An isogeometric analysis approach to gradient‐dependent plasticity

Abstract

Summary Gradient‐dependent plasticity can be used to achieve mesh‐objective results upon loss of well‐posedness of the initial/boundary value problem because of the introduction of strain softening, non‐associated flow, and geometric nonlinearity. A prominent class of gradient plasticity models considers a dependence of the yield strength on the Laplacian of the hardening parameter, usually an invariant of the plastic strain tensor. This inclusion causes the consistency condition to become a partial differential equation, in addition to the momentum balance. At the internal moving boundary, one has to impose appropriate boundary conditions on the hardening parameter or, equivalently, on the plastic multiplier. This internal boundary condition can be enforced without tracking the elastic‐plastic boundary by requiring C1‐continuity with respect to the plastic multiplier. In this contribution, this continuity has been achieved by using nonuniform rational B‐splines as shape functions both for the plastic multiplier and for the displacements. One advantage of this isogeometric analysis approach is that the displacements can be interpolated one order higher, making it consistent with the interpolation of the plastic multiplier. This is different from previous approaches, which have been exploited. The regularising effect of gradient plasticity is shown for 1‐ and 2‐dimensional boundary value problems.

Dispersion and isogeometric analyses of second‐order and fourth‐order implicit gradient‐enhanced plasticity models

“…Figures and indicate that the localisation zone first propagates along the vertical boundary before evolving into a shear band. This is consistent with earlier calculations for the explicit and implicit gradient plasticity models using standard finite elements, meshless methods, and isogeometric analysis . The angle of the shear band is lower than 45 o , which is the theoretical solution for a shear band when using a Tresca plasticity model.…”

“…This is consistent with earlier calculations for the explicit and implicit gradient plasticity models using standard finite elements, 8 meshless methods, 13 and isogeometric analysis. 14 The angle of the shear band is lower than 45 o , which is the theoretical solution for a shear band when using a Tresca plasticity model. Unlike for the Tresca yield function, the intermediate principal stress enters the von Mises yield condition, and this results in a different condition for the onset of localisation, including the angle of shear bands (cf the work of Rudnicki and Rice 35 ).…”

“…This degree of continuity is difficult to satisfy using finite element approaches, and alternative formulations have been put forward. ()…”

“…It is noted that for the second‐order formulation, the load‐displacement curve (after the cusp) has not fully converged with 200 elements. This is because the smallest element size (1 mm) is larger than the length scale considered . However, the fourth‐order formulation shows convergence, which may suggest a stronger regularisation property for the fourth‐order formulation.…”

“…3 It has been attempted to satisfy this requirement using Hermitian finite elements, 12 using meshless methods 13 and, recently, isogeometric analysis. 14 The ability of this explicit gradient plasticity model, in which the yield stress is made a function of the Laplacian of the accumulated plastic strain in addition to the plastic strain itself, to fully regularise the boundary value problem has been demonstrated through 1-dimensional dispersion analyses, 15 spectral analyses, [16][17][18] and shear band simulations. 12,13 On the other hand, implicit gradient plasticity models with second-order gradients do not fully regularise the boundary value problem, as has been demonstrated through spectral analyses 16,17,19 and 3-dimensional simulations.…”