A robust algorithm for configurational‐force‐driven brittle crack propagation with R‐adaptive mesh alignment

Abstract

SUMMARYThe paper considers a variational formulation of brittle fracture in elastic solids and proposes a numerical implementation by a finite element method. On the theoretical side, we outline a consistent thermodynamic framework for crack propagation in an elastic solid. It is shown that both the elastic equilibrium response as well as the local crack evolution follow in a natural format by exploitation of a global Clausius-Planck inequality in the sense of Coleman's method. Here, the canonical direction of the crack propagation associated with the classical Griffith criterion is the direction of the material configurational force which maximizes the local dissipation at the crack tip and minimizes the incremental energy release. On the numerical side, we exploit this variational structure in terms of crack-driving configurational forces. First, a standard finite element discretization in space yields a discrete formulation of the global dissipation in terms configurational nodal forces. As a consequence, the constitutive setting of crack propagation in the space-discretized finite element context is naturally related to discrete nodes of a typical finite element mesh. Next, consistent with the node-based setting, the discretization of the evolving crack discontinuity is performed by the doubling of critical nodes and interface segments of the mesh. Critical for the success of this procedure is its embedding into an r-adaptive crack-segment reorientation procedure with configurational-force-based directional indicator. Here, successive crack releases appear in discrete steps associated with the given space discretization. These are performed by a staggered loading-release algorithm of energy minimization at frozen crack state followed by the successive crack releases at frozen deformation. This constitutes a sequence of positive-definite discrete subproblems with successively decreasing overall stiffness, providing an extremely robust algorithmic setting in the postcritical range. We demonstrate the performance of the formulation by means of representative numerical simulations.

An isogeometric approach to cohesive zone modeling

“…The configurational force models for brittle fracture [18] are examples of such problems. In this contribution we have enhanced the solution space with discontinuities by means of knot insertion, but isogeometric analysis can also be used in combination with the partition of unity method.…”

“…A similar approach can be found in the context of configurational force models for fracture (e.g. [18]). …”

“…A Newton-Raphson solution procedure is then used to solve the non-linear system of equations (18). The integrals (19) and (20) are evaluated in the parameter domain using a Gaussian quadrature, as discussed in Ref.…”

Secret and joy of configurational mechanics: From foundations in continuum mechanics to applications in computational mechanics

“…Taking a material variation of the functional in Eq. (17) and observing the definition in Eq. (18) renders after integration by parts and application of the divergence theorem together with the constitutive relations in Eq.…”

“…Many recent impressive developments are in the application of concepts from configurational mechanics to computational mechanics, see, e.g., [17][18][19][20][21][22]31]. Our own contributions to the field are, e.g., [1,2,4,6,14,[23][24][25][26][27][28][29][30]32].…”