On stress-strain relations for isotropic rigid perfectly plastic solids

Abstract

Abstract.Sufficient conditions under which principal directions of stress and strain rate must coincide are established rigorously. It is the coincidence of these directions which permits a proper interpretation of principal strain rate components in principal stress space.Introduction. A rigid perfectly plastic solid is characterized by a "yield" or "limit" surface /(<r,-,) = 0 in stress component space which encloses allowable stress states. The strain rate tensor must vanish if <ju is inside the surface and, if strain rate component coordinates are superposed upon stress component coordinates, must be normal to the surface if a,,-is on the surface.If the yield surface is that of Mises (that is, if /(<^,) = h^'a^'a > where a',-is the stress deviator tensor) then normality requires that for some scalar X, ei; = X(3//3o-i,) = XaJ,-; coincidence of principal directions of stress and strain rate is then immediate since the principal axes of the stress and stress deviator tensors coincide.More generally, if / is a function of stress invariants which is differentiable in the components of cr,, then the principal directions of cr,,-and d//d<r,,-conicide; by normality, then, the axes of <r,-; and coincide. Alternatively , among others, have taken coincidence for the rigid perfectly plastic Tresca material on the premise of material isotropy but did not supply a proof. Neither of the results quoted above are applicable since the first requires a differentiable yield function and the second a unique correspondence between the tensors of stress and strain rate or their deviators. Moreover, coincidence does not hold for an isotropic frictional material [4], which obeys the verbal hypotheses upon which coincidence for the Tresca material is generally taken.In Sec. 1 a pictorial description of the rigid perfectly plastic Tresca material is given and it is explained why coincidence does not hold for the conceptually similar frictional material. In Sec. 2 it is shown that coincidence must hold for a rigid perfectly plastic material if the question of "whether or not a stress state represents yield" is to be independent of the relative orientation of material and principal stress directions.1. The Tresca and frictional materials. The relation between the tensors of stress and strain rate for the Tresca material is explained in Fig. 1(a) if the principal stress components are ordered as <Ji > cr2 > c3 . Deformation occurs by simple shearing in the direction of the shear stress vector T if the magnitude r of T reaches a critical value k. Since the planes of maximum shearing, called slip planes, are orthogonal, the simple