Partial Opening of a Semiinfinite Crack on the Boundary of an Elastic Strip and a Rigid Wall
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PARTIAL OPENING OF A SEMIINFINITE CRACK ON THE BOUNDARY OF AN ELASTIC STRIP AND A RIGID WALL V. І. Ostryk
UDC 539.3
We study the problems of compression and shear of an elastic strip containing a semiinfinite crack on the boundary of the strip and a rigid wall. The crack is opened in the middle and closed at the ends, where its faces are under the conditions of friction sliding. By the Wiener–Hopf method, we reduce the system of integral equations of the problem to an infinite system of algebraic equations. The stress intensity factor and the crack opening displacements are determined. Keywords: interface crack, elastic strip, friction, sliding contact, Wiener–Hopf method.
Introduction A completely closed semiinfinite crack located on the boundary of an elastic strip and a rigid wall was studied in [1]. It was shown that the crack is completely closed if the ratio of the intensities of shear and compressive loads does not exceed a certain limit value. In what follows, we consider the problem of partial opening of the crack under significant shear loads. The obtained results can be used to find the distribution of hydrogen near the interface crack [2]. Statement of the Problem We generalize the statement of a problem studied in [1]. Consider an elastic strip
– ∞ < x < ∞,
–h≤ y≤h
containing a semiinfinite crack – ∞ < x < 0 between the top edge y = h of the strip and the rigid wall in the case where the crack is open in the middle in a certain interval
– l2 < x < – l1
( l2 > l1 > 0 )
whose ends are not known in advance (see Fig. 1). Outside this interval, the crack faces are in friction contact. The bottom edge y = − h and a part ( 0 ≤ x < ∞ ) of the top edge y = h of the strip are fastened to the walls. As shown in [1], the bottom crack face separates from the top wall if the ratio of the intensities of shear and normal loads q/ p applied to the walls exceeds a certain limit value q∗ . In this case, q∗ = 3.05 and 3.26 for Poisson’s ratio ν = 1/3 if the friction coefficient is equal to µ 0 = 0 and 0.25, respectively. Institute for Applied Physics, National Academy of Sciences of Ukraine, Sumy, Ukraine; e-mail: [email protected]. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 56, No. 1, pp. 94–100, January–February, 2020. Original article submitted April 13, 2017. 1068-820X/20/5601–0097
© 2020
Springer Science+Business Media, LLC
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V. І. OSTRYK
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Fig. 1. Elastic strip with partially opened crack. The boundary conditions of the problem are as follows:
uy
= − δ1 ,
y=h
ux
τ yx
y=h
= 0,
y=h
ux
y=−h
= 0,
y=h
,
τ yx uy
− l1 ≤ x < ∞ ;
0 ≤ x < ∞,
= δ2 ,
y=h
= − µ0 σ y
σy
− ∞ < x ≤ − l2 ,
− ∞ < x < − l2 , y=h
= 0,
y=−h
= 0,
− l1 ≤ x < 0 ;
(1)
− l2 < x < − l1 ;
− ∞ < x < ∞,
where δ1 and δ 2 are the normal and tangential relative displacements of the walls. As in [1], we can write
δ1 1 − 2ν p , = 2h 1 − ν 2G
δ2 2q − µ 0 p , = 2h G
where G is the shear modulus. System of Integral Equations of the Problem and Its Solution We represent the tangential displacements and
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