Anti-plane interaction of a crack and reinforced elliptic hole in an infinite matrix

Anti-plane interaction of a crack with a coated elliptical hole embedded in an infinite matrix under a remote uniform shear load is considered in this paper. Analytical treatment of the present problem is laborious due to the presence of material inhomogeneities and geometric discontinuities. Nevertheless, based on the technique of conformal mapping and the method of analytical continuation in conjunction with the alternating technique, general expressions for displacements and stresses in the coated layer and the matrix are derived explicitly in closed form. By applying the existing complex function solutions for a dislocation, the integral equations for a line crack are formulated and mode-III stress intensity factors are obtained numerically. Some numerical examples are given to demonstrate the effects of material inhomogeneity and geometric discontinuities on mode-III stress intensity factors.     

Analysis of a strengthened weakly conical shell using a complete system of orthogonal functions on an arbitrary contour of its transverse cross-section

Thin-walled weakly conical and cylindrical shells with arbitrary open, simply or multiply closed contour of transverse cross-sections strengthened by longitudinal elements (such as stringers and longerons) are used in the design of wings, fuselages, and ship hulls. To avoid significant deformations of the contour, such structures are stiffened by transverse elements (such as ribs and frames). Various computational models and methods are used to analyze the stress-strain states of such compound structures. In particular, the ground stress-strain states in bending, transverse shear, and twisting of elongated structures are often analyzed with the use of the theory of thin-walled beams [1] based on the hypothesis of free (unconstrained) warping and bending of the contour of transverse cross-sections. In general, the computations with the contour warping and bending constraints caused by the variable load distribution, transverse stiffening elements, and the difference in the geometric and rigidity parameters of the shell cells are usually performed by the finite element method or the superelement (substructure) method [2, 3]. In several special cases (mainly for separate cells of cylindrical and weakly conical shells located between transverse stiffening elements, with the use of some additional simplifying assumptions), efficient variation methods for computations in displacements [4–8] and in stresses [9] were developed, so that they reduce the problem to a system of ordinary differential equations. In the one-and two-term approximations, these methods permit obtaining analytic solutions, which are convenient in practical computations. But for shells with multiply closed contours of transverse cross-sections and in the case of exact computations by using the Vlasov variational method [4], difficulties are encountered in choosing the functions representing the warping and bending of the contour of transverse cross-sections. In [10], in computations of a cylindrical shell with simply closed undeformed contour of the transverse section, warping was represented in the form of expansions in the eigenfunctions orthogonal on the contour, which were determined by the method of separation of variables from a special integro-differential equation. In [11], a second-order ordinary differential equation of Sturm-Liouville type was obtained; its solutions form a complete system of orthogonal functions with orthogonal derivatives on an arbitrary open simply or multiply closed contour of a membrane cylindrical shell stiffened by longitudinal elements. The analysis of such a shell with expansion of the displacements in these functions leads to ordinary differential equations that are not coupled with each other. In the present paper, by using the method of separation of variables, we obtain differential and the corresponding variational equations for numerically determining complete systems of eigenfunctions on an arbitrary contour of a discretely stiffened membrane weakly conical shell and a weakly conical shell with undeformed contour. The obtained systems of eigenfunctions are used to reduce the problem of deformation of shells of these two types to uncoupled differential equations, which can be solved exactly.     

An Exact Solution of Torsion Problem for an Incomplete Torus with Application to Helical Springs

An exact analytical solution for the torsion problem of an incomplete torus with a particular form of non-circular cross-section has been found. The solution extends for the close-coiled helical spring the known solution of the torsion problems for straight cylinders with circular and elliptical cross-sections. The pitch of helix is ignored. The hollow cross-sections of the particular form also demonstrate a closed form of analytical solution. The solution can be used for analysis of helical springs with non-circular wire profile.     

On displacements and stresses in a semi-infinite laminated layer: comparative results

This paper deals with some comparison results for displacements and stresses in a periodically stratified elastic semi-infinite layer determined within the framework of two approaches: (1) based on the homogenized model with microlocal parameters [Woźniak (1987) Int J Eng Sci 25: 483–499; Woźniak (1987) Bull Ac Pol Tech Sci 35: 143–151]; (2) obtained directly from the theory of elasticity. A body is assumed to be composed of n elastic two-component periodically repeated laminae. The perfect mechanical bonding between the layers is assumed. The normal displacements and zero shear stresses on the boundary being a cross-section of the composite component are taken into account. The lateral boundary surfaces are assumed to be rigid fixed. The obtained results from the two approaches are compared and presented in the form of figures.     

