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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Let us consider a case where the point force doublets are distributed arbitrarily on the x-axis. That is the distri bution density for SS , TT and ST types are replaced by arbitrary real density functions such as f ( ξ ), g ( ξ ) and h ( ξ ), respectively, and integrate the complex potentials in the range ξ ∈ L where L is an arbitrary line segment on the x axis such that,

a j 1 2 π i L

2 π i L

2 π i L

a j 1 µ j + a j 2

a j 2 µ j

f ( ξ ) d ξ z j − ξ

g ( ξ ) d ξ z j − ξ

h ( ξ ) d ξ z j − ξ

TT j ( z j ) = −

ST j ( z j ) = −

SS j ( z j ) = −

, ( j = 1 , 2) (43)

, Φ

, Φ

Φ

If the reference point z j = x + µ j y , ( j = 1 , 2) is a point on L , these integral diverge due to a singularity at ξ = x .While, when z j in the region y > 0 or y < 0 approaches as close as possible to a point x on L , Plemelj’s formula can be used to evaluate those integrals. In concrete, when a reference point z j = x + µ j y approaches to a point on the x axis froma positive ( + ) or a negative ( − ) y regions, those integrals result the following limiting values. Φ SS j ( x ) ± = ± a j 1 2 f ( x ) − a j 1 2 π i L f ( ξ ) d ξ x − ξ (44) Φ TT j ( x ) ± = ± a j 2 µ j 2 g ( x ) − a j 2 µ j 2 π i L g ( ξ ) d ξ x − ξ (45) Φ ST j ( x ) ± = ± a j 1 µ j + a j 2 2 h ( x ) − a j 1 µ j + a j 2 2 π i L h ( ξ ) d ξ x − ξ (46) In those equations, the following notation for plus and minus sign is used. Φ SS j ( x ) + = lim y → 0 , y > 0 , x ∈ L − a j 1 2 π i L f ( ξ ) d ξ x + µ j y − ξ = a j 1 2 f ( x ) − a j 1 2 π i L f ( ξ ) d ξ x − ξ (47) Φ SS j ( x ) − = lim y → 0 , y < 0 , x ∈ L − a j 1 2 π i L f ( ξ ) d ξ x + µ j y − ξ = − a j 1 2 f ( x ) − a j 1 2 π i L f ( ξ ) d ξ x − ξ (48) It should be noted that integrals included in the right hand side of Eqs.(44) ∼ (48) have to be interpreted with a Cauchy principal value sense. Based on these limiting values of complex potentials, the gap in displacements between the upper and lower regions of L can be estimated as, ∆ u SS ( x ) = 2 ℜ p 1 Φ SS 1 ( x ) + p 2 Φ SS 2 ( x ) + − 2 ℜ p 1 Φ SS 1 ( x ) + p 2 Φ SS 2 ( x ) − = 2 f ( x ) ℜ p 1 a 11 + p 2 a 21 (49) ∆ v SS ( x ) = 2 ℜ q 1 Φ SS 1 ( x ) + q 2 Φ SS 2 ( x ) + − 2 ℜ q 1 Φ SS 1 ( x ) + q 2 Φ SS 2 ( x ) − = 2 f ( x ) ℜ q 1 a 11 + q 2 a 21 (50) ∆ u TT ( x ) = 2 g ( x ) ℜ p 1 a 12 µ 1 + p 2 a 22 µ 2 (51) ∆ v TT ( x ) = 2 g ( x ) ℜ q 1 a 12 µ 1 + q 2 a 22 µ 2 (52) ∆ u ST ( x ) = 2 h ( x ) ℜ p 1 ( a 11 µ 1 + a 12 ) + p 2 ( a 21 µ 2 + a 22 ) (53) ∆ v ST ( x ) = 2 h ( x ) ℜ q 1 ( a 11 µ 1 + a 12 ) + q 2 ( a 21 µ 2 + a 22 ) (54) Furthermore, using a relationship in Eq.(21), the following relation follows. ∆ u SS ( x ) =∆ v SS ( x ) = 0 (55) ∆ u TT ( x ) = 2 g ( x ) ℜ p 1 a 12 µ 1 + p 2 a 22 µ 2 , ∆ v TT ( x ) = 2 g ( x ) ℜ q 1 a 12 µ 1 + q 2 a 22 µ 2 (56) ∆ u ST ( x ) = 2 h ( x ) ℜ p 1 a 11 µ 1 + p 2 a 21 µ 2 , ∆ v ST ( x ) = 2 h ( x ) ℜ q 1 a 11 µ 1 + q 2 a 21 µ 2 (57) It can be seen that, the distribution of both pair of point forces acting in tangential ( ST type) and in perpendicular ( TT type) to the crack line produce a gap of a displacement both in the tangential ( ∆ u ) and the perpendicular ( ∆ v ) directions to the crack line at the same time and this gap is proportional to a distribution function of the applied pair of point fores g ( x ) and h ( x ).

2.3. Fundamental density function for an isolated line crack problem

When an isolated straight crack of half length b , lying on the x axis in an anisotropic infinite plate and the uniform pressure p 0 or shear stress q 0 is applied along the crack faces, the relative displacement between the upper and lower