A Model of the Short-Term Damageability of a Transversally Isotropic Material

The effective deformation properties and the stress–strain state of a material are determined on the basis of the stochastic equations of elastic theory, which allow for the random nature of microdamages. The problem on the stress-strain state of a porous transversally isotropic material under uniform loading with specified strains is solved using the model proposed. To this end, a numerical–analytical algorithm is proposed. This algorithm is based on the stochastic method of conditional moment functions and the combined iteration method, which is used to solve a transcendental equation. Using this approach, as an example, a nonlinear macrodeformation diagram is plotted and the behavior of a transversally isotropic porous material under biaxial tension is studied     

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.     

Axisymmetric thermoelastoplastic stress-strain state of transversely isotropic laminated shells

The method of successive approximations is used to determine the thermoelastoplastic stress-strain state of isotropic and transversally isotropic laminated shells of revolution under axisymmetric loading. Hill’s theory of plasticity with isotropic hardening is used to describe the deformation of transversely isotropic materials, while the theory of deformation along paths of small curvature is used to describe the deformation of isotropic materials. The elastoplastic stress-strain state of a two-layer cylindrical shell under mechanical and thermal loads is analyzed __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 6, pp. 71–80, June 2006.     

On the stress-strain state of a transversally isotropic body with a paraboloidal inclusion

A closed solution is presented for the three-dimensional problem of the stress-strain state of an unbounded elastic body with a soldered-in transversally isotropic inclusion in the form of a paraboloid of revolution. Here, it is assumed that the body is under axisymmetric tension (compression). A solution of the corresponding problem for a paraboloidal recess is obtained as a special case. Podil’chuk [2, 3] has investigated similar problems for isotropic bodies with an inclusion assuming the form of a paraboloid of revolution or an elliptical paraboloid. Translated from Prikladnaya Mekhanika, Vol. 34, No. 11, pp. 16–22, November, 1998.     

On two approaches to determining the axisymmetric elastoplastic stress-strain state of laminated shells made of isotropic and transversely isotropic bimodulus materials

The paper compares two approaches to determining the axisymmetric elastoplastic stress-strain state of laminated shells made of isotropic and transversely isotropic materials with different elastic moduli in tension and compression. The approaches use different nonlinear constitutive equations describing the elastic deformation of the transversally isotropic bimodulus materials. Both approaches employ the theory of deformation along paths of small curvature and the Kirchhoff-Love hypothesis for the whole laminate to describe the deformation of the isotropic materials. The problem is solved by the method of successive approximations. The solutions of specific problems obtained by the two approaches are analyzed __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 6, pp. 59–69, June 2008.     

The thermoelastic equilibrium of a transversally isotropic medium with an elliptic crack under symmetric loading

The stress intensity factors (SIFs) are evaluated for flat elliptical cracks located in a transversally isotropic material (cracks are assumed perpendicular to the transtropy axis) under an arbitrary load and symmetric temperature. The SIFs for an elliptical crack in a transversally isotropic medium are determined using the formulas (derived by the author in his previous studies) of transition from an isotropic to transversally isotropic material and the relative problem for an isotropic medium. It is proved that these formulas can be employed for an arbitrary homogeneous transversally isotropic material (no matter whether the roots of some characteristic equation of the material are real or complex) with an arbitrary flat crack or a system of coplanar flat cracks, including elliptical ones, under an arbitrary load and symmetric temperature. A transversally isotropic material with two coplanar elliptical cracks is considered as an illustrative example. The dependences of the SIFs on the parameters of cracks and their arrangement at a decreasing temperature are presented. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine Kiev. Translated from Prikladnaya Mekhanika, Vol. 36, No. 4, pp. 96–105, April, 2000.     

Influence of anisotropy and boundary conditions on the thermoelastic equilibrium of nununiformly heated laminated plates

This paper deals with three-dimensional boundary-value problems of the heat transfer and thermoelasticity theory for laminated transversally isotropic rectangular plates affected by a nonuniform temperature field. An exact analytical solution is obtained for the case of both ideal and nonideal thermal and mechanical contacts of adjacent layers. Numerical calculations were performed for a two-layer rectangular plate of arbitrary thickness. The effect of material anisotropy as well as the boundary and contact conditions on the stress-strain state of a nonuniformly heated plate is revealed. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev Translated from Prikladnaya Mekhanika, Vol. 35, No. 11, pp. 96–105, November, 1999.     