The scattering of electroelastic waves by an ellipsoidal inclusion in piezoelectric medium

The scattering problem for a single ellipsoidal piezoelectric inclusion embedded in piezoelectric medium is investigated. Based on the polarization method, the extended displacements are expressed in terms of integral equations, whose kernels are obtained from the Green’s functions of homogenous matrix. In this paper, the 3D dynamic Green’s functions are derived by means of the Radon transform technique. To illustrate the use of the equations, scattering by a piezoelectric, ellipsoidal inhomogeneity in a piezoelectric medium is considered in the low frequency and an asymptotic formula for this scattering cross-section is obtained. Numerical results of the scattering cross-sections are carried out for a spheroidal BaTiO₃-inclusion in a PZT-5H-matrix.     

TRIAL FUNCTION METHOD FOR THE LONGITUDINAL VIBRATION OF RODS WITH VARIABLE CROSS-SECTIONS

给出了一种试探函数法,并研究了变截面杆的纵振动问题. 先给出振动控制方程的特殊函数形式的试探解,然后要求此解满足控制方程,反过来确定了控制方程各种可能的系数函数（即截面变化函数）并得到了控制方程的精确解. 作为例子,给出了一种变截面杆在3 种边界条件下的频率方程,计算出了固有频率. 研究表明,试探函数法简单、直接,适合于研究变截面杆的纵振动问题. 对于杆扭转振动、薄膜振动以及管中波传播等问题,该方法同样有推广应用价值.     

Propagation of electroacoustic waves in the transversely isotropic piezoelectric medium reinforced by randomly distributed cylindrical inhomogeneities

The propagation of electroacoustic waves in a piezoelectric medium containing a statistical ensemble of cylindrical fibers is considered. Both the matrix and the fibers consist of piezoelectric transversely isotropic material with symmetry axis parallel to the fiber axes. Special emphasis is given on the propagation of an electroacoustic axial shear wave polarized parallel to the axis of symmetry propagating in the direction normal to the fiber axis.The scattering problem of one isolated continuous fiber (“one-particle scattering problem”) is considered. By means of a Green’s function approach a system of coupled integral equations for the electroelastic field in the medium containing a single inhomogeneity (fiber) is solved in closed form in the long-wave approximation. The total scattering cross-section of this problem is obtained in closed form and is in accordance with the electroacoustic analogue of the optical theorem.The solution of the one-particle scattering problem is used to solve the homogenization problem for a random set of fibers by means of the self-consistent scheme of effective field method. Closed form expressions for the dynamic characteristics such as total cross-section, effective wave velocity and attenuation factor are obtained in the long-wave approximation.     

Refined beam elements with only displacement variables and plate/shell capabilities

This paper proposes a refined beam formulation with displacement variables only. Lagrange-type polynomials, in fact, are used to interpolate the displacement field over the beam cross-section. Three- (L3), four- (L4), and nine-point (L9) polynomials are considered which lead to linear, quasi-linear (bilinear), and quadratic displacement field approximations over the beam cross-section. Finite elements are obtained by employing the principle of virtual displacements in conjunction with the Unified Formulation (UF). With UF application the finite element matrices and vectors are expressed in terms of fundamental nuclei whose forms do not depend on the assumptions made (L3, L4, or L9). Additional refined beam models are implemented by introducing further discretizations over the beam cross-section in terms of the implemented L3, L4, and L9 elements. A number of numerical problems have been solved and compared with results given by classical beam theories (Euler-Bernoulli and Timoshenko), refined beam theories based on the use of Taylor-type expansions in the neighborhood of the beam axis, and solid element models from commercial codes. Poisson locking correction is analyzed. Applications to compact, thin-walled open/closed sections are discussed. The investigation conducted shows that: (1) the proposed formulation is very suitable to increase accuracy when localized effects have to be detected; (2) it leads to shell-like results in case of thin-walled closed cross-section analysis as well as in open cross-section analysis; (3) it allows us to modify the boundary conditions over the cross-section easily by introducing localized constraints; (4) it allows us to introduce geometrical boundary conditions along the beam axis which lead to plate/shell-like cases.     

Analytical solution for a Mode III crack in a 3D beam lattice

The brittle fracture behavior of an open cell foam is considered. The foam is modeled by an infinite lattice composed of elastic straight-line beam elements (struts) having uniform cross-sections and rigidly connected at the nodal points. The beams are parallel to the three mutually orthogonal lattice vectors thus forming a microstructure with rectangular parallelepiped cells.A semi-infinite Mode III crack is embedded in the lattice and, for the considered antiplane deformation, each node has three degrees of freedom, namely, the displacement parallel to the crack front and two rotations about the axes perpendicular to this direction. The analysis method hinges on the discrete Fourier transform, which allows to formulate the crack problem by means of the Wiener–Hopf equation. Its solution yields closed-form analytical expressions for the forces and the displacements at any cross-section, and, in particular, at the crack plane. An eigensolution for the traction-free crack faces and K-field remote loading is derived from the solution for the loaded crack using a limiting procedure. An analytical expression for the fracture toughness is derived from the eigensolution by comparing the remote stress field and the stresses in the near-tip struts. The obtained expression is found to be consistent with the known analytical and experimental results for Mode I deformation. It appears, that the dependence of the fracture toughness upon shape anisotropy ratio of the lattice material is non-monotonic. The optimal value of this parameter, which provides the maximum crack arresting ability is determined.     

