This paper studies the behavior of flexural waves traveling in carbon nanotubes (CNTs) in a free space and embedded in an elastic matrix. An exact higher-order model for analyzing dynamic behavior of nonlocal elastic beams with circular cross-section is proposed, where shear deformation and rotary inertia are both considered without introducing the shear correction factor. Moreover, traction-free condition at the beam surface is met. Using this model, wave dispersion of CNTs is studied and dispersion relation is obtained for single-walled and double-walled CNTs, respectively. Scale-dependent wave speed is given. The effectiveness and validity of the method are confirmed by comparing obtained numerical results with those based on molecular dynamics simulation, the nonlocal Euler–Bernoulli beam theory and nonlocal Timoshenko beam theory with stress gradient and strain gradient. The effects of the scale coefficient, the surrounding elastic medium and van der Waals force on the phase velocity are expounded.

Transverse vibration of the shear beams containing rotary inertia and with a two-parameter elastic foundation is studied. Using asymptotic analysis of Timoshenko beam theory, we derive explicit characteristic equations of the nonclassical shear beams with Winkler-Pasternak elastic restraint and with both ends linked to translational and rotational springs. The condition of the nonclassical shear beams reducing to the classical ones is found. Natural frequencies of the nonclassical modes are evaluated for free- and pinned-elastically restrained shear beams with or without bracing. The influences of elastic restraint stiffness and rotary inertia on the natural frequencies are discussed. Some extreme cases can be recovered from the present. The obtained results are helpful in the design of a tall frame building.

Free vibration of shear beams is studied when rotational motion is taken into account, while classical shear beams do not consider rotational motion. From a single governing equation of Timoshenko beams, we analytically derive Rayleigh beams and shear beams as two limiting cases of the ratio of reduced shear stiffness to bending stiffness being sufficiently large and small, respectively. Emphasis is placed on the analysis of free vibration of nonclassical shear beams without damping effect. Under the condition of general end restraints, a characteristic equation for nonclassical shear beams with finite rotational inertia is derived in explicit form. A condition that the nonclassical shear beams reduce to the classical ones is found, and classical shear beams may be understood as nonclassical ones with infinite large rotational inertia. Nonclassical natural frequencies and mode shapes are calculated for a standing shear beam on an elastic foundation. Previous results of pinned-free, and free–free shear beams can be taken as special cases of the present analysis. The effects of finite rotational inertia, material properties, geometrical conditions and end restraints on the natural frequencies of shear beams are discussed.Research Highlights► Timoshenko beams can analytically reduce to nonclassical shear beams. ► A closed-form characteristic equation of nonclassical shear beams is derived. ► Rotary inertia has a strong effect on the natural frequencies of shear beams.

The stability analysis of a vertically standing or hanging composite column under end force and distributed axial load is made. The composite column has varying cross-section and variable material properties. The integral equation method is formulated to deal with this problem. Critical buckling load can be evaluated by seeking the lowest eigenvalue of the resulting integral equation. A characteristic equation is derived and it is a polynomial equation. The effects of self-weight and taper ratio on the buckling load are discussed for clamped-free prismatic and non-prismatic columns. As an application, two optimum design problems of freestanding tapered columns against buckling are considered to enhance the load-carrying capacity of cantilevered non-uniform columns. One is devoted to the parameter optimization of given shape profile for a homogeneous heavy column subjected to gravity load and tip load simultaneously under constant weight or volume constraint, and obtained results are very close to the exact ones of the strongest columns; the other is devoted to material tailoring such that the ratio of buckling load to weight reaches maximum for an axially graded inhomogeneous column made of two constituents with uniform cross-section.

