The elastodynamic solution for a solid sphere and dynamic stress-focusing phenomenon

This paper presents an analytical method of solving the elastodynamic problem of a solid sphere.The basic solution of the elastodynamic problem is decomposed into a quasi-static solution satisfying the inhomogeneous compound boundary conditions and a dynamic solution satisfying the homogeneous compound boundary conditions.By utilizing the variable transform,the dynamic equation may be transformed into Bassel equation.By defining a finite Hankel transform,we can easily obtain the dynamic solution for the inhomogeneous dynamic equation.Thereby,the exact elastodynamic solution for a solid sphere can be obtained.From results carried out,we have observed that there exists the dynamic stress-focusing phenomenon at the center of a solid sphere under shock load and it results in very high dynamic stress-peak.     

A theoretical solution for axially symmetric problems in elastodynamics

This paper presents a theoretical solution for the basic equation of axisymmetric problems in elastodynamics. The solution is composed of a quasi-static solution which satisfies inhomogeneous boundary conditions and a dynamic solution which satisfies homogeneous boundary conditions. After the quasi-static solution has been obtained an inhomogeneous equation for dynamic solution is found from the basic equation. By making use of eigenvalue problem of a corresponding homogeneous equation, a finite Hankel transform is defined. A dynamic solution satisfying homogeneous boundary conditions is obtained by means of the finite Hankel transform and Laplace transform. Thus, an exact solution is obtained. Through an example of hollow cylinders under dynamic load, it is seen that the method, and the process of computing are simple, effective and accurate.     

The asymptotic solution of a dynamic buckling problem in elastic columns

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The dynamic indentation of an elastic half-space

A method is given for finding the distribution of traction beneath a conical or a wedge-shaped indentor, moving into an elastic half-space, for the two extreme cases of perfect adhesion between indentor and half-space, and perfectly lubricated indentors. Relations between indenting force, velocity of indentor and area of contact are investigated and found to be fairly insensitive to the friction, although in applications in which the shear traction is required a model with friction would be essential. Mathematically, the paper provides the first accurate solutions of any dynamic mixed boundary value problems with essentially coupled mixed conditions.

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Zusammenfassung Es wird eine Methode zur Bestimmung der Belastung eines keil-oder kegelförmigen Körpers angegeben, der in einen elastischen Halbraum eindringt, wobei die beiden Grenzfäll-vollständige Adhäsion zwischen eindringendem Körper und Halbraum sowie perfekte Schmierung-betrachtet werden. Beziehungen zwischen Eindrückkraft, Eindringgeschwindigkeit und Kontaktfläche werden untersucht. Dabei ergibt sich eine gewisse Unempfindlichkeit gegenüber Reibung, obschon ein Modell unter Berücksichtigung der Reibung dann erforderlich ist, wenn die Schubbeanspruchung benötigt wird. Mathematisch gesehen liefert die vorliegende Arbeit die erste exakte Lösung eines dynamischen gemischten Randwertproblems mit gekoppelten gemischten Bedingungen.

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School of Mathematics, Bath University     

The asymptotic solution to the antiplane shear dynamic crack-tip field in an elastic strain-softening viscoplastic material

ProjectsupportedbytheNationalNaturalScienceFoundationofHeilongiianginP.R.ChinaI.Introductionlnthatengineering,thereisakindofmaterials.Somesofteninganddamagephenomenaa1waysappearwhenthedeformationbecomeslargeenough.Forstudyingthisprocess.asofteningmode1oft…     

The method of solving axisymmetric problems in elastic space by complex function and some examples

This paper proves Love’s stress function of space axisymmetric problem can be represented by choosing two generalized analytic functions of complex variates reasonably^[1], and deduces the expressions of the components of stress displacements and boundary conditions in complex function. To present the feasibility of the method here and examining the truth of the formulae founded in this paper, the problem of circular shaft with globular cavity pressed on the side and pulled at the ends is solved by using power series and the result is the same as that solved by other methods. In the end, the problem of a cone sheared by uniform shear stress on the sideface is solved, and the solution of a cone acted on by gravity is given by converting constant body forces into surface forces.     

