Axisymmetric problem of a nonhomogeneous elastic layer

Summary The paper deals with a theoretical treatment of elastic behavior for a medium with nonhomogeneous materials property, which is defined by the relation , i.e., shear modulus of elasticity G varies with the dimensionless axial coordinate by the power product form, arbitrarily. Fundamental differential equation for such nonhomogeneous medium has been already proposed in [5]. It is given by a second-order partial differential equation. However, it was found that the fundamental equation is not sufficient in general to solve several kinds of boundary-value problems. On the other hand, it is shown in the present paper making use of the fundamental equations system for a nonhomogeneous medium, which has been proposed in our previous paper [7], it is possible to solve axisymmetric problems for a thick plate (layer) subjected to an arbitrarily distributed load or a concentrated load on its surfaces. Numerical calculations are carried out for several cases, taking into account the variation of the nonhomogeneous parameter m. The numerical results for displacements stress and components are shown in graphical form. Accepted for publication 25 March 1997     

An Average-Stretch Full-Network Model for Rubber Elasticity

Two constitutive models that are based on the classical non-Gaussian, Kuhn-Grün probability distribution function are reviewed. It is shown that all chains of a network cell structure comprised of a finite number of identical chains in an affine deformation referred to principal axes may have the same invariant stretch, if and only if the chains are oriented initially along any of eight directions forming the diagonals of a unit cube. The 4-chain tetrahedral and the 8-chain cubic cell structures are familiar admissible models having this property. An easy derivation of the constitutive equation for the Wu and van der Giessen full-network model of initially identical chains arbitrarily oriented in the undeformed state is presented. The constitutive equations for the neo-Hookean model, the 3 -chain model, and the equivalent 4- and 8-chain models are then derived from the Wu and van der Giessen equation. The squared chain stretch of an arbitrarily directed chain averaged over a unit sphere surrounding all chains radiating from a cross-link junction as its center is determined. An average-stretch, full-network constitutive equation is then derived by approximation of the Wu and van der Giessen equation. This result, though more general in that no special chain cell morphology is introduced, is the same as the constitutive equation for the 4- and 8-chain models. Some concluding remarks on extensions to amended models are presented.     

Inclusions in a finite elastic body

Within the framework of 2D or 3D linear elasticity, a general approach based on the superposition principle is proposed to study the problem of a finite elastic body with an arbitrarily shaped and located inclusion. The proposed approach consists in decomposing the initial inclusion problem into the problem of the inclusion embedded in the corresponding infinite body and the auxiliary problem of the finite body subjected to the appropriate boundary loading provided by solving the former problem. Thus, our approach renders it possible to circumvent the difficulty due to the unavailability of the relevant Green function, use various existing solutions for the problem of an inclusion inside an unbounded body and clearly makes appear the finite boundary effects. The general approach is applied and specified in the context of 2D isotropic elasticity. The complex potentials for the problem of an inclusion in an infinite body are given as two boundary integrals, and the boundary integral equation governing the complex potentials for the auxiliary problem is provided. In the important particular situation where a finite body with an arbitrarily shaped and located inclusion is circular, the exact explicit expressions for the complex potentials are derived, leading to those for the strain, stress and Eshelby’s tensor fields inside and outside the inclusion. These results are analytically detailed and numerically illustrated for the cases of a square inclusion placed concentrically, and a circular inclusion located eccentrically, inside a circular body.     

A unified evolution equation for the Cauchy stress tensor of an isotropic elasto-visco-plastic material

In the present study an evolution equation for the Cauchy stress tensor is proposed for an isotropic elasto-visco-plastic continuum. The proposed stress model takes effects of elasticity, viscosity and plasticity of the material simultaneously into account. It is ascribed with some scalar coefficient functions and, in particular, with an unspecified tensor-valued function N, which is handled as an independent constitutive quantity. It is demonstrated that by varying the values and the specific functional forms of these coefficients and N, different known models in non-Newtonian rheology can be reproduced. A thermodynamic analysis, based on the Müller–Liu entropy principle, is performed. The results show that these coefficients and N are not allowed to vary arbitrarily, but should satisfy certain restrictions. Simple postulates are made to further simplify the deduced general results of the thermodynamic analysis. They yield justification and thermodynamic consistency of the existing models for a class of materials embracing thermoelasticity, hypoelasticity and in particular hypoplasticity, of which the thermodynamic foundation is established successively for the first time in literature. The study points at the wide applicability and practical usefulness of the present model in different fields from non-Newtonian fluid to solid mechanics. In this paper the thermodynamic analysis of the proposed evolution-type stress model is discussed, its applications are reported later.      