Elastoplastic Axisymmetric Stress-Strain State of Laminated Shells Made of Isotropic and Transversely Isotropic Materials with Different Moduli

A method is developed for analysis of the elastoplastic stress-strain state of laminated shells of revolution under axisymmetric loading. The shells are made of isotropic and transversally isotropic materials with different moduli. The method is based on the Kirchhoff-Love hypotheses for the whole laminate, the theory of deformation along paths of small curvature (for isotropic materials), and the theory of elasticity with different tensile and compressive moduli (transversely isotropic materials). The problem is solved by the method of successive approximations. Numerical examples are given __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 8, pp. 88–96, August 2005.     

Initiation and stable growth of penny-shaped crack in aging viscoelastic transversally isotropic material

Considered is the long-term cracking of an aging transversally isotropic material containing a Mode I penny-shaped crack under remotely applied tensile stress. The aging material properties are described by the Boltzmann–Volterra’s linear theory for integral operators with non-difference kernels. It applied to wood, concrete, some polymers and rocks. Only the symmetric case is considered where the crack lies in the plane of isotropy. The modified Leonov–Panasyuk–Dugdale’s crack model is used with a constant process zone assuming that the critical opening displacement is the fracture criterion. Volterra’s principle is applied to derive the equations of subcritical crack growth. Numerical calculations are made for subcritical crack growth for the specific example of transversally isotropic material simulating the behavior of reinforced concrete.     

Asymptotic analysis of a 3D elasticity problem for a radially inhomogeneous transversally isotropic hollow cylinder

The method of asymptotic integration of equations of elasticity [1] is used to study the behavior of the solution of a 3D elasticity problem for a radially inhomogeneous transversally isotropic hollow cylinder of small thickness. Under the assumption that the load is sufficiently smooth, the asymptotic method [1] is used to construct inhomogeneous solutions. An algorithm for constructing exact particular solutions of the equilibrium equations is given for loads of specific types in the case where the cylinder lateral surface is loaded by forces polynomially depending on the axial coordinate. Then the homogeneous solutions are constructed. The asymptotic expansions of homogeneous solutions are obtained, and the above analysis is used to explain the character of the stress-strain state.     

On One Technique of Constructing the General Solution to Equilibrium Equations for Nonthin Plates

The mathematical theory of plates based on the expansion of functions into Fourier series in terms of Legendre polynomials is used to state a method for determining the general solution to a system of equilibrium equations describing the stress–strain state of nonthin transversally isotropic plates. The state is assumed symmetric about the median plane     

Delayed fracture of a transversally isotropic viscoelastic material with a crack under a cyclic load

The delayed fracture of a transversally isotropic viscoelastic material due to slow subcritical growth of a flat circular macrocrack of normal separation under a cyclic load is investigated. The analysis is based on the modified δC of fracture and the hypothesis of the constancy of the prefracture zone. The study is made within the framework of the Boltzmann-Volterra theory for bounded resolvent operators of difference type, which describe the transversal isotropy of the strain properties of the material. To determine the analytic form of the kernel for the irrational function of a linear combination of the above-mentioned integral operators, the method of continued fractions is used. Analytical and numerical calculations are carried out for the bounded resolvent operators of difference type with the kernel in the form of Rabotnov's fractional exponential function. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 36, No. 1, pp. 123–129, January, 2000.     

Delayed fracture of a transversally isotropic viscoelastic material with a crack under monotonically increasing loading

The delayed fracture of a transversally isotropic viscoelastic material due to slow subcritical growth of a flat normal-fracture macrocrack with a circular cross-section under monotonically increasing load is examined. The calculations employ the modified δ_C of fracture, which is based on the concept of constancy of the prefailure region. The investigation is carried out within the framework of the Boltzmann-Volterra theory for difference-type bounded resolvent operators, which describe the transversal isotropy of the viscoelastic deformational properties of the material. To find the analytical form of the kernel of an irrational function of a linear combination of the above-mentioned integral operators, the method of operator continued fractions is used. Analytical and numerical calculations are carried out for difference-type bounded resolvent operators with the kernel in the form of Rabotnov's fractional-exponential function. S. P. Timoshenko Institute of Mechanics. National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 35, No. 10, pp. 54–60, October, 1999.     

Axisymmetric thermoelastoplastic stress-strain state of flexible laminated transversely isotropic shells

The paper presents a technique for numerical analysis of the elastoplastic stress-strain state of flexible laminated shells of revolution made of isotropic and transversely isotropic materials and subjected to axisymmetric loading and heating. The technique is based on the Kirchhoff-Love hypotheses for the whole laminate. The deformation of the isotropic materials is described using the theory of deformation along paths of small curvature. The deformation of the transversely isotropic material is described using the flow theory with isotropic hardening. The process of loading is divided into steps at each of which the stress-strain state is determined by the method of successive approximations. A numerical example is given __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 12, pp. 76–86, December, 2006.     

