An interface crack with contact zones in a piezoelectric/piezomagnetic bimaterial

An interface crack with a frictionless contact zone at the right crack tip between two semi-infinite piezoelectric/piezomagnetic spaces under the action of a remote mechanical loading, magnetic and electric fluxes as well as concentrated forces at the crack faces is considered. Assuming that all fields are independent on the coordinate x ₂ co-directed with the crack front, the stresses, the electrical and the magnetic fluxes as well as the derivatives of the jumps of the displacements, the electrical and magnetic potentials are presented via a set of analytic functions in the (x ₁, x ₃)-plane with a cut along the crack region. Two cases of magneto-electric conditions at the crack faces are considered. The first case assumes that the crack is electrically and magnetically permeable, and in the second case the crack is assumed electrically permeable while the open part of the crack is magnetically impermeable. For both these cases due to the above-mentioned representation the combined Dirichlet–Riemann boundary value problems have been formulated and solved exactly. Stress, electric and magnetic induction intensity factors are found in a simple analytical form. Transcendental equations and a closed form analytical formula for the determination of the real contact zone length have been derived for both cases of magnetic conditions in the crack region. For a numerical illustration of the obtained results a bimaterial BaTiO₃–CoFe₂O₄ with different volume fractions of BaTiO₃ has been used, and the influence of the mechanical loading and the intensity of the magnetic flux upon the contact zone length and the associated intensity factors as well as the energy release rate has been shown.     

均布荷载作用下功能梯度简支梁弯曲的解析解

利用应力函数半逆解法,研究了均布载荷作用下、材料属性在厚度上任意变化的功能梯度简支梁弯曲的解析解,给出了各向应力应变与位移的解析显式表达式.首先根据平面应力状态的基本方程,得出了功能梯度梁的应力函数应满足的偏微分方程,并根据应力边界条件得出了各应力分布的表达式;进而根据功能梯度材料的本构方程和位移边界条件,得出了应变和位移的分布.最后,通过将本文的解退化到均质各向同性梁并与经典弹性解比较,证明了本文理论的正确性,并求解了材料组分呈幂律分布的功能梯度梁的应力和位移分布,分析了上下表层材料的弹性模量比λ与组分材料体积分数指数n对应力和位移分布的影响.     

Semi-permeable crack analysis in a 2D magnetoelectroelastic medium based on the EMPS model

For a crack in a magnetoelectroelastic plane under the electrically and magnetically semi-permeable boundary condition, we derive the non-linear analytical solution of the strip electric–magnetic polarization saturation (EMPS) model. Using the extended dislocation theory and integral equation method, we obtain the electric and magnetic yielding zones, as well as the field intensity factor and local J-integral. Adapting an iterative method, numerical examples were performed to analyze the effect of different boundary conditions and the electric–magnetic saturated properties on the electric displacement and magnetic induction in the crack cavity, electric and magnetic yielding zones, stress intensity factor and local J-integral.     

Interaction between a screw dislocation and a viscoelastic piezoelectric bimaterial interface

The complex variable method is employed to derive analytical solutions for the interaction between a piezoelectric screw dislocation and a Kelvin-type viscoelastic piezoelectric bimaterial interface. Through analytical continuation, the original boundary value problem can be reduced to an inhomogeneous first-order partial differential equation for a single function of location z = x + iy and time t defined in the lower half-plane, which is free of the screw dislocation. Once the initial, steady-state and far-field conditions are known, the solution to the first order differential equation can be obtained. From the solved function, explicit expressions are then derived for the stresses, strains, electric fields and electric displacements induced by the piezoelectric screw dislocation. Also presented is the image force acting on the screw dislocation due to its interaction with the Kelvin-type viscoelastic interface. The derived solutions are verified by comparing with existing solutions for the simplified cases, and various interesting features are observed, particularly for those associated with the image force.     

