A method for the determination of the elastic equilibrium of isotropic bodies with curvilinear inclusions. Part 2. Plane problem

Abstract

539.3 Within the framework of the approach proposed in the first part of the work, the two-dimensional boundary-value problem for an isotropic body with noncanonical elastic inclusion is reduced to a finite system of linear algebraic equations. It is shown that the solution of this problem for elastic inclusions with small radius of curvature at the tip and/or cusps describes the intensity and concentration of stresses in the composition. For some special examples, we reveal the influence of elastic properties of the components of the composition and configuration of the inclusions on its local stress-strain state. It is also established that, unlike the method of perturbation of the shape of the boundary, this method is applicable to the determination of the concentration and intensity of stresses in the vicinity of the tips of elastic inclusions with small radii of curvature, including the inclusions whose elastic properties are close to the elastic properties of the matrix.In the first part of the present work [1], we suggested a method for the determination of the stress-strain state of isotropic bodies with curvilinear foreign elastic inclusions in the two-dimensional case. Here, we present some examples of application of this method. For inclusions of noncanonical shapes, the problem was reduced to the solution of a system of linear algebraic equations. It was shown that the investigated approach can be used for the determination of the concentration and intensity of stresses near the tips of inclusions with small radii of curvature. The influence of the configuration and elastic properties of inclusions on the local stress-strain state of the composition is analyzed for several special cases. Reduction of the Two-Dimensional Problem to a Finite System of Linear Algebraic EquationsLet us now return to the statement of the problem from [ 1 ]. We restrict ourselves to the case of a smooth contour L and present basic relationships of the suggested approach for the evaluation of a single complex stress potential ___l (Z). In what follows, the functions of z are denoted by underlined symbols.Let % (~) and ~l (~) be the limiting values on the contour ~/ (in a sense of transformation (8) The functions tp2(• ) and ~2 (0)

A method for the determination of the elastic equilibrium of isotropic bodies with curvilinear inclusions. Part 3. Antiplane deformation

“…Transverse sizes of this body can be regarded as infinite. We present the general form of the function o(~), which conformally maps the exterior domain of a unit circle ), in the parametric plane ~ onto the exterior domain of the inclusion in the physical plane z (z = x + iy) and satisfies conditions (4) in [6], as follows:…”

“…Moreover, Fj(~) is the boundary value (in terms of the notation used in [6]) of the function Fj(z), which is analytic in the domain Sj, on the contour Y, and 9~ is the parameter, cl-62 61+62"…”

“…We multiply boundary condition (5) by the Cauchy-type kemel K0(o, z), which is expandable in series (25) in [6] for z ~ S 2 and is defined by relation (46) in [6] for z ~ S 1 . In view of the condition of holomorphy of the function Fl(z) in the domain S 1 and analyticity of the complex potential F2(z ) the general approach [6], we integrate the boundary condition (5) with respect to circle qt.…”

“…In view of the condition of holomorphy of the function Fl(z) in the domain S 1 and analyticity of the complex potential F2(z ) the general approach [6], we integrate the boundary condition (5) with respect to circle qt. As a result, we arrive at the functional equation for the function FI(z), in the domain S 2 , according to do along the contour of the unit…”

“…Then, in [6,7], we proposed a new approach to the investigation of planar tension-compression of isotropic bodies with elastic inclusions of arbitrary configurations (not only close to a circular or elliptic one) and with arbitrary radius of curvature at a tip. In the present work, we extend this method to longitudinal shear in bodies with inclusion of any form outside a tip, investigate the convergence of numerical calculations, and analyze the influence of the form and elastic characteristics of defects on the local distribution of stresses.…”