Plastic strain increment

Apart from yield criterion, one is interested in the constitutive relations. In the elastic constitutive relation, the stress is related to strain however in the plastic constitutive relation stress can be related to strain-rate or strain-increment. In 1872, M. Levy used an incremental constimtive equation, which was later proposed by von Mises. Levy s paper was not known outside France. Levy-Mises relation considers that the increments of plastic strain increments are in proportion to deviatoric components, i.e.. 

In order for a theory of this type to be useful it is necessary that it is not only descriptive but also predictive. The Hill criterion offers such a possibility in that it allows a prediction of the plastic strain increments at yield, provided sufficient yield stresses have been determined to evaluate F, G, H, L, M and N explicitly. By making the conventional assumption of normality (see Ref. 5, Chapters 2 and 10), it follows that the components... 

We have now discussed in turn, the stresses required to produce yield, the relationship between stress and plastic strain increment, the structural reorientation occurring as a result of yield, and the relationship between constant strain-rate yield and features of non-linear recoverable creep deformation. Theoretical models to describe the behaviour have ranged from single crystal plasticity through to the oriented continuum ideas of plasticity and viscoelasticity. On many points both the experimental data and the interpretations appear almost contradictory and it is therefore helpful to see if any common ground can be established. 

Corresponding system plastic-strain increments are also obtained at the atomic level from the displacement gradients between the four relevant neighboring corner atoms of Delaunay tetrahedra for each external distortion increment and are allocated subsequently as an atomic site average to each Voronoi polyhedral atom environment by a special procedure of double space tessellation developed by Mott et al. (1992) for this purpose, leading eventually to volume averages of strain-increment tensors of all Voronoi atom environments to attain the system-wide strain-inerement tensor. 

It was noted above that Table 1 does not mention dilation, a difficult topic that many geotechnical engineers prefer to ignore. For Mohr-Coulomb models, Plaxis and SAFE allow angles of dilation it to be specified by the user with default values of zero and a limit to the total amount of accumulated dilation. For its default value, CRISP assumes normality, treating the yield surface as a plastic potential, which, for a Mohr-Coulomb model, implies it = cp. This is shown in Figure 19 by the vector of plastic strain increments (5yP) for the plane strain case. All the CRISP analyses reported here use it = shear strains become very large as shear failure occurs. 

Deviatoric stress tensor Norm of deviatoric stress Lode s angle for stress Second invariant of deviatoric stress Third invariant of deviatoric stress Strain increment tensor Elastic strain increment tensor Plastic strain increment tensor Volumetric plastic strain increment tensor... 

Norm of deviatoric plastic strain increment Hardening parameter... 

Let g be a scalar function such that the plastic strain increment de can be obtained as follows ... 

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... 

However, the stress a needed to increase the strain by a plastic strain increment d P l can be stated, for the yield criterion has to be fulfilled. We... 

From the flow rule of normality principle, the following relationship exists between the plastic strain increment and the plastic stress increment ... 

This equation can be interpreted as requiring the normality of the plastic strain increment vector to yield the surface in the hyper-space of n stress dimensions. As before dl is the proportionality constant. 

Figure 2. FEM simulation of the elastic plastic particle constrained by elastic matrix illustrating a layer undergoing a nonzero axial plastic strain increment Asyy.

The Levy-Mises equations define one of a number of possible flow rules that can be derived via an argument that depends upon a concept known as the plastic potential. This idea has been discussed by Hill [ 15]. It is assumed that the components of the plastic strain increment tensor are proportional to the partial derivatives of the plastic potential, which is a scalar function of stress. The flow rule can thus be generated by this differentiation process. We may choose to assume, for a particular form of yield criterion, that the plastic potential has the same functional form as the yield criterion then, the derived flow rule is described as being associated with the yield criterion (or as an associative flow rule). However, this assumption is not obligatory and when it is not true we will be applying a yield criterion together with a non-associated flow rule. This is discussed further by de Souza Neto etal. [19],... 

Figure 12.28 Plastic strain increment by formation of a pair of kinks in a polymer molecule. (Redrawn from Argon, A.S. (1973) A theory for the low-temperature plastic deformation of glassy polymers. Phil. Mag., 28, 839. Copyright (1973) Taylor and Francis.)...

Eq.(l2) shows that if plastic strain increment de is given,... 

On the other hand, the stress-dilatancy equation which constrains the strain is given by Eq.(l3) or (I6). Thus, take a plastic strain increment... 

Plastic strain incremental vector (Normality conditionj ... 

Obviously, a yield criterion is needed to determine the actual magnitudes of plastic strain increments. This theory was first proposed by Prandtl (1925) and Reuss (1930). 

---