Elliptic crack in a space under the action of a heat flow at infinity
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ELLIPTIC CRACK IN A SPACE UNDER THE ACTION OF A HEAT FLOW AT INFINITY M. M. Stadnyk
UDC 539.3
A three-dimensional thermoelastic problem for a body containing a thermally insulated elliptic crack and subjected to the action of a heat flow perpendicular to the plane of the crack at infinity is reduced to a system of two singular integrodifferential equations. The analytic solution of this system is obtained. The formulas for the stress intensity factors are deduced and the influence of the configuration of the crack on the stress intensity factors is analyzed for some important special cases of the problem. Keywords: integrodifferential equations, thermally insulated crack, heat flow.
The solutions of thermoelastic problems for bodies with cracks of different configurations are required for the investigation of their strength by the methods of mechanics of brittle fracture. The thermoelastic problem for thermally insulated circular and tunnel cracks in a body subjected to the action of a heat flow at infinity is solved in [1–6]. In what follows, we solve the thermoelastic problem for a thermally insulated elliptic crack in a body subjected to the action of a heat flow at infinity and obtain representations for the evaluation of the stress intensity factors (SIF). Statement of the Problem and Its Solution Consider a three-dimensional elastic body weakened by a thermally insulated elliptic crack. We choose a system of Cartesian coordinates Oxyz whose origin coincides with the center of the crack and the Oz -axis is perpendicular to the plane of the crack, i.e., the equation of the crack contour has the form x 2 /a 2 + y 2 /b 2 = 1 . At infinity, the body is subjected to the action of a heat flow with intensity q perpendicular to the plane of the crack. The aim of the present work is to determine the SIF along the crack contour. On the basis of the results obtained in [7], the problem is reduced to the following system of two singular integrodifferential equations for the jumps of displacements [ u x ] and u y of the crack surfaces z = ± 0
caused by the action of the heat flow q :
S
[ u x ] R
dd + μ
y x
S
u y dd R y
S
[ u x ] R
dd = ( 1 + μ ) x
[ T ] S
dd , R (1)
S
u y dd + μ R x y
S
[ u x ] R
dd
x
S
u y dd = ( 1 + μ ) R y
[ T ] S
dd , R
Ukrainian National Forestry Engineering University, Lviv, Ukraine; e-mail: [email protected]. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 46, No. 3, pp. 38–41, May–June, 2010. Original article submitted February 11, 2010. 1068-820X/10/4603–0325
© 2010
Springer Science+Business Media, Inc.
325
326
M. M. STADNYK
where = 2 /x 2 + 2 /y 2 , μ is Poisson's ratio of the material of the body, S is the elliptic domain x 2 /a 2 +
y 2 /b 2 1 , R = ( x ) + ( y )2 , is the coefficient of thermal expansion of the body, and [ T ] is the jump of temperature on the crack lips z = ± 0 determined from the integral equa
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