Stress space

The theory is initially presented in the context of small deformations in Section 5.2. A set of internal state variables are introduced as primitive quantities, collectively represented by the symbol k. Qualitative concepts of inelastic deformation are rendered into precise mathematical statements regarding an elastic range bounded by an elastic limit surface, a stress-strain relation, and an evolution equation for the internal state variables. While these qualitative ideas lead in a natural way to the formulation of an elastic limit surface in strain space, an elastic limit surface in stress space arises as a consequence. An assumption that the external work done in small closed cycles of deformation should be nonnegative leads to the existence of an elastic potential and a normality condition. 

Figure 5.2. Particle history and elastic limit surface in stress space.

Perfectly Plastic. When A = Q then / = 0. The elastie limit surfaee in stress space is stationary, and the material is said to be perfectly inelastic. 

Softening. When /I < 0 then / < 0. The elastic limit surface in stress space is moving inward, and the material is said to be softening. 

This ambiguity in the stress space loading criterion may be illustrated by considering a stress-strain plot corresponding to simple tension, as shown schematically in Fig. 5.3. With each point on the stress-strain curve past the initial elastic limit point A, there is associated a point on the elastic limit surface in stress space and a point on the elastic limit surface in strain space. On the hardening portion of the stress strain curve AB, both the stress and the strain are increasing, and the respective elastic limit surfaces are moving... 

When the material is at the ultimate stress point B, inelastic loading will entail a positive strain rate, and the elastic limit surface in strain space will be moving outward. On the other hand, the stress rate at this point is zero, and the elastic limit surface in stress space will be stationary. If the material is perfectly inelastic over a range of strains, then the stress rate will be zero and the elastic limit surface in stress space will be stationary on inelastic loading throughout this range. 

The direction of the stress rate in relation to the elastic limit surface in stress space, expressed in /, cannot be used as an unambiguous indicator of loading or unloading. The proper indicator of inelastic loading in stress space is =/// . 

An analogous normality condition in stress space may be obtained from (5.56) by using (5.22) and (5.26)... 

The normality conditions (5.56) and (5.57) have essentially the same forms as those derived by Casey and Naghdi [1], [2], [3], but the interpretation is very different. In the present theory, it is clear that the inelastic strain rate e is always normal to the elastic limit surface in stress space. When applied to plasticity, e is the plastic strain rate, which may now be denoted e", and this is always normal to the elastic limit surface, which may now be called the yield surface. Naghdi et al. by contrast, took the internal state variables k to be comprised of the plastic strain e and a scalar hardening parameter k. In their theory, consequently, the plastic strain rate e , being contained in k in (5.57), is not itself normal to the yield surface. This confusion produces quite different results. 

In this section, the general inelastic theory of Section 5.2 will be specialized to a simple phenomenological theory of plasticity. The inelastic strain rate tensor e may be identified with the plastic strain rate tensor e . In order to include isotropic and kinematic hardening, the set of internal state variables, denoted collectively by k in the previous theory, is reduced to the set (k, a) where k is a scalar representing isotropic hardening and a is a symmetric second-order tensor representing kinematic hardening. The elastic limit condition in stress space (5.25), now called a yield condition, becomes... 

The choice (5.77) for the evolution equation for the plastic strain sets the evolution equations for the internal state variables (5.78) into the form (5.11) required for continuity. The consistency condition in the stress space description may be obtained by differentiating (5.73), or directly by expanding (5.29)... 

If the work assumption is made, then the normality condition in stress space (5.57) reduces to... 

Since the yield function is independent of p, the yield surface reduces to a cylinder in principal stress space with axis normal to the 11 plane. If the work assumption is made, then the normality condition (5.80) implies that the plastic strain rate is normal to the yield surface and parallel to the II plane, and must therefore be a deviator k = k , k = 0. It follows that the plastic strain is incompressible and the volume change is entirely elastic. Assuming that the plastic strain is initially zero, the spherical part of the stress relation (5.85) becomes... 

Figure 5.7, Isotropic yield surfaces in (a) stress space and (b) strain space.

