PSI - Issue 17

W. Reheman et al. / Procedia Structural Integrity 17 (2019) 850–856

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W Reheman et al. / Structural Integrity Procedia 00 (2019) 000–000

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a)

b)

c) d) Fig. 2. a) Perfectly bonded with wavy interface, b) first step, separation along the interface, c) second step, expanded lower half-space B, d) adjusted remote stresses ∆ σ = ∞ E / 2 to make parts A and B compatible.

a)

b)

c) Fig. 3. a) a point force tangential to the boundary over an area S, b) on coordinates, a) ??? a ( ξ ) should be w ( ξ ) close up on wavy surface

In the second step, the lower part B receives additional strain s . Stress is not a ff ected, but the material fragments A and B does not fit together. Finally the remote stress σ 11 is adjusted to make parts A and B compatible. To maintain equal normal stress σ ∞ , the stress adjustment in part A and B is the same amount but with an opposite sign, σ A 11 = σ ∞ + E s / 2 , σ B 11 = σ ∞ − E s / 2 , σ B 22 = σ A 22 = σ B 12 = σ A 12 = 0 . (3) Because of the symmetries across x 2 = 0, the analysis is reduced to a study of the half-plane B only. The position of the perturbed interface is given by the function w ( ξ ) as indicated in Fig. 3c. From Eq. (3) above the force d F , force per unit of length that act on the upper surface of B at x 1 = ξ is d F = σ B 11 d w , cf. Fig. 3a to c. The resulting stress distribution due to d F as in Fig. 3a, is given by Cerruti, cf. Fung (1965), as the following d σ ρρ = − π 2 d F cos ϕ ρ = − π 2 d F ( x 1 − ξ ) ρ 2 , (4) and all other stress contributions vanish. The polar coordinates are defined in Fig. 3a. The polar coordinate system is attached to the origin of the Cartesian coordinate system, x 1 = x 2 = 0, and ρ = ( x 1 − ξ ) 2 + x 2 2 and ϕ = arctan[ x 2 / ( x 1 − ξ )], according to Fig. 3b. The force is defined as the tangential traction