13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- The differential of the ellipse area modelling the crack advance is: d d d A x y (18) differentiating the coordinates (x, y) according to the new coordinates (a, θ), d ()cos d sin d x b a θ a b θ θ - (19) d sin d cos d y θ a a θ θ + (20) and substituting these expressions on the eq. (18), it is obtained: 2 2 d ( )cos sin d d + A ab a θ b θ a θ (21) The problem that arises in calculating eq. (21) can be found in the previous knowledge of the variation of the parameter b with the crack depth a. The definition of the derivative at a point can be used to this purpose, ( Δ ) ( ) ( ) Δ b a a b a b a a + - (22) Introducing eq. (21) in eq. (13), that allows the computation of the compliance in a cracked round bar subjected to axial tensile loading, it is obtained: 2 π/2 2 2 2 4 0 acos 64 1- = ( )cos sin d d π a h b ν λ Y a ab a θ b θ θ a D E + (23) Introducing eq. (21) in eq. (14), which allows calculating compliance in a cracked round bar subjected to bending loading, it is obtained: 2 π/2 2 2 2 3 0 acos 4096 1- = ()cos + sin d d π a h b ν λ Y a ab a θ b θ θ a D E (24) where f is defined as the dimensionless compliance due to tensile or bending load: π/2 2 2 2 3 0 acos ( )cos sin d d a h b a f Y ab a θ b θ θ a D + (25) The dimensionless compliance value can be calculated incrementally with the crack growth, where the integral, i+1 i π/2 acos = d d a i h a b f R θ a (26) it is solved using the trapezoidal rule (where R is the corresponding expression according to eq. (25)), following the scheme on Fig. 4, dividing every crack increment in eight parts for half of the problem, so they correspond with the coordinate’s isolines (a, θ). (i,j) (i,j+1) (i+1,j) (i+1,j+1) a i+1 ai j j+1 Figure 4. Divisions with the isolines used in the trapezoidal rule
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