Stress-strain state of an anisotropic plate with an elliptic hole and thin rigid inclusions

The stress-strain state of an anisotropic plate containing an elliptic hole and thin, absolutely rigid, curvilinear inclusions is studied. General integral representations of the solution of the problem are constructed that satisfy automatically the boundary conditions on the elliptic-hole contour and at infinity. The unknown density functions appearing in the potential representations of the solution are determined from the boundary conditions at the rigid inclusion contours. The problem is reduced to a system of singular integral equations which is solved by a numerical method. The effects of the material anisotropy, the degree of ellipticity of the elliptic hole, and the geometry of the rigid inclusions on the stress concentration in the plate are studied. The numerical results obtained are compared with existing analytical solutions. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 4, pp. 173–180, July–August, 2007.     

Reinforcement of a plate with a circular hole by an eccentric circular patch

We study the reinforcement of an infinite elastic plate with a circular hole by a larger eccentric circular patch completely covering the hole and rigidly adjusted to the plate along the entire boundary of itself. We assume that the plate and the patch are in a generalized plane stress state generated by the action of some given loads applied to the plate at infinity and on the boundary of the hole. We use the power series method combined with the conformal mapping method to find the Muskhelishvili complex potentials and study the stress state on the hole boundary and on the adhesion line. We consider several examples, study how the stresses depend on the geometric and elastic parameters, and compare the problem under study with the case of a plate with a circular hole without a patch. In scientific literature, numerous methods for reinforcing plates with holes, in particular, with circular holes, have been studied. In the monographs [1, 2], the problem of reinforcing the hole edges by stiffening ribs is solved. Methods for reinforcing a circular hole by using two-dimensional patches pasted to the entire plate surface are studied in [3, 4]. The case of a plate with a circular cut reinforced by a concentric circular patch adjusted to the plate along the boundary of itself or along some other circle was studied in [5, 6]. The reinforcement of an elliptic hole by a confocal elliptic patch was considered in [7].     

Action of an elliptic punch on a transversally isotropic half-space

The 3D contact problem on the action of a punch elliptic in horizontal projection on a transversally isotropic elastic half-space is considered for the case in which the isotropy planes are perpendicular to the boundary of the half-space. The elliptic contact region is assumed to be given (the punch has sharp edges). The integral equation of the contact problem is obtained. The elastic rigidity of the half-space boundary characterized by the normal displacement under the action of a given lumped force significantly depends on the chosen direction on this boundary. In this connection, the following two cases of location of the ellipse of contact are considered: it can be elongated along the first or the second axis of Cartesian coordinate system on the body boundary. Exact solutions are obtained for a punch with base shaped as an elliptic paraboloid, and these solutions are used to carry out the computations for various versions of the five elastic constants. The structure of the exact solution is found for a punch with polynomial base, and a method for determining the solution is proposed.     

Solving the Problem of Bending of Multiply Connected Plates with Elastic Inclusions

This paper describes a method for determining the strain state of a thin anisotropic plate with elastic arbitrarily arranged elliptical inclusions. Complex potentials are used to reduce the problem to determining functions of generalized complex variables, which, in turn, comes down to an overdetermined system of linear algebraic equations, solved by singular expansions. This paper presents the results of numerical calculations that helped establish the influence of rigidity of elastic inclusions, distances between inclusions, and their geometric characteristics on the bending moments occurring in the plate. It is found that the specific properties of distribution of moments near the apexes of linear elastic inclusions, characterized by moment intensity coefficients, occur only in the case of sufficiently rigid and elastic inclusions.     

Indentation of a Rigid Body into an Elastic Plate

The problem of a punch shaped like an elliptic paraboloid pressed into an elastic plate is studied under the assumption that the contact region is small. The action of the punch on the plate is modeled by point forces and moments. The method of joined asymptotic expansions is used to formulate the problem of one–sided contact for the internal asymptotic representation; the problem is solved with the use of the results obtained by L. A. Galin. The coordinates of the center of the elliptic contact region, its dimensions, and the angle of rotation are determined. The moments which ensure translational indentation of the punch are calculated and an equation that relates displacements of the punch to the force acting on it is given.     

The biaxial tension of an elastoplastic space with a prismatic inclusion

The problem on the elastoplastic state of a thick isotropic plate (the case of plane deformation) is solved. A larger prismatic inclusion is made a close fit in a polygonal hole in the plate. The plate is stretched at infinity by constant mutually perpendicular forces. The problem is solved by the small-parameter method and by the theory of ideal plasticity. The axisymmetric state of the plane with a circular hole stiffened by a round ring with a constant force applied to its inner contour is considered as a zero approximation. Some specific shapes of the hole and reinforcing elastic rigid ring are considered. Voronezh University, Russia. Translated from Prikladnaya Mekhanika, Vol. 36, No. 6, pp. 114–120, June, 2000.     

