PSI - Issue 52

Akihide Saimoto et al. / Procedia Structural Integrity 52 (2024) 323–339 A. Saimoto et al. / Structural Integrity Procedia 00 (2023) 000–000

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2.2. Complex potentials for point force doublets

In BFM, a crack is represented by a continuously embedded point force doublets along a contour to be a crack. A point force doublet is a pair of concentrated forces, facing in opposite directions, acting an infinitesimally small distance apart from each other. For 2-dimensional crack problems, 3 components of point force doublets are used in combination as shown in Fig.3. In 3-dimensional problems, 6 components of point force doublets are generally required. The number of point force doublets required corresponds to the number of independent stress components in the dimension of the problem under consideration. Consider an anisotropic infinite plate with multiple straight cracks, where the k -th crack is at an angle β k with the global x -axis. The center of the crack is set at the origin of the local coordinates, the s k -axis is take along the crack line, and the t k -axis is perpendicular to it. The magnitude of a tensile-type point force doublets in the s k direction is denoted by SS , that of the tensile-type point force doublets in the t k direction by TT , and that of the shear-type point force doublets by ST . The complex potentials for those three types of point force doublets can be obtained as follows.

a 11 cos β k + a 12 sin β k 2 π i a 11 cos β k + a 12 sin β k 2 π i

lim δ → 0 log[ z 1 − ζ 1 − δ (cos β k + µ 1 sin β k )] − log( z 1 − ζ 1 ) δ ×

SS 1 ( z 1 ) =

( δ · F s )

Φ

∂ ∂ζ 1

(cos β k + µ 1 sin β k )

log( z 1 − ζ 1 ) × SS

=

( a 11 cos β k + a 12 sin β k )(cos β k + µ 1 sin β k ) 2 π i ( a 21 cos β k + a 22 sin β k )(cos β k + µ 2 sin β k ) 2 π i ( − a 11 sin β k + a 12 cos β k )( − sin β k + µ 1 cos β k ) 2 π i ( − a 21 sin β k + a 22 cos β k )( − sin β k + µ 2 cos β k ) 2 π i a 11 ( µ 1 cos2 β k − sin2 β k ) + a 12 (cos2 β k + µ 1 sin2 β k ) 2 π i a 21 ( µ 2 cos2 β k − sin2 β k ) + a 22 (cos2 β k + µ 2 sin2 β k ) 2 π i SS z 1 − ζ 1 SS z 2 − ζ 2 TT z 1 − ζ 1 TT z 2 − ζ 2

, ( SS = lim δ → 0

( δ · F s ))

(36)

= −

SS 2 ( z 2 ) = − TT 1 ( z 1 ) = − TT 2 ( z 2 ) = − ST 1 ( z 1 ) = − ST 2 ( z 2 ) = −

(37)

Φ

, ( TT = lim δ → 0

( δ · F t ))

(38)

Φ

(39)

Φ

ST z 1 − ζ 1 ST z 2 − ζ 2

, ( ST = lim δ → 0

( δ · F st ))

(40)

Φ

(41)

Φ

δ is an infinitesimal distance between pair of point forces acting in the opposite directions. δ tends to zero at a limit holing a product between δ and the magnitude of point force F s , F t and F st remain a constant at SS , TT and ST , respectively. In particular, when crack lines on a global x axis, the angle between x -axis and k -th crack line, β k , is zero and the complex variable corresponding a source point in the a ffi ne transformed space ζ j , ( j = 1 , 2) can be replaced simply by a real variable ξ such that,

a j 1 µ j + a j 2 2 π i ( z j − ξ )

a j 1 2 π i ( z j − ξ )

a j 2 µ j 2 π i ( z j − ξ )

SS j ( z j ) = −

TT j ( z j ) = −

ST j ( z j ) = −

SS , Φ

TT , Φ

ST , ( j = 1 , 2)

(42)

Φ

Fig. 3. Three types of point force doublets (tension type parallel to crack line SS , tension type perpendicular to crack line TT , shear type ST ).