Antiplane strain of a body undergoing large-rotations

Antiplane strain of a cylindrical elastic body undergoing large rotations under surface load in the absence of body loads is studied. The form of the elastic potential corresponding to this strain is found. The stresses, the strains, and the displacement are expressed in terms of pressure and two independent strains and the pressure is expressed in terms of the linear strain invariant. For the strains and displacement, nonlinear boundary-value problems are formulated and their ellipticity conditions are given. The linear problem for the displacement is obtained by transformation of variables. An example of determining the displacement is considered. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 3, pp. 191–198, May–June, 2007.     

Use of Tikhonov’s regularization method for solving identification problem for elastic systems

We present a method for solving nonlinear inverse problems, which also include identification problems for elastic systems. The problems whose initial data contain an error are usually solved by regularization methods [1–5]. In the present paper, we give preference to Tikhonov’s regularization method, which has been widely used in the recent years in practice to increase the stability of computational algorithms for solving problems in various areas of mechanics [6–9].     

Elastoplastic state of flexible cylindrical shells with a circular hole under axial tension

The elastoplastic state of cylindrical shells with a circular hole is studied considering finite deflections. The material of the shells is isotropic and homogeneous; the load is axial tension. The distribution of stresses, strains, and displacements along the hole boundary and in the zone of their concentration is studied by solving a doubly nonlinear problem. The data obtained are compared with the solutions of the physically and geometrically nonlinear problems and a numerical solution of the linear elastic problem __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 7, pp. 101–109, July 2008.     

Plane strain problem for an incompressible nonlinearly elastic solid

The plane strain of an incompressible body is studied with geometrical and physical nonlinearity and potential forces taken into account. A nonlinear system of equations for strains is obtained in actual variables, and conditions of its ellipticity are derived in terms of the elastic potential. Boundary conditions for strains are found from specified loads. Analytical solutions of the boundary problem in strains and their corresponding stress fields are found for the case of identical elongations. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 2, pp. 217–225, March–April, 2009     

Mechanical Performance of a Recycled Carbon Fibre/PP Composite

A composite made of recycled carbon fibres in recycled polypropylene matrix is studied experimentally to describe the features of the elastic and time dependent nonlinear mechanical behaviour. The properties of the developed material have a large variability to be addressed and understood. It was found that the stress-strain curves in tension are rather nonlinear at low strain rate and the strength is sensitive to strain rate. The elastic properties’ reduction for this composite after loading to high strains is rather limited. More important is that even in the “elastic region” due to viscoelastic effects the slope of loading–unloading curve is not the same and that at higher stress large viscoplastic strains develop and creep rupture is typical. The time and stress dependence of viscoplastic strains was analysed and described theoretically. The viscoelastic response of the composite was analysed using creep compliance, which was found to be slightly nonlinear.     

Stress distribution in physically and geometrically nonlinear thin cylindrical shells with two holes

The elastoplastic state of thin cylindrical shells weakened by two circular holes is analyzed. The centers of the holes are on the directrix of the shell. The shells are made of an isotropic homogeneous material and subjected to internal pressure of given intensity. The distribution of stresses along the hole boundaries and over the zone where they concentrate (when the distance between the holes is small) is analyzed using approximate and numerical methods to solve doubly nonlinear boundary-value problems. The data obtained are compared with the solutions of the physically nonlinear (plastic strains taken into account) and geometrically nonlinear (finite deflections taken into account) problems and with the numerical solution of the linearly elastic problem. The stress-strain state near the two holes is analyzed depending on the distance between them and the nonlinearities accounted for __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 11, pp. 88–95, November 2005.     

Stability of composite shells of revolution

An approach to solving the buckling problem for shells made of a composite material with one plane of elastic symmetry is presented. The approach employs complex Fourier series. The prebuckling stress-strain state is assumed to be geometrically nonlinear. The stability of a cylindrical shell under axial compression and uniform side pressure is analyzed using the Runge-Kutta method with discrete orthogonalization. The numerical results are compared with analytical solutions __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 3, pp. 109–124, March 2008.     

