Fracture of a Semi-Bounded Composite Material with a Near-Surface Penny-Shaped Crack under Compression
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International Applied Mechanics, Vol. 56, No. 2, March, 2020
FRACTURE OF A SEMI-BOUNDED COMPOSITE MATERIAL WITH A NEAR-SURFACE PENNY-SHAPED CRACK UNDER COMPRESSION*
M. V. Dovzhik
The nonclassical problem of fracture mechanics for a near-surface crack is solved in the case of small distances between the free surface and the crack plane. The problem is axisymmetric for a penny-shaped crack. A numerical study for a composite material is carried out as an example. Keywords: near-surface crack, compression along crack, critical stress, composite material Introduction. When the forces acting on a body with a plane crack are parallel to the crack plane, the stress intensity factors predicted by linear fracture mechanics are equal to zero and the Irwin–Griffith failure criteria are inapplicable. In the case of tension and compression of the material along the crack plane, the approach first proposed in [1] is applied. The failure criterion, in this case, is the local loss of stability near the crack described by the three-dimensional linearized theory of elastic stability. According to this approach, the fracture is initiated by local loss of stability near cracks, and the critical compressive loads are determined by solving the appropriate eigenvalue problems using the three-dimensional linearized theory of stability of deformable bodies. The reviews [9–14] and monographs [2–4] provide detailed information on the fracture of materials under compression along the plane of cracks for various locations of interacting cracks. Note that a detailed analysis of approaches to the problems of fracture of materials along cracks was for the first time performed in [9] with a detailed bibliography. The relationship between the critical compressive stress and the distance from the free surface to the plane of a near-surface penny-shaped crack (Fig. 1) was presented in [5, 7, 13, 14] for composite and highly elastic materials. The case where the distance between the free surface and the crack plane tends to zero is of particular interest. This case has been investigated in detail for highly elastic materials [5, 7], but not for composite materials. This issue is of theoretical and applied interest for design of thin interlayers formed after spraying, thermal shock, etc. Here we will use the combined numerical-analytical method proposed in [7] to solve the problem of fracture of a composite half-space compressed along a penny-shaped near-surface crack for small distances between the free surface and the crack. 1. Problem Formulation and Generalized Solution. We consider a penny-shaped crack of radius a in half-space x 3 ³ - h located in the plane x 3 = 0 centered on axis Ox 3 . The initial stresses acting along the crack cause biaxial tension and compression [14]: 0 S 33 = 0,
0 0 0 S 11 = S 22 ¹ 0, u m = d jm ( l j - 1)x j ,
l1 = l 2 ¹ l 3 ,
l j = const,
S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, 3 Nesterova St., Kyiv, Ukraine 03054: [email protected]. Translated from Prikladnaya Mekhanika,
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