Two-dimensional elasticity solution for bending of functionally graded beams with variable thickness

This paper studies the stress and displacement distributions of functionally graded beam with continuously varying thickness, which is simply supported at two ends. The Young’s modulus is graded through the thickness following the exponential-law and the Poisson’s ratio keeps constant. On the basis of two-dimensional elasticity theory, the general expressions for the displacements and stresses of the beam under static loads, which exactly satisfy the governing differential equations and the simply supported boundary conditions at two ends, are analytically derived out. The unknown coefficients in the solutions are approximately determined by using the Fourier sinusoidal series expansions to the boundary conditions on the upper and lower surfaces of the beams. The effect of Young’s modulus varying rules on the displacements and stresses of functionally graded beams is investigated in detail. The two-dimensional elasticity solution obtained can be used to assess the validity of various approximate solutions and numerical methods for the aforementioned functionally graded beams.     

Nonlinear dynamic analysis of Timoshenko beams by BEM. Part I: Theory and numerical implementation

In this two-part contribution, a boundary element method is developed for the nonlinear dynamic analysis of beams of arbitrary doubly symmetric simply or multiply connected constant cross section, undergoing moderate large displacements and small deformations under general boundary conditions, taking into account the effects of shear deformation and rotary inertia. Part I is devoted to the theoretical developments and their numerical implementation and Part II discusses analytical and numerical results obtained from both analytical or numerical research efforts from the literature and the proposed method. The beam is subjected to the combined action of arbitrarily distributed or concentrated transverse loading and bending moments in both directions as well as to axial loading. To account for shear deformations, the concept of shear deformation coefficients is used. Five boundary value problems are formulated with respect to the transverse displacements, to the axial displacement and to two stress functions and solved using the Analog Equation Method, a BEM based method. Application of the boundary element technique yields a nonlinear coupled system of equations of motion. The solution of this system is accomplished iteratively by employing the average acceleration method in combination with the modified Newton–Raphson method. The evaluation of the shear deformation coefficients is accomplished from the aforementioned stress functions using only boundary integration. The proposed model takes into account the coupling effects of bending and shear deformations along the member, as well as the shear forces along the span induced by the applied axial loading.

     

Stability of curvilinear Euler-Bernoulli beams in temperature fields

In this work, stability of thin flexible Bernoulli-Euler beams is investigated taking into account the geometric non-linearity as well as a type and intensity of the temperature field. The applied temperature field T(x,z) is yielded by a solution to the 2D Laplace equation solved for five kinds of thermal boundary conditions, and there are no restrictions put on the temperature distribution along the beam thickness. Action of the temperature field on the beam dynamics is studied with the help of the Duhamel theory, whereas the motion of the beam subjected to the thermal load is yielded employing the variational principles.The heat transfer (Laplace equation) is solved with the use of the finite difference method (FDM) of the third-order accuracy, while the integrals along the beam thickness defining the thermal stress and moments are computed using Simpson's method. Partial differential equations governing the beam motion are reduced to the Cauchy problem by means of application of FDM of the second-order accuracy. The obtained ordinary differential equations are solved with the use of the fourth-order Runge-Kutta method.The problem of numerical results convergence versus a number of beam partitions is investigated. A static solution for a flexible Bernoulli-Euler beam is obtained using the dynamic approach based on employment of the relaxation/set-up method.Novel stability loss phenomena of a beam under the thermal field are reported for different beam geometric parameters, boundary conditions, and the temperature intensity. In particular, it has been shown that stability of the flexible beam during heating the beam surface essentially depends on the beam thickness.     

Elasticity solutions for a transversely isotropic functionally graded circular plate subject to an axisymmetric transverse load qrk

This paper considers the bending of transversely isotropic circular plates with elastic compliance coefficients being arbitrary functions of the thickness coordinate, subject to a transverse load in the form of qr^k (k is zero or a finite even number). The differential equations satisfied by stress functions for the particular problem are derived. An elaborate analysis procedure is then presented to derive these stress functions, from which the analytical expressions for the axial force, bending moment and displacements are obtained through integration. The method is then applied to the problem of transversely isotropic functionally graded circular plate subject to a uniform load, illustrating the procedure to determine the integral constants from the boundary conditions. Analytical elasticity solutions are presented for simply-supported and clamped plates, and, when degenerated, they coincide with the available solutions for an isotropic homogenous plate. Two numerical examples are finally presented to show the effect of material inhomogeneity on the elastic field in FGM plates.     

