Singular integral representation of three-dimensional plasticity fracture problem

The stress field for the three-dimensional plasticity fracture problem is formulated by reducing the formulation to a system of singular integral equations. This will be referred to as the Singular Integral Operators Method (SIOM).An application is given to the determination of the stress field for a double notched tensile specimen in the case of plane stress and plane strain.     

Exact analytical solution of the MHD Jeffery-Hamel flow problem

This article shows that the well known nonlinear boundary value problem namely MHD Jeffery-Hamel flow problem, investigated in recent years by many numerical and semi-analytical approximative methods, is exactly solvable and furthermore, gives analytical exact solution in the implicit form for further physical interpretation.     

A THREE-DIMENSIONAL SOLUTION OF THE PROBLEM OF THREE FIXED CENTRES

ＡＴＨＲＥＥ－ＤＩＭＥＮＳＩＯＮＡＬＳＯＬＵＴＩＯＮＯＦＴＨＥＰＲＯＢＬＥＭＯＦＴＨＲＥＥＦＩＸＥＤＣＥＮＴＲＥＳ￥ＱｉｎＹｉｐｉｎｇ（覃一平）ＮｉｅＺｈａｏｍｉｎｇ（聂昭明）（ＹｕｎｎａｎＯｂｓｅｒｖａｔｏｒｙ，ＡｃａｄｅｍｉａＳｉｎｉｃａ，Ｋｕｎ...     

Exact solution of rotating FGM shaft problem in the elastoplastic state of stress

Plane strain analytical solutions to estimate purely elastic, partially plastic and fully plastic deformation behavior of rotating functionally graded (FGM) hollow shafts are presented. The modulus of elasticity of the shaft material is assumed to vary nonlinearly in the radial direction. Tresca’s yield criterion and its associated flow rule are used to formulate three different plastic regions for an ideal plastic material. By considering different material compositions as well as a wide range of bore radii, it is demonstrated in this article that both the elastic and the elastoplastic responses of a rotating FGM hollow shaft are affected significantly by the material nonhomogeneity.     

Exact quasi-steady solution of the problem of hydraulic fracturing of a permeable formation

An exact solution of the problem of hydraulic fracturing in a permeable medium with continuous fluid injection in a partially penetrated formation is constructed using the Perkins-Kern fracture model. The amount of fluid leakage from the fracture is determined using the pressure field of the fluid filtrate defined by the Shchelkachev equation (of the piezoconductivity type). Universal profiles of the fluid pressure in the fracture and the rate of fluid flow from it are obtained. It is shown that at the Perkins-Kern fracture tip, there is a dramatic increase in the leakage from the fracture.     

Exact solution of 3D Timoshenko beam problem: problem-dependent formulation

Problem-dependent interpolation functions for displacements and rotations are obtained from the exact analytical solution of the 3D Timoshenko beam problem by introducing a full set of boundary conditions. The developed methodology allows us to derive a new solution that coincides with the classical result of the engineering beam theory. In addition, the proposed interpolation enables exact strain recovery at any point within the problem domain.     

Exact statement of the instability problem for frames and its direct numerical solution

A general approach for the systematic evaluation of the critical buckling load and the determination of the buckling mode is presented. The Navier-Bernoulli beam model is considered, having the possibility of variable cross-section under any type of load (including pressures and thermal loading). With this purpose, the equilibrium equations of each beam element in its deformed configuration under the hypothesis of infinitesimal strains and displacements is considered, resulting in a system of differential equations with variable coefficients for each element. To obtain the nonlinear response of the frame, one should impose the compatibility of displacements and the equilibrium of forces and moments in each beam-end, also in the deformed configuration. The solution is obtained by requiring that the total variation of potential energy is zero at the instant of buckling. The objective of this work is to develop a systematic method to determine the critical buckling load and the bucklingmode of any frame without using the common simplifications usually assumed in matrix analysis or finite element approaches. This way, precise results can be obtained regardless of the discretization done.     

Exact solution of the Muskat–Leibenzon problem for a growing elliptic bubble

The exact solution of the two-phase time-dependent Hele–Shaw problem (in other words, the plane Muskat–Leibenzon problem) in which a fluid occupied an unbounded channel is displaced by another fluid incoming through a slitted cut in the channel. In this case the interface between the phases, namely, fluids of different viscosity, evolves as an ellipse whose area and eccentricity vary continuously.     

Asymptotic solution of the axisymmetric problem of ideal fluid flow in the neighborhood of cusped cavities

A denumerable set of axisymmetric solutions of the equations of motion of an ideal fluid in the neighborhood of a cavity closure point, including an infinitely remote point, is obtained. This family of solutions is similar to the well-known family of corresponding two-dimensional solutions (1), (2).Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 84–87, July–August, 1990.     

Reproduction of solutions of bidimensional ideal plasticity

The symmetries of a system of differential equations allowed the transformation of its solutions to a solution of this system. New analytical exact solutions of a system of two-dimensional ideal plasticity equations were constructed from two well-known solutions, that for a circular cavity stressed by normal pressure, and Prandtl's solution for a block compressed between perfectly rough plates, for the case where the thickness of the block was rather small. A mechanical sense of new solutions was discussed.     

Exact solution of problem of deflection of an anisotropic plate with an aperture

The exact solution of the problem of the deflection of an anisotropic plate weakened by an aperture is known only for the case in which the aperture has the shape of a circle or an ellipse [1, 2]. An exact solution has not been derived for any other aperture shapes. Approximate methods [3–6] which are widespread for the case of multiply connected anisotropic plates [7] are applied to the determination of the bending moments in an anisotropic plate near an aperture differing little from an elliptical or circular one.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 168–177, September–October, 1977.