°. A similar transformation can be made for finite strains, but the rotation is then not 45°. All states of homogeneous strain can thus be regarded as pure if suitable axes are chosen. 

For simple shear a = 0, a = 1 corresponds to pure straining motion. In Figure 2 Bentley and Leal s results are summarized ... 

Mechanical spectroscopy is an ideal technique for investigation of the viscoelastic properties of materials. It involves the application of a sinusoidal oscillation of strain (7) and frequency (oj) to the material. A perfectly elastic material will have a stress wave exactly in phase with the applied strain wave. A purely viscous material will have a stress wave exactly 90° out of phase with the applied strain since at the maximum deflection the rate of strain (shear rate) is zero. This is illustrated in Figure 2.2 [5]. 

Figure 36 is representative of creep and recovery curves for viscoelastic fluids. Such a curve is obtained when a stress is placed on the specimen and the deformation is monitored as a function of time. During the experiment the stress is removed, and the specimen, if it can, is free to recover. The slope of the linear portion of the creep curve gives the shear rate, and the viscosity is the appHed stress divided by the slope. A steep slope indicates a low viscosity, and a gradual slope a high viscosity. The recovery part of Figure 36 shows that the specimen was viscoelastic because relaxation took place and some of the strain was recovered. A purely viscous material would not have shown any recovery, as shown in Figure 16b. 

Fluids without any sohdlike elastic behavior do not undergo any reverse deformation when shear stress is removed, and are called purely viscous fluids. The shear stress depends only on the rate of deformation, and not on the extent of derormation (strain). Those which exhibit both viscous and elastic properties are called viscoelastic fluids. 

In crystals with the LI2 structure (the fcc-based ordered structure), there exist three independent elastic constants-in the contracted notation, Cn, C12 and 044. A set of three independent ab initio total-energy calculations (i.e. total energy as a function of strain) is required to determine these elastic constants. We have determined the bulk modulus, Cii, and C44 from distortion energies associated with uniform hydrostatic pressure, uniaxial strain and pure shear strain, respectively. The shear moduli for the 001 plane along the [100] direction and for the 110 plane along the [110] direction, are G ooi = G44 and G no = (Cu — G12), respectively. The shear anisotropy factor, A = provides a measure of the degree of anisotropy of the electronic charge... 

Depending on the relative velocity in opposite pairs of the rollers, flow can be either purely rotational (X = — 1), simple shear (X = 0) or hyperbolic straining (X = 1, shown on this figure)... 

FIGURE 25.6 Examples of variable ampUtude fatigue crack growth test signals applied to pure shear specimens to investigate the effects of (a) load severity, (b) load sequence, (c) R-ratio, and (d) dwell periods on crack growth rates. A, B, and C denote peak strain levels. 

The two-network method has been carefully examined. All the previous two-network results were obtained in simple extension for which the Gaussian composite network theory was found to be inadequate. Results obtained on composite networks of 1,2-polybutadiene for three different types of strain, namely equibiaxial extension, pure shear, and simple extension, are discussed in the present paper. The Gaussian composite network elastic free energy relation is found to be adequate in equibiaxial extension and possibly pure shear. Extrapolation to zero strain gives the same result for all three types of strain The contribution from chain entangling at elastic equilibrium is found to be approximately equal to the pseudo-equilibrium rubber plateau modulus and about three times larger than the contribution from chemical cross-links. 

At the instant of contact between a sphere and a flat specimen there is no strain in the specimen, but the sphere then becomes flattened by the surface tractions which creates forces of reaction which produce strain in the specimen as well as the sphere. The strain consists of both hydrostatic compression and shear. The maximum shear strain is at a point along the axis of contact, lying a distance equal to about half of the radius of the area of contact (both solids having the same elastic properties with Poisson s ratio = 1/3). When this maximum shear strain reaches a critical value, plastic flow begins, or twinning occurs, or a phase transformation begins. Note that the critical value may be very small (e.g., in pure simple metals it is zero) or it may be quite large (e.g., in diamond). 

A straightforward estimate of the maximum hardness increment can be made in terms of the strain associated with mixing Br and Cl ions. The fractional difference in the interionic distances in KC1 vs. KBr is about five percent (Pauling, 1960). The elastic constants of the pure crystals are similar, and average values are Cu = 37.5 GPa, C12 = 6 GPa, and C44 = 5.6 GPa. On the glide plane (110) the appropriate shear constant is C = (Cu - C12)/2 = 15.8 GPa. The increment in hardness shown in Figure 9.5 is 14 GPa. This corresponds to a shear flow stress of about 2.3 GPa. which is about 17 percent of the shear modulus, or about C l2n. 

The total contribution of the shear strain to the fibre strain is the sum of the purely or immediate elastic contribution involving the change in angle, A0e=0o- , occurring immediately upon loading of the fibre at f=0, and the time-dependent or viscoelastic and plastic contribution A0(f)=0(f)-0o [7-10]. According to the continuous chain model for the extension of polymer fibres, the time-dependent shear strain during creep can be written as... 

Ctjki is a fourth order tensor that linearly relates a and e. It is sometimes called the elastic rigidity tensor and contains 81 elements that completely describe the elastic characteristics of the medium. Because of the symmetry of a and e, only 36 elements of Cyu are independent in general cases. Moreover only 2 independent rigidity constants are present in Cyti for linear homogeneous isotropic purely elastic medium Lame coefficient A and /r have a stress dimension, A is related to longitudinal strain and n to shear strain. For the purpose of clarity, a condensed notation is often used... 

---