13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- ( ) I I I I A u m m u ε = ⊗ + ⊗ 2 1 (13) The summation rule is implied in the above equation, and I m is ( ) IK IK IJ IJ I L L n n m + = 2 1 (14) where IJ L and IKL are the length of edges IJ and IK, IJ n and IK n denote the external normal of edges IJ and IK, respectively. Then, for an isotropic material, the strain energy in the element is reduced to ( ) ( ) ⎥⎦ ⎤ ⎢⎣ ⎡ + + − = ε ε U ε : : 2 21 1 2ν ν ν AE WD (15) in which E is Young’s modulus, ν is Poisson’s ratio. Substituting (13) into (15) we have ( ) ( ) ( ) ( )( )⎥⎦ ⎤ ⎢⎣ ⎡ + + + − = ⋅ ⋅ ⋅ ⋅ I J J I IJ J I I I D m A E W u m u u u m u m 2 1 2 2 4 1 ν ν ν (16) where J I IJ m m m⋅ = (17) From Equation (16), we can obtain the force acting on this element at node I J IJ I D I W K u u f ⋅ = ∂ ∂ = (18) where ( ) ⎥⎦ ⎤ ⎢⎣ ⎡ ⊗ + + ⊗ + − = I J IJ J I IJ m A E m m U m m K ν ν ν 1 2 2 2 1 (19) Here, IJ K is a second-order tensor that is called the stiffness matrix. The characteristics of the stiffness matrix IJ K compared with the traditional FEM are as: (1) This expression of stiffness matrix IJ K can easily be extended to apply for arbitrary polygonal elements problem in two dimensions or arbitrary polyhedral element problem in three dimensions. (2) The expression of the stiffness matrix IJ K is a precise expression, and it is not necessary to introduce the Gauss’ integral for calculating the stiffness coefficient at a point. (3) This expression of IJ K can be used for calculating the stiffness of various elements with a unified method. (4) This expression of stiffness matrix IJ K can be used in any coordinate system. (5) The method of constructing the stiffness matrix does not regulate the introduction of interpolation. The model of the base force element method will be used to analysis the damage problem for recycled aggregate concrete and be used to analysis the relationships of meso-structure and macroscopic mechanical performance of recycled concrete. 4. Random Aggregate Model for RAC Based on the Fuller grading curve, Walraven J.C et al (1981) put the three dimensional grading curve into the probability of any point which located in the sectional plane of specimens, and its expression as follow:
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