Body-Fitted Adjoint Formulation of the Euler Equations

The shape optimization problem governed by the Euler equations is posed in a fixed reference plane. The boundary control is exerted by a parametric mapping from the physical plane to the reference fixed plane. The adjoint equations are derived in such fixed plane. By using this approach remeshing is unnecessary; furthermore, as in many practical applications the parametric mapping can be easily differentiated, the computation of mesh sensitivities is avoided.     

Effects of middle plane curvature on vibrations of a thickness-shear mode crystal resonator

We study the effects of a small curvature of the middle plane of a thickness-shear mode crystal plate resonator on its vibration frequencies, modes and acceleration sensitivity. Two-dimensional equations for coupled thickness-shear, flexural and extensional vibrations of a shallow shell are used. The equations are simplified to a single equation for thickness-shear, and two equations for coupled thickness-shear and extension. Equations with different levels of coupling are used to study vibrations of rotated Y-cut quartz and langasite resonators. The influence of the middle plane curvature and coupling to extension is examined. The effect of middle plane curvature on normal acceleration sensitivity is also studied. It is shown that the middle plane curvature causes a frequency shift as large as 10⁻⁸ g⁻¹ under a normal acceleration. These results have practical implications for the design of concave–convex and plano-convex resonators.     

Complex variable function method for the scattering of plane waves by an arbitrary hole in a porous medium

Based on the complex variable function method, a new approach for solving the scattering of plane elastic waves by a hole with an arbitrary configuration embedded in an infinite poroelastic medium is developed in the paper. The poroelastic medium is described by Biot's theory. By introducing three potentials, the governing equations for Biot's theory are reduced to three Helmholtz equations for the three potentials. The series solutions of the Helmholtz equations are obtained by the wave function expansion method. Through the conformal mapping method, the arbitrary hole in the physical plane is mapped into a unit circle in the image plane. Integration of the boundary conditions along the unit circle in the image plane yields the algebraic equations for the coefficients of the series solutions. Numerical solution of the resulting algebraic equations yields the displacements, the stresses and the pore pressure for the porous medium. In order to demonstrate the proposed approach, some numerical results are given in the paper.     

Three-dimensional crack in the interior of a half-space

In this note, integral equations for the problem of an internal plane crack of arbitrary shape in a three-dimensional elastic half-space are derived. The crack plane is assumed to beparallel to the free surface. Use is made of Mindlin's point force solution in the interior of a semi-infinite solid in deriving the integral equations for the problem.     

Elastic equations for a cylindrical section of a tree

Considering a cylindrical section of a tree subjected to loads independent of x₃ as a relaxed Saint-Venant's problem, it was shown that plane sections remain plane. Since plane sections remain plane, the displacement equations for the neutral fiber derived using either the relaxed Saint-Venant's problem or elementary beam theory are equivalent. The stresses in the plane of the transverse cross-section were found to equal to zero. Therefore, it is appropriate to use elementary beam theory to estimate the three-dimensional stress functions when the wood is considered to be homogeneous. In addition the three-dimensional displacement equations allow the required elastic coefficients in cylindrical coordinates to be measured from full size samples.     

Conservation integrals and determination of HRR singularity fields

The angular distribution functions of HRR singularity fields are analyzed via conservation integrals. Two functional equations are proved for the angular distribution functions and can be used for their solutions. The detailed forms of the functional equations and the final governing equations for solutions are given for the cases of plane strain and plane stress. Accurate numerical results are also given for some typical parameters and the equivalence of different governing equations is proved.     

Meshless analysis of plasticity with application to crack growth problems

The governing equations for classical rate-independent plasticity are formulated in the framework of meshless method. The special J₂ flow theory for three-dimensional, two-dimensional plane strain and plane stress problems are presented. The numerical procedures, including return mapping algorithm, to obtain the solutions of boundary-value problems in computational plasticity are outlined. For meshless analysis the special treatment of the presence of barriers and mirror symmetries is formulated. The crack growth process in elastic–plastic solid under plane strain and plane stress conditions is analyzed. Numerical results are presented and discussed.     

On solitary wave diffraction by multiple,in-line vertical cylinders

The interaction of solitary waves with multiple, in-line vertical cylinders is investigated. The fixed cylinders are of constant circular cross section and extend from the seafloor to the free surface. In general, there are N of them lined in a row parallel to the incoming wave direction. Both the nonlinear, generalized Boussinesq and the Green–Naghdi shallow-water wave equations are used. A boundary-fitted curvilinear coordinate system is employed to facilitate the use of the finite-difference method on curved boundaries. The governing equations and boundary conditions are transformed from the physical plane onto the computational plane. These equations are then solved in time on the computational plane that contains a uniform grid and by use of the successive over-relaxation method and a second-order finite-difference method to determine the horizontal force and overturning moment on the cylinders. Resulting solitary wave forces from the nonlinear Green–Naghdi and the Boussinesq equations are presented, and the forces are compared with the experimental data when available.     

