M. P. Savruk

Distribution of stresses over the contour of a rounded V-shaped notch under antiplane deformation

The solution of the antiplane problem of the theory of elasticity for a plane with semiinfinite rounded V-shaped notch is obtained by the method of singular integral equations. On this basis, we deduce the relationships between the stress concentration and stress intensity factors for sharp and rounded notches. The obtained solution is compared with the well-known solution of a similar problem for a hyperbolic notch. It is shown that both the radius of rounding of the notch tip and the shape of its neighborhood strongly affect the distribution of stresses on the boundary contour.

Stress Concentration near Holes in the Elastic Plane Subjected to Antiplane Deformation

By the method of singular integral equations, we determine the solutions of antiplane problems of the theory of elasticity for a plane weakened by curvilinear holes with rounded and sharp vertices. The stress intensity factors at sharp vertices are found as a result of the limit transition from rounded vertices by using the relation between the stress intensity factors and stress concentration factors at the sharp and rounded tips of a V-shaped notch. We also obtain numerical results for the holes of various configurations: elliptic, in the form of physical slots, oval, rhombic, and rectangular.

Distribution of stresses near V-shaped notches in the complex stressed state

By the method of singular integral equations, we obtain the solution of a two-dimensional problem of the elasticity theory for a plane containing a semiinfinite rounded V-notch under complex loading. On the basis on this solution, we establish the relationships between the stress intensity factors at the tip of the sharp V-notch and the maximum stresses and their gradients at the tip of the corresponding rounded notch. For finite bodies with V-notches, the presented solutions are obtained as asymptotic dependences for small radii of rounding of the tips. The deduced relationships can be used to perform the limit transitions and find the stress intensity factors at the tips of sharp V-notches on the basis of the solutions obtained for the corresponding rounded stress concentrators. The efficiency of the method is illustrated for the problem of determination of the stress intensity factors at the vertices of a rectangular hole in the elastic plane.

Plane Eigenvalue Problems of the Theory of Elasticity for Orthotropic and Quasiorthotropic Wedges

We construct the characteristic equations for the plane eigenvalue problems of the theory of elasticity for orthotropic wedges with symmetric and antisymmetric stress distributions relative to their bisectors. Singular stresses are expressed via the stress intensity factors at the tip of the wedge. The corresponding results for a quasiorthotropic wedge were obtained by passing to the limit as the roots of the characteristic equation approach each other. We approximately determine the shape of the plastic zone near the tip of an orthotropic wedge under symmetric and antisymmetric loads.

Curvilinear Cracks in the Anisotropic Plane and the Limit Transition to the Degenerate Material

We establish the relationship between the singular integral equations of the first basic problem of the plane theory of elasticity for an anisotropic body with curvilinear cracks in the auxiliary and principal complex planes. By the limit transition, we construct integral equations of the problem for a degenerate anisotropic medium with cracks in the case where the complex roots of the characteristic equation are multiple. The formulas for the determination of the stressed state and stress intensity factors at the crack tips are obtained from the solutions of the integral equations.

STRESS CONCENTRATION NEAR SHARP AND ROUNDED V-NOTCHES

We present a survey of investigations of the stress concentration in elastic bodies weakened by sharp and rounded V-notches. Special attention is given to the unified approach to the study of stress distributions near the indicated stress concentrators in the case where the stress intensity factors at sharp tips of the notches are determined according to the stress concentration factors in the vicinity of a rounded notch with small radius of curvature. We consider two-dimensional bodies with internal acute-angled holes and edge V-notches.

Distribution of Stresses Near V-Notches in an Orthotropic Plane Under Symmetric Loads

The solution of a plane problem of elasticity theory for an orthotropic plane with semiinfinite rounded V-notch under the action of symmetric loads is obtained by the method of singular integral equations. By using this solution, we deduce the relations between the stress intensity factor at the sharp tip of the V-notch and the normal stresses at the tip of the corresponding rounded notch. For bounded bodies with V-notches, the indicated solution is obtained as an asymptotic relation for small radii of rounding of their tips. The presented relation can be used in the limit transitions in order to find the stress intensity factor at the tips of sharp V-notches according to the solutions for the corresponding rounded stress concentrators.

Plane Problem of the Theory of Elasticity for a Quasiorthotropic Body with Cracks

We write basic relations of the plane problem of the theory of elasticity for a quasiorthotropic body. The integral representations for the complex stress potentials are constructed for a quasiorthotropic plane in terms of the jumps of displacements on open curvilinear contours. The first basic problem for a plane with cracks is reduced to singular integral equations. We find the asymptotic distribution of stresses near the tip of a curvilinear crack. The analytic solution of the problem is obtained for an arbitrarily oriented rectilinear crack. We numerically compute the stress intensity factors for a parabolic crack and analyze the influence of the ratio of the basic moduli of elasticity of the material on their behavior.

Stress Concentration Near an Elliptic Hole or a Parabolic Notch in a Quasiorthotropic Plane

We consider the problem of stress distribution in an infinite quasiorthotropic plane containing an elliptic hole whose contour is free of external forces and a homogeneous stressed state is imposed at infinity. The solution of the problem is obtained with the help of the boundary transition from the known analytic solution for an elliptic hole in an orthotropic plane in the case where the roots of the characteristic equation approach each other. In the boundary case where the major semiaxis of the ellipse tends to infinity, these results yield the stress distribution in a plane weakened by a parabolic notch for two main types of deformation (symmetric tension and transverse shear).

Influence of the Anisotropy of Materials on the Distribution of Stresses Near Parabolic Notches

We consider the problems of distribution of stresses in an infinite anisotropic plane containing a parabolic notch for three main types of deformation in the case where the asymptotics of the stress field is specified at infinity (including stress intensity factors at the tip of the corresponding semiinfinite crack). The solutions of these problems are obtained with the help of the limit transition from the known analytic solutions obtained for an elliptic hole in the anisotropic plane for three types of loading at infinity (symmetric tension, transverse and longitudinal shear). These results generalize the well-known data on the distribution of stresses near narrow rounded notches in the isotropic plane and reflect the influence of anisotropy of the material on the stress concentration.