From (A.81), /3T, = k, and this equation implies that the yield surface in stress space is a circular cylinder of radius k, shown in a FI plane projection in Fig. 5.7(a). The corresponding yield surface in strain space may be obtained by inserting the deviatoric stress relation (5.86) into the yield function (5.92)... 

The yield surface in strain space is a ciruclar cylinder normal to the 11 plane with radius k/2fi and axis offset from the origin by e ", as shown in Fig. 5.7(b). It may be seen that, if the yield function in stress space is independent of pressure, then the yield function in strain space is independent of volume change and vice versa. 

The elastic limit conditions in strain space (5.1) and stress space (5.25) become... 

The elastic limit function in stress space may be translated into spatial terms by using (5.151) and (A.39) in (5.135)... 

Tsai and Wu postulated that a failure surface in six-dimensional stress space exists in the form... 

The terms that are linear in the stresses are useful in representing different strengths in tension and compression. The terms that are quadratic in the stresses are the more or less usual terms to represent an ellipsoid in stress space. However, the independent parameter F,2 is new and quite unlike the dependent coefficient 2H = 1/X in the Tsai-Hill failure criterion on the term involving interaction between normal stresses in the 1- and 2-directions. 

The various octants in principal stress space are described by considering conditions of tension (+) and compression (—) and their combination. Convention usually takes the upper right octant extending out from the page in Figure 15 to be the tension-tension-tension (+ H—h)... 

To visualize the stress states at which failure occurs, a failure surface may be constructed. The six stress components may be resolved into three orthogonal principal stresses. Plotting failure in principal stress space, any stress state which exists in the bounded space containing the... 

Figure 23. Various failure surfaces in principal stress space (a) cylinder, (b) cone, and (c) paraboloid (110)...

Other coordinate systems may be used for failure surface representations in addition to stress space. Blatz and Ko (11) indicate that either stress (commonly used because the failure surface concept was originally applied to metals, for which stress and strain are more simply related. Viscoelastic materials, on the other hand, may show a multitude of strain values at a given stress level, depending on test conditions. 

Figure 25. Failure surface in principal stress space for a composite solid...

Here F, denotes the tensile strength of the material, Fc is the compressive strength, and Ybc represents equal biaxial compression strength of the material. The reader is directed to Palko25 where the specific forms of B and A are presented. This model is termed a three-parameter model, referring to the three material strength parameters (F FC, and Ybc) used to characterize the model. Failure is defined when / = 0, and the multiaxial criterion is completely defined in the six-dimensional stress space. 

Note The sample area was 0.20 cm2, and the zero-stress -spacing was 550 A. 

For macroscopically isotropic polymers, the Tresca and von Mises yield criteria take very simple analytical forms when expressed in terms of the principal stresses cji, form surfaces in the principal stress space. The shear yield surface for the pressure-dependent von Mises criterion [Eqs (14.10) and (14.12)] is a tapering cylinder centered on the applied pressure increases. The shear yield surface of the pressure-dependent Tresca criterion [Eqs (14.8) and (14.12)] is a hexagonal pyramid. To determine which of the two criteria is the most appropriate for a particular polymer it is necessary to determine the yield behavior of the polymer under different states of stress. This is done by working in plane stress (ct3 = 0) and obtaining yield stresses for simple uniaxial tension and compression, pure shear (di = —CT2), and biaxial tension (cti, 0-2 > 0). Figure 14.9 shows the experimental results for glassy polystyrene (13), where the... 

Powder Yield Loci For a given shear step, as the applied shear stress is increased, the powder will reach a maximum sustainable shear stress "U, at which point it yields or flows. The functional relationship between this limit of shear stress "U and applied normal load a is referred to as a yield locus, i.e., a locus of yield stresses that may result in powder failure beyond its elastic limit. This functional relationship can be quite complex for powders, as illustrated in both principal stress space and shear versus normal stress in Fig. 21-36. See Nadia (loc. cit.), Stanley-Wood (loc. cit.), and Nedderman (loc. cit.) for details. Only the most basic features for isotropic hardening of the yield surface are mentioned here. 

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