Configurational forces in the virtual element method with applications to cracks

The Virtual Element Method (VEM) constitutes a generalization of the Finite Element Method, allowing for spatial discretizations with arbitrary, even non-convex polygonal elements. This offers significant advantages, particularly for problems involving moving discontinuities, local stress concentrations, or multiple phases with distinctly different dimensions. The theory of configurational forces (CF) represents a very flexible basis to quantify driving forces for virtual defect displacements within a thermodynamical framework, and at the same time provides an indicator for the local mesh quality in numerical calculations. In combination, the VEM and the CF theory thus constitute an effective basis for numerical simulations of crack growth within the context of a classical approach with strong discontinuities and sharp crack tips. Challenges to cope with are attributable to the lacking knowledge of shape functions within virtual elements, the numerical integration on arbitrary polygonal elements, and the requirement of an automated generation of polygonal meshes. The weak form of the energy-momentum balance is derived from a variational approach, uniquely separating nodal CFs acting at free surfaces from those driving the defect. Furthermore, internal degrees of freedom, which are inherent to higher-order virtual elements, have to be interpreted within the context of CFs. Unphysical, so-called spurious, CFs are exploited for an improved mesh design at crack tips, eventually generating accurate results of crack tip loading and crack path simulation.