Benchmark analytic solution of the dynamic response of a spherical shell composed of a transverse isotropic elastic material

Developing benchmark analytic solutions for problems in solid and fluid mechanics is very important for the purpose of testing and verifying computational physics codes. In order to test the numerical results of physics codes, we consider the geometrically linear dynamic sphere problem. We present an exact solution for the dynamic response of a spherical shell composed of a linearly elastic material exhibiting transverse isotropic symmetry. The solution takes the form of an infinite series of eigenfunctions. We demonstrate, both qualitatively and quantitatively, the convergence of the computed benchmark solution under spatial, temporal, and eigenmode refinement.     

General solution for dynamical problem of infinite beam on an elastic foundation

Based on the theory of Euler-Bernoulli beam and Winkler assumption for elasticfoundation,a mathematical model is presented.By using Fourier transformation for spacevariable,Laplace transformation for time variable and convolution theorem for theirinverse transformations,a general solution for dynamical problem of infinite beam on anelastic foundation is obtained.Finally,the cases of free vibration,impulsive response andmoving load are also discussed.     

Kelvin states in an inhomogeneous elastic space

Summary An explicit solution to Kelvin's problem in linear elastostatics is given when the indefinite elastic medium in isotropie, but not homogeneous. As a constitutive law, it is assumed that the tangent modulus is a constant, while varies with the distance from the origin according to an exponential law.

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Sommario Viene data una soluzione formale del problema di Kelvin in elastostatica quando il mezzo elastico indefinito è isotropo, ma non omogeneo. Come legge costitutiva, si assume costante il modulo tangenziale , mentre varia con la dlstanza dall'origine secondo una legge esponenziale.

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On leave at Istituto di Scienza delle Costruzioni. Pisa (Italy).     

Surface instability of an elastic half space with material properties varying with depth

If a body with a stiffer surface layer is loaded in compression, a surface wrinkling instability may be developed. A bifurcation analysis is presented for determining the critical load for the onset of wrinkling and the associated wavelength for materials in which the elastic modulus is an arbitrary function of depth. The analysis leads to an eigenvalue problem involving a pair of linear ordinary differential equations with variable coefficients which are discretized and solved using the finite element method.The method is validated by comparison with classical results for a uniform layer on a dissimilar substrate. Results are then given for materials with exponential and error-function gradation of elastic modulus and for a homogeneous body with thermoelastically induced compressive stresses.     

不确定性移动载荷激励下弹性简支梁的动态响应边界分析及应用

不确定性移动载荷激励下的弹性梁振动是土木、机械和航空航天等工程领域普遍存在的一类重要问题。在许多实际工程中,不确定移动载荷的样本测试数据有限或测试成本较高,本文引入区间过程模型对此类动态不确定性参数进行描述,提出了一种求解不确定移动载荷激励下弹性梁振动响应边界的非随机振动分析方法。首先,介绍了确定性移动载荷激励下弹性梁的振动微分方程及其解析求解方法;其次,引入区间过程模型,以上下边界函数的形式对不确定性移动载荷进行度量,进而基于模态叠加法发展出弹性梁振动响应边界求解的非随机振动分析方法;最后,将上述非随机振动分析方法应用于车桥耦合振动问题。     

Torsion of elastic shaft of revolution embedded in an elastic half space

The problem of torsion of elastic shaft of revolution embedded in an elastic half space is studied by the Line-Loaded Integral Equation Method (LLIEM). The problem is reduced to a pair of one-dimensional Fredholm integral equations of the first kind due to the distributions of the fictitious loads "Point Ring Couple (PRC) "and "Point Ring Couple in Half Space (PRCHS) "on the axis of symmetry in the interior and external ranges of the shaft occutied respectively. The direct discrete solution of this integral equations may be unstable, i.e. an ill-posed case occurs. In this paper, such an ill-posed Fredholm integral equation of first kind is replaced by a Fredholm integral equation of the second kind with small parameter, which provides a stable solution. This method is simpler and easier to carry out on a computer than the Tikhonov’s regularization method for ill-posed problems. Numerical examples for conical, cylindrical, conical-cylindrical, and parabolic shafts are given.     

