On isoperimetric inequalities in plasticity
Hence, if P„) = 0 on each boundary, J vanishes for all admissible choices of Ta/>. Hence, by the converse theorem mentioned above, t'af is derivable from a single valued displacement. The same is then necessarily true of = ea0 + t'afi, and hence <raf, remains a solution for the stress when Poisson's ratio is changed. On the other hand, if P"' does not vanish on some boundaries, a suitable choice of ra(3 can always be made to render J non-zero. But this would necessarily imply that the strains e'a/j (and hence e*0) are not derivable from a single valued displacement, whence <ra0 would certainly not constitute a solution for the new material. The present theorem can be useful in simplifying the initial formulation of some problems. For example, the choice v = $ in conjunction with a total stress-strain law of plasticity permits the use of a single formula for the sum of the elastic and plastic components of strain; in other problems, the choice v = 0 might be more appropriate. In addition, the present theorem may conceivably have significance in connection with photoplasticity. Acknowledgment. The financial support of the Office of Naval Research under Contract Nonr 1866(02) is gratefully acknowledged.