13th International Conference on Fracture June 16–21, 2013, Beijing, China -6- 0 1 2 i c P c c r r = + + (13) where ( 0,1,2) ic i = are constants. The Mode I and Mode II stress intensity factors that will be used in the estimation of crack growth can be determined by applying a displacement correlation technique, which makes use of the nodal displacement at four locations A, B, E and D and the crack tip (Figure 3). Figure 3. The crack tip geometry and the node locations. The stress intensity factors are given by I II 0 4[ ( ) ( )] [ ( ) ( )] 2 4[ ( ) ( )] [ ( ) ( )] ( 1) u B u D u E u A K K u B u D u E u A k l η η η η ξ ξ ξ ξ μ π − + − ⎧ ⎫ ⎪ ⎬ = ⎨ − + − ⎭ + ⎪⎩ (14) where (3 4 ) k ν = − and 0l is the length of the crack tip element ξ and η are the local coordinates at the crack tip. 3.2. Modelling of Crack Extension The boundary element approach can be used to examine the crack extension during indentation. The stress state necessary to initiate crack nucleation can be obtained by using integral results for the stress state associated with the mixed boundary value problem defined by (1). The results presented by Harding and Sneddon [27] can be used for this purpose. The axisymmetric stress state is { } { } 0 0 1 2 2 0 1 1 1 0 1 1 2 4 ( , ) 2 4 4 ( ) ( , ) ( 2 ) ( 2 ) 4 ( , ) 2 zz rz r z J J a r z J J J a r z J a θθ μ λ μ σ ξ π λ μ λμ μ λ μ σ π λ μ ρ λ μ μ μ λ μ σ ξ π λ μ ⎛ ⎞ Δ + =− + ⎜ + ⎟ ⎝ ⎠ ⎧ ⎫ Δ Δ + =− − ⎨ − ⎬ + + ⎩ ⎭ ⎛ ⎞ Δ + =− ⎜ + ⎟ ⎝ ⎠ % (15) etc…., where ( 1) 0 2 2 1 2 2 2 2 2 2 2 sin( ) ( ) 1 ; tan ; ( 1) 4 2 tan ( 1) m n p n m J p p e J p dp r R ξ ρ ξ θ ξ ρ ξ ξ ξ φ ρ ξ ∞ − − − = = + = = + − + = + − ∫ % % % (16) ( ) mJ x is the Bessel function of the first kind of order mand the dimensionless coordinates are / r a ρ=% and / z a ξ= . The maximum local tensile stress within the elastic geomaterial, in the
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