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51
For this claim he did not give an account to material symmetries in an invariant form, but
the claim is correct since all stress eigenvalues are uniquely determined by means of its
proper invariants. Accepting associate flow rule and using the identity T :߲T ݂ ൌ 1, we
arrive at evolution equations for plastic strain increments݀ߝ௉ ൌ ݀ߣ ߲T ݂ ൌ ߪത ݀ ߝ ҧ
௉߲T ݂ ൌ ߪത ܼ ߲ T ݂ ሺ߲T ݂: ݀Tሻ
or, equivalently ݂݀ ൌ ݀ ߣ ߲݂ ߲ߪ௜௝ ൌ ߪത݀ ߝҧ
௉ ߲݂ ߲ߪ௜௝ ൌ ߪത ܼ ߲݂ ߲ߪ௜௝ ߲݂ ߲ߪ ௞௟ ݀ ߪ௞௟ (22)
Here the use has been made of the following definitions݂݀ ൌ ߲݂ ߲ߪ௜௝ ݀ߪ௜௝ ൅ ߲݂ ߲ߝ௉ ݀ߝҧ
௉ ൌ (23)
݀መ ܹ௉ ൌ ߪ௜௝ ݀ߝ௜௝ ൌ ݀ ߪ ߣ௜௝ ߲݂ ߲ߪ௜௝ ൌ ݀ ߣ and ݀ߝ ҧ
௉ ׷ൌ ݀መ ܹ௉ ߪത (24)
for consistency condition, elementary plastic work and equivalent plastic strain
increment. At this point Hill's stability postulate, Ref. /19/, is invoked stating in its local
form that at a bifurcation point (Fig. 2) the following equality holdsTሶ ׷ ߝሶ
௉ ൌ ݎݐሼߝሶ
௉ · T·ߝሶ
௉ ሽ or ߪሶ ௜௝ߝ ሶ
௣௜௝ ൌ ߪ௜௝ߝ ሶ
௣௝௞ ߝ ሶ
௣௜௟ ߜ௞௟ (25)
for rigid plastic materials (i.e. materials where elastic strain is so smaller than plastic
strain that it may be neglected i.e. ߝ ൎ ߝ ௉). In above equality Tሶ is time rate of true stress,
whereas ߝሶ
௉ is time rate of Eulerian true plastic strain tensor. Combining above equations
we arrive at the following critical subtangent already written by Hill /29/ and by Ref. /21/
in terms of principal stresses.ܼு ൌ ݂݀ ݀ߪത ߪ௠௡ ߲݂ ߲
ߪ௠௡ ൬ߪ௜௝ ߲݂ ߲ߪ௜௞ ߲݂ ߲ߪ௝௟ ߜ௞௟ ൰ିଵ (26)
Equating this with the definition of subtangent from the “universal” hardening curveܼ ൌ ߪത ݀ߝҧ
௣ ݀ߪത (27)
we obtain limit strains ߝ ҧ
௨ (“uniform” strains) for diverse loading conditions. It should be
noted that from Eq. (26) there follows previous expression for Z. Obviously less general
expressions for uniaxial tension, equibiaxial tension etc. follow automatically reducing
the corresponding stress components to zero. However, we must not forget that critical
subtangent approach is restricted heavily by assumption on existence of universal curveߪത ൌ ߪതሺߝ ҧ
௉ሻ excluding from Eq. (25) mixed stress-plastic strain invariants responsible for
kinematic and more general types of hardening.
4. Korhonen, 1978, used for orthotropic materials from Hill's yield condition:2݂ ׷ൌ 1 ݄ሺߝ ҧሻ 2 3 ߪതଶ ൌ ܨ ሺߪଶ െ ߪଷሻଶ ൅ ܩሺߪଷ െ ߪଵሻଶ ൅ ܪሺߪଵ െ ߪଶሻଶ ൌ 1 ሺ28 ሻ
He further obtained the critical diffuse subtangent in the following form (with
identities ܴଵ ܩ ܪ ؠ⁄ , ܴଶ ܨ ܪ ؠ⁄ ) ܼு ൌ √2 √3 ሺܴଵ ൅ ܴଶ ൅ ܴଵܴଶሻଵ ଶ⁄ ሼ݉ଶሺ1 ൅ ܴଶሻܴଵ െ 2ܴ݉ଵܴଶ ൅ ሺ1 ൅ ܴଵሻܴଶሽଷ ଶ⁄
݉ଷ ܴଵଶሺ1 ൅ ܴଶሻଶ െ ݉ଶܴଵଶܴ ଶሺ2 ൅ ܴଶሻ െ ܴ݉ଶଶܴଵሺ2 ൅ ܴଵሻ ൅ ܴଶଶሺ1 ൅ ܴଵሻଶ (29)
for plane stress with ratio of principal stresses equals to ݉ ൌ ߪଶ ߪଵ ⁄ which with notations ܿ ؠ 1 ൅ 1 ܴଵ, ܾ ؠ 1 ൅ 1 ܴଶ, ⁄ ⁄ may be written in the following more compact form