Boussinesq Problem Research Articles

Analysis of a rubber cone tensioned by a concentrated force

Using the constitutive equation of a rubber-like materials given by Gao (1997), this paper investigates the problem of a cone under tension of a concentrated force at its apex. Under consideration is the axial-symmetry case and the large strain is taken into account. The stress strain fields near the apex are obtained by both asymptotic analysis and finite element calculation. The two results are consistent well. When the cone angle is 180°, the solution becomes that of non-linear Boussinesq's problem for tension case.

General solutions of problems of the theory of elasticity and boundary-value problems

The vector potentials of the displacements of the general solutions of static Boussinesq and Papkovich problems are presented in a form which leads to the splitting of the vector equations of the potentials in cylindrical and spherical coordinates into two scalar potentials. The solutions of the equations of the scalar potentials for finite bodies of canonical form contain orthogonal systems of functions on the coordinate surfaces in the region occupied by the body considered, including its boundary surfaces. One thereby creates the prerequisites for converting the boundary conditions into infinite systems of linear algebraic equations after expanding the stresses or displacements, specified on the boundary surfaces, in orthogonal functions of the equations of the potentials.

Asymptotic analysis of the nonlinear Boussinesq problem for a kind of incompressible rubber material (compression case)

A half rubber space compressed by a concentrated force normal to the surface is asymptotically analyzed based on large strain elastic theory. The material domain near the singular point is divided into an expanding domain and a shrinking domain. The asymptotic equations are derived and solved individually for both domains. The solutions to expanding and shrinking domains are further assembled together so that there is only one free parameter left which can indicate the amplitude of the singular stress and strain field. Finally, the amplitude parameter of the field is determined by the given concentrated force.

An Asymptotic Finite Deformation Analysis for an Isotropic Incompressible Hyperelastic Half-Space Subjected to a Tensile Point Load

The nonlinearly elastic Boussinesq problem is to find the deformation produced in a homogeneous, isotropic, elastic half-space by a point force normal to the undeformed boundary, using the exact equations of elasticity. In this paper, we consider a tensile point load and address this core problem within the context of incompressible finite elasticity. The governing equations are first derived from a material formulation of the equilibrium equations for nominal stresses, and then, assuming that the well-known linear Boussinesq solution is valid at large distances from the point load, some implications of conservation laws of elastostatics when applied to the entire half-space are examined. Finally, asymptotic forms for solutions near the point load are hypothesized, and asymptotic tests providing a simple way of determining if an isotropic homogeneous material is capable of sustaining a tensile point load are developed. The results are then applied to several particular constitutive models for incompressible nonlinearly elastic materials.

半無限弾性体応力を求めるBoussinesq問題への境界条件の追加とその適用

半無限弾性体応力を求めるBoussinesq問題への境界条件の追加とその適用

On Boussinesq's problem

This note presents an elementary procedure for obtaining the solution to Boussinesq's problem for the loading of an isotropic elastic halfspace by a concentrated normal load.

Direct Proof of the Asymptotic Expression of the Loading Love Numbers

AbstractThe loading Love numbers h'n, l'n and k'n are used to represent Earth's deformation under surface load of punctual mass. When n → ∞, the asymptotic values of h'n, nl'n and nk'n are all constants. Farrell (1972) obtained these results by comparing the solution of the Boussinesq problem and that of the spherical harmonic series development of Earth's deformation under surface load of punctual mass or obtaining these results, Farrel has made use of the Hankel transforms and the asymptotic relation between Legendre functions and Bessel functions near the pole .In this paper, the same results are derived out directly from the differential equations satisfied by the loading Love numbers Only the knowledge of differential equations and the asymptotic expression of Legendre functions are used. The method of this paper is relatively more straightforward and simpler, and easier to be understood.

Green's function based stress diffusion solutions in the porous elastic half space for time varying finite reservoir loads

The formulas for the simulation of stresses and pore pressure in a porous elastic half space for a finite reservoir surface load using the Green's function approach are reported. Solutions are given under the decoupled approximation of the governing partial differential equations. In this case results of the classical Boussinesq problem may be imported directly to calculate the stresses. However, for the calculation of pore pressure an inhomogeneous diffusion equation is solved.

General nonlocal solution of the elastic half space loaded by a concentrated force P perpendicular to the boundary

The main purpose of this paper is to develop the results introduced in Artan (1996) and to find a general nonlocal linear elastic solution for Boussinesq problem. The general nonlocal solution given Artan (1996) is valid only when the distance to the boundary is greater than one atomic measure. The nonlocal stress field presented in this paper is valid for the whole half plane.

Singular Solutions in Second-Order Elasticity

The problems of a concentrated point force in an infinite medium (Kelvin's problem), or applied normal to the boundary of a half-space (Boussinesq's problem), as well as the corresponding problems for a concentrated couple, are solved to second order for incompressible isotropic elastic solids. The solutions are based on a simplified form of the second-order equations, obtained previously, and use of the Love's function representation for axisymmetric problems in the linear theory.