Analysis of a Griffith crack at the interface of two piezoelectric materials under anti-plane loading

The main objective of this work is the contribution to the study of the piezoelectric structures which contain preexisting defect (crack). For that, we consider a Griffith crack located at the interface of two piezoelectric materials in a semi-infinite plane structure. The structure is subjected to an anti-plane shearing combined with an in-plane electric displacement. Using integral Fourier transforms, the equations of piezoelectricity are converted analytically to a system of singular integral equations. The singular integral equations are further reduced to a system of algebraic equations and solved numerically by using Chebyshev polynomials. The stress intensity factor and the electric displacement intensity factor are calculated and used for the determination of the energy release rate which will be taken as fracture criterion. At the end, numerical results are presented for various parameters of the problem; they are also presented for an infinite plane structure.     

The effect of rotation and initial stress on the propagation of waves in a transversely isotropic elastic solid

In this paper the equations governing small amplitude motions in a rotating transversely isotropic initially stressed elastic solid are derived, both for compressible and incompressible linearly elastic materials. The equations are first applied to study the effects of initial stress and rotation on the speed of homogeneous plane waves propagating in a configuration with uniform initial stress. The general forms of the constitutive law, stresses and the elasticity tensor are derived within the finite deformation context and then summarized for the considered transversely isotropic material with initial stress in terms of invariants, following which they are specialized for linear elastic response and, for an incompressible material, to the case of plane strain, which involves considerable simplification. The equations for two-dimensional motions in the considered plane are then applied to the study of Rayleigh waves in a rotating half-space with the initial stress parallel to its boundary and the preferred direction of transverse isotropy either parallel to or normal to the boundary within the sagittal plane. The secular equation governing the wave speed is then derived for a general strain–energy function in the plane strain specialization, which involves only two material parameters. The results are illustrated graphically, first by showing how the wave speed depends on the material parameters and the rotation without specifying the constitutive law and, second, for a simple material model to highlight the effects of the rotation and initial stress on the surface wave speed.     

Displacement function and simplifying of plane elasticity problems of two-dimensional quasicrystals with noncrystal rotational symmetry

The paper systematically investigates the plane elasticity problems of two-dimensional quasicrystals with noncrystal rotational symmetry. First, applying their independent elastic constants, the equilibrium differential equations of the problems in terms of displacements are derived and it is found that the plane elasticity of pentagonal quasicrystals is the same as that of decagonal. Then by introducing displacement functions, huge numbers of complicated partial differential equations of the problems are simplified to a single or a pair of partial differential equations of higher order, which is called governing equations, such that the problems can be easily solved. Finally, some solving methods of these governing equations obtained are introduced briefly.     

Finite volume methods for laminar and turbulent flows using a penalty function approach

A penalty function, finite volume method is described for two-dimensional laminar and turbulent flows. Turbulence is modelled using the k-? model. The governing equations are discretized and the resulting algebraic equations are solved using both sequential and coupled methods. The performance of these methods is gauged with reference to a tuned SIMPLE-C algorithm. Flows considered are a square cavity with a sliding top, a plane channel flow, a plane jet impingement and a plane channel with a sudden expansion. A sequential method is employed, which uses a variety of dicretization practices, but is found to be extremely slow to converge; a coupled method, evaluated using a variety of matrix solvers, converges rapidly but, relative to the sequential approach, requires larger memory.     

Axisymmetric mixed flows in magnetogasdynamics

In contrast with conventional gasdynamics, in magnetogasdynamics there are several types of mixed flows. A detailed study of such plane flows was first made by Kogan [1]. After this, intensive work was done on the magnetogasdynamic mixed flows [2–13], with the plane case being considered in all the studies except [9]. In [9] the equations of the possible mixed flows for the axisymmetric case were obtained in terms of the disturbance velocity components.The axisymmetric mixed flows are studied in detail in the present paper. The exact equations of motion are obtained for the velocity potential and the streamfunction, and the corresponding approximate equations are obtained for all the transitional regimes (transonic, hypercritical, trans-Alfvenic, transonic-trans-Alfvenic). Simple particular solutions are obtained for these approximate equations.For greater generality the entire study is made simultaneously for the plane and axisymmetric cases.The author wishes to thank S. V. Fal'kovich for his interest in the study and for valuable discussions.     

