In Appendix E we have examined Drucker’s postulate for the stability of the material undergoing plastic deformations. To develop the plastic constitutive equations or the associated flow rule it is necessary to assume that a yield function exists, i. e.
f (ац) = k
As discussed in Chapter 3, when referred to the multi-dimensional stress space, the convex yieldfunction with a unique normal at each point identifies the boundary between elastic states in the material for which f (ai.) < k and plastic states for which f (ai.) = k. For the present purposes we shall restrict attention to non-strain hardening materials. Consider the inequalities given by (E.7) in relation to a vector space consisting of the stress tensor and the strain rate vector. The expression related to plastic energy dissipation rate can be visualised as the scalar product of two vectors (ai. — a-j) and eP. In order for the first inequality of (E.7) to be satisfied, the included angle between the vectors (aij — a.) and the plastic strain rate vector eP. should be acute. This condition will hold for any aj. located either within the yield surface or on the yield surface itself.
Consider the point B in Figure F.1, which is located on the yield surface f (aij) = k, and assume that the associated flow rule with the governing normality condition gives the plastic strain rate vector, which will therefore be normal to the yield surface. Now consider the tangent plane to the yield surface at this point. We are assured, by the convexity of the yield surface, that any stress point aij will lie to one side of this tangent plane. The line of action of the vector (ai. — a.) must therefore make an acute angle with eP.. Therefore for the rate of plastic energy dissipation to be positive definite we must have
Figure F.1. Geometrical representation of the stability postulate.
Other arguments for this form of the associated flow rule are also given by R. Hill and T. Y. Thomas.
The second inequality of (E.7) dealing with the requirement for material stability can be written as
= XT—- — 0 (R4)
The state of either loading, neutral loading or unloading of the plastic medium can be defined in relation to the direction of the stress increments in the following manner:
d j > 0; loading
— ffij = 0; neutral loading (F.5)
d<Tij < 0; unloading
In the case of neutral loading, the loading path follows the yield surface itself and the unloading process results in no plastic deformation. The condition (F.4) implies that for plastic deformations to occur the scalar multiplier must be non-negative; i. e.
X — 0 (F.6)
If the stress state satisfies the yield condition but with X = 0, then there is no plastic deformation. Also for a perfectly plastic material, there is no essential difference between the processes of loading and neutral loading since in the stress space (df /doij)dij = 0, whenever dij lies on the yield surface.
In the preceding we have focused on the application of Drucker’s stability postulate to the development of the associated flow rule for failure surfaces that have a unique normal at each point on the surface. Let us now focus attention on situations where the failure surface can have either edges or corners along which the orientation of the unit normal is not determined uniquely. Examples of such failure surfaces can include the Tresca and Coulomb failure criteria. We can extend the definition of the associated flow rule to cover such non-singular boundaries (Figure F.2). When considering non-singular failure surfaces, the associative flow rule should be modified to include several (say n) intersecting surfaces at a point. In such a case, the associated flow rule can be written as
where the derivatives dfa/doij are linearly independent in view of the fact that the failure
Figure F.2. Convex failure surface with non-singular points.
p df1 df2
Sf: = M— + Л2— = к
1 1 d*ij dffij
surfaces themselves are independent. The complete plastic strain rate is found according to (F.7) and the resultant of these plastic strain rates will be contained within the region obtained by surfaces spanning the unit normals to the segments of the yield surfaces, which intersect at the singular points. For example, referring to Figure F.2, where n = 2, the plastic strain rate at the corner C is given by
where 0 < ф < 1. It should be noted that although the direction of the plastic strain rate is not unique, the energy dissipation rate is uniquely determined, since for a given plastic strain rate the corresponding stress is unique. For example, considering a perfectly plastic material, which obeys the associated flow rule, the energy dissipation rate is given by
D = j (F.9)
If we realize that, at failure, the stresses are uniquely determined by the failure criterion, then the dissipation function can be expressed solely in terms of the plastic strain rate, i. e. D = D(sf ), and such a representation can be used to present an inverse form of the associated flow rule (F.3), where now the stresses can be expressed in terms of the strain rates.