Strains pure shear

Here, Wo is a characteristic level of back stress that primarily affects the initial slope of the uniaxial stress versus remanent strain curve, and m is another hardening parameter that controls how abruptly the strain saturation conditions are reached. Figure 2a illustrates the predictions of the effective stress versus the effective remanent strain from the constitutive law for uniaxial compression, pure shear strain, pure shear stress and uniaxial tension. It is interesting to note that the shear strain and shear stress curves do not coincide. This feature is due to the fact that the material can strain more in tension than in compression, and has been confirmed in micromechanical simulations. Figure 2b illustrates the uniaxial stress versus remanent strain hysteresis curves for two sets of the material parameters Wq and m. 

Figure 2. (a) Effective stress versus effective remanent strain curves for the model material described in Section 2 in uniaxial compression, pure shear strain, pure shear stress and uniaxial tension tests, (b) Uniaxial stress versus remanent strain hysteresis loops for the model material illustrating the effect of the hardening parameter In both cases notice the asymmetry in the remanent strains that can be achieved in tension versus compression. 

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... 

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. 

The stress-strain relations for some special cases of biaxial defonnation are derived from Eqs. (13) to (15) in the following way. Strip biaxial extension of incompressible material is defined as the mode of deformation in which one of the Xj, say X2, is kept at unity, while the other, Xt, varies. This deformation is also called pure shear . We have for it ... 

Short term stress-strain properties In pure shear ... 

Data can be obtained from tests in uniaxial tension, uniaxial compression, equibiaxial tension, pure shear and simple shear. Relevant test methods are described in subsequent sections. In principle, the coefficients for a model can be obtained from a single test, for example uniaxial tension. However, the coefficients are not fully independent and more than one set of values can be found to describe the tension stress strain curve. A difficulty will arise if these coefficients are applied to another mode of deformation, for example shear or compression, because the different sets of values may not be equivalent in these cases. To obtain more robust coefficients it is necessary to carry out tests using more than one geometry and to combine the data to optimize the coefficients. 

A good approximation to pure shear is obtained with a strip test piece strained perpendicular to its length as shown in Figure 8.18. 

The viscoelastic behaviour of rubbers is not linear stress is not proportional to strain, particularly at high strains. The non-linearity is more pronounced in tension or compression than in shear. The result in practice is that dynamic stiffness and moduli are strain dependent and the hysteresis loop will not be a perfect ellipse. If the strain in the test piece is not uniform, it is necessary to apply a shape factor in the same manner as for static tests. This is usually the case in compression and even in shear there may be bending in addition to pure shear. Relationships for shear, compression and tension taking these factors into account have been given by Payne3 and Davey and Payne4 but, because the relationships between dynamic stiffness and the basic moduli may be complex and only approximate, it may be preferable for many engineering applications to work in stiffness, particularly if products are tested. 

Figure 12.2 Schematic representations of (a) pure shear test (b) plane strain compression test and (c) three-point bending test.

Using this concept, Erwin [9] demonstrated that the upper bound for the ideal mixer is found in a mixer that applies a plane strain extensional flow or pure shear flow to the fluid and where the surfaces are maintained ideally oriented during the whole process this occurs when N = 00 and each time an infinitesimal amount of shear is applied. In such a system the growth of the interfacial areas follows the relation given by... 

Planar extensional flow or pure shear flow is extensional flow with the same but opposite rates of strain in two directions in the third direction, there is no flow ... 

The (1 -f- c2)2 term arises from the assumption that specimens deform isotropically in width and thickness. Volume strain vs. longitudinal strain curves are shown for three specimens of ABS 1 in Figure 7. For comparison, theoretical curves for total cavitation, i.e., c2 = 0, and pure shear, i.e., AV = 0, are also shown. Volume strain vs. longitudinal strain curves, together with nominal stress vs. nominal strain curves, for ABS 1, ABS 2, and ABS 3 are shown in Figure 8. Because of the size of the strains involved, it is not possible to approximate Equation 3 with an expression which contains only first-order powers of strain when calculating volume strains. 

Another alternative is the plane strain compression test, shown in Figure 14.5c. The advantage displayed by this experiment is that the area of the specimen remains constant over the test and therefore = a . This test can be classified as a pure shear test as only two of the three sample dimensions are changed. 

In the ideal case of a Hookean body, the relationship between stress and strain is fully linear, and the body returns to its original shape and size, after the stress applied has been relieved. The proportionality between stress and strain is quantified by the modulus of elasticity (unit Pa). The proportionality factor under conditions of normal stress is called modulus of elasticity in tension or Young s modulus E), whereas that in pure shear is called modulus of elasticity in shear or modulus of rigidity (G). The relationships between E, G, shear stress, and strain are defined by ... 

The spontaneous strain for long-range ordering in ilmenite is approximately a pure shear (with s- - and 33 having opposite sign and Sy 0). It seems reasonable to assume that short-range ordering, which is often an important feature of such transitions, will... 

Note that the simple Hooke s law behavior of the stress in a solid is analogous to Newton s law for the stress of a fluid. For a simple Newtonian fluid, the shear stress is proportional to the rate of strain, y (shear rate), whereas in a Hookian solid, it is proportional to the strain, y, itself. For a fluid that shares both viscous and elastic behavior, the equation for the shear stress must incorporate both of these laws— Newton s and Hooke s. A possible constitutive relationship between the stress in a fluid and the strain is described by the Maxwell model (Eq. 6.3), which assumes that a purely viscous damper described by Eq. 6.1 and a pure spring described by Eq. 6.2 are connected in series (i.e., the two y from Eqs. 6.1 and 6.2 are additive). 

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]. 

From this figure we learn that pure deformation without any change in size (pure shear) and change in size without any change in shape of the body (dilatation or compression) have to be considered. For the latter case the tensor of strain becomes Uj,j 8 (Landau Lifschitz 1953). Any deformation can be given as the sum of pure shear and dilatation deformations. Therefore, we get... 

For small strains, this is simply the tangent of the angle of deformation. In pure shear, Hooke s law is expressed as... 

In a majority of cases, a body under stress experiences neither pure shear nor pure dilatation. Generally, a mixture of both occurs. Such a situation is exemplified by uniaxial loading which, of course, may be tensile or compressive. Here a test specimen is loaded axially resulting in a change in length, AL. The axial strain, e, is related to the applied stress in an elastic deformation by Hooke s law ... 

Pure shear is represented in Fig. 5 and is defined as a homogeneous strain in which one of the principal extensions is zero and the volume is unchanged. If the extension ratio A] = a while At = 1. then is /a. 

Shear strain is measured by the magnitude of the angle representing the displacement of a certain plane relative to the other, due to the application of a pure shear stress, such as a in Figure 3.1c. The corresponding shear strain y may be taken equal to the ratio aa /ab ( = tan a). A shear strain is produced in torsion, when, for example, a circular rod is twisted by tangential forces, as shown in... 

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