On a representation of the solution of the linear elasticity equations

A new representation of the stress tensor in the linear theory of elasticity is proposed. The representation satisfies the equilibrium equations and the compatibility conditions for strains. In this representation, the stress tensor is expressed in terms of a harmonic vector. The second boundary-value problem for an elastic half-space and elastic layer is considered as an example.Translated from Prikladnaya Mekhanika, Vol. 40, No. 11, pp. 85–91, November 2004.This revised version was published online in April 2005 with a corrected cover date.     

Thermoelastic Solution for Stresses

A new solution to the three-dimensional thermoelastic problem for stresses is proposed. It satisfies the static equation and compatibility condition. The stress tensor is expressed in terms of a harmonic vector. With such a solution, the boundary conditions for some elastic bodies can be satisfied quite easily__________Translated from Prikladnaya Mekhanika, Vol. 41, No. 3, pp. 46–54, March 2005.     

Solution of the elastic boundary value problems for a layer with tunnel stress raisers

A novel procedure for solving three-dimensional problems for elastic layer weakened by through-thickness tunnel cracks has been developed and is presented in this paper. This procedure reduces the given boundary value problem to an infinite system of one-dimensional singular integral equations and is based on a system of homogeneous solutions for a layer. Integral representations of single- and double-layer potentials are used for metaharmonic and harmonic functions entering in the singular integral equations. These representations provide a continuous extendibility of the stress vector while allowing a jump in the displacement vector in the transition through the cut.Expanding the potential and biharmonic solutions in the Fourier series over the thickness coordinate yields the integral representations of the displacement vector and stress tensor. The problem of reducing a denumerable set of the integral equations of the given boundary value problem to one-to-one correspondence with the set of unknown densities appearing in the Fourier’s coefficient representations has been settled efficiently. Numerical investigations show a rapid convergence of the proposed reduction procedure as applied to the solution of the infinite system of one-dimensional integral equations. Numerical examples illustrate the proposed method and demonstrate its advantages.     

Flexural vibrations of a thin circular elastic inclusion in an unbounded body under the action of a plane harmonic wave

The interaction of a plane harmonic longitudinal wave with a thin circular elastic inclusion is considered. The wave front is assumed to be parallel to the inclusion plane. Since the inclusion is thin, the matrix-inclusion interface conditions (perfect bonding) are formulated on the mid-plane of the inclusion. The bending displacements of the inclusion are determined from the bending equation for a thin plate. The problem is solved using discontinuous Lamé solutions for harmonic vibrations. Therefore, the problem can be reduced to the Fredholm equation of the second kind for a function associated with the discontinuity of normal stresses on the inclusion. The equation obtained is solved by the method of mechanical quadratures using Gaussian quadrature formulas. Approximate formulas for the stress intensity factors are derived. Results from a numerical analysis of the dependence of the SIFs on the dimensionless wave number and the stiffness of the inclusion are presented __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 5, pp. 16–21, May 2008.     

Dynamic stress intensity factors of two 3D rectangular cracks in a transversely isotropic elastic material under a time-harmonic elastic P-wave

The dynamic stress intensity factors (DSIFs) of two 3D rectangular cracks in a transversely isotropic elastic material under an incident harmonic stress wave are investigated by generalized Almansi’s theorem and the Schmidt method in the present paper. Using 2D Fourier transform and defining the jumps of displacement components across the crack surface as the unknown functions, three pairs of dual integral equations are derived. To solve the dual integral equations, the jumps of the displacement components across the crack surfaces are expanded in a series of Jacobi polynomials. Numerical examples are provided to show the effects of the geometric shape of the rectangular crack, the characteristics of the harmonic wave and the distance between two rectangular cracks on the DSIFs of the transversely isotropic elastic material.     

Complete Asymptotic Expansions of Solutions of the System of Elastostatics in the Presence of an Inclusion of Small Diameter and Detection of an Inclusion

We consider the system of elastostatics for an elastic medium consisting of an imperfection of small diameter, embedded in a homogeneous reference medium. The Lamé constants of the imperfection are different from those of the background medium. We establish a complete asymptotic formula for the displacement vector in terms of the reference Lamé constants, the location of the imperfection and its geometry. Our derivation is rigorous, and based on layer potential techniques. The asymptotic expansions in this paper are valid for an elastic imperfection with Lipschitz boundaries. In the course of derivation of the asymptotic formula, we introduce the concept of (generalized) elastic moment tensors (Pólya–Szegö tensor) and prove that the first order elastic moment tensor is symmetric and positive (negative)-definite. We also obtain estimation of its eigenvalue. We then apply these asymptotic formulas for the purpose of identifying with high precision the order of magnitude of the diameter of the elastic inclusion, its location, and its elastic moment tensors.     

