Problems of a cut in a three-dimensional elastic wedge

The Ritz variational method is applied to problems of a crack (a cut) in the middle half-plane of a three-dimensional elastic wedge. The faces of the elastic wedge are either stress-free (Problem A) or are under conditions of sliding or rigid clamping (Problems B and C respectively). The crack is open and is under a specified normal load. Each of the problems reduces to an operator integrodifferential equation in relation to the jump in normal displacement in the crack area. The method selected makes it possible to calculate the stress intensity factor at a relatively small distance from the edge of the wedge to the cut area. Numerical and asymptotic solutions [Pozharskii DA. An elliptical crack in an elastic three-dimensional wedge. Izv. Ross Akad. Nauk. MTT 1993;(6):105–12] for an elliptical crack are compared. In the second part of the paper the case of a cut reaching the edge of the wedge at one point is considered. This models a V-shaped crack whose apex has reached the edge of the wedge, giving a new singular point in the solution of boundary-value problems for equations of elastic equilibrium. The asymptotic form of the normal displacements and stress in the vicinity of the crack tip is investigated. Here, the method employed in [Babeshko VA, Glushkov YeV, Zinchenko ZhF. The dynamics of Inhomogeneous Linearly Elastic Media. Moscow: Nauka; 1989] and [Glushkov YeV, Glushkova NV. Singularities of the elastic stress field in the vicinity of the tip of a V-shaped three-dimensional crack. Izv. Ross Akad. Nauk. MTT 1992;(4):82–6] to find the operator spectrum is refined. The new basis function system selected enables the elements of an infinite-dimensional matrix to be expressed as converging series. The asymptotic form of the normal stress outside a V-shaped cut, which is identical with the asymptotic form of the contact pressure in the contact problem for an elastic wedge of half the aperture angle, is determined, when the contact area supplements the cut area up to the face of the wedge.     

The three-dimensional problem of a thin inclusion in a composite elastic wedge

The three-dimensional problem of a thin rigid elliptic inclusion in the middle of a composite elastic wedge is investigated. The wedge consists of three connected wedge-shaped layers connected by a sliding clamp, in which the layer containing the inclusion is incompressible. The outer faces of the composite wedge are also under sliding-clamp conditions. The inclusion is completely bonded to the elastic medium in the contact region. Using Fourier and Kontorovich–Lebedev transformations, a system of integral equations of the problems are derived for the shear contact stresses. A regular asymptotic method is used to solve this system. Calculations are carried out. The results can be used for calculations on the strength of rubber-metal articles and structures having a corner line.     

Problems of cuts in a composite elastic wedge

Problems of strip and elliptical cuts (tensile cracks) in the middle of a three-layer elastic wedge are investigated in a three-dimensional formulation. Free or rigid clamping conditions or the stress-free condition are stipulated on the outer surfaces of the composite wedge. The problems are assumed to be symmetrical about the plane of the cut. The wedge-shaped layer containing the cut is incompressible and hinged along both faces with two other layers. The integral equations of the problems with respect to the opening of the cut are derived. Inverse operators are obtained for the operators occurring in the kernels of these equations. The relation between problems on cuts and the corresponding contact problems for a composite wedge of half the aperture angle is used. The method of paired integral equations is used for the case of a strip cut emerging from the edge of the wedge. The problems are reduced to Fredholm integral equations of the second kind in certain auxiliary functions, in terms of the values of which the normal stress intensity factors are expressed. A regular asymptotic solution is constructed for the case of an elliptic cut.     

Three-dimensional contact problems with friction for a composite elastic wedge

Contact problems for a composite elastic wedge in the form of two joined wedge-shaped layers with different aperture angles joined by a sliding clamp, where the layer under the punch is incompressible, are studied in a three-dimensional formulation. Conditions for a sliding or rigid clamp or the absence of stresses are set up on one face of the composite wedge. The integral equations of the problems are derived taking account of the friction forces perpendicular to the edge of the wedge. The method of non-linear boundary integral equations of the Hammerstein type is used when the contact area is unknown. A regular asymptotic solution is constructed for an elliptic contact area. By virtue of the incompressibility of the material of the layer in contact with the punch, this solution retains the well known root singularity in the boundary of the contact area when account is taken of friction.     

