Applicability of the ?? ? criterion for prediction of a curved crack trajectory

Panasyuk, V. V.; Zboromirskii, A. I.; Ivanitskaya, G. S.; Yarema, S. Ya.

doi:10.1007/bf01522791

Cited by 5 publications

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“…In order to find the shape of the crack development trajectory we use the o~-criterion of failure [4,5] ..whose application for an elastically brittle material gives good agreement with the descriPtion for experimental data, e.g., [6,7]. This means that in each step of crack advance only the change in the local stress field is considered in the vicinity of the crack lip caused by its advance and bending, and the effect of dynamic factors is ignored.…”

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Crack emergence trajectories at a free surface

Alekseeva¹,

Martynyuk²

1991

Soviet Mining Science

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In the Institute of Mining, Siberian Branch of the Academy of Sciences of the USSR, under the leadership of Chernov [i-3] technology is being developed for rock breaking using normal separation cracks.With this aim, a borehole is drilled at some distance from the free surface and a small nucleating slot is created with a special tool.A working liquid is fed into the borehole under pressure.On reaching a critical pressure determined by the properties of the material being broken and the size and position of the concentrator, the initial crack increases in size. By pumping the liquid and maintaining the pressure required for further crack growth, pieces of rock separate from the rock mass.We must evaluate the size and shape of the separated fragment in relation to the main parameters, i.e., initial size of the nucleating slot, curvature in its shape, distance from the free surface and degree of penetration of the working liquid into a growing crack.In order to explain the qualitative features of the process under plane strain conditions a study has been made of quasistatic hydraulic-rupture crack propagation close to the free surface. It is assumed that the hydraulic-rupture crack is characterized by zero tangential stresses along its length and normal stresses that are constant along its length.In order to find the shape of the crack development trajectory we use the o~-criterion of failure [4,5] ..whose application for an elastically brittle material gives good agreement with the descriPtion for experimental data, e.g., [6,7]. This means that in each step of crack advance only the change in the local stress field is considered in the vicinity of the crack lip caused by its advance and bending, and the effect of dynamic factors is ignored. This last assumption is correct because the hydraulic-rupture crack growth rate is much less than the velocity of Rayleigh surface waves.In solving the problem, we use the method of integral singular equations similar to that described in monographs [5,[8][9][10], and in the last of these this method is generalized for the case of smooth curvilinear slits. Statement of the Problem.Let an isotropic elastic body occupy a lower half-plane Im z ~ 0, z = x + iy, and it contains N smooth curvilinear slits Lk, k = i, N. With each slit L k we connect its local coordinate system Xk0kYk, k = I--?--N. The connection between coordinates of the points of a plane in the main and local coordinate systems is given by the expressionwhere o k is the angle between axes 0kX k and 0x, and Xk ~ yk ~ are coordinates of the origin of the k-th local system in the main system. We assume that the shape of each slit L k in its local coordinate system is known and described by a parametric equationThen solution of the problem in elasticity theory when stresses are specified along the peripheries of the slits as:(a positive sign relates to the upper edge, and a minus sign to the lower edge) is reduced to finding the solution of g'k(~), k = i, N of a set of N singular equations in the form [9, ii]:Institute ...

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Crack emergence trajectories at a free surface

Alekseeva¹,

Martynyuk²

1991

Soviet Mining Science

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Quasistatic edge crack growth with non-self-balancing stresses at the edges

Datsishin

Marchenko

1992

Mater Sci

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Emergence paths of cracks on a free surface during wedging

Efimov¹,

Martynyuk²,

Sher³

1995

J Appl Mech Tech Phys

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The present paper is a continuation of [1], where we mainly examined the propagation of hydraulic fracturing cracks near free surfaces. A hydraulic fracturing crack is defined on the assumption that the tangential stresses at its boundaries are zero, while the normal stresses are constant along the length. In mining, one usually employs a technique in which the working tool is a wedge. One therefore needs an adequate description of wedging.Here the wedging is simulated by the application of a pair of localized forces whose directions are dependent on the angle of the wedge and the coefficient of friction between the wedge and the medium. The quasistatic emergence paths on a rectilinear boundary have been compared with the experimental paths derived from impacts of wedges at various angles on lucite specimens. We examined how the major parameters affected the path shape: inclination angle of nuclear crack, inclination angle of wedge axis, wedge angle, and coefficient of fraction. The algorithm used to calculate the quasistatie paths is justified for describing impact wedging, where in the experiments, the mean speed of the crack was 200-250 m/see. We used the singular integral equation method [2,4].We simulate the action of the wedge on the medium. By 2/3 we denote the wedge angle and by a0 the deviation of the wedge axis from the vertical, which in general may not coincide with the initial nuclear crack direction. We take the area of contact between the wedge and the material as much less than the size of the crack, and then the normal and tangential stresses can be replaced by resultant localized forces N and T, which are directed respectively along the normal and along the tangent to the boundaries of the wedge. We assume that [ T ] = k t ] N I (kt = tan 3, is the coefficient of friction between the metal and the medium). The crack in wedging thus develops under the localized forces [ F I = IF I I = [ F21, applied to the edges, which emerge on the free surface, with the direction of the forces determined by the wedge angle, coefficient of friction, and inclination angle of the wedge axis.The [2, 3] solution method works well when the load varies smoothly along the crack. We therefore displace the points of application of the localized forces from the edges of the crack into the body, but leave their directions unchanged. We assume that the localized forces F 1 and F 2 are applied at the points z 1 = (e I --ie2)e i~0 and z 2 = -(e I + ie2)e ic~o, where e 1 and e 2 are the coordinates of the points of application in the Xl'OYl', coordinate system, which is related to the direction of the wedge axis (Fig. la). The localized forces are applied not to the edges of the crack but are referred to the surfaces at internal points z 1 and z2, which is unimportant because test calculations with crack length L > 4e 1 (e 1 = e 2) gave stress intensity coefficients at the vertex of the crack differing from standard values [2, 3] by less than 2 %.Quasistatic Formulation. Consider an isotropic elastic half-plane y < 0 containi...

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Applicability of the ?? ? criterion for prediction of a curved crack trajectory

Cited by 5 publications

References 1 publication

Crack emergence trajectories at a free surface

Crack emergence trajectories at a free surface

Quasistatic edge crack growth with non-self-balancing stresses at the edges

Emergence paths of cracks on a free surface during wedging

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