3D‐FEM formulations of limit analysis methods for porous pressure‐sensitive materials

Abstract

SUMMARYThe first purpose of this paper is the numerical formulation of the three general limit analysis methods for problems involving pressure‐sensitive materials, that is, the static, classic, and mixed kinematic methods applied to problems with Drucker–Prager, Mises–Schleicher, or Green materials. In each case, quadratic or rotated quadratic cone programming is considered to solve the final optimization problems, leading to original and efficient numerical formulations. As a second purpose, the resulting codes are applied to non‐classic 3D problems, that is, the Gurson‐like hollow sphere problem with these materials as matrices. To this end are first presented the 3D finite element implementations of the static and kinematic classic methods of limit analysis together with a mixed method formulated to give also a purely kinematic result. Discontinuous stress and velocity fields are included in the analysis. The static and the two kinematic approaches are compared afterwards in the hydrostatic loading case whose exact solution is known for the three cases of matrix. Then, the static and the mixed approaches are used to assess the available approximate criteria for porous Drucker–Prager, Mises–Schleicher, and Green materials. Copyright © 2013 John Wiley & Sons, Ltd.

Static limit analysis and strength of porous solids with hill orthotropic matrix

“…The FE computations were conducted on hollow sphere subjected at its external boundary to homogeneous strain rate conditions. As in the theoretical model, axisymmetric strain rate loadings are considered such that: (30) and eq (37) and that proposed by Benzerga et al [4] dicted envelope remains inside the yield envelopes obtained from the numerical FE results, as expected. However, one can note that the criterion obtained by [4], by using a kinematic approach provides closer upper bounds when compared to FE results.…”

“…Those bounds are obtained using respectively static and kinematic FE approaches and based on the solution to the corresponding conic programming problems 10 . In Fig 4 the yield surfaces projected in the plane (Σ 11 − Σ 33 , Σ m ) corresponding to the established criteria (30) and (37) are compared to the bounds given by the static and kinematic codes from [38]. In all quadrants, the theoretical criteria predict a macroscopic yield envelope inside the numerical lower bound.…”

“…In summary, the assessment of the proposed model can be considered successful; 10 the readers interested by this original numerical approach and its application to various porous materials can refer to [37]. (37)and eq (30) and by FE [35] 20 moreover it must be emphasized that, to the best of our knowledge, it is the first criterion which accounts for the effect of the sign of J 3 in a context of ductile porous materials with an anisotropic matrix.…”

“…(37)and eq (30) and by FE [35] 20 moreover it must be emphasized that, to the best of our knowledge, it is the first criterion which accounts for the effect of the sign of J 3 in a context of ductile porous materials with an anisotropic matrix.…”

Numerical limit analysis and plasticity criterion of a porous Coulomb material with elliptic cylindrical voids

“…Due to the bad conditioning of the classic kinematic approach experienced for porous Drucker-Prager materials in [18] (where the usual allowed number of elements was 672...), we have elaborated a new mixed kinematic approach based on affine velocity fields in the triangle elements (Lagrange P1). Another motivation to do this was induced by the high level of performance allowed by the mixed approach for the hollow spheroid problems investigated in [22] and [23]. Such level of performance will be verified in the following tests, resulting in lower and upper bounds almost undistinguishable (a posteriori verified) obtained in a couple of seconds for each point on a laptop computer.…”