Issue 29

N.A. Nodargi et alii, Frattura ed Integrità Strutturale, 29 (2014) 111-127; DOI: 10.3221/IGF-ESIS.29.11 115     p ˆ ˆ ˆ arctan g g         σ σ ε σ (20) This is a scalar equation in the unknown   ˆ 0, 3    σ , which implicitly defines a function   p ˆ    σ ε . By applying Dini’s theorem, its first derivative turns out to be:   p 2 2 ˆ d d g g g g g       σ ε (21) where g and its derivatives are computed at   p ˆ    σ ε . Noting that p ˆ     σ ε can be assumed to belong to   3, 3    , by (20) it follows that     p p ˆ ˆ 2 2 2 2 cos , sin g g g g g g                σ σ ε ε (22) whence, recalling (18) and (21), p ( ) D     and its first and second derivatives with respect to p    easily follow:         p p p 4 2 2 3 2 2 2 2 3 2 2 2 1 , , g g g g g g D D D g g g g g g g g g g                           ε ε ε    (23) where the derivatives on the right-hand sides of (23) are performed with respect to ˆ  σ and computed at   p ˆ    σ ε . Finally the gradient and the Hessian of   p D  ε are straightforwardly obtained from (16):   p p p 0 p p p p p p p p p p 0 y y D c D D D c D D D D                                                            ε ε ε ε ε ε ε ε ε ε ε ε        (24) Simple expressions for the gradient and the Hessian of p   ε and p    are given in Appendix A. The same argument holds if kinematic hardening is present. Deviatoric yield function, isotropic hardening The yield function is assumed to be       0 ˆ ˆ ˆ ˆ ˆ i y i , , , f q g c q          σ σ σ σ σ (25) whence the dissipation function follows:             p p p ˆ ˆ ˆ i ˆ ˆ y i 0 p ˆ ˆ ˆ i i i , , , , sup cos q g c q q D σ σ σ σ σ σ σ σ ε ε ε ε                           (26) This implies:           p p p 0 i y 0 p i y i i i 0, , sup q D c q D q                   ε ε ε ε  (27) Accordingly, the dissipation function turns out to be:         p p 0 p p p p 0 y i i p i y i if , if otherwise c D D c D c D                                ε ε ε ε ε ε ε    (28)