The fundamental solution of Mindlin plates resting on an elastic foundation in the Laplace domain and its applications

The aim of this study is to investigate the method of fundamental solution (MFS) applied to a shear deformable plate (Reissner/Mindlin’s theories) resting on the elastic foundation under either a static or a dynamic load. The complete expressions for internal point kernels, i.e. fundamental solutions by the boundary element method, for the Mindlin plate theory are derived in the Laplace transform domain for the first time. On employing the MFS the boundary conditions are satisfied at collocation points by applying point forces at source points outside the domain. All variables in the time domain can be obtained by Durbin’s Laplace transform inversion method. Numerical examples are presented to demonstrate the accuracy of the MFS and comparisons are made with other numerical solutions. In addition, the sensitivity and convergence of the method are discussed for a static problem. The proposed MFS is shown to be simple to implement and gives satisfactory results for shear deformable plates under static and dynamic loads.     

On the stability of an elastic column positioned on an elastic half space

Summary Stability of a heavy elastic column loaded by a concentrated force at the top is analysed. It is assumed that the column is fixed to a rigid circular plate that is positioned on a homogeneous, isotropic, linearly elastic half-space. The constitutive equations for the column are assumed in the form that allows axial compressibility and takes into account the influence of shear stresses. It is shown that eigenvalues of the linearized equations determine the bifurcation points of the full non-linear system of equilibrium equations. Also, the type of bifurcation at the lowest eigenvalue is examined and shown that it could be both super-and sub-critical. The post-critical shape of the column is determined by numerical integration of the equilibrium equations. Received 13 June 1998; accepted for publication 12 November 1998     

Second-order effect of an elastic circular shaft during torsion

In this paper we deal with the second-order effect of an elastic circular shaft duringtorsion.The analysis is based on the method of co-moving coordinates and the strain-rotation decomposition theorem in continuum mechanics.By using asymptotic expansionmethods,we comfirm that the effect of axial elongation and distortion of plane cross-section exists in an elastic circular shaft during large torsion and give the expressions ofthe axial force and the torque.     

Hardy space infinite elements for time-harmonic two-dimensional elastic waveguide problems

We consider time-harmonic linear elasticity equations in domains containing two-dimensional semi-infinite strips. Since for such problems there exist modes with different signs of group and phase velocity, standard perfectly matched layer (PML) as well as standard Hardy space infinite element methods fail.We apply a recently developed infinite element method for a physically correct discretization of such waveguide problems which is based on a Laplace transform in propagation direction. In the Laplace domain the space of transformed solutions can be separated into a sum of a space of incoming and a space of outgoing functions where both function spaces are certain Hardy spaces. The Hardy space is chosen such that the construction of a simple infinite element is possible.The method does not use a modal separation and works on domains of frequencies. On those domains the involved operators are frequency independent and hence lead to linear eigenvalue problems when computing resonances. Numerical experiments containing convergence tests and resonance problems are included.     

The existence of solution to the finite elastodynamics with mixed boundary conditions

I.IntroductionIn[5,6]Ebinetelconsideredcarefullytheinitial-valueproblemsoftheimcompressibleelastodynamicsforpartfreeboundary,partdisplacementboundaryconstraints.Theinitialvalueproblemsofthequasilinearhyperbolicpartialdifferentialequationsrelatedwiththeabovefiniteelastodynamicshavebeenalsoconsideredin[3,4,8,9]bymanyauthorswhentheelasticitytensorsatisfysomeconditions.Someimportantconclusionsabouttheexistenceofsolutiontoabovesystems(equations)wereobtained.Inthispresentpapertheexistenceofsolution…     

流体饱和多孔弹性柱体动力响应的微分求积法

基于多孔介质混合物理论,在小变形的假设下,建立了两相不可压流体饱和多孔介质弹性空间轴对称问题的控制方程。在空间域和时间域内分别采用微分求积方法和二阶向后差分格式来离散控制方程,给出了处理对称轴处奇异性条件的方法,并在空间离散后采用消元法来缩减未知量,提高计算速度。作为应用,分析了流体饱和弹性多孔介质圆柱体的动力响应,考察了所布节点数对数值结果的影响。     

The dynamic indentation of an elastic half-space by a rigid punch

The two-dimensional contact problem between a rigid die and an elastic half-space is considered. A numerical method of solution is proposed which involves an iterative process which is continued until the correct solution is obtained according to certain criteria. The method is general enough and can handle punches of arbitrary shape as well as time-dependent indentation velocities. The treatment is unified for subsonic, transonic and supersonic indentations. The numerical procedure is checked with analytical results which are known in several special cases and good agreement is obtained. Results are presented for the smooth as well as frictional indentation by a wedge-shaped die and for a smooth parabolic punch.