Issue 29

N.A. Nodargi et alii, Frattura ed Integrità Strutturale, 29 (2014) 111-127; DOI: 10.3221/IGF-ESIS.29.11 117 A convenient initial guess is computed by substituting p p p ,      e e e u with deviatoric unit p  e u , in (34). Exploiting the degree-one positive homogeneity of the dissipation function, the Taylor expansion about p 0    e of the corresponding Euler-Lagrange condition results in a scalar algebraic equation for the Lode angle p   e u . Once computed p  e satisfying (35), the algorithmic consistent elastoplastic tangent is given by:   ep 1 sph dev dev dev dev t W W W         I I      (37) Remark. The present formulation, when specialized to particular choices of yield function and elastic/hardening potentials, gets simplified. As an example, we consider a linear elastic and hardening behavior with von Mises criterion. The Helmoltz free energy and the hardening potentials are given by: e e 2 e 2 2 2 k k k k i i i i 1 1 1 ( ) (tr ) G , ( ) , ( 2 ) 3 2 K H k H k        ε ε e α α     (38) the yield function and the relevant dissipation function are given by:     0 p p ˆ ˆ ˆ ˆ i y i y0 , ( ), , , f c q D q c             σ σ σ σ ε ε   (39) The Euler-Lagrange Eq. (35) becomes:       0 p e,trial p trial p trial p 1 k k, 1 i i, 1 y y0 p 2 2 1 3 n n n G k k c c                        e e e α e ε 0 e     (40) Introducing the deviatoric trial back-stress trial e,trial trial 1 1 k k, 1 2 2 3 , n n n G k      s e α  noting that the latter results to be parallel to p  e and multiplying (40) by p p   e e   , the following solution for p  e is obtained:   trial trial p p 1 trial y0 i i, 1 p p 1 trial k 1 , (2 2 3) n n n n c k c G k                     s ε e e s e s            (41) As a consequence, the well-known expression (see, e.g., [35]) for the algorithmic consistent elastoplastic tangent results: trial trial p p 1 1 ep dev trial trial trial trial 2 k i 1 1 1 1 2 1 2 2 2 2 2 3 n n n n n n G G G G G k c K k                                  e e s s I I s s s s                     (42) NUMERICAL TESTS n this section we assess the numerical reliability and robustness of the proposed state update algorithm. Its convergence properties are explored by a single increment test and compared with the performances of the classical return mapping approach. The numerical results prove that the two strategies have complementary convergence properties. Then, we consider the integration of a non-proportional stress-strain loading history of a material point, and finite element simulations. Single increment test In [17] Armero and Pérez-Foguet assess the converge properties of return mapping algorithms by using the following single increment test. Assuming the initial state to be virgin, a total strain ε (coinciding with the elastic trial strain e,trial ε ) is applied to the Gauss point. The total strain is defined in terms of its Haigh-Westergaard coordinates in the deviatoric plane, namely  ε and  ε . A von Mises–Tresca yield function with 20 m  (see Appendix B) is adopted and this choice corresponds to a yield surface with high curvature points close to °, 0 0 6 °   σ and low curvature in between. Because of the symmetry of the resulting yield surface, only Lode angle °, ° 0 0 ] [ 3   ε need to be considered. Isotropic linear elastic constitutive law, with bulk modulus 164.206 GPa K  and shear modulus 89.1938 GPa G  , and perfect plasticity, with yield stress y0 0.45 GPa   , are considered. I