ABSTRACT
The theory of plasticity describes the relation between load and deformation for many metals at higher stresses. The theory is rate independent: the same strains and stresses are obtained if the same loads are applied more rapidly (as long as inertial forces are negligible). The role of time in the constitutive equations is merely as a convenient bookkeeping parameter, and the occurrence of time derivatives does not indicate that such things as creep or relaxation can occur. The following is a summary of the basic equations.*
The material is initially elastic:
τ ε ε τ( ) ( ) , ( ) ( ) .t c t t c tkm kmrs rs km kmrs rs= = −1
(8.1)
For an isotropic material,
cijkm = μ(δikδjm + δimδjk) + λδijδkm (8.2)
and
c
=
+ + −1
1 2
(8.3)
When the stress-or equivalently, the strain-reaches a critical magnitude, the material yields and incremental deformations may be inelastic. If the magnitude of the stress then decreases, the incremental deformations are elastic. If the magnitude of the stress increases, the magnitude of the strain increases with apparently reduced moduli in a rate-independent manner. The additional strain,
ε ε τP( ) ( ) ( )t t c tij ij ijkm km= −
(8.4)
is called the plastic strain. Experiments on metals show that the volume change is elastic to a high degree of approximation, so that
= 0, (8.5)
and we will consider only this case. Typical criteria for yielding have the form
f(τ(t),εP(t),β(t),κ(t)) ≤ 0 (8.6)
where the tensor β is called the back stress and the scalar κ is called the hardening parameter. Both parameters depend on the history of plastic strain. The yield function f is normalized so that it is negative before yield, and yield occurs when f = 0. For fixed εP, β, and κ, the condition f = 0 defines a surface in stress space, which is called the yield surface. Experimental evidence shows that this surface is closed and convex, and we will also assume that it is smooth. The tensor
f =
∂ ∂
(8.7)
is an outward normal to the yield surface. Suppose the material is at the point of yield so that f(t) = 0. An increment of stress inward from the yield surface reduces the magnitude of f and so an elastic increment in strain occurs. An increment in stress that is outward from the surface expands the yield surface and plastic strains occur. Let us define
fˆ ij ij= η τ .(8.8)
At yield, f = 0, we have three possibilities:
(1) Elastic unloading: fˆ < 0 and therefore εP = 0. (2) Plastic loading: fˆ > 0 and therefore εP ≠ 0. (3) A neutral process: fˆ = 0 and εP = 0.
