## 71 Review of yield behavior of fully dense metals

Fully dense metals deform plastically at constant volume. Because of this, the yield criterion which characterizes their plastic behavior is independent of mean stress. If the metal is isotropic (i.e. has the same properties in all directions) its plastic response is well approximated by the von Mises criterion: yield occurs when the von Mises effective stress oe attains the yield value Y.

The effective stress, ae, is a scalar measure of the deviatoric stress, and is defined such that it equals the uniaxial stress in a tension or compression test. On writing the principal stresses as (oi, on, am), ae can be expressed by

When the stress a is resolved onto arbitrary Cartesian axes not aligned with the principal axes of stress, a has three direct components (a11, <o22, o33) and three shear components (o12, o23, o31), and can be written as a symmetric 3 x 3 matrix, with components oj. Then, the mean stress om, which is invariant with respect to a rotation of axes, is defined by

where the repeated suffix, here and elsewhere, denotes summation from 1 to 3. The stress a can be decomposed additively into its mean component om and its deviatoric (i.e. shear) components Sij, giving aij = Sij + VmSij (7.3)

where Sij is the Kronecker delta symbol, and takes the value Sij = 1 if i = j, and Sij = 0 otherwise. The von Mises effective stress, ae, then becomes oe2 = | StjStj (7.4)

In similar fashion, the strain rate ¿ is a symmetric 3 x 3 tensor, with three direct components (S11, ¿22, ¿33) and three shear components (¿12, ¿23, ¿31). The volumetric strain rate is defined by

and the strain rate can be decomposed into its volumetric part sm and deviatoric part ¿ij according to

The strain rate ¿ can be written as the sum of an elastic strain rate ¿E and a plastic strain rate ¿P. In an analogous manner to equation (7.6a), the plastic strain rate can be decomposed into an deviatoric rate ¿P' and a mean rate

Now, for fully dense metallic solids, plastic flow occurs by slip with no change of volume, and so the volumetric plastic strain rate ¿m = ¿Pk equals zero. Then, a useful scalar measure of the degree of plastic straining is the effective strain rate ¿e, defined by

where the factor of | has been introduced so that ¿e equals the uniaxial plastic strain rate in a tension (or compression) test on an incompressible solid. In conventional Prandtl-Reuss J2 flow theory, the yield criterion is written

and the plastic strain rate ¿P, is normal to the yield surface \$ in stress space, and is given by

l] dOij

(This prescription for the plastic strain rate enforces it to be incompressible, that is, ¿m = 0.) The hardening rate is specified upon assuming that the effective strain rate ¿e scales with the effective stress rate oe according to

where the hardening modulus, h, is the slope of the uniaxial stress versus plastic strain curve at a uniaxial stress of level àe. In all the above, true measures of stress and strain are assumed.

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