随机损伤过程和依赖于时间的损伤概率分布

本文用马尔柯夫过程描述一般损伤现象,将确定性损伤发展方程随机化为随机微分方程,并得其相应的 Fokker-Planck 方程的封闭解,该解即依赖于时间的损伤概率分布,其包含了某些对可靠性工程有意义的特殊情况。     

KBM unified method for solving an nth order non-linear differential equation under some special conditions including the case of internal resonance

The Krylov-Bogoliubov-Mitropolskii (KBM) unified method is used for obtaining the approximate solution of an nth order (n?4) ordinary differential equation with small non-linearities when a pair of eigen-values of the unperturbed equation is multiple (approximately or perfectly) of the other pair or pairs. The general solution can be used arbitrarily for over-damped, damped and undamped cases. In a damped or undamped case, one of the natural frequencies of the unperturbed equation may be a multiple of the other. Thus, the solution also covers the case of internal resonance which is an interesting and important part of non-linear oscillation. The determination of the solution is very simple and easier than the existing procedures developed by several authors (both in methods of averaging and multiple time scales) especially to tackle the case of internal resonance. The method is illustrated by an example of a fourth-order differential equation. The solution shows a good agreement with numerical solution in all of the three cases, e.g. over-damped, damped and undamped.     

轴对称载荷下正交异性圆环壳的一般解

本文导出了在Kirchhoff假设下受轴对称载荷时、材料正交异性的圆环壳的复变量方程,给出了该方程的一般解,该解可用于α=a/R<1,其中a为环壳截面的半径,R为环壳的总体半径,文中并举例说明其应用,结果表明本文所提出的解是很有效的。     

Parallel large eddy simulation of turbulent flow around MIRA model using linear equal‐order finite element method

A parallel large eddy simulation code that adopts domain decomposition method has been developed for large‐scale computation of turbulent flows around an arbitrarily shaped body. For the temporal integration of the unsteady incompressible Navier–Stokes equation, fractional 4‐step splitting algorithm is adopted, and for the modelling of small eddies in turbulent flows, the Smagorinsky model is used. For the parallelization of the code, METIS and Message Passing Interface Libraries are used, respectively, to partition the computational domain and to communicate data between processors. To validate the parallel architecture and to estimate its performance, a three‐dimensional laminar driven cavity flow inside a cubical enclosure has been solved. To validate the turbulence calculation, the turbulent channel flows at Re_τ = 180 and 1050 are simulated and compared with previous results. Then, a backward facing step flow is solved and compared with a DNS result for overall code validation. Finally, the turbulent flow around MIRA model at Re = 2.6 × 10⁶ is simulated by using approximately 6.7 million nodes. Scalability curve obtained from this simulation shows that scalable results are obtained. The calculated drag coefficient agrees better with the experimental result than those previously obtained by using two‐equation turbulence models. Copyright © 2007 John Wiley & Sons, Ltd.     

Vibration of arbitrarily shaped membranes with elastical supports at points

This paper presents a new method for solving the vibration of arbitrarily shaped membranes with elastical supports at points. The reaction forces of elastical supports at points are regarded as unknown external forces acting on the membranes. The exact solution of the equation of motion is given which includes terms representing the unknown reaction forces. The frequency equation is derived by the use of the linear relationship of the displacements with the reaction forces of elastical supports at points. Finally the calculating formulae of the frequency equation of circular membranes are analytically performed as examples and the inherent frequencies of circular membranes with symmetric elastical supports at two points are numerically calculated.     

Plane Strain Bending of Cylindrical Sectors of Admissible Compressible Hyperelastic Materials

In the theory of nonlinear elasticity of rubber-like materials, if a homogeneous isotropic compressible material is described by a strain–energy function that is a homogeneous function of the principal stretches, then the equations of equilibrium for axisymmetric deformations reduce to a separable first-order ordinary differential equation. For a particular class of such strain–energy functions, this property is used to obtain a general parametric solution to the equilibrium equation for plane strain bending of cylindrical sectors. Specification of the arbitrary function that appears in such strain–energy functions yields some parametric solutions. In some cases, the parameter can be eliminated to yield closed-form solutions in implicit or explicit form. Other possible forms for the arbitrary constitutive function that are likely to yield such solutions are also indicated.     

Uniqueness of positive solutions of Δu−u+up=0 in Rn

We establish the uniqueness of the positive, radially symmetric solution to the differential equation u–u+u^p=0 (with p>1) in a bounded or unbounded annular region in R ⁿ for all n1, with the Neumann boundary condition on the inner ball and the Dirichlet boundary condition on the outer ball (to be interpreted as decaying to zero in the case of an unbounded region). The regions we are interested in include, in particular, the cases of a ball, the exterior of a ball, and the whole space. For p=3 and n=3, this a well-known result of Coffman, which was later extended by McLeod & Serrin to general n and all values of p below a certain bound depending on n. Our result shows that such a bound on p is not needed. The basic approach used in this work is that of Coffman, but several of the principal steps in the proof are carried out with the help of Sturm's oscillation theory for linear second-order differential equations. Elementary topological arguments are widely used in the study.     