An analysis of the stability of circular cylindrical columns/beams composed of functionally graded materials is made where shear deformation is taken into account. In this study, a new approach is carried out. Different from the assumption of uniform shear stress at the cross-section adopted in the Timoshenko beam theory, proposed model provides a new approach for treating the problem. Based on the traction-free surface condition, two coupled governing equations for the deflection and rotation are derived, and a single governing equation is further obtained. A comparison of buckling loads derived from the proposed circular column model and the Timoshenko and Euler–Bernoulli theories of beams is made. Moreover, the effects of radial gradient on buckling loads of elastic columns with circular cross-section made of functionally graded materials are elucidated. Finally, the stability of double-walled carbon nanotubes is considered and critical stress is determined and compared with existing results. The results obtained by the proposed model show very good agreement with the results of the Timoshenko beam theory or Reddy–Bickford beam theory.

This paper studies buckling behavior of non-uniform fiber columns under axially distributed compressive load and presents a new approach for determining buckling load. For free-built-in fibers, a Fredholm equation is derived from solving the governing equation with end constraints. Critical load and critical length can be evaluated by seeking the lowest eigenvalue of the resulting equation. For the cases of the cross-sectional moment of inertia and axially distributed load as power functions with respect to the axial distance from the free end, a necessary condition for buckling is derived, and a polynomial characteristic equation is then obtained and solved. The effect of the weight and axial profile of tapered fibers on the critical length is discussed.

Dynamic analysis of a crack embedded in a magnetoelectric material is made when subjected to in-plane mechanical, electric and magnetic impacts. The Laplace and Fourier transforms are applied to reduce the associated initial- and mixed-boundary value problem to dual integral equations, and then to singular integral equations with Cauchy kernel. By numerically solving the resulting equation, the dynamic field intensity factors as well as CODs, and energy release rates near the crack tip are evaluated and presented graphically. The effects of applied magnetic and electric impacts on crack growth are discussed. Obtained results show that, different from the static results, applied magnetic and electric impacts can strongly affect dynamic stress intensity factors.

This paper studies crack extension resulting from a closed crack in compression. The crack-tip field of such a crack contains a singular field relative to KII and non-singular T-stresses Tx and Ty parallel and perpendicular to the crack plane, respectively. Using a modified maximum tensile stress criterion with the singular and non-singular terms, the kinking angle at the onset of crack growth is determined by a two parameter field involving the mode-II stress intensity factors and T-stresses, and at fracture initiation a wing crack may be created at an arbitrary angle from 0° to 90°. A compressive Ty increases the kinking angle and reinforces apparent mode-II fracture toughness, while a compressive Tx decreases the kinking angle and enhances apparent mode-II fracture toughness. The direction and resistance of fracture onset is strongly affected by T-stresses as well as frictional stress. The von Mises effective stress is determined for small-scale yielding near the crack tip. The effective stress contour shape exhibits a marked asymmetrical behavior unless 2Tx = Ty ≤ 0 for plane stress state. Coulomb friction between two crack faces generally increases the kinking angle, shrinks the size enclosed by the effective stress contour and enhances apparent fracture toughness. Field evidence and experimental observations of many phenomena involving the growth of closed cracks in compression agree well with theoretical predictions of the present model.

This paper has twofold aims. One is to study the dynamic response of a magnetoelectroelastic half-space with functionally graded coating containing crack at the interface when subjected to sudden impacts. Two different loading positions, where the material and crack surfaces are loaded respectively, are considered. By using the integral transform method, the problem is reduced to solving singular integral equations. Obtained numerical results show that the overshoots of dynamic fracture parameters are strongly amplified or reduced depending on negative or positive gradient, respectively for the case of the material surface being loaded suddenly. This implies that a functionally graded coating with a positive gradient index is preferable in engineering design due to its capability of preventing the structure from cracking. The second objective is to give a comparison of relevant dynamic parameters such as the intensity factors of stress and strain, energy release rate, and energy density factor, and their features are elucidated under dynamic combined loadings. It indicates that the strain intensity factor can overcome the drawbacks of the rest parameters, and may be chosen as an effective fracture parameter, while three others cannot be adopted as fracture criteria to describe the feature of onset of crack growth.