Interaction of plane nonstationary waves with a thin elastic inclusion under smooth contact conditions

We solve the problem of determining the stress state near a thin elastic inclusion in the form of a strip of finite width in an unbounded elastic body (matrix) with plane nonstationary waves propagating through it and with the forces exerted by the ambient medium taken into account. We assume that the matrix is in the plane strain state, and the smooth contact conditions are realized on both sides of the inclusion. The method for solving this problem consists in using the integral Laplace transform with respect to time and in representing the stress and displacement images in terms of the discontinuous solution of Lamé equations in the case of plane strain. As a result, the initial problem is reduced to a system of singular integral equations for the transforms of the unknown stress and displacement jumps. To invert the Laplace transform, we use a numerical method based on replacing the Mellin integral by the Fourier series. As a result, we obtain approximate formulas for calculating the stress intensity factors (SIF) for the inclusion, which are used to study the SIF time-dependence and its influence on the values of the inclusion rigidity. We also studied the possibility of considering the inclusions of higher rigidity as absolutely rigid inclusions.     

Influence of the viscoelastic properties of a composite on the stress distribution around an elliptic hole in a plate

The time variation in the stresses around an elliptic hole in a composite plate is studied. Solutions that characterize the effect of the time dependence of the relaxation moduli of the composite components on stresses are obtained. The solutions in the time domain are obtained from the elastic–viscoelastic analogy and the corresponding elastic solutions for the effective moduli of the composite and the stress field around an elliptic hole in an anisotropic plate. The inverse Laplace transformation is carried out by an effective numerical method     

Effect of friction in a contact problem for a plate with a pin

A solution is obtained for a contact problem concerning the tension of a rectangular elastic plate with a circular hole into which a rigid stationary pin has been inserted. There is a small gap between the hole and the pin, which is of circular cross section. Friction acts in the contact region in accordance with the Coulomb law. The finite-element method and the Boussinesq principle are used to determine the load that realizes a specified contact region. Two variants of boundary conditions on the contour of the hole are examined. Lavrent'ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 4, pp. 184–192, July–August, 1998.     

Formation of a contact zone during compression of a plate with an elliptic hole

The problem of compression of a thin plate with an elliptic hole is considered. It is assumed that increasing the distant compressive load can lead to contact of opposite regions of the boundaries of the ellipse. The problem is solved within the framework of a modified Leonov-Panasyuk-Dugdale model and an elastoplastic analog of the Griffith problem for an ellipse using the Goodier and Kanninen model. The critical fracture parameters providing an equilibrium configuration of the system are determined from a sufficient strength criterion representing a system of two equations, one of which specifies the absence of partial overlapping of the upper and lower surfaces of the contact zone, and the other is a deformation criterion of critical opening of the ellipse. The compression-induced deformation of the boundaries of ellipses with various curvature radii at the top is shown by the example of annealed copper having nanostructure.     

Bending of an Elastic Anisotropic Plate with a Circular Opening Stiffened Over Finite Areas

A contact problem is solved for an infinite anisotropic plate weakened by a circular opening, stiffened by inclusions of variable stiffness, and subjected to bending. For the unknown contact force of interaction between the plate and an inclusion, an integro-differential equation is derived and then reduced to an infinite system of linear algebraic equations. The system is analyzed for regularity.     

On the anti-plane shear of an elliptic nano inhomogeneity

The elastic field of an elliptic nano inhomogeneity embedded in an infinite matrix under anti-plane shear is studied with the complex variable method. The interface stress effects of the nano inhomogeneity are accounted for with the Gurtin–Murdoch model. The conformal mapping method is then applied to solve the formulated boundary value problem. The obtained numerical results are compared with the existing closed form solutions for a circular nano inhomogeneity and a traditional elliptic inhomogeneity under anti-plane. It shows that the proposed semi-analytic method is effective and accurate. The stress fields inside the inhomogeneity and matrix are then systematically studied for different interfacial and geometrical parameters. It is found that the stress field inside the elliptic nano inhomogeneity is no longer uniform due to the interface effects. The shear stress distributions inside the inhomogeneity and matrix are size dependent when the size of the inhomogeneity is on the order of nanometers. The numerical results also show that the interface effects are highly influenced by the local curvature of the interface. The elastic field around an elliptic nano hole is also investigated in this paper. It is found that the traction free boundary condition breaks down at the elliptic nano hole surface. As the aspect ratio of the elliptic hole increases, it can be seen as a Mode-III blunt crack. Even for long blunt cracks, the surface effects can still be significant around the blunt crack tip. Finally, the equivalence between the uniform eigenstrain inside the inhomogeneity and the remote loading is discussed.     

Slow nonstationary vertical motions of a die on the surface of an elastic half-space

We consider the dynamic contact problem on vertical motions of an absolutely rigid body on an elastic half-space. We assume that the contact region does not vary during the motion and there is no friction under the die bottom. We construct an approximate solution of the problem under the assumption that the variation in the contact pressure under the die bottom on the time interval in which the Rayleigh wave runs the distance equal to the contact area diameter is small. Computational formulas are obtained for the cases of circular and elliptic dies.     

