PSI - Issue 2_A

Stepanova Larisa et al. / Procedia Structural Integrity 2 (2016) 793–800 Author name / Structural Integrity Procedia 00 (2016) 000–000

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3

) / , m

(

e A ψ σ ψ = − & (4) where ψ & denotes the time derivative, while A and m are material constants. The solution of Eqs. 1 – 4 should satisfy the traditional traction free boundary conditions on the crack surfaces ( ) ( ) , 0, , 0. r r r θθ θ σ θ π σ θ π = ± = = ± = The mixed-mode loading can be characterized in terms of the mixity parameter p M which is defined as ( ) ( ) ( ) 0 2 / arctan lim , 0 / , 0 . p r r M r r θθ θ π σ θ σ θ → = = = The mixity parameter p M equals 0 for pure mode II; 1 for pure mode I, and 0 1 p M < < for different mixities of modes I and II. Thus, for combine-mode fracture the mixity parameter p M completely specifies the near-crack-tip fields for a given value of the creep exponent. By postulating the Airy stress function ( ) , r χ θ expressed in the polar coordinate system, the stress components state are expressed as: , rr θθ σ χ = , 2 , / , / rr r r r θθ σ χ χ = + , ( ) , / , r r r θ θ σ χ = − . As for the asymptotic stress field at the crack tip 0 r → , one can postulate the Airy stress function and the continuity parameter as ( ) ( ) 1 1 0 0 , ( ), , 1 ( ) j j j j j j r r f r r g λ γ χ θ θ ψ θ θ ∞ ∞ + + = = = = − ∑ ∑ . (5) First consider the leading terms of the asymptotic expansions (5): 1 ( , ) ( ) r r f λ χ θ θ + = , 1 ψ = , where λ is indeterminate exponent and ( ) f θ is an indeterminate function of the polar angle, respectively. In view of the asymptotic presentation for the Airy stress potential (5) the asymptotic stress field at the crack tip is derived as follows 1 ( , ) ( ) ij ij r r λ σ θ σ θ − = % , where 1 λ − denotes the exponent representing the singularity of the stress field, and will be called the stress singularity exponent hereafter. According to Eq. 2 the asymptotic strain field as 0 r → takes the form ( 1) ( , ) ( ). n ij ij r Br λ ε θ ε θ − = % The compatibility condition in Eq. 1 results in the nonlinear forth-order ordinary differential equation (ODE) for the function ( ) f θ : ( ) [ ] { } [ ] { } [ ] [ ] [ ] { [ ] ( ) [ ] 2 2 2 2 4 2 2 2 2 2 2 2 1 ( 1)(2 ) 2 / 2 2 6 ( 1) 1 ( 1) ' ( 1)( 3) ( 1)(2 ) 2 ( 1) ( 1)( 2) 2 ( 1) ' ( 1) ( 1) ( 1) ' ( 1) / 2 ( 1) ' ( 1) IV e e e e e f f n f f f n n f hf f f n n h f f n f f f f f f f f f ff ff f f λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ ′′ ′′ − + − + + + − + − + + − − × ′′ ′′ ′′′ ′′ ′′ × + − + + − + + + + + + + + + + ′′ ′ ′′′ + + + − + − + + + ( ) ( ) [ ] ( ) } [ ] [ ] [ ] [ ] 4 2 2 2 4 4 1 2 ( 1) ( 1) 3 2( 1) ( 1)(2 ) 2 / 2 ( 1) ( 1)(2 ) 2 ( 1) 1 ( 1) ( 1)(2 1) 0, e e e e f f f f f f f f f n f h f f nf f f n nf f f λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ ′ ′ + + − − ′′ ′′ ′ ′ ′′ ′ ′′ − + + + + + + − + − + − ′′ ′′ − − + − + + − + − + − − = (6) where the following notations are adopted [ ] [ ] ( ) [ ] [ ] [ ] [ ] 2 2 2 2 2 2 2 2 2 ( 1) ( 1) ( 1) 1 3 , ( 1) ( 1) ( 1) ( 1) ( 1) / 2 ( 1) ( 1) / 2 3 . e f f f f f f f f h f f f f ff f f f f f f f f λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ ′ ′ ′ ′ = + + + + − + + + + = + + × ′ ′′ ′ ′ ′′ ′ ′ ′ ′′ × + + + + − + + + − + + + + The boundary conditions imposed on the function ( ) f θ follow from the traction free boundary conditions on the crack surfaces: ( ) 0, ( ) 0 f f θ π θ π ′ = ± = = ± = . (7) 3. Numerical solution of the nonlinear eigenvalue problem. Computational scheme. Eigenvalues and eigenfunctions Thus the eigenfunction expansion method results in the nonlinear eigenvalue problem: it is necessary to find eigenvalues λ leading to nontrivial solutions of Eq. 6 satisfying the boundary conditions (Eq. 7). Therefore, the