13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- can be found by resolving the following equation: 0 2 sin 2 1 cos 2 1 3 max ( ) 0 2 0 ⎟ ′ ′ ′ = ⎠ ⎞ ⎜ ⎝ ⎛ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ′ − = − ∫ ∫ − R Ic r c rdrd r K R r R f R π π θ θ θ π π σ (9) with ( ) ( ) θ θ θ θ θ ′ + ′ ′ ′ = ′ ′ + ′ = + ′ cos sin tan sin cos 0 2 2 0 r r r r r r r By using the non-local scheme (4) and with R calculated from Equation (9), the damage model (6) can directly be used to predict the crack growth. To this end, we just need to find the point ( ) 0 0 , θ r near the crack tip where the non-local principal stress is maximal: when the non-local stress attaints the material strength, the crack grows to the point( ) 0 0 , θ r according to the Griffith-Irwin criterion. The above-mentioned damage model was implemented into a Fast Fourier Transfer (FFT) code. The iterative method on the basis of FFT was originally proposed by for homogenizing linear and nonlinear composites [17]. The FFT-based formulation for a periodic heterogeneous cell with damage was developed according to the original FFT scheme [18]. Since an element in a structure is linearly elastic before its complete damage, the method of crack propagation evaluation used in this work is very similar to that adopted in linear fracture mechanics: An elastic calculation is first carried out for a cracked structure, and then crack progression and the corresponding remote load are determined according to the damage criterion (6). This procedure is then repeated after each small crack progression in the structure 4: Numerical examples and discussion In this section, we will carry out a series of numerical calculations in order to assess the performance of the present fracture model in predicting the crack evolution in brittle and quasi-brittle materials. First, we will verify its accuracy in predicting crack growth with a cell containing a central crack under pure mode-I loading. Then cells with more complicated microstructure will be considered in order to evaluate its efficiency and potentiality. 4.1: Cells with a central crack The first numerical example is a plane stress plate containing a central crack. In the numerical simulations, the dimension of the plate is 2h×2h =10×10mm2 with a central crack of different sizes, namely a = 0, 0.1, 0.3, 0.5, 1, 2, 3 and 4mm, here a = 0 represents a non-cracked plate. The stress intensity factors of such cracks can be found in any handbook of stress intensity factors. The mechanical properties of the material are: Young’s modulus E = 3000 MPa, Poisson’s ratio ν = 0.3, the ultimate stress σc = 72MPa, the critical stress intensity factor 36MPa mm = Ic K , the non-local action radius R=0.105mm and the radius of the damaged zone rd=0.05mm. In order to investigate
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