Three-dimensional contact problems for an elastic wedge with a coating

Three-dimensional contact problems for an elastic wedge, one face of which is reinforced with a Winkler-type coating with different boundary conditions on the other face of the wedge, are investigated. A power-law dependence of the normal displacement of the coating on the pressure is assumed. The contact area, the pressure in this region, and the relation between the force and the indentation of a punch are determined using the method of non-linear boundary integral equations and the method of successive approximations. The results of calculations are analysed for different values of the aperture angle of the wedge, the relative distance of the punch from the edge of the wedge, the ratio of the radii of curvature of the punch (an elliptic paraboloid), and the non-linearity factors of the coating. The results obtained are compared with the solutions of similar problems for a wedge without a coating.     

The interaction of punches on the face of an elastic wedge

The interaction of two punches, which are elliptic in plan, on the face of an elastic wedge is investigated in a three-dimensional formulation for different types of boundary conditions on the other face. The wedge material is assumed to be incompressible. An asymptotic solution is obtained for punches which are relatively distant from one another and from the edge of the wedge. For the case when the punches are arranged relatively close to the edge of the wedge (or reach the edge, the contact area is unknown) the numerical method of boundary integral equations is used. The mutual effect of the punches is estimated by means of calculations. The asymptotic solution of the generalized Galin problem, concerning the effect of a concentrated force applied on the edge of the three-dimensional wedge on the contact pressure distribution under a circular punch relatively far from the edge, is obtained.     

Application of the method of direct cutting-out to the solution of the problem of longitudinal shear of a wedge with thin heterogeneities of arbitrary orientation

The method of direct cutting-out consists of modeling of a finite body, in particular, with thin heterogeneities, using a much simpler problem for a bounded or a partially bounded body with thin heterogeneities located in the same manner and the presence of additional cracks or absolutely rigid inclusions of fairy large length, which are modeled by the boundary conditions of a bounded body. The method is tested on the problems of antiplane deformation of a symmetrically loaded crack in a wedge with free faces and an absolutely rigid inclusion placed with some tension in a wedge with restrained faces. For an elastic inclusion, we construct generalized conditions of interaction, which enable us to unify the procedure of giving different boundary conditions in the case of using the method of direct cutting-out.     

A linear approximation for the regular reflection of a weak shock at a wedge satisfying sonic condition

The problem of shock reflection by a wedge, which the flow is dominated by the unsteady potential flow equation, is a important problem. In weak regular reflection, the flow behind the reflected shock is immediately supersonic and becomes subsonic further downstream. The reflected shock is transonic. Its position is a free boundary for the unsteady potential equation, which is degenerate at the sonic line in self-similar coordinates. Applying the special partial hodograph transformation used in [Zhouping Xin, Huicheng Yin, Transonic shock in a nozzle I, 2-D case, Comm. Pure Appl. Math. 57 (2004) 1-51; Zhouping Xin, Huicheng Yin, Transonic shock in a nozzle II, 3-D case, IMS, preprint (2003)], we derive a nonlinear degenerate elliptic equation with nonlinear boundary conditions in a piecewise smooth domain. When the angle, which between incident shock and wedge, is small, we can see that weak regular reflection as the disturbance of normal reflection as in [Shuxing Chen, Linear approximation of shock reflection at a wedge with large angle, Comm. Partial Differential Equations 21 (78) (1996) 1103-1118]. By linearizing the resulted nonlinear equation and boundary conditions with above viewpoint, we obtain a linear degenerate elliptic equation with mixed boundary conditions and a linear degenerate elliptic equation with oblique boundary conditions in a curved quadrilateral domain. By means of elliptic regularization techniques, delicate a priori estimate and compact arguments, we show that the solution of linearized problem with oblique boundary conditions is smooth in the interior and Lipschitz continuous up to the degenerate boundary.     

