PSI - Issue 39
L.V. Stepanova et al. / Procedia Structural Integrity 39 (2022) 735–747 Author name / Structural Integrity Procedia 00 (2019) 000–000
738
4
{
}
) ( 1 1
)
2
(
2 λ ′′ − − + + + − − × 2 ( 1)( 3) n n e f f f
2 f f e
IV
n
{
} ( 2
(
)
(
)
)
′′
′
f + + ′′′
′ ′′
2 λ − + + f f ′′ 1
2 λ × − + 1
2 λ
2 λ
1
4
f f
f
f f
−
{
(
)
(
)
(
)
2
′
2 λ f f + + − + − ′′′ ′′ 1 1 f
′′
2
2 λ
2 λ
( 1)
1
n f
f
f
+ −
−
+
e
) } (
(
) )
(10)
′′
′ ′′′
2 λ − + + − × 2 1 2( 1) e f f n f ′′
2 λ
2
4
f
f f
+
+
{
} (
)
(
)
(
′
′ ′′
′
f + + ′′′
′′
′
′
2 λ × − + 1
2 λ
2 λ
2 λ
4
1
1
f
f f
f
f f
f
−
+
+
−
{
}
(
)
(
) (
′′
′
f + + ′′′
′ ′′ ′
2 λ
2 λ
2
2 λ − +
( 1) C n f
1
1
4
f f
f
f f f
+ −
−
+
1
e
(
)
)
′′
2 λ f f − + + − f ′′ 4 1 1
′′
4 C f f
4
2 λ
0,
2 C f
f
+
−
=
1
e
e
e
where the following notations ( ) 2 2 2 1 e f f f λ ′′ = − + +
[
]
[
]
2 2 4 , f λ ′
4 ( 1) 1 , n λ λ − +
( 1) ( 1) 2 n n λ λ = − − +
C
C
=
1 are adopted. The equation is derived for plane strain conditions. The fourth order nonlinear differential equation (10) with boundary conditions following from the traditional traction-free boundary conditions ( ) ( ) 0, 0. f f θ π θ π ′ = ± = = ± = (11) defines a nonlinear eigenvalue problem in which the constant λ is the eigenvalue and ( ) f θ is the corresponding eigenfunction. For plane stress conditions the compatibility condition (5) results in the nonlinear ordinary differential equation 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 ′ ′′ + + − − ′′ ′′ ′′ ′ ′′′ ′ ′′′ − + + + + + + − + − + − ′′ ′′ − − + − + + − + − + − − = (12) 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 (11). The order of the stress singularity is the eigenvalue and the angular variations of the field quantities correspond to the eigenfunctions. The solution of Hutchinson and Rice and Rosengren is the most important achievement of nonlinear fracture mechanics. Equations (10) and (12) provide the circumferential distribution function of the solution. The function is governed by very complicated nonlinear differential equations of the fourth order. Thus, the eigenfunction expansion method results in the nonlinear eigenvalue problem: it is necessary to find eigenvalues leading to nontrivial solutions of Eqs. (10) and (12) satisfying the boundary conditions (11). The eigenvalue corresponding to the HRR problem is well known / ( 1). n n λ = + (13) 3. Eigenspectrum of the nonlinear eigenvalue problem The further development of fracture mechanics required analysis of eigenspectra and orders of singularity at a crack tip for power-law materials (Meng and Lee (1998), Stepanova (2008), Stepanova (2009)). In (Meng and Lee (1998)) the necessity of introducing higher or lower order singular terms to more correctly describe the asymptotic fields of 2
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