Issue 29

N.A. Nodargi et alii, Frattura ed Integrità Strutturale, 29 (2014) 111-127; DOI: 10.3221/IGF-ESIS.29.11 116 whence it is finite provided that   p p i 0 c D         ε ε  . In case of hardening material, recalling (25), the solution 0 i y q   of the supremum in (27) is unfeasible. Hence it can be assumed that (16) prevails, complemented with     p p 0 p i y 1 c D D           ε ε ε  (29) The same treatment holds if also kinematic hardening is present. STATE UPDATE ALGORITHM n this section, under the assumption of deviatoric yield function, we introduce a reduced formulation for the incremental minimization problem (8) and discuss the state update algorithm proposed for its solution. The following common choice for the free energy function e ( , )  ε α is made:       e e , W    ε α ε α  (30) where W and  , respectively, denote the elastic and hardening potentials, and may account for nonlinear elastic and/or hardening behavior. The contributions from kinematic and isotropic hardening are distinguished in the latter, as follows:       k k i i    α α    (31) As a consequence, substituting k  α [resp., i   ] from (11) [resp.(29)], the incremental energy problem (8) is transformed into:               0 p p e,trial p trial p trial p p 1 k k, 1 i i, 1 y 0 inf n n n W D D                     ε ε ε ε α ε ε ε    (32) Under isotropy assumption, the elastic potential is decomposed as:       e e e sph dev tr W W W   ε ε e (33) where sph W and dev W denote the spherical and deviatoric parts of W , e e denotes the deviatoric part of e ε , and tr denotes the trace operator. Moreover, without loss of generality, the kinematic hardening potential is assumed to depend only on the deviatoric part of k . α Therefore the incremental energy problem (32) is further reduced to:                   0 p e,trial e,trial p trial p trial p p sph 1 dev 1 k k, 1 i i, 1 y tr inf n n n n W W D D                    e ε e e α e ε e    (34) where p  e is the deviatoric part of p  ε . Recalling that the dissipation function   p D  e is singular at the origin p   e 0 , the state update algorithm first computes an elastic predictor. In case the corresponding state is plastically admissible, the step is elastic and the infimum in (34) is attained at the origin. Conversely, having excluded the singularity, a smooth incremental energy minimization problem can be solved. By introducing the linear projector operator on the deviatoric tensor subspace dev / 3    I I   , the Euler-Lagrange equation of (34) is:             0 0 e,trial p trial p trial p p dev dev 1 k k, 1 i i, 1 y y , 1 t n n n W D D                          e e α e e e 0    (35) where the gradients of dev W , k  , and D are taken, respectively, with respect to e e , k α , and p  e . Eq. (35) is solved by the Newton-Raphson method, using the consistent tangent:                     0 0 0 0 e,trial p trial p trial p p dev dev 1 k k, 1 i i, 1 y y trial p 2 p p i i, 1 y y dev 1 . t n n n n W D D D D D                                      e e α e e e e e e       (36) I