Elasticity solutions for plane anisotropic functionally graded beams

This paper considers the plane stress problem of generally anisotropic beams with elastic compliance parameters being arbitrary functions of the thickness coordinate. Firstly, the partial differential equation, which is satisfied by the Airy stress function for the plane problem of anisotropic functionally graded materials and involves the effect of body force, is derived. Secondly, a unified method is developed to obtain the stress function. The analytical expressions of axial force, bending moment, shear force and displacements are then deduced through integration. Thirdly, the stress function is employed to solve problems of anisotropic functionally graded plane beams, with the integral constants completely determined from boundary conditions. A series of elasticity solutions are thus obtained, including the solution for beams under tension and pure bending, the solution for cantilever beams subjected to shear force applied at the free end, the solution for cantilever beams or simply supported beams subjected to uniform load, the solution for fixed–fixed beams subjected to uniform load, and the one for beams subjected to body force, etc. These solutions can be easily degenerated into the elasticity solutions for homogeneous beams. Some of them are absolutely new to literature, and some coincide with the available solutions. It is also found that there are certain errors in several available solutions. A numerical example is finally presented to show the effect of material inhomogeneity on the elastic field in a functionally graded anisotropic cantilever beam.     

The derivation of a thick and thin plate formulation without ad hoc assumptions

For the plate formulation considered in this paper, appropriate three-dimensional elasticity solution representations for isotropic materials are constructed. No a priori assumptions for stress or displacement distributions over the thickness of the plate are made. The strategy used in the derivation is to separate functions of the thickness variable z from functions of the coordinates x and y lying in the midplane of the plate. Real and complex 3-dimensional elasticity solution representations are used to obtain three types of functions of the coordinates x, y and the corresponding differential equations. The separation of the functions of the thickness coordinate can be done by separately considering homogeneous and nonhomogeneous boundary conditions on the upper and lower faces of the plate. One type of the plate solutions derived involves polynomials of the thickness coordinate z. The other two solution forms contain trigonometric and hyperbolic functions of z, respectively. Both bending and stretching (or in-plane) solutions are included in the derivation.     

On the three-dimensional vibrations of elastic prisms with skew cross-section

Three-dimensional (3-D) free vibration of an elastic prism with skew cross-section is investigated using an elasticity-based variational Ritz procedure. Specifically, the associated energy functional minimized in the Ritz procedure is formulated using a simple coordinate mapping to transform the solid skew elastic prism into a unit cube computational domain. The displacements of the prism in each direction are approximately expressed in the form of variable separation. As an enhancement to conventional use of algebraic polynomials trial series in related solid body vibration studies in the associated literature, the assumed skew prism displacement, u, v and w in the computational ξ–η–ζ skew coordinate directions, respectively, are approximated by a set of generalized coefficients multiplied by a finite triplicate Chebyshev polynomial series and boundary functions in ξ–η–ζ to ensure the satisfaction of the geometric boundary conditions of the prism. Upon invoking the stationary condition of the Lagrangian energy functional for the skew elastic prism with respect to the assumed generalized coefficients, the usual characteristic frequency equations of natural vibrations of the skew elastic prism are derived. Upper bound convergence of the first eight non-dimensional frequencies accurate to four significant figures is achieved by using up to 10–15 terms of the assumed skew prism displacement functions. First known 3-D vibration characteristics of skew elastic prisms are examined showing the effects of varying prism length ratios (ranging from skew solids to skew slender beams), as well as, varying cross-sectional side ratios and skewness, which collectively can serve as benchmark studies against which vibration modes predicted by classical Euler and shear deformable skew beam theories as well as alternative methodologies used in elastic prism vibrations of mechanical and structural components.     