A linear approximation for the regular reflection of a weak shock at a wedge

The problem of shock reflection by a wedge in the flow dominated by the unsteady potential flow equation is an important problem. In weak regular reflection, the flow behind the reflected shock is immediately supersonic and becomes subsonic further downstream. The reflected shock is transonic. Its position is a free boundary for the unsteady potential equation, which is degenerate at the sonic line in self-similar coordinates. Applying the special partial hodograph transformation used in [Zhouping Xin, Huicheng Yin, Transonic shock in a nozzle I, 2-D case, Comm. Pure Appl. Math. LVII (2004) 1-51; Zhouping Xin, Huicheng Yin, Transonic shock in a nozzle II, 3-D case, IMS, preprint, 2003], we derive a nonlinear degenerate elliptic equation with nonlinear boundary conditions in a piecewise smooth domain. When the angle between incident shock and wedge is small, we can see the weak regular reflection as the disturbance of normal reflection as in [Chen Shuxing, Linear approximation of shock reflection at a wedge with large angle, Comm. Partial Differential Equations 21(78) (1996) 1103-1118]. By linearizing the resulted nonlinear equation and boundary conditions with the above viewpoint in [Chen Shuxing, Linear approximation of shock reflection at a wedge with large angle, Comm. Partial Differential Equations 21(78) (1996) 1103-1118], we obtain a linear degenerate elliptic equation with mixed boundary conditions in a curved quadrilateral domain. By means of elliptic regularization techniques, a delicate a priori estimate and compact arguments, we show that the solution of the linearized problem is smooth in the interior and Lipschitz continuous up to the degenerate boundary.     

Integral equations of dynamic problems for multilayered media containing a system of cracks

A new method of determining the dynamic characteristics of multilayered semi-bounded media with defects of the inclusion or crack type at the layer interfaces [1] is used to solve antiplane problems. Systems of integral equations of the corresponding boundary-value problems are constructed and the properties of their kernels are investigated. The dispersion curves of the determinants and matrix elements of these systems are analysed as functions of the number of layers and their elastic and geometric characteristics.     

Action of an elliptic stamp moving at a constant speed on an elastic half-space : PMM vol. 42, no. 6, 1978, pp 1074–1079

The action of a rigid stamp moving at a constant speed, on the boundary of an elastic half-space, is investigated. It is assumed that the frictional forces between the stamp and the surface of the half-space are absent. The integral equation obtained in [1] yields formulas for the pressure, for the case when the area of contact between the stamp and the half-space has an elliptic form.     

Limits of relaxed dirichlet problems involving a non symmetric dirichlet form

In this paper we study the convergence of solutions of a sequence of relaxed Dirichlet problems relative to non-symmetric Dirichlet forms. The techniques rely on the study of the behaviour of the solutions of the adjoint problems, as suggested by G. Dal Maso and A. Garroni in [16] in the case of linear elliptic operators of second order with bounded measurable coefficients. In particular we prove a compactness results due to Mosco [31] in the symmetric case. Entrata in Redazione il 18 gennaio 1999     

Three-dimensional contact problem for a prestressed elastic body : PMM vol. 42, no. 6, 1978, pp 1080–1084

A half-space of an incompressible neo — Hookean [1,2] material subjected to a homogeneous bi-axial tension or compression along its boundary, is considered. A small deformation caused by the action of a smooth rigid stamp on the boundary of the half-space is superimposed on the initial finite deformation. An integral equation is obtained for the contact pressure. A solution of this equation is obtained for an inclined elliptic stamp with a flat base, and for an elliptic stamp with a curved base, for the cases when the extension coefficients in two directions are either identical, or differ little from each other. The influence of the inital loading on the distribution of the contact pressure, the displacement of the stamp and the form of the contact zone, is analysed.     

New Proofs of Monotonicity of Period Function for Cubic Elliptic Hamiltonian

In [1] S.-N. Chow and J. A. Sanders proved that the period function is monotone for elliptic Hamiltonian of degree 3. In this paper we significantly simplify their proof, and give a new way to prove this fact, which may be used in other problems.     

