A Note on Equivalence of the Postulates of Isotropic Hardening

Theoretical plasticity of textured aggregates

“…6(e ij ) = BVLp{(o%e i ,)/<r, <j>{a%) = c}, (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12) by virtue of the inequality ".…”

“…Gotoh incorrectly assumes in advance that the function can only involve a like polynomial and duly finds its constniction impossible. This failure tempts him to announce non-existence in general (7), p. 512. and this can be inserted either in (2)(3)(4)(5)(6)(7)(8)(9)(10)(11) as the work-equivalent measure of strain-rate. It is reiterated that e need by no means govern the current rate of hardening.…”

“…Some obvious modifications are needed when plastic deformations are treated as volume-preserving, as is well justified for metals. The surface is now merely convex, in that expression (3 Since tensor fi ijkl is only semi-definite, it no longer has a unique inverse; in fact any 7a $ki can D e added to a ijkl without affecting (3)(4)(5)(6). This indeterminacy is henceforward removed by arbitrarily imposing ij «-» Id symmetry on a iJld .…”

“…From (3-2) we now have e u / e = {(? + h ) °"u -hcr 22 -gcr 33 }/(r, e i3 /e = l(T 23 /<r, etc., (3)(4)(5)(6)(7)(8) and, from (3-6) with the imposed symmetrization, 9 ) Finally, from (3-4) with a iikl read from (3-9), the work-equivalent measure is (fg + gh + hf) I " m i…”

“…Correspondingly, in assessments of (3-7), a and the quadratic coefficients are held to depend only on the one variable; often, indeed, the coefficient ratios are presumed to be completely independent of path, implying geometric similarity of the family of surfaces. However, such limitations are not inherent in (3)(4)(5)(6)(7) and (3)(4)(5)(6)(7)(8); the functional path-dependences of <r and the coefficients can in principle be arbitrarily complex. This was my position in 1948 (l), when (3)(4)(5)(6)(7) and (3)(4)(5)(6)(7)(8) were first proposed, and it remains so now.…”