An energy‐conserving scheme for dynamic crack growth using the eXtended finite element method

Abstract

SUMMARYThis paper proposes a generalization of the eXtended finite element method (X-FEM) to model dynamic fracture and time-dependent problems from a more general point of view, and gives a proof of the stability of the numerical scheme in the linear case. First, we study the stability conditions of Newmark-type schemes for problems with evolving discretizations. We prove that the proposed enrichment strategy satisfies these conditions and also ensures energy conservation. Using this approach, as the crack propagates, the enrichment can evolve with no occurrence of instability or uncontrolled energy transfer. Then, we present a technique based on Lagrangian conservation for the estimation of dynamic stress intensity factors for arbitrary 2D cracks. The results presented for several applications are accurate for stationary or moving cracks.

Direct estimation of generalized stress intensity factors using a three‐scale concurrent multigrid X‐FEM

“…Another possible extension of this work is the simulation of dynamic crack propagation. Dynamic stress factors will be extracted in the same way that the static case, up to a multiplicative factor that depends on the crack velocity (see [30]). We will need to define a projection to transfer kinematic fields from patch to the X-FEM grid (and vice-versa) during the propagation.…”

An explicit dynamics extended finite element method. Part 1: Mass lumping for arbitrary enrichment functions

“…In our approach, we follow the same strategy as in Réthoré et al [34] and Prabel et al [30], that is we enrich all the mechanical fields (displacement, velocity, acceleration). The discrete problem can therefore be written as a classical finite element dynamic problem given in Eqs.…”

“…Réthoré et al [34] proved that to achieve energy conservation and numerical stability when the crack is growing, the following strategy should be adopted. Once the criteria for crack growth is met, all the "old" enrichments are retained and new ones, associated to the crack extension, are introduced.…”

“…Belytschko et al [3] were the first to point the problem of possibly null critical time step with X-FEM and therefore used the implicitexplicit element by element scheme of Hughes and Liu [17]. Réthoré et al [33,34] and Prabel et al [30] circumvent this problem by using an implicit time integrator with a specific enrichment strategy to ensure energy conservation. De Borst et al [8] and Remmers et al [32] also experienced the critical time step problem with discontinuous enrichment.…”

Stability of incompressible formulations enriched with X-FEM

“…Proper enrichment of the finite element basis makes it possible to model crack, material inclusions and holes with non-conforming meshes. The X-FEM method has been used for the simulation of a wide variety of problems such as fracture mechanics problems (2D [18][19][20], 3D [21][22][23], plates [24,25], cohesive zone modeling [26,27], dynamic fracture [28], nonlinear fracture mechanics [29][30][31]), holes [32,33], but also material inclusions [33,34] or multiple phase flows [35]. Here, we focus on the application of this method to mixed formulations for the treatment of holes, material inclusions and cracks in the incompressible limit.…”