13th International Conference on Fracture June 16–21, 2013, Beijing, China -7- hydraulic fracture [18]. 4.2 Element implementation As shown in Figure 3, the two-dimensional 4-node plane strain channel and tip elements have been constructed for the hydraulic fracture problem. Each node has the standard displacement degrees of freedom Iu . The additional degree of freedom Ia and l Ib are assigned to the four nodes of channel and tip elements, respectively. In addition, the virtual degree of freedom of fluid pressure has been assigned to nodes 3 and 4 so as to represent the internal fluid pressure within the crack. It should be pointed out that nodes 3 and 4 physically do not have fluid pressure degrees of freedom because here the fluid flow is confined within the crack. So the integral calculation of the related element matrixes and equivalent nodal forces (e.g. Eq. (24)) must be correctly carried out along the true crack path within the element. Figure 3. 2-D 4-node plane strain hydraulic fracture elements So, the active degrees of freedom for the channel element are 1 1 2 2 3 3 4 4 Standard T 1 1 2 2 3 3 4 4 3 4 18 Heaviside Enriched Coupled ˆ x y x y x y x y x y x y x y x y u u u u u u u u aaaaaaaapp eu (31) and for the tip element the Heaviside enriched degrees of freedom Ia need to be replaced by the crack tip field enriched degrees of freedom l Ib . According to the element connectivity and the arrangement of nodal degrees of freedom, the bilinear shape functions are used to approximate the displacement field, and the linear shaped functions are used to approximate the fluid pressure field. Gauss quadrature is used to calculate the system matrix and equivalent nodal force. Since the discontinuous enrichment functions are introduced in approximating the displacement field, integration of discontinuous functions is needed when computing the element stiffness matrix and equivalent nodal force. In order to ensure the integral accuracy, it is necessary to modify the quadrature routine. Both the channel and tip elements are partitioned by the crack surface into two quadrature sub-cells where the integrands are continuous and differentiable. Then Gauss integration is carried out by a loop over the sub-cells to obtain an accurate integration result. Due to the flexibility, the user subroutine of UEL provided in the finite element package ABAQUS [20] has been employed in implementing the proposed elements in Fortran code. The main purpose of UEL is to provide the element stiffness matrix as well as the right hand side residual vector, as need in a context of solving the discrete system of equations. 4 1 2 3 Crack Channel element 1 2 3 4 r θ CrackTip Tip element
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