13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- three-dimensional elastic frictional contact problems between the crack surfaces in the framework of XFEM. The structure of the paper is given as follows. Firstly, the variational formulation for frictional contact problem and the contact conditions are reviewed in Section 2. Secondly, the details of computation of discontinuities across contact surfaces and the equivalent nodal force due to the contact force acting at the contact surface are described in the framework of XFEM in Section 3. In Section 4, after the B- differentiable equations method for frictional contact problem is introduced we give the formulation and solution procedure of the combined XFEM and BDEM for frictional contact problem. Two numerical examples are presented in Section 5 for demonstrate the feasibility and accuracy of the proposed method. The concluding remarks are given in the last section. 2. Problem formulation 2.1. General description of the problem Consider a body Ω∈Rn, (n=2,3) embedded with two crack surfaces 1 cΓ and 2 cΓ which are also taken as the contact surfaces. We denote by Γ the outside boundary of Ω. Γ is composed by Γu on which prescribed displacements are imposed, Γσ on which prescribed tractions are imposed and the crack surfaces, i.e. Γ=Γu∪Γσ∪ 1 cΓ ∪ 2 cΓ . The crack surfaces may intersect the boundary Γ with points in 2D case or lines in 3D case. In the following sections, the variables with superscripts ‘1’ and ‘2’ indicate that they are related to 1 cΓ and 2 cΓ respectively. On the two crack surfaces, the displacements and tractions are denoted as , ( 1,2) = u t i i c i c respectively. It should be noted that since the contact surfaces are embedded in the element, they are assumed to be coincident initially, i.e., no initial gap exists between the contact surfaces. The quasi-static loading by a body force bf and given traction t on Γσ are assumed. The equilibrium equation and boundary conditions are described as follows. ( ) c f + = Ω Γ\ 0 in div b σ (1) 2 2 2 1 1 1 on on on on c c c c c c u Γ ⋅ = Γ ⋅ = Γ = Γ ⋅ = n t σ n t σ u u n t σ σ σ (2) Where, σ is the Cauchy stress tensor, uis the prescribed displacement vector on boundary uΓ . nσ is the unit normal vector to the boundary σΓ . i cn and i ct (i=1,2) are the unit normal vectors to the contact surface i cΓ and contact stresses on i cΓ respectively. 2.2. Frictional contact constraint formulation by B-differentiable equation According to the assumption of small displacement and small strain, a contact pair consists of the two points with the same coordinates on 1 cΓ and 2 cΓ , which are denoted as 1 cx and 2 cx respectively. A local coordinate system nab is established on 2 cΓ as shown in Figure 1. Therein, n is the normal vector and a and b are tangential vectors to the contact surface 2 cΓ .
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