Seepage consolidation of elastic half-space under an axisymmetric load

The process of seepage consolidation of elastic saturated half-space under the action of a normal load on its surface is investigated assuming both incompressibility of fluid and skeleton grains and independence of the total skeleton stresses on time. Analytic representations for the fluid pressure and the half-space surface settlement are found when the half-space is loaded by a concentrated force. The maximum settlement is also found for a uniform loading of the surface over the circle area.     

Asymptotic modeling of the impact of a spherical indenter on an elastic half-space

The impact of a rigid sphere on a homogeneous, isotropic elastic half-space in the absence of friction and adhesion is considered. The influence of the superseismic stage immediately following the moment of first contact upon the impact process is investigated in the frame of the Hertzian impact theory. The first order asymptotic approximation for the contact force in a three-dimensional dynamic contact problem with the slowly moving contact zone boundary is obtained and the corresponding asymptotic model of impact is developed. The motion of the indenter as it indents and rebounds from the elastic medium is analytically described. Explicit formulas are derived for the peak indentation depth, contact time, and rebound velocity as functions of the initial impact velocity, indenter mass, and characteristics of the elastic half-space.     

The displacement discontinuity method in the analysis of open cracks

The application of the displacement discontinuity numerical technique to the solution of some problems of fracture mechanics is demonstrated in the hypothesis of homogeneous and elastic material. The fracture is supposed to be free from traction and is represented by a set of constant displacement discontinuity elements, except for two parabolic elements, located at each crack tip, in order to simulate the singularity of the solution near the crack tips. On the basis of the stress and displacement field determined by the displacement discontinuity method, the stress intensity factors for mode I and II are computed according to the method of the displacements. Three examples are provided to verify the validity of the formulation.

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Sommario Lo scopo del presente lavore è di illustrare l'applicazione del metodo numerico della Displacement Discontinuity alla soluzione di alcuni problemi di meccanica della frattura, nell'ipotesi di materiale omogeneo ed elastico. La frattura è supposta aperta ed è rappresentata da una linea di elementi a discontinuità di spostamento costante, con l'eccezione di due speciali elementi parabolici, ubicati agli apici, al fine di simulare la singolarita' del campo tensionale. Sulla base del campo degli sforzi e degli spostamenti cosi determinati, vengono ricavati i fattori di concentrazione degli sforzi in modo I e II mediante il metodo degli spostamenti. Vengono inoltre riportati tre esempi di calcolo, effettuati al fine di verificare la validità del procedimento proposto.

     

非饱和弹性半空间地基的稳态响应半解析法研究

应用半解析法研究简谐荷载下非饱和弹性半空间地基的稳态响应。基于非饱和土的动力控制方程以及非饱和弹性半空间的边界条件,建立地基层单元的半解析函数,应用加权残数法得到在简谐荷载下非饱和弹性半空间地基的稳态响应半解析方程。对半解析方程求解,得到了竖向简谐荷载作用下非饱和弹性地基水平位移和竖向位移幅值,数值分析了饱和度和地基深度等参数对孔压和位移幅值的影响。研究结果表明,应用本文方法研究非饱和弹性半空间地基的稳态响应是切实有效的。     

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 axisymmetric problem of propagation of a normal stress discontinuity in a semi-infinity elastic medium

The displacements at the free surface due to sudden creation of a normal stress discontinuity over a circular area which expands uniformly after creation, inside a semi-infinite isotropic elastic medium are studied. Applying Cagniard's method as modified by De Hoop, the displacement components are derived in the integral forms. Their approximate evaluations are carried out in order to obtain the displacement components at the time of their first arrival.     

Some axisymmetric problems for an elastic inhomogeneous thick-walled tube

An equilibrium differential equation for an axisymmetric problem is reduced to an integrable form under the assumption that the shear modulus is continuously differentiable and Poisson’s ratio is constant. A procedure of successive approximations is proposed for the case of a compressible material, and the Lamé problem is solved exactly for the case of an incompressible material. A piecewise continuous variation of the Lamé parameter as a function of radius is considered. Several examples of determining the stress-tensor components are given for various cases of inhomogeneity.     

An ALE method for simulations of axisymmetric elastic surfaces in flow

The dynamics of membranes, shells, and capsules in fluid flow has become an active research area in computational physics and computational biology. The small thickness of these elastic materials enables their efficient approximation as a hypersurface, which exhibits an elastic response to in-plane stretching and out-of-plane bending, possibly accompanied by a surface tension force. In this work, we present a novel arbitrary Lagrangian-Eulerian (ALE) method to simulate such elastic surfaces immersed in Navier-Stokes fluids. The method combines high accuracy with computational efficiency, since the grid is matched to the elastic surface and can, therefore, be resolved with relatively few grid points. The focus of this work is on axisymmetric shapes and flow conditions, which are present in a wide range of biophysical problems. We formulate axisymmetric elastic surface forces and propose a discretization with surface finite-differences coupled to evolving finite elements. We further develop an implicit coupling strategy to reduce time step restrictions. We show in several numerical test cases that accurate results can be achieved at computational times on the order of minutes on a single core CPU. We demonstrate two state-of-the-art applications which to our knowledge cannot be simulated with any other numerical method so far: we present first simulations of the observed shape oscillations of novel microswimming shells and the uniaxial compression of the cortex of a biological cell during an AFM experiment.     

Torsion of a hemisphere embedded in an elastic half-space

This paper deals with the stress distribution in a homogeneous isotropic elastic hemisphere embedded in a semi-infinite homogeneous isotropic elastic medium when a rigid circular disc is clamped to the plane face of the hemisphere and the stresses are caused by the rotation of the disc through an angle . The problem is reduced to the solution of a Fredholm integral equation of the second kind in the auxiliary function (t). An analytical expression for the torque T required to rotate the die through an angle is obtained in terms of (t). The Fredholm integral equation is solved numerically, and the numerical values of T are graphed.This work has been supported by the National Research Council of Canada through NRC-Research Grant No. A4177.     

Rayleigh waves in an orthotropic elastic half-space overlaid by an elastic layer with spring contact

Meccanica - In this paper, the propagation of Rayleigh waves in an orthotropic elastic half-space overlaid by an orthotropic elastic layer of arbitrary uniform thickness is investigated. The layer...     

An analytical solution of an axisymmetric problem of deforming an isotropic half-space with an elastic fixed boundary under the action of a distributed load

An analytical solution to the axisymmetric problem on the action of a distributed load on an isotropic half-space when the load is given by a function dependent on the radial coordinate is obtained. The surface of the half-space is elastically fixed outside the circular domain of load application, the shear stresses are absent along the entire boundary, and the stresses vanish at infinity. At the boundary and inside the elastic half-space, the solutions are represented by the formulas for the stress tensor components and for the displacement vector components.