The Sub-Harmonic Bifurcation of Stokes Waves

Steady periodic water waves on the free surface of an infinitely deep irrotational flow under gravity without surface tension (Stokes waves) can be described in terms of solutions of a quasi-linear equation which involves the Hilbert transform and which is the Euler-Lagrange equation of a simple functional. The unknowns are a 2π-periodic function w which gives the wave profile and the Froude number, a dimensionless parameter reflecting the wavelength when the wave speed is fixed (and vice versa). Although this equation is exact, it is quadratic (with no higher order terms) and the global structure of its solution set can be studied using elements of the theory of real analytic varieties and variational techniques. In this paper it is shown that there bifurcates from the first eigenvalue of the linearised problem a uniquely defined arc-wise connected set of solutions with prescribed minimal period which, although it is not necessarily maximal as a connected set of solutions and may possibly self-intersect, has a local real analytic parametrisation and contains a wave of greatest height in its closure (suitably defined). Moreover it contains infinitely many points which are either turning points or points where solutions with the prescribed minimal period bifurcate. (The numerical evidence is that only the former occurs, and this remains an open question.) It is also shown that there are infinitely many values of the Froude number at which Stokes waves, having a minimal wavelength that is an arbitrarily large integer multiple of the basic wavelength, bifurcate from the primary branch. These are the sub-harmonic bifurcations in the paper's title. (In 1925 Levi-Civita speculated that the minimal wavelength of a Stokes wave propagating with speed c did not exceed 2πc ²/g. This is disproved by our result on sub-harmonic bifurcation, since it shows that there are Stokes waves with bounded propagation speeds but arbitrarily large minimal wavelengths.) Although the work of Benjamin & Feir} and others [9, 10] has shown Stokes waves on deep water to be unstable, they retain a central place in theoretical hydrodynamics. The mathematical tools used to study them here are real analytic-function theory, spectral theory of periodic linear pseudo-differential operators and Morse theory, all combined with the deep influence of a paper by Plotnikov [36]. Accepted: December 6, 1999     

Three-dimensional elastic behavior of a nonhomogeneous layer

Summary  This paper deals with the theoretical treatment of a three-dimensional elastic problem governed by a cylindrical coordinate system (r,θ,z) for a medium with nonhomogeneous material property. This property is defined by the relation G(z)=G ₀(1+z/a) ^m where G ₀,a and m are constants, i.e., shear modulus of elasticity G varies arbitrarily with the axial coordinate z by the power product form. We propose a fundamental equation system for such nonhomogeneous medium by using three kinds of displacement functions and, as an illustrative example, we apply them to an nonhomogeneous thick plate (layer) subjected to an arbitrarily distributed load (not necessarily axisymmetric) on its surfaces. Numerical calculations are carried out for several cases, taking into account the variation of the nonhomogeneous parameter m. The numerical results for displacement and stress components are shown graphically. Received 10 May 1999; accepted for publication 15 August 1999     

A Pressurized Functionally Graded Hollow Cylinder with Arbitrarily Varying Material Properties

The elastic analysis of a pressurized functionally graded material (FGM) annulus or tube is made in this paper. Different from existing studies, this study deals with an axisymmetrical FGM hollow cylinder or disk with arbitrarily varying material properties. A simple and efficient approach is suggested, which reduces the associated problem to solving a Fredholm integral equation. The resulting equation is approximately solved by expanding the solution as series of Legendre polynomials. The stresses and displacements can be represented in terms of the solution to the equation. For radius-dependent Young’s modulus, numerical results of the distribution of the radial and circumferential stresses are presented graphically. Our results indicate that change in the gradient of the FGM tube does not produce a substantial variation of the radial stress, but strongly affects the distribution of the hoop stress. In particular, the hoop stress may reach its maximum at an internal position or at the outer surface when the tube is internally pressurized. The results obtained are helpful in designing FGM cylindrical vessels to prevent failure.      

A unified Krylov-Bogoliubov method for solving second-order non-linear systems

A unified theory is presented for obtaining the transient response of second-order non-linear systems by the Krylov-Bogoliubov method. The method is a generalization of Bogoliubov's asymptotic method and covers all three cases when the roots of the corresponding linear equation are real, complex conjugate, or pure imaginary. It is shown that by suitable substitution for the roots in the general result, that the solution corresponding to each of the three cases can be obtained. The solution for the equation governing the motion of a simple pendulum with and without damping derived from the general solution reduces to that obtained by Popov's [4] method.     