The Boussinesq problem for soils with bounded non-homogeneity

The response of a compressible continuously non-homogeneous elastic soil to a static vertical point load on its surface is analytically investigated by using classical integral transform techniques and the extended power series method for obtaining the solution in the transform domain. The non-homogeneity is described by means of a depth-function which is non-zero at the surface and bounded at infinity and is capable in modelling both increasing and decreasing soil stiffness with depth. The influence of non-homogeneity on the displacements and stresses at the surface and in the interior is examined over a wide range on the governing parameters. © 1998 John Wiley & Sons, Ltd.

The Boussinesq problem for soils with bounded non‐homogeneity

The Boussinesq problem for soils with bounded non‐homogeneity

An equilibrium solution for the Boussinesq problem for a mixture of an elastic solid and a fluid in an infinite half-space

Using the linearized theory for a binary mixture of an isotropic elastic solid and viscous fluid, an equilibrium solution is obtained for the Boussinesq problem of the mixture in an infinite half-space.

Efficient multigrid version of the finite-element method for the solution of problems of the theory of elasticity

We propose an economic version of the multigrid finite-element method based on the concept of substructures for computing large complex finite-element systems. The product of matrices derived by the incomplete decomposition of stiffness matrices of substructures is used as a relaxation operator of the multigrid process. The main advantages of the described method are its explicit decomposition, high rate of convergence, and considerable economy in memory and arithmetic operations. The efficiency of the proposed version of the finite-element method as compared with the well-known methods is illustrated by the solution of several problems, namely, the axially symmetric and three-dimensional Boussinesq problems and the three-dimensional problem of fracture mechanics about stretching a plate with a lateral crack. It follows from the numerical examples that the indicated approach provides a considerably higher rate of convergence and requires substantially less memory for the solution. The proposed method enables the solution of large problems with personal computers for which the calculation rate and memory capacity play the decisive role in the restriction of the order of problems.

Use of in situ measurements of ground inclination for determining shallow elastic parameters

We present the application of high-resolution ground inclination measurements to estimate some elastic properties in shallow stratified terrain. The direct problem for the solution of vertical and radial displacements under static loads is illustrated by means of the well-known Boussinesq problem for the elastic half-space. More realistic (stratified) configurations are taken into account by means of the stiffness matrix method proposed by Kausel and Roesset. In order to test our procedure some inclination surveys were performed in the virgin Texcoco Lake zone in Mexico City. Results are presented for the shear wave velocity and depth of the uppermost strata. A good fit between theoretical and observed inclinations was obtained. This leads to improved estimates of the ground parameters studied. Comparisons with results from independent seismic wave refraction profiles and cone resistance values validate our procedure. The agreement between the results from measurements of the ground surface inclination and results from other common, established techniques shows that the former technique provides a good estimate of ground elastic parameters. Nous montrons comment des mesures à haute résolution du pendage permettent d'estimer certains paramètres d'élasticité de couches superficielles stratifiées. Le probléme direct de la résolution des déplacements verticaux et radiaux sous l'effet de charges statiques est illustré à l'aide de la formule bien connue de Boussinesq pour la masse élastique semi-infinie. On tient compte de configurations stratifiées plus réalistes en utilisant la matrice de rigidité proposée par Kausel et Roesset (1981). Pour Vérifier notre méthodologie, nous avons effectué des relevés de pendage dans la zone vierge du lac Texcoco à Mexico. Nous présentons les résultats obtenus pour la vitesse de l'onde de cisaillement et la profondeur des couches superficielles. Les valeurs théoriques du pendage concordent bien avec les valeurs observées, ce qui veut dire que notre méthode permet d'estimer avec plus de précision les paramétres du terrain étudié.Elie est validée par des comparaisons avec les résultats d'études ind6pendantes de profils de réfraction des ondes sismiques et des valeurs de résistance en pointe. La concordance entre les résultats des mesures du pendage des couches superficielles et les résultats d'autres techniques bien établies montre que les premiéres offrent une bonne estimation des paramétres d'élasticité du sol.

A continuum theory for the thermomechanics of solidification

A general theory for the phenomenon of solidification is presented in which the coupling between the thermal and kinematical fields is fully taken into account. The resulting model descsribes the liquid region as an ordinary Newtonian liquid and the solid phase as an elastic material. In the case of multi-component solidification it allows for the existence of a mixed region separating the pure phases whose behavior is modeled as a non-linear viscoelastic material. After a preliminary analysis of the jump conditions across the singular surface separating the two phases, the strong interdependence between the thermomechanical field, the geometry of the singular surface and the freezing temperature θ f is examined in detail. A simple one-dimensional problem (Boussinesq problem) has been discussed to show how only a dynamical theory can predict with reasonable accuracy the final shape of the solid.

