13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- Equation (7) expresses the α T by strain energy directly. Thus, αu is just the conjugate variable of α T . It can be seen that the mechanics problem can be completely established by means of α T and αu . 3. Model of BFEM with Triangular Element We will derive explicit expressions for stiffness matrices of a triangular element now, base on the concept of “base forces”. Consider a triangular element with boundary S as shown in Figure 2. Figure 2. A triangular element For the small displacement case, the real strain ε can be replaced by ε . We can obtain the average stress in element as A AA d 1 ∫ = ε ε (8) in which A is the area of element. Substituting Equation (4) into Equation (8), we have ( ) A AA d 2 1 ∫ = ⊗ + ⊗ α α α αu P P u ε (9) Using Green’ theorem, Equation (9) becomes ( ) s AS d 2 1 ∫ = ⊗ + ⊗ u n n u ε (10) where n is the current normal of boundary S. When the element is small enough, Equation (10) can be written as ( ) ∑ = ⊗ + ⊗ = 3 1 2 1 i i i i i iL A u n n u ε (11) where iL is the length of edge ( ) 1,2,3 = i i , in denotes the external normal of edge ( ) 1,2,3 = i i , iu is the displacement of geometric center of edge ( ) 1,2,3 = i i . Further, we assume that any edge of the triangular element in the deformation process keeps its edges straight lines. Then, we can obtain the following expression for iu : ( ) J I iu u u = + 2 1 (12) where Iu and Ju denote the displacements of both ends of edge ( ) 1,2,3 = i i , respectively. Substituting Equation (12) into Equation (11) yields K I J Iu Ku Ju
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