This paper analyzes the transient response of the dynamic stress intensity factor for an interfacial crack of a functionally graded piezoelectric material (FGPM) coated on the surface of a homogeneous piezoelectric substrate. Different from previous analyses, this study mainly considers a realistic situation when electromechanical loadings are suddenly applied at the material surface. Obtained results are compared with those when the crack surfaces are directly loaded by the same impacts. By using the integral transform method, the problem is reduced to solving two singular integral equations. It is found that dynamic stress intensity factors are significantly amplified and reduced depending on the negative and positive gradient for electromechanical impacts at the material surface. Positive or negative electric impact also decreases or increases the overshoot of the dynamic stress intensity factor. It is suggested that designing an FGPM with a positive gradient index is safer than a negative gradient index.

T-stress as an important parameter characterizing the stress field around a cracked tip has attracted much attention. This paper concerns the T-stress near a cracked tip in a magnetoelectroelastic solid. By applying the Fourier transform, we solve the associated mixed boundary-value problem. Adopting crack-faces electromagnetic boundary conditions nonlinearly dependent on the crack opening displacement, coupled dual integral equations are derived. Then, the closed-form solution for the T-stress is obtained. A comparison of the T stresses for a cracked magnetoelectroelastic solid and for a cracked purely elastic material is made. Obtained results reveal that in addition to applied mechanical loading, the T-stress is dependent on electric and magnetic loadings for a vacuum crack.

The transient response of a crack embedded in a functionally graded material (FGM) layer sandwiched between two dissimilar elastic layers is analyzed under anti-plane shear impact loads. The material properties of the FGM is assumed to be an exponential function of the thickness. The problem is reduced to a singular integral equation via the Fourier and Laplace transforms, and the resulting equation is solved numerically. Dynamic stress intensity factors are determined by a numerical inversion of the Laplace transform. The results indicate that in addition to the crack geometry, the dynamic stress intensity factors depend upon the material properties and the crack position. Moreover, there is a significant error in the obtained dynamic stress intensity factors for applied impacts acting at the material surface and at the crack surfaces.

Nanoindentation is a powerful technique for measuring the mechanical properties of materials such as elastic modulus, hardness, fracture toughness and interfacial adhesion, especially for some structures in micron thickness. Conventional nanoindentation techniques produce large plastic deformation directly affecting the elastic properties to be measured. As compared to conventional nanoindentation, the cross-sectional nanoindentation lowers the influence of plastic deformation on measured elastic properties, and measured data are more reliable. For the cross-sectional nanoindentation, a modified theoretical model based on the elastic plate theory with elastically restrained edges is presented to evaluate the energy release rate of an interfacial delamination between a thin film and an elastic substrate. A closed-form solution is given and the predicted interfacial adhesion is in satisfactory agreement with existing experimental data.

The propagation of piezoelectric gap waves between a piezoceramic half-space and a piezoceramic plate is investigated in the framework of the theory of linear piezoelectricity. Anti-plane shear electroacoustic waves are emphasized. Secular equation for describing dispersion relation is given. The present solutions reduce to the existing results such as the well-known Bleustein–Gulyaev waves for a piezoceramic half-space with an unelectroded surface or electroded surface, the waves propagating in a piezoceramic plate, and the gap waves between two piezoceramic half-spaces. Numerical results are carried out for BaTiO3, PZT-4, and PZT-5H piezoelectric ceramics.

This paper presents a novel method for analyzing steady thermal stresses in a functionally graded hollow cylinder. The thermal and thermoelastic parameters are assumed to arbitrarily vary along the radial direction of the hollow cylinder. The boundary value problem associated with a thermoelastic problem is converted to a Fredholm integral equation. By numerically solving the resulting equation, the distribution of the thermal stresses and radial displacement is obtained. The numerical results obtained are presented graphically and the influence of the gradient variation of the material properties on thermal stresses is investigated. It is found that appropriate gradient can make the distribution of thermal stresses more gentle in the whole structure.