Propagation of unsteady waves in an elastic layer

We consider a plane problem of propagation of unsteady waves in a plane layer of constant thickness filled with a homogeneous linearly elastic isotropic medium in the absence of mass forces and with zero initial conditions. We assume that, on one of the layer boundaries, the normal stresses are given in the form of the Dirac delta function, the tangential stresses are zero, and the second boundary is rigidly fixed. The problem is solved by using the Laplace transform with respect to time and the Fourier transform with respect to the longitudinal coordinate. The normal displacements at an arbitrary point are obtained in the form of finite sums.     

Thermoelastostatic equilibrium of a spatial quadrant: Green’s function and solution in integrals

In this study, a new Green??s function and a new Green-type integral formula for a 3D boundary value problem (BVP) in thermoelastostatics for a quarter-space are derived in closed form. On the boundary half-planes, twice mixed homogeneous mechanical boundary conditions are given. One boundary half-plane is free of loadings and the normal displacements and the tangential stresses are zero on the other one. The thermoelastic displacements are subjected by a heat source applied in the inner points of the quarter-space and by mixed non-homogeneous boundary heat conditions. On one of the boundary half-plane, the temperature is prescribed and the heat flux is given on the other one. When the thermoelastic Green??s function is derived, the thermoelastic displacements are generated by an inner unit point heat source, described by ??-Dirac??s function. All results are obtained in elementary functions that are formulated in a special theorem. As a particular case, when one of the boundary half-plane of the quarter-space is placed at infinity, we obtain the respective results for half-space. Exact solutions in elementary functions for two particular BVPs for a thermoelastic quarter-space and their graphical presentations are included. They demonstrate how to apply the obtained Green-type integral formula as well as the derived influence functions of an inner unit point body force on volume dilatation to solve particular BVPs of thermoelasticity. In addition, advantages of the obtained results and possibilities of the proposed method to derive new Green??s functions and new Green-type integral formulae not for quarter-space only, but also for any canonical Cartesian domain are also discussed.     

Torsion of cylindrical bodies with varying strain properties

We analyze the results of experimental studies of effective strain properties of damaged, porous, and other inhomogeneous materials and study the main laws of their behavior under strain. We consider the possible versions of constitutive relations taking account of the dependence of the properties of the media under study on the loading conditions or the strain conditions and the relations between the shear and bulk strains. Since the traditional statement of the torsion problems for bodies with such properties cannot be used, we analyze the strain consistency equations and the relations between the strains and displacements in cylindrical coordinates and obtain expressions for the displacements in an appropriate generalized form, which can be used not only for the torsion problems. We study how the distributions of displacements, strains, and stresses under torsion depend on the parameter characterizing the susceptibility of the material strain properties to variations in the stress state type. We show that, in the case of torsion of a cylinder of circular cross-section, there is no deplanation of the cross-section, just as in the classical solution, but the distributions of displacements, strains, and stresses significantly differ from the well-known solutions.     

Deformation gradient based kinematic hardening model

A kinematic hardening model applicable to finite strains is presented. The kinematic hardening concept is based on the residual stresses that evolve due to different obstacles that are present in a polycrystalline material, such as grain boundaries, cross slips, etc. Since these residual stresses are a manifestation of the distortion of the crystal lattice a corresponding deformation gradient is introduced to represent this distortion. The residual stresses are interpreted in terms of the form of a back-stress tensor, i.e. the kinematic hardening model is based on a deformation gradient which determines the back-stress tensor. A set of evolution equations is used to describe the evolution of the deformation gradient. Non-dissipative quantities are allowed in the model and the implications of these are discussed. Von Mises plasticity for which the uniaxial stress–strain relation can be obtained in closed form serves as a model problem. For uniaxial loading, this model yields a kinematic hardening identical to the hardening produced by isotropic exponential hardening. The numerical implementation of the model is discussed. Finite element simulations showing the capabilities of the model are presented.     

