Deformation criterion of the limiting state of a cracked plate under biaxial loading
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DEFORMATION CRITERION OF THE LIMITING STATE OF A CRACKED PLATE UNDER BIAXIAL LOADING M. M. Stadnyk
UDC 539.3
The experimentally established dependence of the critical crack-tip opening displacement on the type of loading [1, 2] indirectly shows that the crack-tip opening displacement also depends on the type of loading. The analytic representation of the crack-tip opening displacement for the case of uniaxial tension of a plate is wellknown [3]. In what follows, we present a theoretically substantiated relation taking into account the influence of the biaxiality of loading on the crack-tip opening displacement. The influence of the biaxiality of loading on KI is studied in [4]. Consider an infinite plate containing a rectilinear crack of length 2a stretched at infinity by uniformly distributed mutually perpendicular forces p (normal to the crack line) and q. On the basis of the deformation δccriterion, it is necessary to determine the critical values of forces p* and q* for which the limiting equilibrium state of the plate is attained. First, we assume that the plate is weakened by an elliptic hole with axes 2a and 2c ( a >> c ). The origin of a coordinate system x O z is placed at the center of the hole and the Ox-axis is directed along the axis of the ellipse of length 2a parallel to the forces q. The problem is reduced to the solution of the integrodifferential equation [5] qd3 cπ [u˜ z ]*′ dt = − 2 p π d1 + , t x a − −a a
G
∫
| x | ≤ a,
(1)
for the derivative of the jump of displacements [u˜ z ]* of the crack edges – a ≤ x ≤ a, z = ± 0, subjected to the action of stresses translated from the surfaces of the elliptic hole z = ± h = ±c 1 −
2
x 2 , a
where G is the shear modulus, μ is Poisson’s ratio of the material of the plate, d1 =
1+ κ , 4
d3 =
κ −1 , 2
κ = 3 – 4μ for the case of plane deformation or k =
3−μ 1+ μ
Ukrainian National University of Forestry Engineering, Lviv, Ukraine. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 44, No. 2, pp. 126–128, March–April, 2008. Original article submitted May 25, 2007. 1068–820X/08/4402–0305
© 2008
Springer Science+Business Media, Inc.
305
306
M. M. STADNYK
for the plane stressed state, and u˜ z = uz – uz0 , where uz0 is the displacement of the surfaces z = ± h in the homogeneous body and u˜ z is the perturbed displacement caused by the defect. As a result of the solution of Eq. (1), we obtain qcd3 ⎞ a 2 − x 2 [u˜ z ]* = ⎛⎝ 2 pd1 − , a ⎠ G
| x | ≤ a,
(2)
provided that 2 pd1 −
qcd3 > 0. a
Relation (2) implies that the jump of displacements [u˜ z ]* under the conditions of biaxial tension of the plate by the forces p and q can be found from the jump of displacements [u˜ z ]* in the case of uniaxial tension p = p1 , q = 0, by the change p1 = p −
qcd3 > 0. 2 d1a
(3)
For a crack with c = 0, relation (2) implies that the jump of displacements [u˜ z ]* is independent of the forces q for p = 0. If p > 0, then the crack opens, i.e., c > 0, and [u˜ z ]* is no longer independent of the forces q. Assume that the plate contains a crack of
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