Dynamic analysis of two collinear electro-magnetically dielectric cracks in a piezoelectromagnetic material is made under in-plane magneto-electro-mechanical impacts. Generalized semi-permeable crack-face boundary conditions are proposed to simulate realistic opening cracks with dielectric. Ideal boundary conditions of a combination of electrically permeable or impermeable and magnetically permeable or impermeable assumptions are several limiting cases of the semi-permeable dielectric crack. Utilizing the Laplace and Fourier transforms, the mixed initial-boundary-value problem is reduced to solving singular integral equations with Cauchy kernel. Dynamic intensity factors of stress, electric displacement, magnetic induction and crack opening displacement (COD) near the inner and outer crack tips are determined in the Laplace transform domain. Numerical results for a special magnetoelectroelastic solid are calculated to show the influences of the dielectric permittivity and magnetic permeability inside the cracks on the crack-face electric displacement and magnetic induction. By means of a numerical inversion of the Laplace transform, the variations of the normalized intensity factors of stress and COD are discussed against applied magnetoelectric impact loadings and the geometry of the cracks for fully impermeable, vacuum, fully permeable cracks and shown in graphics.

A novel method for determining an approximate solution to an integral equation with fixed singularity is presented. This integral equation is encountered in solving a cruciform crack. On the basis of Taylor’s series for the unknown function, the integral equation can be transformed to a system of linear equations for the unknown and its derivatives when neglecting a sufficiently small quantity. Moreover, the nth-order approximation obtained is exact for a solution of a polynomial of degree less than or equal to n. The proposed method is simple, fast, and can be performed by symbolic computation using any personal computer. A test example is given to indicate the efficiency of the method. This method is also applicable to a variety of integral equations.

The elastostatic problem of a mode-I crack embedded in a bimaterial with an imperfect interface is investigated. The crack is in proximity to and perpendicular to the imperfect interface, which is governed by linear spring-like relations. The Fourier transform is applied to reduce the associated mixed-boundary value problem to a singular integral equation with Cauchy kernel. By numerically solving the resulting equation, stress intensity factors near both crack tips are evaluated. Obtained results reveal that the stress intensity factors in the presence of the imperfect interface vary between that with a perfect interface and that with a completely debonding interface. Moreover, an increase in the interface parameters decreases the stress intensity factors. In particular, when crack approaches to the weakened interface closer, the stress intensity factors become larger for a sliding interface, and become larger or smaller for a Winkler interface, depending on the crack lying in a stiffer or softer material. The influences of the imperfection of the interface on the stress intensity factors for a bimaterial composed of aluminum and steel are presented graphically.

This paper presents a new and efficient approach for determining the solution of Riccati differential equation. The Riccati equation is first converted to a second-order linear ordinary differential equation, and then to a Volterra integral equation. By solving the resulting Volterra equation by means of Taylor’s expansion, the approximate solution of Riccati differential equation is obtained, which can be achieved by symbolic computation. The accuracy of approximate solution can be further improved with the increase of the order of approximations. An error analysis is given. Test examples demonstrate the effectiveness of the method. A comparison between the present results with previous results is made, inferring that the suggested method is not only enough accurate but also quite stable.

A class of Volterra integral equations are solved in this paper, where the involved kernel may be weakly singular, singular, and hypersingular. For various cases, existence and uniqueness of such Volterra integral equations are established. Furthermore, by reducing such equations to ordinary differential equations, analytic solutions have been determined explicitly. When restricting the solutions to certain special function classes, the number of suitable solutions is given.

The electroelastic analysis of two bonded dissimilar piezoelectric ceramics with a crack perpendicular to and terminating at the interface is made. By using Fourier integral transform, the associated boundary value problem is reduced to a singular integral equation with generalized Cauchy kernel, the solution of which is given in closed form. Results are presented for a permeable crack under anti-plane shear loading and in-plane electric loading. Obtained results indicate that the electroelastic field near the crack tip in the homogeneous piezoelectric ceramic is dominated by a traditional inverse square-root singularity, while the electroelastic field near the crack tip at the interface exhibits the singularity of power law r−α, r being distance from the interface crack tip and α depending on the material constants of a bi-piezoceramic. In particular, electric field has no singularity at the crack tip in a homogeneous solid, whereas it is singular around the interface crack tip. Numerical results are given graphically to show the effects of the material properties on the singularity order and field intensity factors.