Physically and Geometrically Nonlinear Deformation of Spherical Shells with an Elliptic Hole

The elastoplastic state of thin spherical shells with an elliptic hole is analyzed considering that deflections are finite. The shells are made of an isotropic homogeneous material and subjected to internal pressure of given intensity. Problems are formulated and a numerical method for their solution with regard for physical and geometrical nonlinearities is proposed. The distribution of stresses (strains or displacements) along the hole boundary and in the zone of their concentration is studied. The results obtained are compared with the solutions of problems where only physical nonlinearity (plastic deformations) or geometrical nonlinearity (finite deflections) is taken into account and with the numerical solution of the linearly elastic problem. The stress—strain state in the neighborhood of an elliptic hole in a shell is analyzed with allowance for nonlinear factors __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 6, pp. 95–104, June 2005.     

Independent‐Stress Method for Analysis of Nonlinear Antiplane Strain

The stress field in a cylindrical body under antiplane strains is studied using the nonlinear theory of elasticity in actual variables under assumptions of the absence of body forces and weak nonlinearity of the elastic potential. The stresses are determined by solving the nonlinear boundaryvalue problem for two independent stresses in polar coordinates of the physical and stress planes. Analytical solutions of the nonlinear problems are obtained. The effect of potential nonlinearity is studied. It is shown that the nonlinear problem can be solved using the harmonicequation solution corresponding to the linear potential.     

Nonlinear Multigrid Methods for Numerical Solution of the Unsaturated Flow Equation in Two Space Dimensions

Picard and Newton iterations are widely used to solve numerically the nonlinear Richards’ equation (RE) governing water flow in unsaturated porous media. When solving RE in two space dimensions, direct methods applied to the linearized problem in the Newton/Picard iterations are inefficient. The numerical solving of RE in 2D with a nonlinear multigrid (MG) method that avoids Picard/Newton iterations is the focus of this work. The numerical approach is based on an implicit, second-order accurate time discretization combined with a second-order accurate finite difference spatial discretization. The test problems simulate infiltration of water in 2D unsaturated soils with hydraulic properties described by Broadbridge–White and van Genuchten–Mualem models. The numerical results show that nonlinear MG deserves to be taken into consideration for numerical solving of RE.     

Mechanics of crack propagation in materials with initial (residual) stresses (review)

Major results on the mechanics of crack propagation in materials with initial (residual) stresses are analyzed. The case of straight cracks of constant width that propagate at a constant speed in a material with initial (residual) stresses acting along the cracks is examined. The results were obtained, based on linearized solid mechanics, in a universal form for isotropic and orthotropic, compressible and incompressible elastic materials with an arbitrary elastic potential in the cases of finite (large) and small initial strains. The stresses and displacements in the linearized theory are expressed in terms of analytical functions of complex variables when solving dynamic plane and antiplane problems. These complex variables depend on the crack propagation rate and the material properties. The exact solutions analyzed were obtained for growing (mode I, II, III) cracks and the case of wedging by using methods of complex variable theory, such as Riemann–Hilbert problem methods and the Keldysh–Sedov formula. As the initial (residual) stresses tend to zero, these exact solutions of linearized solid mechanics transform into the respective exact solutions of classical linear solid mechanics based on the Muskhelishvili, Lekhnitskii, and Galin complex representations. New mechanical effects in the dynamic problems under consideration are analyzed. The influence of initial (residual) stresses and crack propagation rate is established. In addition, the following two related problems are briefly analyzed within the framework of linearized solid mechanics: growing cracks at the interface of two materials with initial (residual) stresses and brittle fracture under compression along cracks     

Ellipticity conditions of the static equations of nonlinear elasticity

The relations of the nonlinear model of the theory of elasticity are considered. The Cauchy and the strain gradient tensors are taken to be the characteristics of the stress-strain state of a body. Sufficient conditions under which the static equations of elasticity are of elliptic type are established. These conditions are expressed in the form of constraints imposed on the derivatives of the elastic potential with respect to the strain-measure characteristics. The cases of anisotropic and isotropic bodies are treated, including the case where the Almansi tensor is taken to be the strain measure. The plane strain of a body is investigated using actual-state variables. Novosibirsk State University, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 2, pp. 196–203, March–April, 1999.     

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.     

Inelastic Deformation of Flexible Spherical Shells with Two Circular Openings

Elastoplastic analysis of thin-walled spherical shells with two identical circular openings is carried out with allowance for finite deflections. The shells are made of an isotropic homogeneous material and subjected to internal pressure of known intensity. The distributions of stresses (strains or displacements) along the contours of the openings and in the zone of their concentration are studied by solving doubly nonlinear boundary-value problems. The solution obtained is compared with the solutions that account for only physical nonlinearity (plastic deformations) and only geometrical nonlinearity (finite deflections) and with a numerical solution of the linearly elastic problem. The stress–strain state near the two openings is analyzed depending on the distance between the openings and the nonlinear factors accounted for     

A SOLUTION PROCEDURE FOR LOWER BOUND LIMIT AND SHAKEDOWN ANALYSIS BY SGBEM

The symmetric Galerkin boundary element method (SGBEM) instead of the finite element method is used to perform lower bound limit and shakedown analysis of structures. The self-equilibrium stress fields are constructed by a linear combination of several basic self-equilibrium stress fields with parameters to be determined. These basic self-equilibrium stress fields are expressed as elastic responses of the body to imposed permanent strains and obtained through elastic-plastic incremental analysis. The complex method is used to solve nonlinear programming and determine the maximal load amplifier. The limit analysis is treated as a special case of shakedown analysis in which only the proportional loading is considered. The numerical results show that SGBEM is efficient and accurate for solving limit and shakedown analysis problems. Project supported by the National Natural Science Foundation of China (No. 19902007), the National Foundation for Excellent Doctorial Dissertation of China (No. 200025) and the Basic Research Foundation of Tsinghua University.     

The stress state of a laminated nonlinearly elastic thick-walled spherical shell

This paper deals with spatial axisymmetric boundary-value problems of the physically nonlinear theory of elasticity for piecewise-homogeneous spherical bodies. The passage to dimensionless characteristics of the stress-strain state allows us to extract a physical dimensionless small parameter in the nonlinear state equations. The solution of nonlinear equilibrium equations and boundary-value problems is searched for in the form of series in positive degrees of the small parameter. This approach allows reducing the stated physically nonlinear boundary-value problem to a sequence of corresponding linear nonhomogeneous problems. A specific analytical solution and numerical results are obtained for a two-layer nonlinearly elastic spherical shell under bilateral pressure. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 35, No. 12, pp. 26–32, December, 1999.     

On the stress state of a piezoceramic body with a flat crack under symmetric loads

A static-equilibrium problem is solved for an electroelastic transversely isotropic medium with a flat crack of arbitrary shape located in the plane of isotropy. The medium is subjected to symmetric mechanical and electric loads. A relationship is established between the stress intensity factor (SIF) and electric-displacement intensity factor (EDIF) for an infinite piezoceramic body and the SIF for a purely elastic material with a crack of the same shape. This allows us to find the SIF and EDIF for an electroelastic material directly from the corresponding elastic problem, not solving electroelastic problems. As an example, the SIF and EDIF are determined for an elliptical crack in a piezoceramic body assuming linear behavior of the stresses and the normal electric displacement on the crack surface __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 11, pp. 67–77, November 2005.     

Domain decomposition methods applied to solve frictionless-contact problems for multilayer elastic bodies

A parallel Dirichlet–Dirichlet domain-decomposition algorithm for solving frictionless-contact problems for elastic bodies made of composite materials is proposed and justified. Numerical results that demonstrate the effectiveness of the approach and its software implementation are presented     

Elastoplastic state of flexible cylindrical shells with two circular holes

The elastoplastic state of thin cylindrical shells with two equal circular holes is analyzed with allowance made for finite deflections. The shells are made of an isotropic homogeneous material. The load is internal pressure of given intensity. The distribution of stresses along the hole boundary and in the stress concentration zone (when holes are closely spaced) is analyzed by solving doubly nonlinear boundary-value problems. The results obtained are compared with the solutions that allow either for physical nonlinearity (plastic strains) or geometrical nonlinearity (finite deflections) and with the numerical solution of the linearly elastic problem. The stresses near the holes are analyzed for different distances between the holes and nonlinear factors.Translated from Prikladnaya Mekhanika, Vol. 40, No. 10, pp. 107–112, October 2004.     

Inelastic deformation of flexible cylindrical shells with an elliptic hole

The elastoplastic state of isotropic homogeneous cylindrical shells with elliptic holes and finite deflections under internal pressure is studied. Problems are formulated and numerically solved taking into account physical and geometrical nonlinearities. The distribution of stresses (displacements, strains) along the boundary of the hole and in the zone of their concentration is analyzed. The data obtained are compared with the numerical solutions of the physically nonlinear, geometrically nonlinear, and linear problems. The stress-strain state of cylindrical shells in the neighborhood of the elliptic hole is analyzed with allowance for nonlinear factors __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 5, pp. 46–54, May 2007.