PSI - Issue 52

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

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A. Saimoto et al. / Structural Integrity Procedia 00 (2023) 000–000

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reference point

source point

Fig. 2. Point force solution (left), limiting value of stresses caused by uniform force in x direction of magnitude ρ x (middle), limiting value of stresses caused by uniform force in y direction of magnitude ρ y (right).

the midpoint of the line segment on which the distributed force acts from the region of positive y-coordinates can be estimated as follows. σ + x = −ℜ ( a 11 µ 2 1 + a 21 µ 2 2 ) ρ x + ( a 12 µ 2 1 + a 22 µ 2 2 ) ρ y = 1 2 c 16 c 11 ρ x + c 12 c 11 ρ y (26) (28) On the other hand, the stress components at the limiting point approaching the midpoint of the line segment on which the distributed force acts from the region of negative y-coordinates are as follows. σ − x = ℜ ( a 11 µ 2 1 + a 21 µ 2 2 ) ρ x + ( a 12 µ 2 1 + a 22 µ 2 2 ) ρ y = − 1 2 c 16 c 11 ρ x + c 12 c 11 ρ y (29) (31) Combining these results, it can be seen that in the limit of the proximity of parallel line segments with uniform, mutually opposing distribution forces acting in opposite directions, the stress components in the region between the line segments becomes as follows. ∆ σ x = σ − x − σ + x = − c 16 c 11 ρ x + c 12 c 11 ρ y , ∆ σ y = σ − y − σ + y = ρ y , ∆ τ xy = τ − xy − τ + xy = ρ x (32) These components of stress “jump” correspond to components of strain “jump” through Eq.(6) as (33) When an infinitesimal distance between the parallel line segments on which the distributed force acts is denoted by δ , the above strain jump may correspond to the following relative displacements in the x and y directions ∆ u and ∆ v between those line segments. ∆ u = δ × ∆ γ xy = c 11 c 66 − c 2 16 c 11 ( ρ x × δ ) + c 11 c 26 − c 12 c 16 c 11 ( ρ y × δ ) (34) ∆ v = δ × ∆ ε y = c 11 c 26 − c 12 c 16 c 11 ( ρ x × δ ) + c 11 c 22 − c 2 12 c 11 ( ρ y × δ ) (35) That is, it is understood that the region sandwiched between two straight lines on which a uniform density of dis tributed force acts produces a relative displacement proportional to the product of the density of the distributed force ρ x and ρ y and the distance between the two lines δ . This corresponds to the fact that pairs of distributed point forces (force doublets) produce relative displacements along their lines of action. σ + y = −ℜ ( a 11 + a 21 ) ρ x + ( a 12 + a 22 ) ρ y = − ρ y 2 τ + xy = ℜ ( a 11 µ 1 + a 21 µ 2 ) ρ x + ( a 12 µ 1 + a 22 µ 2 ) ρ y = − ρ x 2 (27) σ − y = ℜ ( a 11 + a 21 ) ρ x + ( a 12 + a 22 ) ρ y = ρ y 2 τ − xy = −ℜ ( a 11 µ 1 + a 21 µ 2 ) ρ x + ( a 12 µ 1 + a 22 µ 2 ) ρ y = ρ x 2 (30) ∆ ε x = 0 , ∆ ε y = c 11 c 26 − c 12 c 16 c 11 ρ x + c 11 c 22 − c 2 12 c 11 ρ y , ∆ γ xy = c 11 c 66 − c 2 16 c 11 ρ x + c 11 c 26 − c 12 c 16 c 11 ρ y