Interaction of an Ellipsoidal Inclusion with an Elliptic Crack in an Elastic Material under Triaxial Tension

The interaction of an elastic ellipsoidal inclusion with an elliptic crack in an infinite elastic medium under triaxial loading is analyzed. The stress state in the elastic space is represented as a superposition of the principal state and perturbed states, which are due to the presence and interaction of the inclusion and the crack. The analytical solution of the problem is found using the method of equivalent inclusion, the potential of an inhomogeneous ellipsoid, and a system of harmonic functions for an elliptic crack. The effect of triaxial loading on the stress intensity factors is analyzed     

Timoshenko inclusions in elastic bodies crossing an external boundary at zero angle

The paper concerns an analysis of equilibrium problems for 2D elastic bodies with a thin Timoshenko inclusion crossing an external boundary at zero angle. The inclusion is assumed to be delaminated, thus forming a crack between the inclusion and the body. We consider elastic inclusions as well as rigid inclusions. To prevent a mutual penetration between the crack faces, inequality type boundary conditions are imposed at the crack faces.Theorems of existence and uniqueness are established. Passages to limits are investigated as a rigidity parameter of the elastic inclusion going to infinity.     

带裂纹的椭圆孔口问题的应力分析

断裂现象与材料和结构中的孔洞、缺口或裂纹等缺陷密切相关,这是因为缺陷附近的应力集中明显.该文利用复变方法,通过保角映射研究了带裂纹的椭圆孔洞的平面弹性问题,给出了应力强度因子的解析解.并由此计算了两互相垂直的裂纹问题.     

Piezothermoelastic analysis of a piezoelectric material with an elliptic cavity under uniform heat flow

Summary The problem of a two-dimensional piezoelectric material with an elliptic cavity under a uniform heat flow is discussed, based on the modified Stroh formalism for the piezothermoelastic problem. The exact electric boundary conditions at the rim of the hole are introduced in the analysis. Expressions for the elastic and electric variables induced within and outside the cavity are derived in closed form. Hoop stress around the hole and electric fields in the hole are obtained. The limit situation when the hole is reduced to a slit crack is discussed, and the intensity factors for the problem are obtained. Received 14 April 1998; accepted for publication 25 June 1998     

Stress and electric displacement analyses in piezoelectric media with an elliptic hole and a small crack

In this paper, the interactions between an elliptic hole and an arbitrary distributed small crack in plane piezoelectric medium, which are often happened in engineering problems, are discussed. The Green’s functions in a piezoelectric plate with an elliptic hole for a generalized line dislocation and a generalized line force are presented. The small crack is represented by unknown continuous distributed dislocations. By considering traction free conditions on the surface of the small crack, the problem is then reduced to a group of singular integral equations which are solved by using a special numerical technique. Accuracy of the present method is confirmed by comparing the numerical results with those in literatures for PZT-4 when the elliptic hole is degenerated into a crack. The generalized stress intensity factors of cracks and the generalized stress on the edge of the elliptic hole are shown graphically. It is shown that the small crack may have shielding or amplifying effects on the main elliptic hole or crack, which depends on the location and orientation of the small crack. The hole near a crack can significantly reduce the stress intensity factor of the crack. The direction of the electric field is important to shielding effect.     

A new solution for thermopiezoelectric solid with an insulated elliptic hole

A new solution is obtained for thermal analysis of insulated elliptic hole embedded in an infinite thermopiezoelectric plate. In contrast to our previous results, the present formulation is based on the use of exact electric boundary conditions at the rim of the hole, thus avoiding the common assumption of electric impermeability. Using Lekhnitskii's formulation and conformal mapping, the elastic and electric fields can be expressed in a closed form in terms of complex potentials. The solutions for the crack problem are obtained by setting the minor axis of the ellipse approach to zero. As a consequence, the stress and electric displacement (SED) intensity factors and strain energy release rate can be derived analytically. One numerical example is considered to illustrate the application of the proposed formulation and compare with those obtained from impermeable model. The work was performed with the support of Australian Research Council Foundation.     

Physically and geometrically nonlinear deformation of conical shells with an elliptic hole

The elastoplastic state of conical shells weakened by an elliptic hole and subjected to finite deflections is studied. The material of the shells is assumed to be isotropic and homogeneous; the load is constant internal pressure. The problem is formulated and a technique for numerical solution with allowance for physical and geometrical nonlinearities is proposed. The distribution of stresses, strains, and displacements along the hole boundary and in the zones of their concentration is studied. The solution obtained is compared with the solutions of the physically and geometrically nonlinear problems and a numerical solution of the linear elastic problem. The stress-strain state around an elliptic hole in a conical shell is analyzed considering both nonlinearities __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 2, pp. 69–77, February 2008.