Studying the harmonic vibrations of a cylindrical shell made of a nonlinear elastic piezoelectric material

The paper examines the harmonic vibrations of an infinitely long thin cylindrical shell made of a nonlinear elastic piezoceramic material and subjected to periodic electric loading. Amplitude-frequency characteristics are plotted for different amplitudes of the load. Points of these characteristics are analyzed for stability. The transients occurring while harmonic vibrations attain the steady state are studied __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 4, pp. 101–106, April 2008.     

Solving the three-dimensional equations of the linear theory of elasticity

The three-dimensional Lamé equations are solved using Cartesian and curvilinear orthogonal coordinates. It is proved that the solution includes only three independent harmonic functions. The general solution of equations of elasticity for stresses is found. The stress tensor is expressed in both coordinate systems in terms of three harmonic functions. The general solution of the problem of elasticity in cylindrical coordinates is presented as an example. The three-dimensional stress–strain state of an elastic cylinder subjected, on the lateral surface, to arbitrary forces represented by a series of eigenfunctions is determined. An axisymmetric problem for a finite cylinder is solved numerically     

Interaction of an Ellipsoidal Inclusion with an Elliptic Crack in an Elastic Material under Triaxial Tension

The interaction of an elastic ellipsoidal inclusion with an elliptic crack in an infinite elastic medium under triaxial loading is analyzed. The stress state in the elastic space is represented as a superposition of the principal state and perturbed states, which are due to the presence and interaction of the inclusion and the crack. The analytical solution of the problem is found using the method of equivalent inclusion, the potential of an inhomogeneous ellipsoid, and a system of harmonic functions for an elliptic crack. The effect of triaxial loading on the stress intensity factors is analyzed     

The evolution of Hooke's law due to texture development in FCC polycrystals

Initially isotropic aggregates of crystalline grains show a texture-induced anisotropy of both their inelastic and elastic behavior when submitted to large inelastic deformations. The latter, however, is normally neglected, although experiments as well as numerical simulations clearly show a strong alteration of the elastic properties for certain materials. The main purpose of the work is to formulate a phenomenological model for the evolution of the elastic properties of cubic crystal aggregates. The effective elastic properties are determined by orientation averages of the local elasticity tensors. Arithmetic, geometric, and harmonic averages are compared. It can be shown that for cubic crystal aggregates all of these averages depend on the same irreducible fourth-order tensor, which represents the purely anisotropic portion of the effective elasticity tensor. Coupled equations for the flow rule and the evolution of the anisotropic part of the elasticity tensor are formulated. The flow rule is based on an anisotropic norm of the stress deviator defined by means of the elastic anisotropy. In the evolution equation for the anisotropic part of the elasticity tensor the direction of the rate of change depends only on the inelastic rate of deformation. The evolution equation is derived according to the theory of isotropic tensor functions. The transition from an elastically isotropic initial state to a (path-dependent) final anisotropic state is discussed for polycrystalline copper. The predictions of the model are compared with micro–macro simulations based on the Taylor–Lin model and experimental data.     

A dynamic problem for a prestressed compressible layered half-space

The linearized theory of elasticity for prestressed bodies is used to solve a stationary plane problem for a prestressed two-layer half-space under a surface load moving with constant velocity. The half-space is assumed to be compressible and to have an arbitrary elastic potential. The Fourier transform is used to obtain the fundamental solution of the problem for different contact conditions and load velocities. A compressible material with a harmonic elastic potential is considered as an example __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 4, pp. 35–55, April 2008.     

General invariant representations of the constitutive equations for isotropic nonlinearly elastic materials

This paper develops general invariant representations of the constitutive equations for isotropic nonlinearly elastic materials. Different sets of mutually orthogonal unit tensor bases are constructed from the strain argument tensor by using the representation theorem and corresponding irreducible invariants are defined. Their relations and geometrical interpretations are established in three dimensional principal space. It is shown that the constitutive law linking the stress and strain tensors is revealed to be a simple relationship between two vectors in the principal space. Relative to two different sets of the basis tensors, the constitutive equations are transformed according to the transformation rule of vectors. When a potential function is assumed to exist, the vector associated with the stress tensor is expressed in terms of its gradient with respect to the vector associated with the strain tensor. The Hill’s stability condition is shown to be that the scalar product of the increment of those two vectors must be positive. When potential function exists, it becomes to be that the 3 × 3 constitutive matrix derived from its second order derivative with respect to the vector associated with the strain must be positive definite. By decomposing the second order symmetric tensor space into the direct sum of a coaxial tensor subspace and another one orthogonal to it, the closed form representations for the fourth order tangent operator and its inversion are derived in an extremely simple way.     