The interaction between a penny-shaped crack and a spherical inclusion under torsion

The axisymmetric interaction problem of an elastic spherical inclusion with a penny-shaped crack in an elastic space under torsion is considered. The superposition and reflection methods [3]-[4] are used to solve the mixed boundary value problem in question. With the help of the dual integral equations technique and appropriate re-expansion of the eigenfunction, the problem is reduced to an infinite system of linear algebraic equations of the second kind. The matrix elements of that system decrease exponentially along the rows and the columns. Its unique solution is proved to exist in a proper class of sequences and is shown to be represented by a convergent, in the vicinity of the origin, power series in a geometric parameter, equal to the ratio of the radius of the inclusion to its distance from the crack. This procedure provides an efficient formula for the stress intensity factor.     

准各向同性复合材料弹性和强度的各向异性效应(Ⅱ)──应用与细观力学分析*

文献中尚未见到针对准各向同性复合材料的各向异性效应对复合材料结构影响的分析。本文在第(Ⅰ)部分[1]所提出的强度准则模型的基础上,给出了复合材料各向异性特性在含椭圆孔和单个裂纹问题中的具体应用。在椭圆孔问题中分析了远场载荷随材料几何参数变化的规律;在含裂纹问题中分析了裂纹扩展方向随裂纹方向的变化规律。最后,用细观力学方法分析了一类三轴编织复合材料的弹性本构方程和强度准则,以及各向异性效应,得到了与实验和第(Ⅰ)部分理论模型相一致的结果。     

An asymptotic method in contact problems

A modification of the “smal λ” singular asymptotic method of solving the integral equations of mixed problems in continuum mechanics [1] is proposed in the case of a special behaviour of the symbol of the kernel encountered, for example, in contact problems of the theory of elasticity for cylindrical and conical bodies [2–4]. Contact problems for elastic cylindrical bodies are considered as an example.     

Static and extending interface cracks with contact zones in anisotropic as well as in piezoelectric bimaterials

An interface crack with an electrically permeable and mechanically frictionless contact zone in a piezoelectric bimaterial under the action of a remote mixed mode mechanical loading as well as thermal and electrical fields is considered in the first part of this paper. By use of the matrix‐vector representations of thermal, mechanical and electrical fields via sectionally‐holomorphic functions the problems of linear relationships are formulated and solved exactly both for an electrically permeable and an electrically impermeable interface crack. For these cases the transcendental equations and clear analytical formulas are derived for the determination of the contact zone lengths and the associated fracture mechanical parameters. A plane strain problem for a crack with a frictionless contact zone at the leading crack tip extending stationary along an interface of two semi‐infinite anisotropic spaces with a subsonic speed under the action of various loading is considered in the second part of this paper. By introducing of a moving coordinate system connected with the crack tip and by using the formal similarity of static and propagating crack problems the combined Dirichlet‐Riemann boundary value problem is formulated and solved exactly for this case as well and a transcendental equation is obtained for the determination of the real contact zone length. It is found that the increase of the crack speed leads to an increase of the real contact zone length and the correspondent stress intensity factors which increase significantly for a quasi‐Rayleigh wave speed.     

Antiplane periodic contact problems fora layer non-uniform along its thickness

Antiplane periodic contact problems for an elastic layer with a shear modulus which variesexponentially along its thickness are considered. The problems are reduced to an integral equation of the first kind with an irregular, periodic, difference kernel. A method which has been described previously [1,2] is used for the approximate solution of this equation.     

Quaternions and applications

This note gives an overview of two novel applications of Quaternions which appeared in [1]–[3]: First, the evaluation of certain three dimensional real integrals without integrating with respect to the real variables. This is the generalisation of the well-known Cauchy Residue Theorem from the case of two dimensions to the case of four dimensions. Second, the solution of boundary value problems for linear elliptic PDEs in four dimensions. This is the extension of some of the results of [4] from two to four dimensions. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)