Issue 29

N.A. Nodargi et alii, Frattura ed Integrità Strutturale, 29 (2014) 111-127; DOI: 10.3221/IGF-ESIS.29.11 118 Fig. 1 compares the performances of the return mapping and incremental energy minimization algorithms in terms of number of iterations needed for convergence. In Fig. 1(a) it is clear that when the elastic trial state in stress space is close to high curvature points of the yield surface, i.e. for 0      σ ε , the return mapping algorithm has convergence difficulty. As depicted in Fig. 1(b), for the same elastic trial state the proposed state update algorithm needs 1-3 iterations to reach convergence. A converse situation happens around 30      σ ε . This result is explained observing that high curvature points in the yield surface correspond to low curvature points in the graph of the dissipation function, and vice- versa (see Appendix B). (a) (b) Figure 1 : Single increment test. Number of iterations needed for convergence for Von Mises–Tresca yield surface ( m =20) by: (a) return mapping algorithm [17]; (b) proposed incremental energy minimization algorithm. Pointwise mixed stress-strain test In order to explore the algorithm effectiveness, we consider a pointwise mixed stress-strain test as reported in [36]. In particular, a non-proportional stress-strain history is adopted. The two controlled strain components, x  and xy  , follow the loading history described in Tab. 1, whereas the four controlled stress components, y  , z  , xz  , yz  , are held identically equal to zero. The strains are varied proportionally to the first yielding value in a uniaxial loading history y0  . Isotropic linear elastic constitutive law with Young modulus 100 MPa E  and Poisson coefficient 0.3   is assumed. We consider a von Mises yield function (see Appendix B) characterized by yield stress y0 15 MPa   , linear kinematic and isotropic with hardening modulus k 10 MPa H  and i 10 MPa H  respectively. In Fig. 2 the numerical solution, expressed in terms of the stress-strain history, proves to be in perfect agreement with the analytical one (for example, see [35]). Time unit 0 1 2 3 4 5 6 7 x y0 /   0 5 5 -5 -5 5 5 0 xy y0 /   0 0 5 5 -5 -5 0 0 Table 1 : Pointwise mixed stress-strain test. Controlled strain history: in between two next time units, strain components vary linearly.