Propagation of Harmonic Waves in an Orthotropic Cylindrical Shell on Elastic Foundation

An equation is derived, using Timoshenko shell theory, to analyze axisymmetric strain fields in an orthotropic cylindrical shell on an elastic foundation. Also a dispersion equation is derived to study the natural harmonic waves in a shell depending on the properties of the elastic foundation. The wave velocities computed by the numerical method proposed are in agreement with the analytical solutions, which confirms the reliability of the results     

Spectral decomposition of the compliance fourth-rank tensor for orthotropic materials

Summary The compliance tensor related to orthotropic media is spectrally decomposed and its characteristic values are determined. Further, its idempotent tensors are estimated, giving rise to energy orthogonal states of stress and strain, thus decomposing the elastic potential in discrete elements. It is proven that the essential parameters, required for a complete characterisation of the elastic properties of an orthotropic medium, are the six eigenvalues of the compliance tensor, together with a set of three dimensionless parameters, the eigenangles θ, ϕ and ω. In addition, the intervals of variation of these eigenangles with respect to different values of the elastic constants are presented. Furthermore, bounds on Poisson's ratios are obtained by imposing the thermodynamical constraint on the eigenvalues to be strictly positive, as specified from the positive-definite character of the elastic potential. Finally, the conditions are investigated under which a family of orthotropic media behaves like a transversely isotropic or an isotropic one. Received 5 January 1999; accepted for publication 22 June 1999     

A mathematical theory of materials with elastic range and the definition of back stress tensor

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考虑温度效应的弹性损伤一般理论

现有的各种损伤理论基本上都是关于等温问题的 ,且在不同程度上依赖于某些经验假设。本文在严格的不可逆热力学理论基础之上 ,建立了考虑温度效应的弹性损伤一般理论。推导出热弹性各向同性与各向异性损伤材料全部本构方程的一般形式 ,其中包括应力 应变关系、熵密度方程、损伤对偶张量表达式、热 固 损伤耦合的热传导方程和损伤演化方程。它们的特殊形式包含了等温各向同性与各向异性弹性损伤的本构方程     

Harmonic elastic inclusions in the presence of point moment

We employ conformal mapping techniques to design harmonic elastic inclusions when the surrounding matrix is simultaneously subjected to remote uniform stresses and a point moment located at an arbitrary position in the matrix. Our analysis indicates that the uniform and hydrostatic stress field inside the inclusion as well as the constant hoop stress along the entire inclusion–matrix interface (on the matrix side) are independent of the action of the point moment. In contrast, the non-elliptical shape of the harmonic inclusion depends on both the remote uniform stresses and the point moment.     

The Stress State of an Elastic Medium with an Elliptic Crack and Two Ellipsoidal Cavities

This paper is a study into the interaction of two triaxial ellipsoidal cavities whose surfaces are under different pressures with an elliptic crack in an infinite elastic medium. The stress state in the elastic space is represented by a superposition of perturbed states due to the presence and interaction of the cavities and the crack. The exact solution of the problem is constructed by using a modified method of equivalent inclusion, the potential of an inhomogeneous ellipsoid, and a system of harmonic functions for the elliptic crack. A numerical analysis is carried out to find how the geometry of the cavities and the crack, the distance between them, and the pressure on their surfaces affect the stress intensity factors     

A NOTE ON MORI-TANAKA＇S METHOD

Explicit expressions of Mori-Tanaka＇s tensor for a transversely isotropic fiber rein- forced UD composite are presented. Closed-form formulae for the effective elastic properties of the composite are obtained. In a 3D sense, the resulting compliance tensor of the composite is symmetric. Nevertheless, the 2D compliance tensor based on a deteriorated Mori-Tanaka＇s tensor is not symmetric. Nor is the compliance tensor defined upon a deteriorated 2D Eshelby＇s tensor. The in-plane effective elastic properties given by those three approaches are different. A detailed comparison between the predicted results obtained from those approaches with experimental data available for a number of UD composites is made.     

具有弹性放大器的双稳态压电俘能器建模及参数影响分析

利用广义Hamilton变分原理,建立了具有弹性放大器的双稳态压电俘能系统BPH+EM的动力学方程。考虑谐波激励,采用调和平衡法获得了BPH+EM系统的位移、输出电压和功率的解析解。利用求得的解析解,讨论了BPH+EM系统扩大能量俘获的频率范围和提高能量俘获效率的机理,研究了弹性放大器的刚度质量比对BPH+EM系统的动力性能影响规律。当弹性放大器的刚度质量比趋于无限大时,具有弹性放大器的双稳态压电俘能系统退化为双稳态压电俘能系统BPH。弹性放大器的刚度质量比趋于0但不等于0时,BPH+EM的俘能效率低于BPH。结果表明,在合适的刚度质量比范围内,BPH+EM的俘能效率显著优于BPH。研究结果为BPH+EM系统的优化设计提供了理论指导。