Solidification of a binary mixture with arbitrary heat flux and initial conditions

The problem of solidification of a binary mixture in a semi-infinite region with arbitrarily prescribed initial temperature and composition subject to an arbitrary heat flux at its surface is studied. The liquidus and solidus lines of the phase diagram are used to relate the freezing temperature and the composition of the mixture. Solidification, depending on the prescribed data, could occur immediately or at a later time. In the latter case, there is a period of presolidification. Thus the initial condition of the subsequent solidification cannot be assigned arbitrarily; in particular, it cannot be taken as a uniform state. The conditions for occurrence of these cases are studied and specified. The exact solutions for each of these are found. Existence, uniqueness and convergence of the series solutions are also considered and proved.     

A new meshless method for steady-state heat conduction problems in anisotropic and inhomogeneous media

Summary A new meshless method is developed to analyze steady-state heat conduction problems with arbitrarily spatially varying thermal conductivity in isotropic and anisotropic materials. The analog equation is used to construct equivalent equations to the original differential equation so that a simpler fundamental solution of the Laplacian operator can be employed to take the place of the fundamental solutions related to the original governing equation. Next, the particular solution is approximated by using radial basis functions, and the corresponding homogeneous solution is solved by means of the virtual boundary collocation method. As a result, a new method fully independent of mesh is developed. Finally, several numerical examples are implemented to demonstrate the efficiency and accuracy of the proposed method. The numerical results show good agreement with the actual results.This work was supported by the National Natural Science Foundation of China (No. 10472082) and Australian Research Council.     

Love solutions in the linear inhomogeneous transversely isotropic theory of elasticity

A general Love solution for the inhomogeneous transversely isotropic theory of elasticity with the elastic constants dependent on the coordinate z is proposed. This result may be considered as a generalization of the Love solutions we recently derived for the inhomogeneous isotropic theory of elasticity. The key steps of deriving the Love solution for the classical linear homogeneous transversely isotropic theory of elasticity are described for further use of the derivation procedure, which is then generalized to the inhomogeneous transversely isotropic case. Some particular cases of inhomogeneity traditionally used in the theory of elasticity are also examined. The significance of the derived solutions and their importance for the modeling of functionally graded materials are briefly discussed     

Complete Characterization and Synthesis of the Response Function of Elastodynamic Networks

The response function of a network of springs and masses, an elastodynamic network, is the matrix valued function W(ω), depending on the frequency ω, mapping the displacements of some accessible or terminal nodes to the net forces at the terminals. We give necessary and sufficient conditions for a given function W(ω) to be the response function of an elastodynamic network, assuming there is no damping. In particular we construct an elastodynamic network that can mimic a suitable response in the frequency or time domain. Our characterization is valid for networks in three dimensions and also for planar networks, which are networks where all the elements, displacements and forces are in a plane. The network we design can fit within an arbitrarily small neighborhood of the convex hull of the terminal nodes, provided the springs and masses occupy an arbitrarily small volume. Additionally, we prove stability of the network response to small changes in the spring constants and/or addition of springs with small spring constants.     

General stress-strain relations in non-linear theory of viscoelasticity for large deformations

Conclusion General phenomenoligical stress-strain relations in non-linear theory of visco-elasticity for large deformations have been presented.In the first place, according to V. V. Novozhilov 1 we express the generalized equilibrium equation for large deformations in the Lagrange representation, and we apply the generalized Hamilton's principle to the equation of energy conservation, which denotes that the sum of the elastic energy and the dissipative energy is equal to the work done by the body force and the surface on the substance; so that we obtain the required general stress-strain relations in comparison with the above two equations.On the condition that the elastic potential is a function only of the strain, and the dissipation function is a function of the rate of strain and of strain; such a substance is reduced to the Voigt material necessarily, and the stresses which act on the substance are given by the sum of elastic- and viscous stresses, and the stress-strain relations are reduced to the so-called Lagrangian form.If elongations, shears and angles of rotation are small and also the strains and rates of strain are sufficiently small, the stress-strain relations are expressed by a linear Voigt model constituting a Hookian spring in parallel with a Newtonian dashpot.Non-linearity in the theory is classified into two groups i. e. the geometrical non-linearity and the physical non-linearity. The former is introduced into the theory through the definition of the generalized strain and of the generalized stress and through the equilibrium equation for large deformation, and the latter through the general stress-strain relations.The main result of this paper is that the general stress-strain relations in viscoelasticity are deduced necessarily from the physically appropriate assumptions.