Plastic isotropic constants 99

Figure 6-43. The modulus of elasticity, the shear modulus, and Poisson s ratio for an isotropic material such as fiber matt-reinforced plastics. Because constants do not vary with the angle of load, the ratio of modulus Ei at any angle to E at any reference direction is unity. The shear modulus is a constant proportion of E, and Poisson s ratio is constant.

The Z-direction is perpendicular to the page. For simplicity the material is assumed to be isotropic, ie same properties in all directions. However, in some cases for plastics and almost always for fibre composites, the properties will be anisotropic. Thus E and v will have different values in the x, y and z direction. Also, it should also be remembered that only at short times can E and v be assumed to be constants. They will both change with time and so for long-term loading, appropriate values should be used. 

In the perfectly elastic, perfectly plastic models, the high pressure compressibility can be approximated from static high pressure experiments or from high-order elastic constant measurements. Based on an estimate of strength, the stress-volume relation under uniaxial strain conditions appropriate for shock compression can be constructed. Inversely, and more typically, strength corrections can be applied to shock data to remove the shear strength component. The stress-volume relation is composed of the isotropic (hydrostatic) stress to which a component of shear stress appropriate to the... 

Fig. 2.8. Idealized elastic/perfectly plastic solid behavior results in a stress tensor in which there is a constant offset between the hydrostatic (isotropic) loading and shock compression. Such behavior is only an approximation which may not be appropriate in many cases.

For those not familiar with this type information recognize that the viscoelastic behavior of plastics shows that their deformations are dependent on such factors as the time under load and temperature conditions. Therefore, when structural (load bearing) plastic products are to be designed, it must be remembered that the standard equations that have been historically available for designing steel springs, beams, plates, cylinders, etc. have all been derived under the assumptions that (1) the strains are small, (2) the modulus is constant, (3) the strains are independent of the loading rate or history and are immediately reversible, (4) the material is isotropic, and (5) the material behaves in the same way in tension and compression. 

Isotropic material In an isotropic material the properties at a given point are the same, independent of the direction in which they are measured (Fig. 3-19). The term isotropic means uniform. As one moves from point-to-point in this type of homogeneous plastic the material s composition remains constant. Also, the smallest samples of material... 

Indeed, it has been observed that the onset of yielding of isotropic polymers is approximately constant, 0.02< [<0.025, which implies that 0.04shear yield strain, the plastic shear deformation of the domain satisfies a plastic shear law. For temperatures below the glass transition temperature, the continuous chain model enables the calculation of the tensile curve of a polymer fibre up to about 10% strain [6]. Figure 7 shows the observed stress-strain curves of PpPTA fibres with different moduli compared to the calculated curves. 

Each block is modeled as linear, isotropic, homogeneous and elastic medium and subdivided with a mesh of constant-strain triangle finite-difference elements. Key factors affecting the hydraulic behaviour of fractures such as opening, closure, sliding and dilation of fractures are modeled by an elasto-perfectly plastic constitutive model of a fracture. A step-wise non-linear normal stress-normal closure relationship is adopted with a linear Mohr-Coulomb failure for shear (Figure 3). 

This relation contains two competing terms the first term represents plastic hardening as a function of the volumetric part of plastic strain, the second term describes chemical softening due to an increase in contaminant concentration. Let us consider the plastic response to an increase in contaminant concentration at constant isotropic stress. The condition p =pc =0 in equation (7) implies... 

Plastic products with biaxial orientation (such as the wall of a PET carbonated drink bottle) have a greater resistance to puncturing than initially isotropic or uniaxially oriented film, because there is no weak direction in the plane of the film. In such anisotropic films, the value of cr — rr to cause thinning is larger than [Pg.245]

In practice of rock engineering projects, the generalized HB criterion is suited to a homogeneous, isotropic and massive rock with few discontinuities or to a heavily jointed rock mass. When a cavern or cavern group is constructed in the heavily jointed rock mass, it has been observed that the surrounding rock failure depends on mutual separation and slip of discontinuities. For this case in the elasto-plastic analysis, it can be accepted that the three parameters of cr, m- and GSI m the generalized HB criterion are fixed to be constants, but D is treated as a variable related to the confining pressure... 

More constants are needed to define the behavior of non-isotropic materials. Shah V (1998) Handbook of plastics testing technology. John Wiley and Sons, New York. 

It is important to appreciate that plasticity is different in kind from elasticity, where there is a unique relationship between stress and strain defined by a modulus or stiffness constant. Once we achieve the combination of stresses required to produce yield in an idealized rigid plastic material, deformation can proceed without altering stresses and is determined by the movements of the external constraints, e.g. the displacement of the jaws of the tensometer in a tensile test. This means that there is no unique relationship between the stresses and the total plastic deformation. Instead, the relationships that do exist relate the stresses and the incremental plastic deformation, as was first recognized by St Venant, who proposed that for an isotropic material the principal axes of the strain increment are parallel to the principal axes of stress. 

The flow rule (2.302) implies that the direction of the plastic strain increment deP is normal to the surface g = constant, and coincides with the stress a. For isotropic materials this can be described as follows. We introduce the unit tensors (see Sect. 2.8.3) as... 

If we perform undrained triaxial tests (namely, void ratio e=constant) on isotropically and normally consolidated clay at several confining pressures p, the stress-strain behavior is schematically shown in Fig. 6.2. It is observed that at the final stage of loading (namely, the failure state) the stress ratio rj = q/p becomes constant qf = q/p )f = M), which is referred to as the critical state, which means that the deformation is developed under a constant volumetric plastic strain and a constant shear stress at the critical state q = Mp. ... 

---