The two-dimensional stressed state of a multiconnected anisotropic body with cavities and cracks

Abstract

We present a method of determining the two-dimensional generalized stress-strain state and the stress intensity factors for an anisotropic body with cylindrical cavities and plane cracks. The method is based on the use of generalized complex potentials, conformal mappings, the method of least squares, and numerical passage to the limit to determine the stress intensity factors. We apply the method to study the stress-strain state and the change in stress intensity factors as functions of the geometric and elastic characteristics of an orthotropic cylinder with one or two cracks, an infinite anisotropic body with elliptic cavities and cracks, and an infinite body with a curvilinear cavity. Five figures. Six tables. Bibliography: 7 titles

The two-dimensional electroelasticity problems for multiconnected bodies situated under electric potential difference action

“…Finding of the constants c li ; c l3 is a matter of some difficulty. So far investigators (Kaloerov and Goryanskaya, 1998;Kaloerov et al, 2007;Glushchenko and Khoroshev, 2008;Khoroshev, 2010) had avoided it by differentiation of Eq. (2.5) and (2.6) on an angular position s of a contour L l .…”

“…A rectilinear cut is an ellipse with a zero-valued semiaxis b l . We will solve the problem following the numerically-analytical technique firstly introduced by Kaloerov and Goryanskaya, 1998 and developed also in Kaloerov et al (2007); Kaloerov and Baeva (2001); Glushchenko and Khoroshev (2008); Khoroshev (2010). As it was shown in Kaloerov and Goryanskaya (1998), the holomorphic in S k and single-valued functions U k0 z k ð Þ are expressed in the form We expand the holomorphic inside L k0 functions U kg 0 z k ð Þ into the Taylor series (Kaloerov and Goryanskaya, 1998) …”

“…introduced and investigated four generalized complex potentials, obtained boundary conditions for their determination, allocated the logarithmic peculiarities of the functions for solving the two-dimensional problems of electroelasticity for arbitrary multiconnected cylindrical piezoelectric bodies. Based on the numerically-analytical technique (Kaloerov and Goryanskaya, 1998) which consist in use of conformal mapping, functions expansions in the Taylor and the Laurent series and the least-squares method for satisfying boundary conditions in differential form, there are given solutions of the two-dimensional (and plane) electroelastic problems for bodies (plates) with finite number of holes and cracks in Kaloerov andBaeva (2001, 2003). Using this technique, investigations of the electroelastic states of concrete bodies and plates with holes and cracks under mechanical forces, remote or concentrated electric loading may be found in the generalizing monograph .…”

Thermostressed State of an Anisotropic Plate with Holes and Cracks

“…Let a l and b l be the semiaxes of the ellipses, x l 0 and y l 0 be the coordinates of their centers, and ϕ l be the angle of the semiaxis Ox. Conformally mapping the exterior of a unit disk | | ζ kl ≥ 1 onto the exterior of the ellipses L kl [5], to which L l are transformed by the affinities (3.5), we express the functions (3.19) as follows [4]: Similarly, satisfying the condition∂ ∂ = I a kln σ / 0for the functional (5.6), we obtain the system for the coefficients a kln :…”

Determining the intensity factors for stresses, electric-flux density, and electric-field strength in multiply connected electroelastic anisotropic media

“…The use of the linear conjugation method to solve problems of elasticity for multiply connected anisotropic bodies that have holes besides cracks (even if they lie along one straight line) involves severe mathematical and computational difficulties. Therefore, an approximate method was proposed in [7,9] to determine the SIFs for an arbitrary multiply connected domain with arbitrarily arranged rectilinear cuts. The method employs numerical limiting processes to find derivatives of complex potentials or even normal and tangential stresses in small neighborhood of crack tips.…”