Issue 29

N.A. Nodargi et alii, Frattura ed Integrità Strutturale, 29 (2014) 111-127; DOI: 10.3221/IGF-ESIS.29.11 114 whence it follows that p k    α ε (11) and         i i p p i i i ˆ , ˆ , 0 ˆ , sup q f q D q           σ σ ε σ ε (12) With the assumption that the yield function is isotropic, it can be represented, with a slight abuse of notation, as   ˆ ˆ ˆ i , , , f q    σ σ σ , where ˆ  σ , ˆ 0   σ ,   ˆ 0, 3    σ are the Haigh-Westergaard coordinates of ˆ σ (see Appendix A). Since the maximum of the scalar product of two symmetric tensors is attained when they are coaxial (see, e.g., [34]), in (12) it is possible to assume that ˆ σ is coaxial with p  ε and to express their scalar product making use of the Haigh-Westergaard representation. Accordingly, the dissipation function is given by         p p p ˆ ˆ ˆ i ˆ ˆ ˆ i p ˆ ˆ ˆ i i i , , , , , , 0 , sup cos( ) q f q D q                          σ σ σ σ σ σ σ σ σ ε ε ε ε (13) where p   ε , p 0    ε ,   p 0, 3     ε are the Haigh-Westergaard coordinates of p  ε . Deviatoric yield function, perfect plasticity or kinematic hardening The yield function is assumed to be of the form     0 ˆ ˆ ˆ ˆ ˆ y , , f g c         σ σ σ σ σ (14) where g is a 2 C , positive, even, periodic function with period 2 3  , such that 0 g g    . The latter condition guarantees that curves of polar equation   ˆ ˆ const g    σ σ are convex. Moreover, 0 y  and c are positive constants; usually, 2 3 c  : in that case, 0 y  can be interpreted as the initial yield limit in tension if   0 1 g  , or in compression if   3 1 g   . According to (14), the dissipation function is given by           p p p ˆ ˆ ˆ ˆ ˆ y 0 p ˆ ˆ ˆ , , sup cos g c D                     σ σ σ σ σ σ σ σ ε ε ε ε (15) This implies:     p p p 0 p y 0, D c D           ε ε ε ε  (16) where         p p ˆ ˆ ˆ ˆ ˆ ˆ { , } 1 sup cos g D              σ σ σ σ σ σ ε ε  (17) or, equivalently,         p p ˆ ˆ ˆ cos sup D g                    σ σ ε ε σ  (18) It is worth noting that D  is the support function of the convex set of polar coordinates       ˆ ˆ ˆ ˆ , : 1 g      σ σ σ σ . Because the supremum is attained on the boundary of that set, a simple computation yields         p p ˆ ˆ ˆ ˆ sin cos 0 g g              σ σ σ σ ε ε (19) which can be recast as: