One formulation of the problem of elastoplastic separation

The problem of the beginning of motion of a cut in a plane under symmetric external loading is considered. The material lying on the cut continuation forms a layer (interaction layer). A transition to a plastic state within the layer is assumed to be possible. The behavior of the layer is described by an ideally elastoplastic model, and the plane outside the layer is assumed to be linearly elastic. A system of boundary integral equations for determining the stress-strain state is derived. Based on this system, a discrete model of separation of the layer material is constructed under the assumption of a constant stress-strain state in the element of the interaction layer. The distribution of stresses in the pre-fracture zone is determined. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 4, pp. 187–195, July–August, 2009     

Finding the elastic strain limit at the tip region of a physical cut with arbitrarily loaded faces

A model describing the stress-strain state in the neighborhood of a physical cut with an arbitrary distribution of external load along its faces is presented. The stress-strain state of a material layer bounded by the continuations of the cut faces is considered. The interaction between the layer and the external half-planes leads to a closed system of integrodifferential equations for the mean stress components in the layer, which splits into two equations for the mean normal stresses and an equation for the mean shear stress. Numerical solutions of the system for the cases of symmetric and antisymmetric loading of the faces by concentrated forces are given. Conditions for the transition of the tip region of the cut to the state of plasticity and fracture are considered.     

Solution of one problem of fracture mechanics

This paper considers a model for the opening-mode fracture separation process based on the introduction of an interaction layer. This layer is defined as the region of localization of the fracture process. The stress-strain state of the layer material is uniform in the cross section of the layer. A study is made of the deformation of a double-cantilever beam weakened by a notch whose width is equal to the thickness of the interaction layer. The problem is solved in a linearly geometrical approximation. The thickness of the interaction layer is estimated, and a method for solving the formulated problem is proposed. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 4, pp. 121–127, July–August, 2007.     

Creep of axisymmetrically loaded plates with allowance for damage accumulation in their material

The Rabotnov kinetic creep theory was used to calculate the stress-strain state and damage accumulation in the material of axisymmetrically loaded circular and ring plates at any time before the beginning of fracture. It is shown that the solution of the problem can be reduced to solving the same problem under the assumption of steady-state creep of the material. The unsteady creep problem is solved by multiplying the known solution of the steady-state creep problem by certain functions of the coordinates and time, which are determined from a corresponding system of equations. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 5, pp. 157–168, September–October, 2006.     

Mesomechanics of deformation and short-term damage of linear elastic homogeneous and composite materials

The structural theory of microdamage of homogeneous and composite materials is generalized. The theory is based on the equations and methods of the mechanics of microinhomogeneous bodies with stochastic structure. A single microdamage is modeled by a quasispherical pore empty or filled with particles of a damaged material. The accumulation of microdamages under increasing loading is modeled as increasing porosity. The damage within a single microvolume is governed by the Huber-Mises or Schleicher-Nadai failure criterion. The ultimate strength is assumed to be a random function of coordinates with power-law or Weibull one-point distribution. The stress-strain state and effective elastic properties of a composite with microdamaged components are determined using the stochastic equations of elasticity. The equations of deformation and microdamage and the porosity balance equation constitute a closed-form system of equations. The solution is found iteratively using conditional moments. The effect of temperature on the coupled processes of deformation and microdamage is taken into account. Algorithms for plotting the dependences of microdamage and macrostresses on macrostrains for composites of different structure are developed. The effect of temperature and strength of damaged material on the stress-strain and microdamage curves is examined __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 6, pp. 3–42, June 2007.     

Microdamage of a nonlinear elastic material in combined stress state

The structural theory of short-term damage is used to study the coupled processes of deformation and microdamage of a physically nonlinear material in a combined stress state. The basis for the analysis is the stochastic elasticity equations for a physically nonlinear porous medium. Damage in a microvolume of the material is assumed to occur in accordance with the Huber-Mises failure criterion. The balance equation for damaged microvolumes is derived and added to the macrostress-macrostrain relations to produce a closed-form system of equations. It describes the coupled processes of nonlinear deformation and microdamage of the porous material. Algorithms are developed for calculating the dependence of microdamage on macrostresses and macrostrains and plotting stress-strain curves for a homogeneous material under either biaxial normal loading or combined normal and tangential loading. The plots are analyzed depending on the type of stress state __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 11, pp. 30–39, November 2006.     

Radial expansion of a granular medium in spherical and cylindrical layers

An exact solution that describes the fields of displacements and stresses in an expanding spherical layer is constructed within the framework of the theory of small strains of a granular medium with rigid particles. For finite strains, the problem reduces to a nonlinear system of ordinary differential equations, which is solved by numerical methods. Similar solutions are found in the problem for a cylindrical layer. Based on these solutions, the effect of the dilatancy of the granular medium on the stress-strain state near expanding cavities is found. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 3, pp. 190–196, May–June, 2009.     

Analysis of the axisymmetric thermoelastoplastic state of branched laminated transversally isotropic shells

A technique is developed for determination of the axisymmetric thermoelastoplastic stress-strain state of branched laminated transversally isotropic shells of revolution under loads that cause the meridional stress state and torsion. The method is based on the rectilinear-element hypotheses for the whole package of layers. To describe the processes of active elastoplastic deformation of a transversally isotropic material, deformation-type equations, which are constructed without recourse to the plastic-potential existence condition, are used. The scalar functions in the constitutive equations depend on the shear-strain rate and temperature. The solution of the problem is reduced to numerical integration of systems of differential equations. An example of determination of the elastoplastic state of a two-layer cylindrical shell stiffened with a rigid ring support is presented. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 36, No. 4, pp. 125–131, April, 2000.     

Short-term microdamage of a physically nonlinear particulate material under a combination of normal and tangential loads

The structural theory of short-term damage is generalized to the case where the undamaged components of a particulate composite deform nonlinearly under loads that induce a compound stress state. The basis for this generalization is the stochastic elasticity equations for a particulate composite with porous components whose skeletons deform nonlinearly. Damage in a microvolume of the material is assumed to occur in accordance with the Huber-Mises failure criterion. Balance equations for damaged microvolume are derived for the physically nonlinear materials of the components. Together with the macrostress-macrostrain relationship for a particulate composite with porous nonlinear components, they constitute a closed-form system of equations. This system describes the coupled processes of physically nonlinear deformation and microdamage. Algorithms for calculating the microdamage-macrostrain relationship and plotting stress-strain curves are proposed. Such curves are plotted for the case where the composite is subjected to a combination of normal and tangential loads, and microdamages occur in the linearly hardened matrix and do not in the linearly elastic inclusions. The stress-strain curves are examined depending on the volume fraction of inclusions and presence of tangential stresses __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 12, pp. 48–57, December, 2006.     

STRESS-STRAIN STATE OF ELASTOPLASTIC BODIES WITH CRACK

A physical cut model is used to describe the changes in the stress-strain state(SSS)in elastoplastic bodies weakened by cracks. The distance between the crack edges is considered to be finite in contrast to the mathematical cut. The interactive layer with a thickness limited by the possibility of using the hypothesis of continuity is distinguished on the physical cut extension.Distribution of stresses and strains over the layer thickness is constant and does not depend on the geometry of the boundary between the cut and the interactive layer. The relationship between stresses and strains is determined by the deformation plasticity theory. The problem of plane strain or plane stress state of an arbitrary finite body weakened by a physical cut is reduced to solving a system of two variational equations for displacement fields in the body parts adjacent to the interactive layer. The proposed approach eliminates the singularity in stress distribution in contrast to the mathematical cut model. Use of local strength criteria allows us to determine the time, point and direction of the fracture initiation. Possibilities of the proposed model are illustrated by solving the problems of determining the SSS of a rectangular body weakened by a physical cut under symmetric and antisymmetric loadings.     

Edge effects in the stress state of a thin elastic interlayer

The edge effects in the stress state of an interlayer in stretching and shearing by rigid slabs are studied. On the basis of the equations of momentless and moment elastic layers, we solve problems modeling qualitatively the stress-strain state in the “soft” layer between two “rigid” layers. Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 2, pp. 189–195, March–April, 1999.     

Stability of multilayered shells of revolution with allowance for geometric nonlinearity of the subcritical state

The finite-difference method and the Trefftz-Reissner variational principle are used to obtain a system of equations in mixed from to describe the stability and geometric nonlinearity of composite shells of revolution. Methods are developed and an algorithm is proposed to calculate the components of the geometrically nonlinear subcritical stress-strain state and to use those components to determine the “upper” critical values for shells with zero Gaussian curvature loaded by uniform external pressure, an axisymmetric load, or a combination of these loads. The stability of cylindrical, conical, and compound shells under uniform pressure is examined for different support conditions. Linear and nonlinear methods of determining the subcritical stress-strain state are compared and their effect on the critical loads is estimated. Ukrainian Transportation Institute and the Ukrainian Academy of Water Management, Kiev, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 35, No. 6, pp. 60–66, June, 1999.     

Thermoelastoplastic deformation of noncircular cylindrical shells

A method to determine the nonstationary temperature fields and the thermoelastoplastic stress-strain state of noncircular cylindrical shells is developed. It is assumed that the physical and mechanical properties are dependent on temperature. The heat-conduction problem is solved using an explicit difference scheme. The temperature variation throughout the thickness is described by a power polynomial. For the other two coordinates, finite differences are used. The thermoplastic problem is solved using the geometrically nonlinear theory of shells based on the Kirchhoff-Love hypotheses. The theory of simple processes with deformation history taken into account is used. Its equations are linearized by a modified method of elastic solutions. The governing system of partial differential equations is derived. Variables are separated in the case where the curvilinear edges are hinged. The partial case where the stress-strain state does not change along the generatrix is examined. The systems of ordinary differential equations obtained in all these cases are solved using Godunov's discrete orthogonalization. The temperature field in a shell with elliptical cross-section is studied. The stress-strain state found by numerical integration along the generatrix is compared with that obtained using trigonometric Fourier series. The effect of a Winkler foundation on the stress-strain state is analyzed Translated from Prikladnaya Mekhanika, Vol. 44, No. 8, pp. 79–90, August 2008.     

Methods of determining the stress-strain state and fracture toughness of gas- and oil-line pipe (survey)

Conclusion We have surveyed the existing methods of determining the stress-strain state and fracture toughness of gas and oil lines. The discussion illustrated the promise of methods of evaluating the stress-strain state that are based on allowance for the strain-hardening of the material of the pipe or plastic indicators attached to its surface and deformed along with it. The first method, being destructive, can be used to evaluate the effect of different shaping methods on the strength of the pipe material, to analyze accidental pipeline fractures, and to generate a data bank on the stress-strain state of sections of a pipeline that have lost their longitudinal stability and are scheduled for replacement. Information obtained from the first method can also be used to establish correlations between the geometric parameters of such sections (deflection, curvature) and their stress-strain state. This will make it possible, with the use of reliable statistical methods, to predict the stress level on curved sections of pipelines without having to cut out samples for testing and analysis. Translated from Prikladnaya Mekhanika, Vol. 33, No. 8, pp. 3–30, August, 1997.     

Analysis of the thermoelastoplastic nonaxisymmetric stress-strain state of laminated circular cylindrical shells

A technique for analysis of the nonaxisymmetric thermoelastoplastic stress-strain state of laminated circular cylindrical shells is developed. It is assumed that the layers in a laminated package do not slip and separate relative to each other. The problem is solved using the geometrically linear theory of shells that is based on the Kirchhoff-Love hypotheses. The equations of thermoplasticity are written in the form of the method of additional strains. The order of the obtained system of partial differential equations is reduced with the help of trigonometric series in the cyclic coordinate. The systems of ordinary differential equations thus obtained are solved by Godunov's method of discrete orthogonalization. As an example, the nonaxisymmetric thermoelastoplastic stress-strain state of a two-layer cylindrical shell is considered. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 36, No. 2, pp. 105–110, February, 2000.     

Evolution of the joint motion of two viscous heat-conducting fluids in a plane layer under the action of an unsteady pressure gradient

A study is made of an invariant solution of the equations of a viscous heat-conducting fluid, which is treated as unidirectional motion of two such fluids in a plane layer with a common boundary under the action of an unsteady pressure gradient. A priori estimates of the velocity and temperature are obtained. The steady state is determined, and it is shown (under some conditions on the pressure gradient) that, at larger times, this state is the limiting one. For semiinfinite layers, a solution in closed form is obtained using the Laplace transform. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 4, pp. 94–107, July–August, 2008.     

Elastic plane with a physical cut loaded by an antisymmetric system of forces

The problem of determining the stress-strain state of an elastic plane with a physical cut loaded by an antisymmetric system of forces is posed and solved under the condition that the tangential stresses are homogeneous across the thickness of the layer continuing the cut.     

On finite-element solution of spatial thermoviscoelastoplastic problems

A finite-element algorithm is proposed for the analysis of the thermoviscoelastoplastic stress-strain state of bodies under complex loading (thermal and mechanical). It is assumed that an arbitrary element of the body deforms along a rectilinear or slightly curved path. The three-dimensional stress-strain state of the body’s elements is determined using the iterative method of additional strains. The technique is tested by analyzing the three-dimensional viscoelastic stress-strain state of a hollow cylinder and the thermoplastic state of a disk __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 5, pp. 16–25, May 2006.     

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.     

Transient dynamic behavior of a closed gas-filled shell subjected to internal blast loading

The paper concerns the numerical solution of a problem on the forced vibration of a shell system under a blast load. The mathematical model of the transient dynamic process in the given multicomponent system is based on the use of a system of hydrodynamic equations to describe the motion of the detonation products and air and equations from shell theory to determine the stress-strain state of the shell system. The hydrodynamic equations are integrated in moving grids by the method developed by S. K. Godunov. The shell-theory equations are integrated by the integro-interpolation method. The distribution of the loads on the shells, the distribution of the maximum strains along the generatrix of the shell system, and the transformations of the detonation products at characteristic points are presented. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 34, No. 3, pp. 76–82, March, 1999.