Three-dimensional analysis of transient thermal stresses in functionally graded plates

An analytical solution is presented for three-dimensional thermomechanical deformations of a simply supported functionally graded (FG) rectangular plate subjected to time-dependent thermal loads on its top and/or bottom surfaces. Material properties are taken to be analytical functions of the thickness coordinate. The uncoupled quasi-static linear thermoelasticity theory is adopted in which the change in temperature, if any, due to deformations is neglected. A temperature function that identically satisfies thermal boundary conditions at the edges and the Laplace transformation technique are used to reduce equations governing the transient heat conduction to an ordinary differential equation (ODE) in the thickness coordinate which is solved by the power series method. Next, the elasticity problem for the simply supported plate for each instantaneous temperature distribution is analyzed by using displacement functions that identically satisfy boundary conditions at the edges. The resulting coupled ODEs with variable coefficients are also solved by the power series method. The analytical solution is applicable to a plate of arbitrary thickness. Results are given for two-constituent metal-ceramic FG rectangular plates with a power-law through-the-thickness variation of the volume fraction of the constituents. The effective elastic moduli at a point are determined by either the Mori–Tanaka or the self-consistent scheme. The transient temperature, displacements, and thermal stresses at several critical locations are presented for plates subjected to either time-dependent temperature or heat flux prescribed on the top surface. Results are also given for various volume fractions of the two constituents, volume fraction profiles and the two homogenization schemes.     

Nonlinear dynamic analysis of Timoshenko beams by BEM. Part II: applications and validation

In this two-part contribution, a boundary element method is developed for the nonlinear dynamic analysis of beams of arbitrary doubly symmetric simply or multiply connected constant cross section, undergoing moderate large displacements and small deformations under general boundary conditions, taking into account the effects of shear deformation and rotary inertia. In Part I the governing equations of the aforementioned problem have been derived, leading to the formulation of five boundary value problems with respect to the transverse displacements, to the axial displacement and to two stress functions. These problems are numerically solved using the Analog Equation Method, a BEM based method. In this Part II, numerical examples are worked out to illustrate the efficiency, the accuracy and the range of applications of the developed method. Thus, the results obtained from the proposed method are presented as compared with those from both analytical and numerical research efforts from the literature. More specifically, the shear deformation effect in nonlinear free vibration analysis, the influence of geometric nonlinearities in forced vibration analysis, the shear deformation effect in nonlinear forced vibration analysis, the nonlinear dynamic analysis of Timoshenko beams subjected to arbitrary axial and transverse in both directions loading, the free vibration analysis of Timoshenko beams with very flexible boundary conditions and the stability under axial loading (Mathieu problem) are presented and discussed through examples of practical interest.

     

Solution of a mixed dynamic problem on the displacements given on the boundary of a half-plane lying on the surface of an isotropic homogeneous elastic half-space

We consider the problem on the motion of an isotropic elastic body occupying the half-space z ≥ 0 on whose boundary, along the half-plane x ≥ 0, the horizontal components of displacement are given, while the remaining part of the boundary is stress-free. We seek the solution by the method of integral Laplace transforms with respect to time t and Fourier transforms with respect to the coordinates x, y; the problem is reduced to a system of Wiener-Hopf equations, which can be solved by the methods of singular-integral equations and circulants. We invert the integral transforms and reduce the solution to the Smirnov-Sobolev form. We calculate the tangential stress intensity coefficients near the boundary z = 0, x = 0, |y| < ∞ of the half-plane. The circulant method for solving the Wiener-Hopf system was proposed in [1]. A static problem similar to that considered in the present paper was solved earlier. The Hilbert problem was reduced to a system of Fredholm integral equations in [2]. In the present paper, we solve the above problem by reducing the solution to quadratures and a quasiregular system of Fredholm integral equations. We give a numerical solution of the Fredholm equations and calculate the integrals for the tangential stress intensity coefficients.     

