Issue 29

N.A. Nodargi et alii, Frattura ed Integrità Strutturale, 29 (2014) 111-127; DOI: 10.3221/IGF-ESIS.29.11 120 Stretching of a perforated rectangular plate A classical benchmark for the implementation of plasticity models is the stretching of a thin perforated plate along its longituindal axis (see [10], for instance). This problem is solved under the usual assumption of plane stress, which is straightforwardly imposed within the present formulation. To this end, it suffices to minimize the incremental energy functional in (34) also with respect to the total strain component normal to the stress plane. The plate is 20 mm wide, 36 mm high, 1 mm thick and presents a circular hole of radius 10 mm at its center. The geometry and boundary conditions are depicted in Fig. 3(a). The material is supposed to be isotropic linear elastic, with Young modulus 70 GPa E  and Poisson coefficient 0.2   . A von Mises yield criterion (see Appendix B), with yield stress y0 0.243 GPa   and linear isotropic hardening of modulus i 0.2 GPa H  , is adopted. Because of the symmetry, only one quarter of the plate is considered in the analysis by applying the appropriate boundary condition on the symmetry edges. The finite element mesh adopted is constituted by 576 OPT triangular elements (see [38], for instance), with a total number of 325 nodes. The analysis is conducted under displacement control, assigning a vertical displacement on the top constrained edge. In Fig. 3(b) the diagram of the total reaction force versus the prescribed displacement is reported. The numerical solution is in good agreement with the reference one [10]. In Fig. 3(c) the evolution of the accumulated equivalent plastic strain p p 0 2 3 T dt   ε ε  corresponding to different load levels is shown. Figure 3 : Pinching of a cylindrical shell with diaphragms. (a) Geometry, boundary conditions and finite element mesh; (b) Deformed configuration; (c) Comparison of load-displacement diagram between the proposed algorithm and reference solution (see [39]). Pinching of a cylindrical shell with diaphragms In this section we consider the application of the proposed state update algorithm to a geometrically nonlinear problem. In particular, we refer to the problem of a cylindrical shell, constrained by rigid diaphragms at end-sections, and pinched by concentrated forces in the mid-section [39]. The cylinder has radius of 300 consistent units (c.u.), while its total length is of 600 c.u. The initial shell thickness is 3 c.u. Fig. 4(a) shows the geometry and boundary conditions (we assume that the presence of rigid diaphragms constrains the in-plane degrees of freedom of the cylinder end sections). We assume the material to be isotropic linear elastic, with Young modulus 3000 E  c.u. and Poisson coefficient 0.3   . A von Mises yield criterion (see Appendix B), with yield stress y0 24.3   c.u. and linear isotropic hardening of modulus i 300 H  c.u., is adopted, under plane-stress assumption. Due to symmetry, only one-eighth of the cylinder is considered in the analysis by applying appropriate boundary conditions on the symmetry edges. The geometrical nonlinearity is treated by a corotational approach [40, 41], and the