Two-dimensional analysis of progressive delamination in thin film electrodes

By employing the two-dimensional analysis, i.e., plane strain and plane stress, a semi-analytical method is developed to investigate the interfacial delamination in electrodes. The key parameters are obtained from the governing equations, and their effects on the evolution of the delamination are evaluated. The impact of constraint perpendicular to the plane is also investigated by comparing the plane strain and plane stress. It is found that the delamination in the plane strain condition occurs easier, indicating that the constraint is harmful to maintain the structure stability. According to the obtained governing equations, a formula of the dimensionless critical size for delamination is provided, which is a function of the maximum volumetric strain and the Poisson’s ratio of the active layer.     

Formulation of Problems in the General Kirchhoff—Love Theory of Inhomogeneous Anisotropic Plates

In this paper we study the procedure of reducing the three-dimensional problem of elasticity theory for a thin inhomogeneous anisotropic plate to a two-dimensional problem in the median plane. The plate is in equilibrium under the action of volume and surface forces of general form. À notion of internal force factors is introduced. The equations for force factors (the equilibrium equations in the median plane) are obtained from the thickness-averaged three-dimensional equations of elasticity theory. In order to establish the relation between the internal force factors and the characteristics of the deformed middle surface, we use some prior assumptions on the distribution of displacements along the thickness of the plate. To arrange these assumptions in order, the displacements of plate points are expanded into Taylor series in the transverse coordinate with consideration of the physical hypotheses on the deformation of a material fiber being originally perpendicular to the median plane. The well-known Kirchhoff—Love hypothesis is considered in detail. À closed system of equations for the theory of inhomogeneous anisotropic plates is obtained on the basis of the Kirchhoff—Love hypothesis. The boundary conditions are formulated from the Lagrange variational principle.     

Plane strain and generalized plane stress problems for fibre-reinforced materials

In this paper the basic equations governing the plane strain or generalized plane stress deformations of a linear elastic material reinforced by a single family of parallel inextensible fibres are deduced. It is found that a single system of equations will cover all cases. The solutions for plane, half-plane and strip problems are evaluated and compared with those for an ideal fibre-reinforced material.     

非局部平面应变和平面应力问题界定及其精确性讨论

在非局部弹性理论框架下对平面应变和平面应力状态重新界定.首先,分别在其相应简化假设下推导控制方程,并与经典局部情况进行比较.然后,引入变形协调条件对两类非局部平面问题的精确性进行讨论.其中,对于非局部平面应力状态,通过应变协调方程的Fourier变换形式来进行研究,使问题得以简化.通过以上分析,最终得到一些有价值的结论.     

A numerical analysis of partial slip problems under Hertzian contacts

In this study the tangential partial slip problems in Hertzian contact regions are treated by a numerical technique. The tangential loading may include tangential forces in the contact plane and a twisting moment normal to the contact plane. The Coulomb’s law of friction and the property that the direction of friction must oppose the relative motion lead to nonlinear equations. The Newton-Raphson method is utilized to solve these nonlinear equations. Numerical results for tangential tractions and sizes of stick and slip zones may be determined, and they agree with existing analytical results for circular contacts.     

Interaction of plane elastic harmonic waves with a perfectly bonded elastic inclusion

The interaction of plane harmonic waves with a thin elastic inclusion in the form of a strip in an infinite body (matrix) under plane strain conditions is studied. It is assumed that the bending and shear displacements of the inclusion coincide with the displacements of its midplane. The displacements in the midplane are found from the theory of plates. The priblem-solving method represents the displacements as discontinuous solutions of the Lamé equations and finds the unknown discontinuities solving singular integral equations by the numerical collocation method. Approximate formulas for the stress intensity factors at the ends of the inclusion are derived     

Longitudinal propagation in a layered magneto-ionic medium (Epstein profile)

Summary The longitudinal propagation and reflection of a plane electromagnetic wave in a horizontally stratified magneto-ionic medium is considered. In this case Maxwell's equations reduce to two uncoupled ordinary second-order differential equations, describing the propagation of two elliptically polarized plane waves. The electron density of the medium is assumed to vary with the vertical Cartesian coordinatez according to the Epstein law. Rigorous solutions of the relevant differential equations can be obtained either in the form of hypergeometric functions or in the form of an integral representation. The reflection coefficients of both waves are then expressed in terms of gamma functions. The following quantities are considered in detail in their dependence on the parameters involved: the modulus of the reflection coefficient, the phase delay time and the group delay time. Some numerical results are given.