Abstract
This paper investigates the variational finite element formulation and its numerical implementation of the damage evolution in solids, using a new discrete embedded discontinuity approach. For this purpose, the kinematically optimal symmetric (KOS) formulation, which guarantees kinematics, is consistently derived. In this formulation, rigid body motion of the parts in which the element is divided is obtained. To guarantee equilibrium at the discontinuity surfaces, the length of the discontinuity is introduced in the numerical implementation at elemental level. To illustrate and validate this approach, two examples, involving mode-I failure, are presented. Numerical results are compared with those reported from experimental tests. The presented discontinuity formulation shows a robust finite element method to simulate the damage evolution processes in quasi-brittle materials, without modifying the mesh topology when cohesive cracks propagate.
Highlights
In the discrete modelling of fractures, the cohesive forces across a crack are modelled as a progressive degradation of the mechanical properties of the material (Figure 1), and a stress-displacement relationship is used [9]
All embedded discontinuity approaches have a common basic underlying; the process zone is incorporated into the variational formulation of the finite element model, by an enhancement of the relevant fields interpolation, which bears the required discontinuity only where it appears without affecting the main fields [4, 5, 18]
Enforcement of tractions equilibrium on the surface defined by equation (62) makes the kinematically optimal symmetric (KOS) formulation strongly dependent on the mesh configuration since the discontinuity must be parallel to one side of the element
Summary
Crossed by a discontinuity Γd, dividing the body domain Ω in two subdomains Ω+ and Ω−. For a two-dimensional case, these concepts can be properly generalized by introducing the vector ⟦u⟧ containing the components of jump displacements associated with the discontinuity Γd: u. For this case, the matrix HΓd is derived by a straightforward generalization of its counterpart in equations (6) and (7). E strain kinematic is derived, in a finite element framework, by means of the displacement field given by equation (14) as ε Bu + Bc⟦u⟧,. Equations (14) and (15) define the displacement and strain fields, respectively, corresponding to the discrete approach of the present embedded discontinuity formulation [14, 33]. Where n+ denotes the number of nodes belonging to the subdomain Ω+
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