Short-Term Microdamageability of Fibrous Composites with Transversally Isotropic Fibers and a Microdamaged Binder

The theory of microdamageability of fibrous composites with transversally isotropic fibers and a microdamaged isotropic porous matrix is proposed. Microdamages in the matrix are simulated by pores filled with particles of the destroyed material that resist compression. The criterion of damage in the matrix microvolume is taken in the Schleicher–Nadai form. It accounts for the difference between the ultimate tensile and compressive loads. The ultimate strength is a random function of coordinates with Weibull distribution. The stress–strain state and effective properties of the material are determined from the stochastic equations of the elastic theory for a fibrous composite with porous components. The equations of deformation and microdamage are closed by the equations of porosity balance in the matrix. Nonlinear diagrams of the concurrent processes of deformation of fibrous materials and microdamage of the matrix are plotted. The effect of the physical and geometrical parameters on them is studied     

The iterative theory of deformation of laminated shells

The equations of the iteration theory of nonshallow transversally isotropic laminated shells, which account for all the components of the stress-strain state (SSS) and describe the inner SSS, potential, and vortex boundary effects, are obtained. The equations are based on the method of expansion of SSS into series in transverse coordinate and the method of variation with respect to the state being determined. The order of the equations does not depend on the number of layers and expansion terms that approximate the displacement and stress. The accuracy of the solution for the inner SSS and boundary effects is estimated. Pridneprovskaya State Academy of Building and Architecture, Dnepropetrovsk, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 35, No. 11, pp. 40–45, November, 1999.     

The Micromechanics of Short-Term Damageability of Fibrolaminar Composites

A theory of microdamageability is constructed for fibrous laminated composites consisting of transversally isotropic fibers and a microdamaged isotropic porous binder. Microdamages in the binder are simulated by pores filled with compression-resisting particles of the destroyed material. Damage in a microvolume of the binder is described by the Schleicher–Nadai strength criterion, which allows for the difference between the ultimate tensile and compressive loads. The ultimate strength is a random function of coordinates with the Weibull distribution. The stress–strain state and effective characteristics of the material are determined by solving the stochastic equations of elastic theory for a fibrous laminated composite with a porous binder. The equations of deformation and microdamageability are closed by the equations of porosity balance in the binder. Nonlinear diagrams of the concurrent processes of deformation of the fibrous laminated material and microdamage of the matrix for various physical and geometrical parameters are constructed     

Stress-strain state of a flexible spherical shell with an eccentric circular hole

The stress-strain state of thin flexible spherical shells weakened by an eccentric circular hole is analyzed. The shells are made of an isotropic homogeneous material and subjected to internal pressure. A problem formulation is given, and a method of numerical solution with allowance for geometrical nonlinearity is outlined. The distribution of displacements, strains, and stresses along the hole boundary and in the region of their concentration is examined. The data obtained are compared with numerical solutions of the linear problem. The stress-strain state around the eccentric circular hole is analyzed with allowance for geometrical nonlinearity __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 10, pp. 92–98, October 2007.     

On the representation of the solutions of the equilibrium equations of a piezoelectric transversally-isotropic spherical shell

A method is proposed for constructing equilibrium equations for thickness-polarized transversally isotropic piezoceramic shells. The method is based on Fourier expanding the required functions in Legendre polynomials. The appropriate system of differential equations is formed for the expansion coefficients as functions of two independent variables. The equilibrium equations are given in particular for transversally isotropic spherical shells. A method is given for constructing the general solution in the first approximation. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 35, No. 7, pp. 59–68, July, 1999.     

Viscoelastic Deformation of a Reinforced Plate with a Crack

The deformation of a viscoelastic reinforced composite is studied. The composite has an axis of elastic symmetry and consists of transversally isotropic fibers and a viscoelastic matrix, which differ by the volume concentration and mechanical characteristics. The material is modeled by a transversely isotropic homogeneous linearly viscoelastic medium with some averaged characteristics. A plate fabricated from the composite in question is weakened by a through mode I crack and is subjected to constant tensile forces. The viscoelastic properties of the matrix material are described by a convolution operator. The Volterra principle is used to derive expressions for the viscoelastic characteristics and crack opening. The irrational function of the integral operator that describes the crack opening is expanded into an operator continued fraction and is represented as the sum of base operators