Deformation shear

Material parameters defined by Equations (1.11) and (1.12) arise from anisotropy (i.e. direction dependency) of the microstructure of long-chain polymers subjected to liigh shear deformations. Generalized Newtonian constitutive equations cannot predict any normal stress acting along the direction perpendicular to the shearing surface in a viscometric flow. Thus the primary and secondary normal stress coefficients are only used in conjunction with viscoelastic constitutive models. 

Until now we have restricted ourselves to consideration of simple tensile deformation of the elastomer sample. This deformation is easy to visualize and leads to a manageable mathematical description. This is by no means the only deformation of interest, however. We shall consider only one additional mode of deformation, namely, shear deformation. Figure 3.6 represents an elastomer sample subject to shearing forces. Deformation in the shear mode is the basis... 

Figure 3.6 Definition of variables to define the shear deformation of an elastic body.

Power. There are two main ways to measure the power deUvered by the driver to the pump. The first method is to install a torque meter between the pump and the driver. A torque meter is a rotating bat having a strain gauge to measure shear deformation of a torqued shaft. Discussion of the principle of torque meter operation is available (16). The benefit of this method is direct and accurate measurements. The power deUveted to the pump from the driver is calculated from torque, T, and speed (tpm) in units of brake horsepower, ie, BHP (eq. 4a) when Tis in lbs-ft, and kW (eq. 4b) when T is N-m. 

Extensional viscosity that results purely from shear deformation seems to be of less interest, but has been measured (108). The theology of several different polymer melts in terms of shear viscosity and uniaxial and biaxial extensional viscosity has been compared (231). Additional information on the measurement of extensional viscosity are also available (105,238—240). 

The behavior of constitutive equations may be investigated by prescribing particular deformations. Consider tbe simple homogeneous shear deformation... 

That this is not always the case should be expected. In fact, if it was not for heterogeneous localization of some flow phenomena, it would be very diflicult to initiate secondary explosives, or to effect shock-induced chemical reactions in solids. Heterogeneous shear deformation in metals has also been invoked as an explanation for a reduction in shear strength in shock compression as compared to quasi-isentropic loading. We present here a brief discussion of some aspects of heterogeneous deformation in shock-loaded solids. 

Bai [48] presents a linear stability analysis of plastic shear deformation. This involves the relationship between competing effects of work hardening, thermal softening, and thermal conduction. If the flow stress is given by Tq, and work hardening and thermal softening in the initial state are represented... 

Twinning is itself a kind of solid solid phase transformation with the first and seeond phases having the same erystal strueture (henee the same pressure-volume response) but with prineipal erystallographie orientation at some angle to the original orientation. This results in a eontribution to plastie shear deformation under both quasi-statie and shoek-wave eonditions [61]. 

The factor 3 appears because the viscosity is defined for shear deformation - as is the shear modulus G. For tensile deformation we want the viscous equivalent of Young s modulus . The answer is 3ri, for much the same reason that = (8/3)G 3G - see Chapter 3.) Data giving C and Q for polymers are available from suppliers. Then... 

Once values for R , Rp, and AEg are calculated at a given strain, the np product is extracted and individual values for n and p are determined from Eq. (4.19). The conductivity can then be calculated from eq. (4.18) after the mobilities are calculated. The hole mobility is the principal uncertainty since it has only been measured at small strains. In order to fit data obtained from elastic shock-loading experiments, a hole-mobility cutoff ratio is used as a parameter along with an unknown shear deformation potential. A best fit is then determined from the data for the cutoff ratio and the deformation potential. 

Fig. 4.10. The conductivity of uniaxially compressed (111) and (100) high purity germanium crystals leads to a determination of the shear deformation potential for the designated valley minima in the energy band (after Davison and Graham [79D01]).

The shear deformation potential for the (111) and (100) valley minima determined by fits to the data of Fig. 4.10 are shown in Table 4.5 and compared to prior theoretical calculations and experimental observations. The deformation potential of the (111) valley has been extensively investigated and the present value compares favorably to prior work. The error assigned recognizes the uncertainty in final resistivity due to observed time dependence. The distinguishing characteristic of the present value is that it is measured at a considerably larger strain than has heretofore been possible. Unfortunately, the present data are too limited to address the question of nonlinearities in the deformation potentials [77T02]. 

Although the [100] data are quite limited, the shear deformation potential determined is the only measurement for this valley in germanium. At atmospheric pressure and small strains the (100) valley minimum is well above the (111) valley minima and not accessible for measurement. In the present... 

Chapter 4. Physical Properties Under Elastic Shock Compression Table 4.5. Shear deformation potentials. (See Davison and Graham [79D01].)... 

An important implication of the presence of the shear-extension coupling coefficient is that off-axis (non-principal material direction) tensile loadings for composite materials result in shear deformation in addition to the usual axial extension. This subject is investigated further in Section 2.8. At this point, recognize that Equation (2.97) is a quantification of the foregoing implication for tensile tests and of the qualitative observations made in Section 1.2. 

Accordingly, we have supposedly found the shear modulus G.,2. However, a relationship such as Equation (2.107) does not exist for strengths because strengths do not transform like stiffnesses. Thus, this experiment cannot be relied upon to determine S, the ultimate shear stress (shear strength), because a pure shear deformation mode has not been excited with accompanying failure in shear. Accordingly, other approaches to obtain S must be used. 

The in-plane shear modulus of a lamina, G12. is determined in the mechanics of materials approach by presuming that the shearing stresses on the fiber and on the matrix are the same (clearly, the shear deformations cannot be the samel). The loading Is shown in the representative volume element of Figure 3-15. By virtue of the basic presumption,... 

On a microscopic scale, the deformations are shown in Figure 3-15. Note that the matrix deforms more than the fiber in shear because the matrix has a lower shear modulus. The total shearing deformation is... 

Obviously, the classical lamination theory stresses in Pagano s example converge to the exact solution much more rapidly than do the displacements as the span-to-thickness ratio increases. The stress errors are on the order of 10% or less for S as low as 20. The displacements are severely underestimated for S between 4 and 30, which are common values for laboratory characterization specimens. Thus, a practical means of accounting for transverse shearing deformations is required. That objective is attacked in the next section. 

The results shown in Figure 6-21 for the present shear-deformation approach versus classical lamination theory are quite similar qualitatively to the comparison between the exact cylindrical bending solution and classical lamination theory in Figure 6-17. 

Whitney and Pagano [6-32] extended Yang, Norris, and Stavsky s work [6-33] to the treatment of coupling between bending and extension. Whitney uses a higher order stress theory to obtain improved predictions of a, and and displacements at low width-to-thickness ratios [6-34], Meissner used his variational theorem to derive a consistent set of equations for inclusion of transverse shearing deformation effects in symmetrically laminated plates [6-35]. Finally, Ambartsumyan extended his treatment of transverse shearing deformation effects from plates to shells [6-36]. 

Eric Reissner, The Effect of Transverse Shear Deformation on the Bending of Elastic Plates, Journal of Applied Mechanics, June 1945, pp. A-69-77. 

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