Implementing image stresses in a 3D dislocation simulation

A three-dimensional (3D) dislocation simulation has been developed in the last few years in order to fill the gap between atomistic simulations and the more macroscopic approaches. The specific role of such a simulation is to combine all mechanisms responsible for the hardening (e.g. the forest mechanism) and the multiplication of dislocations which typically occur in three dimensions. Many problems involve the presence of interfaces, which can come from cracks, oxide layers, particles etc. The present work deals with a method to tackle such problems in a 3D simulation. It is firstly restricted to the image stress case. A method of calculating the image stress field of dislocation segments due to the presence of a semi-infinite free surface has been proposed by Gosling and Willis. An alternative method is proposed here based on the Boussinesq problem of point loading in a half-space. It is firstly shown on simple cases that the methods are equivalent. The latter one is used in the present simulation, and the image stress field due to populations of dislocations is investigated. From a practical point of view, we calculate the depths within which all dislocations have to be taken into account, in order to get reasonable accuracy on the image stresses, and we also investigate how this stress field decays in space.

COMPUTATIONAL STRATEGIES FOR ITERATIVE SOLUTIONS OF LARGE FEM APPLICATIONS EMPLOYING VOXEL DATA

SUMMARY FE-models for structural solid mechanics analyses can be readily generated from computer images via a ‘voxel conversion’ method, whereby voxels in a two- or three-dimensional computer image are directly translated to elements in a FE-model. The fact that all elements thus generated are the same creates the possibilities for fast solution algorithms that can compensate for a large number of elements. The solving methods described in this paper are based on an iterative solving algorithm in combination with a uniqueelement Element-by-Element (EBE) or with a newly developed Row-by-Row (RBR) matrix-vector multiplication strategy. With these methods it is possible to solve FE-models on the order of lo5 3-D brick elements on a workstation and on the order of lo6 elements on a Cray computer. The methods are demonstrated for the Boussinesq problem and for FE-models that represent a porous trabecular bone structure. The results show that the RBR method can be 3.2 times faster than the EBE method. It was concluded that the voxel conversion method in combination with these solving methods not only provides a powerful tool to analyse structures that can not be analysed in another way, but also that this approach can be competitive with traditional meshing and solving techniques.

ON THE GENERALIZATION OF CERRUTI'S PROBLEM IN AN ELASTIC HALF-SPACE

A singular boundary-value problem of an elastic half-space subjected to a force vector at one point of the surface is solved. The force vector has three components which are two tangential and one normal forces to the surface. Solutions to the problem are expressed in orthogonal curvilinear coordinates and are applied to rectangular Cartesian, cylindrical and spherical coordinates, as examples of the orthogonal curvilinear coordinates. The expressions for displacement and stress components are demonstrated in these coordinate systems. They are coincident with the solutions of Cerruti's and Boussinesq's problems when the three components of the force vector are specialized.

The green element method

AbstractThis paper discusses an element‐by‐element approach of implementing the Boundary Element Method (BEM) which offers substantial savings in computing resource, enables handling of a wider range of problems including non‐linear ones, and at the same time preserves the second‐order accuracy associated with the method. Essentially, by this approach, herein called the Green Element Method (GEM), the singular integral theory of BEM is retained except that its implementation is carried out in a fashion similar to that of the Finite Element Method (FEM). Whereas the solution procedure of BEM couples the information of all nodes in the computational domain so that the global coefficient matrix is dense and full and as such difficult to invert, that of GEM, on the other hand, involves only nodes that share common elements so that the global coefficient matrix is sparse and banded and as such easy to invert. Thus, GEM has the advantage of being more computationally efficient than BEM. In addition, GEM makes the singular integral theory more flexible and versatile in the sense that GEM readily accommodates spatial variability of medium and flow parameters (e.g., flow in heterogeneous media), while other known numerical features of BEM—its second‐order accuracy and ability to readily handle problems with singularities are retained by GEM.A number of schemes is incorporated into the basic Green element formulation and these schemes are examined with the goal of identifying optimum schemes of the formulation. These schemes include the use of linear and quadratic interpolation functions on triangular and rectangular elements. We found that linear elements offer acceptable accuracy and computational effort. Comparison of the modified fully implicit scheme against the generalized two‐level scheme shows that the modified fully implicit scheme with weight of about 1·25 offers a marginally better approximation of the temporal derivative. The Newton–Raphson scheme is easily incoporated into GEM and provides excellent results for the time‐dependent non‐linear Boussinesq problem. Comparison of GEM with conventional BEM is done on various numerical examples, and it is observed that, for comparable accuracy, GEM uses less computing time. In fact, from the numerical simulations carried out, GEM uses between 15 and 45 per cent of the simulation time of BEM.