### A deformation bounding theorem for flow-law plasticity

Philip G. Hodge
1966 Quarterly of Applied Mathematics
It is shown that the elastic principle of minimum complementary energy can be generalized to provide a bounding theorem for an elastic/perfectly-plastic material. For certain special classes of problems the bounding principle becomes a true minimum principle. 1. Introduction. A minimum theorem in continuum mechanics proves that among all members of a certain class, the complete solution to a given boundary value problem minimizes a certain functional. A bounding theorem, on the other hand,
more » ... s that the value of the functional associated with any member of a given class is not less than the value associated with the complete solution. Obviously, all minimum theorems are bounding theorems, but the converse is true if and only if the complete solution is a member of a given class. The principles of minimum potential or complementary energy in elasticity and the theorems of limit analysis in plasticity are examples of minimum theorems, whereas recent work by Martin1 has used a class of static solutions in proving bounding theorems for dynamic problems. The present paper proves a bounding theorem for a functional of the total deformation of an elastic/perfectly-plastic material with a flow-law type of constitutive equation. We consider the usual type of boundary value problem in which we are given a region V bounded by a surface S. The entire surface is assumed to be composed of two types of sub-surfaces: SD on which either the displacement vector u is prescribed or the traction T vanishes, and ST on which either T is prescribed or u = 0. In addition the body force f is given throughout V. If the material in V is elastic, then for any stress state <i we may define the complementary energy density U.(*) = J *(4)-d6, and the total complementary energy by n. = f Ue -f T u. ( J v J sd The well-known theorem of minimum complementary energy then states that among all equilibrium states of stress the actual one minimizes Uc . For an elastic/perfectly-plastic material, the theorem is, in general, not valid. The purpose of the present note is to point out that by placing certain additional re-*