Fracture-mechanical assessment of electrically permeable interface cracks in piezoelectric bimaterials by consideration of various contact zone models

Summary An interface crack with an artificial contact zone at the right-hand side crack tip between two piezoelectric semi-infinite half-planes is considered under remote mixed-mode loading. Assuming the stresses, strains and displacements are independent of the coordinate x ₂, the expression for the displacement jumps and stresses along the interface are found via a sectionally holomorphic vector function. For piezoceramics of the symmetry class 6 mm and for electrically permeable crack faces, the problem is reduced to a combined Dirichlet-Riemann boundary value problem which can be solved analytically. Further, analytical expressions for the stresses, electrical displacements, derivatives of elastic displacement jumps, stress and electrical intensity factors are found at the interface. Real contact zone lengths and the well-known oscillating solution are derived from the obtained solution as well. Analytical relationships between the fracture-mechanical parameters of various models are found, and recommendations are suggested concerning the application of numerical methods to the problem of an interface crack in the discontinuity area of a piezoelectric bimaterial. Received 16 March 1999; accepted for publication 31 May 1999     

A new method for solving Biot＇s consolidation of a finite soil layer in the cylindrical coordinate system

A new method is developed to solve Biot＇s consolidation of a finite soil layer in the cylindrical coordinate system. Based on the governing equations of Biot＇s consolidation and the technique of Laplace transform, Fourier expansions and Hankel transform with respect to time t, coordinate θ and coordinate r, respectively, a relationship of displacements, stresses, excess pore water pressure and flux is established between the ground surface （z = 0） and an arbitrary depth z in the Laplace and Hankel transform domain. By referring to proper boundary conditions of the finite soil layer, the solutions for displacements, stresses, excess pore water pressure and flux of any point in the transform domain can be obtained. The actual solutions in the physical domain can be acquired by inverting the Laplace and the Hankel transforms.     

Elasticity solution of clamped-simply supported beams with variable thickness

This paper studies the stress and displacement distributions of continuously varying thickness beams with one end clamped and the other end simply supported under static loads. By introducing the unit pulse functions and Dirac functions, the clamped edge can be made equivalent to the simply supported one by adding the unknown horizontal reactions. According to the governing equations of the plane stress problem, the general expressions of displacements, which satisfy the governing differefitial equations and the boundary conditions attwo ends of the beam, can be deduced. The unknown coefficients in the general expressions are then determined by using Fourier sinusoidal series expansion along the upper and lower boundaries of the beams and using the condition of zero displacements at the clamped edge. The solution obtained has excellent convergence properties. Comparing the numerical results to those obtained from the commercial software ANSYS, excellent accuracy of the present method is demonstrated.     

Computations of hydrodynamic coefficients, displacement-amplitude ratios and forces for a floating cylinder due to wave diffraction and radiation

Computations of the hydrodynamic coefficients, displacement-amplitude ratios and loadings on floating vertical circular cylinder due to diffraction and radiation are presented here. The boundary value problem (BVP) is solved in terms of diffraction potential and three potentials due to radiation, two translational motions about x-axis (surge) and about z-axis (heave), one rotational motion about y-axis (pitch). The analytical expressions for the hydrodynamic coefficients, displacement-amplitude ratios and loadings for this case were obtained previously by Bhatta and Rahman [1]. In this paper, we present the computational aspects of those analytical results for different depth to radius and draft to radius ratios. JMSL (Java Mathematical and Statistical Library) is used to compute special functions and solve complex matrix equations.     

Piezoelectric study on circularly cylindrical layered media

In this paper, an analytical solution in series form for the problem of a circularly cylindrical layered piezoelectric composite consisting of N dissimilar layers is presented within the framework of linear piezoelectricity. Each layer of the composite is assumed to be transversely isotropic with respect to the longitudinal direction (x₃ direction), and the composite is subject to arbitrary electromechanical singularities infinitely extended in a direction perpendicular to the x₁–x₂ plane such that only in-plane electric fields and out-of-plane displacement are produced. The alternating technique in conjunction with the method of analytical continuation is applied to derive the general multilayered media solution in an explicit series form, whose convergence is guaranteed numerically. The distributions of the shear stress and electric field are found to be dependent on the material combinations and the magnitude and position of the electromechanical singularities. An exactly closed form solution is obtained and discussed graphically for a practical example.