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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Lekhnitskii (1968) introduced a set of complex potentials Φ 1 ( z 1 ) and Φ 2 ( z 2 ) in which z j = x + µ j y , ( j = 1 , 2) are new complex variables corresponding to a reference point ( x , y ) in an a ffi ne transformed planes. By using these potentials, elastic fields such as components of stress, strain, displacement and resultant force are expressed as follows.

2  j = 1

2  j = 1

2  j = 1

µ 2 j Φ ′ j ( z j ) , σ y = 2 ℜ

Φ ′ j ( z j ) , τ xy = − 2 ℜ

µ j Φ ′ j ( z j )

σ x = 2 ℜ

(15)

2  j = 1

2  j = 1

2  j = 1

2 j + c 26 − c 66 µ j ) Φ ′ j ( z j )

p j Φ ′ j ( z j ) , ε y = 2 ℜ

q j µ j Φ ′ j ( z j ) , γ xy = 2 ℜ

ε x = 2 ℜ

( c 16 µ

(16)

2  j = 1

2  j = 1

q j Φ j ( z j )

(17)

p j Φ j ( z j ) , v = 2 ℜ

u = 2 ℜ

and

2  j = 1

2  j = 1

P x = 2 ℜ

µ j Φ j ( z j ) , P y = − 2 ℜ

Φ j ( z j )

(18)

where the operator ℜ stands for a real part of the complex number and the prime sign ( ′ ) represents the first derivative with respect to argument z j . The complex constants p j and q j are defined as follows. p j = c 11 µ 2 j + c 12 − c 16 µ j , q j = c 12 µ j + c 22 /µ j − c 26 , ( j = 1 , 2) (19) 2.1. Complex potentials for point force

The complex potentials for an isolated force acting in the x and y directions with magnitudes of X and Y respec tively, at a source point ( ξ,η ) in an anisotropic infinite plate is expressed as follows.

a j 1 X + a j 2 Y 2 π i

Φ j ( z j ) = (20) where z j = x + µ j y and ζ j = ξ + µ j η, ( j = 1 , 2). In this equation, a jk ( j , k = 1 , 2) are new complex constants defined by log( z j − ζ j ) , ( j = 1 , 2)

   

   

   

=    

   

   

a 11 a 12 a 21 a 22 a 11 a 12 a 21 a 22

− 1 0 0 1 0 0 0 0

µ 1 µ 2 µ 1 µ 2 1 1 1 1 p 1 p 2 p 1 p 2 q 1 q 2 q 1 q 2

(21)

The above equation corresponds to the condition that the resultant force appearing along a closed contour which internally includes the source point is balanced with the applied force and there is no discrepancy in displacements along the contour. Solving this system of equation gives a jk explicitly as

µ 1 ( µ 1 − µ 2 )( µ 1 − µ 1 )( µ 1 − µ 2 )  − µ 1 ( µ 1 − µ 2 )( µ 1 − µ 1 )( µ 1 − µ 2 )  − µ 2 ( µ 2 − µ 1 )( µ 2 − µ 2 )( µ 2 − µ 1 )  − µ 2 ( µ 2 − µ 1 )( µ 2 − µ 2 )( µ 2 − µ 1 )  −

+ µ 1 + µ 2 + µ 2  1 − µ 1 µ 2

c 12 c 22 

c 16 c 11 c 12 c 11 c 16 c 11 c 12 c 11

(22)

a 11 =

+ µ 1 µ 2  1 − µ 2

c 26 c 22 

+ µ 2 ( µ 1 + µ 2 ) 

a 12 =

(23)

+ µ 2 + µ 1 + µ 1  1 − µ 2 µ 1

c 12 c 22 

(24)

a 21 =

+ µ 1 ( µ 2 + µ 1 ) 

+ µ 2 µ 1  1 − µ 1

c 26 c 22 

a 22 =

(25)

Consider the case of a uniformly distributed force of density ρ x and ρ y for each x and y directions acting on a certain length on the x axis. After a simple calculation, the stress components of appearing at a limiting point approaching