圆弧形裂纹问题中的应力对数奇异性

研究了无限大板上的一条圆孤形裂纹, 又在裂纹表面作用有反对称载荷. 换言之, 裂纹两侧表面的载荷是大小相等方向相同的. 上述问题可用复变函数方法来解决. 应力和位移分量通过两个复位函数来表示. 经过一系列推导, 此问题可归结为复变函数的黎曼-希尔巴德(Riemann-Hilbert) 问题, 并且可用闭合形式得出解答. 裂纹端的应力强度因子用通常方法定出. 在裂纹端邻域, 得到的复位函数中有对数函数部分. 由这个对数函数部分, 可以定义和得出裂纹端的对数奇异性, 此对数奇异性系数用闭合型式得出.     

Elastic waves in piezoceramic cylinders of sector cross-section

The propagation of elastic waves in piezoceramic cylindrical waveguides of circular cross-sections with sector cut is investigated on the basis of the linear theory of electroelasticity. Dispersion functions are obtained from boundary conditions in an analytical form of functional determinants for each value of the generalized wave number. A selected set of numerical results including real, imaginary and complex branches of full dispersion spectrums with various symmetry of wave movements is presented to describe the essential characteristics of the waves. Leading effects of spectrums transformation by change of waveguide’s angular measure are enlightened, and wave asymptotic behavior is analyzed. The variation of the cross-section is considered as a mechanism to control the dispersion characteristics of waveguides.     

Crack edge displacement and elastic constant determination for an orthotropic material

A method for determining the elastic constants of an isotropic material, based on crack edge displacement data, is extended to an orthotropic material. Complex potentials are used to obtain the stresses and displacements for plane strain. Mode I crack problems in three mutually orthogonal planes are considered and solved. In particular, the expressions of crack edge displacements are obtained in an explicit form. An iterative statistical identification method, based on a Bayesan approach, is used to identify the elastic constants of an orthotropic medium from the Mode I crack displacements measured from the mid-point of the crack. Some graphics are displayed to illustrate the convergence of the pertinent parameters and the approach of the analytical displacements to their experimental values.     

A general solution for static piezothermoelastic problems of crystal class 6mm solids

In this paper, a general solution for three-dimensional static piezothermoelastic problems of crystal class 6mm solids is presented. The general solution involves four piezoelastic potential functions and a piezothermoelastic potential function, of which four piezoelastic potential functions are governed by weighted harmonic differential equations. Compared with the general solution given by Ashida et al., in which seven potential functions are introduced, the general solution proposed in the present paper is more rigorously derived. Moreover, it has a simple form and is convenient for application. As an illustrative example, the problem of a pyroelectric half-space subjected to axisymmetric heating is studied. Numerical results of displacements stresses, electric potential and electric displacements are obtained for a cadmium selenide half-space. Thermally induced displacements and stress distribution are compared with those obtained for the same material without piezoelectric and pyroelectric effects. Project supported by the National Natural Science Foundation of China (19872060).     

Closed-form solutions for two anti-plane collinear cracks in a magnetoelectroelastic layer

In this paper we develop closed form solutions for anti-plane mechanical and in-plane electric and magnetic fields for two collinear cracks in magneto-electro-elastic layer of finite thickness under the conditions of permeable crack faces using integral transform method. The anti-plane mechanical shear or displacement and in-plane electrical and magnetic loading are applied to the top and bottom surfaces of the layer for the two cases considered. Expressions for shear stresses, electric displacements and magnetic inductions in the vicinity of the cracks are derived as well as intensity factors for two cracks in magneto-electro-elastic layer. Numerical results for stress intensity factors and energy release rate are shown graphically.     

Axisymmetric displacement boundary value problem for a penny-shaped crack

Summary A solution is derived from equations of equilibrium in an infinite isotropic elastic solid containing a penny-shaped crack where displacements are given. Abel transforms of the second kind stress and displacement components at an arbitrary point of the solid are known in the literature in terms of jumps of stress and displacement components at a crack plane. Limiting values of these expressions at the crack plane together with the boundary conditions lead to Abel-type integral equations, which admit a closed form solution. Explicit expressions for stress and displacement components on the crack plane are obtained in terms of prescribed face displacements of crack surfaces. Some special cases of the crack surface shape functions have been given in the paper.     

Transversely isotropic solid with elliptical inclusions or cracks

Consider the elastostatic problem of a transversely isotropic space embedded with an inclusion in the form of a thin rigid sheet with an elliptical opening. The sheet is given an infinitesimal tangential shift along an arbitrary direction in the plane. By means of Fourier transforms, the problem is reduced to a system of coupled two-dimensional integro-differential equations. Closed-form solutions are derived by using the Ferrers–Dyson and Galin theorems. Explicit expressions for the displacements and stresses are obtained.