13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- concrete in uniaxial tension test are simulated using the BFEM. The simulation result agrees with the test result. This research method is the new way for investigating fracture mechanism and numerical simulation of mechanics properties of recycled aggregate concrete. 2. Basic Equation Consider a two-dimensional domain of solid medium, let ( ) 1,2 =α αx denote the Lagrangian coordinate system, where P and Q the position vectors of a material point before and after deformation, respectively. Two triads for original and current configurations can be defined as: α α x∂ ∂ = P P , α α x∂ ∂ = Q Q , (1) Let u denotes the displacement of a point, then u Q P = − (2) The gradient of displacement αu can be written as: α α α α Q P u u = − ∂ ∂ = x (3) Then, the Green strain ε can be written as ) ( 2 1 i i i iu P P u ε = ⊗ + ⊗ (4) In order to describe the stress state at a point Q, a parallelogram with the edges 2 2 1 d ,d Q Q 1 x x is shown in Figure 1. Define, 0 , d d d 1 → = + α α α α x x T T (5) where 3=1 for indexes. Quantities ( 1,2) =α α T are called the base forces at point Q in the two-dimensional coordinate system αx . According to the definitions of various stress tensors, the relation between the base forces and various stress tensors can be given. The Cauchy stress is α α T Q σ = ⊗ QA 1 . (6) Figure 1. Base Forces on a plane element Further, the base forces are given as follows α α α ρ ρ u u T ∂ ∂ = ∂ ∂ = W A W A P Q 0 (7) in which W is the strain energy density, 0ρ is the mass density before deformation. Q 1x 2x 1 dx 2 dx 2 dT 1 dT 1Q 2Q
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