In this contribution, a crack propagation within brittle materials is presented by means of a novel technique, the virtual element method (VEM) [1]. In comparison to FEM, VEM allows usage of arbitrary polygonal (2D) and polyhedral (3D) elements. Such a method can be extremely useful for simulation of crack propagation problems, which may be providing an alternative approach for XFEM and remeshing. Based on the stress intensity factor, calculated by the interaction integral, the direction of the crack propagation is predicted by using the maximum circumferential stress criterion. For domain discretization, we used arbitrary polygonal elements with arbitrary number of nodes. Without changing the element formulations, the crack path is constructed by splitting the elements in the direction of growth. Also, if the crack growth stops within the element, this can be easily accomplished. Finally, a numerical example is provided to demonstrate the performance of the model.
Constitutive formulationConsider an elastic body Ω ⊂ R 2 bounded by Γ, which is subdivided into Neumann boundary conditions on Γ N , Dirichlet boundary conditions on Γ D and a traction-free discontinuity interface on Γ c . The body satisfy the equation of equilibriumwhere ∇ is the gradient operator and f the body force vector per unit volume. The boundary conditions are given bywhere n is the outward unit normal vector, u the displacement vector,ū the prescribed displacement on Γ D , and t the surface traction on Γ N . For a homogeneous isotropic linear elastic material the strain energy density function W can be expressed aswhere is the infinitesimal strain tensor, σ the Cauchy stress tensor, λ and µ are the Lamé constants. The variational formulation of the equilibrium equation (1) can be obtained by using the principle of stationary elastic potential
VEM formulationIn VEM, the displacement u consists of the sum of a projected displacement u H and a residual term (u − u H ). The projected displacement is defined by a polynomial function of degree m on each edge k of the virtual element Ω e . For the linear case m = 1 the function may be expressed in matrix form as
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