Boussinesq Problem Research Articles

Surface effect in axisymmetric Hertzian contact problems

Surface effect in three different axisymmetric Hertzian contact models is investigated in this paper with a recently developed elastic theory for nanostructured materials, including a Boussinesq problem, contact problem between a rigid flat-ended cylindrical indenter and an elastic half space as well as contact problem between a rigid spherical indenter and an elastic half space. With the help of the Love's strain function method and Hankel integral transformation, closed-form solutions of the stress and displacement fields at the surface of an elastic half space subjected to a concentrated force are achieved, based on which the interface tractions and displacements in the three different axisymmetric contact problems can be further obtained. It is found that surface effect in these contact problems can be characterized only by an intrinsic length, i.e., the ratio of the bulk surface energy density to the bulk shear modulus of the indented material. When the contact radius is comparable with the intrinsic length, surface effect is much obvious, leading to a serious deviation between the two solutions predicted respectively by the theoretical model developed for nanomaterials and the classical contact model. A more interesting phenomenon is about surface effect on the indentation hardness, which is found to increase with the reduction of the indenter radius when the external load is fixed, or to increase with the decrease of the external load when the indenter radius keeps unchanged. All the results in this paper should be helpful not only for deep understanding of the surface effect on nano-contact behaviors but also for further revealing the nature of surface effect of nano-indentation hardness.

A divergence-free low-order stabilized finite element method for a generalized steady state Boussinesq problem

In this work we propose and analyze a new stabilized finite element method for the coupled Navier–Stokes/temperature (or Boussinesq) equations. The method is built using low order conforming elements for velocity and temperature, and piecewise constant elements for pressure. With the help of the lowest order Raviart–Thomas space, a lifting of the jumps of the discrete pressure is built in such a way that when this lifting is added to the conforming velocity field, the resulting velocity is solenoidal (at the price of being non-conforming). This field is then fed to the momentum and temperature equations, guaranteeing that the convective terms in these equations are antisymmetric, without the need of altering them, thus simplifying the analysis of the resulting method. Existence of solutions, discrete stability, and optimal convergence are proved for both the conforming velocity field, and its corresponding divergence-free non-conforming counterpart. Numerical results confirm the theoretical findings, as well as the gain provided by the solenoidal discrete velocity field over the conforming one.

One‐level and two‐level finite element methods for the Boussinesq problem

Previous works on the convergence of numerical methods for the Boussinesq problem were conducted, while the optimal L2‐norm error estimates for the velocity and temperature are still lacked. In this paper, the backward Euler scheme is used to discrete the time terms, standard Galerkin finite element method is adopted to approximate the variables. The MINI element is used to approximate the velocity and pressure, the temperature field is simulated by the linear polynomial. Under some restriction on the time step, we firstly present the optimal L2 error estimates of approximate solutions. Secondly, two‐level method based on Stokes iteration for the Boussinesq problem is developed and the corresponding convergence results are presented. By this method, the original problem is decoupled into two small linear subproblems. Compared with the standard Galerkin method, the two‐level method not only keeps good accuracy but also saves a lot of computational cost. Finally, some numerical examples are provided to support the established theoretical analysis.

New Solution of Axisymmetric Contact Problem of Elasticity

We consider two problems of elasticity, namely, the Boussinesq problem about the action of a lumped force on a half-space and the related problem about the interaction of the half-space with a cylindrical rigid punch with plane base. In the classical statement, these problems have singular solutions. In the Boussinesq problem, the displacement under the action of the force is infinitely large, and in the punch problem, the infinitely large variable is the pressure on the punch boundary. In the present paper, these problems are solved with the use of relations of generalized elasticity derived regarding a medium element of small but finite dimensions rather than a traditional infinitesimal element. The structure parameter of the medium contained in the solutions can be determined experimentally. The obtained generalized solutions of the problems under study are regular.

FIRST PRINCIPLES DERIVATION OF A STRESS FUNCTION FOR AXIALLY SYMMETRIC ELASTICITY PROBLEMS, AND APPLICATION TO BOUSSINESQ PROBLEM

In this work, a stress function is derived from first principles to describe the behaviour of three dimensional axially symmetric elasticity problems involving linear elastic, isotropic homogeneous materials. In the process, the fifteen governing partial differential equations of linear isotropic elasticity were reduced to the solution of the biharmonic problem involving the stress function. thus simplifying the solution process. The stress function derived was found to be identical with the Love stress function. The stress function was then applied to solve the axially symmetric problem of finding the stress fields, strain fields and displacement fields in the semi-infinite linear elastic, isotropic homogeneous medium subject to a point load P acting at the origin of coordinates also called the Boussinesq problem. The results obtained in this study for the stresses and displacements were exactly identical with those from literature, as obtained by Boussinesq.http://dx.doi.org/10.4314/njt.v36i3.15

FIRST PRINCIPLES DERIVATION OF A STRESS FUNCTION FOR AXIALLY SYMMETRIC ELASTICITY PROBLEMS, AND APPLICATION TO BOUSSINESQ PROBLEM

In this work, a stress function is derived from first principles to describe the behaviour of three dimensional axially symmetric elasticity problems involving linear elastic, isotropic homogeneous materials. In the process, the fifteen governing partial differential equations of linear isotropic elasticity were reduced to the solution of the biharmonic problem involving the stress function. thus simplifying the solution process. The stress function derived was found to be identical with the Love stress function. The stress function was then applied to solve the axially symmetric problem of finding the stress fields, strain fields and displacement fields in the semi-infinite linear elastic, isotropic homogeneous medium subject to a point load P acting at the origin of coordinates also called the Boussinesq problem. The results obtained in this study for the stresses and displacements were exactly identical with those from literature, as obtained by Boussinesq. http://dx.doi.org/10.4314/njt.v36i3.15

Analytical Solutions for Axisymmetric Normal Loadings Acting on a Particulate Composite Modeled as a Mixture of Two Linear Elastic Solids

AbstractConstitutive equations, based on continuum mechanics and representing behavior of a mixture of two elastic solids, can be used for modeling of materials such as particulate composites. In this study, the behavior of continuum of a mixture occupying half-space under axisymmetric loads is calculated using Fourier and Hankel transform methods. For this purpose, Love's strain functions are used and the general solution of problem under proper boundary conditions is presented. By applying the results obtained to the sinusoidal distributed vertical load and Boussinesq problems that require the use of Cartesian and cylindrical coordinate systems, displacement vectors, diffusive force vector and components of stress tensors are calculated. At the end of the study, the experimental results of a special particulate composite are used to check the accuracy of the solutions obtained.

Analysis of a conforming finite element method for the Boussinesq problem with temperature-dependent parameters

In this paper we analyze a conforming finite element method for the numerical simulation of non-isothermal incompressible fluid flows subject to a heat source modeled by a generalized Boussinesq problem with temperature-dependent parameters. We consider the standard velocity–pressure formulation for the fluid flow equations which is coupled with a primal-mixed scheme for the convection–diffusion equation modeling the temperature. In this way, the unknowns of the resulting formulation are given by the velocity, the pressure, the temperature, and the normal derivative of the latter on the boundary. Hence, assuming standard hypotheses on the discrete spaces, we prove existence and stability of solutions of the associated Galerkin scheme, and derive the corresponding Cea’s estimate for small and smooth solutions. In particular, any pair of stable Stokes elements, such as Hood–Taylor elements, for the fluid flow variables, continuous piecewise polynomials of degree ≤k+1 for the temperature, and piecewise polynomials of degree ≤k for the boundary unknown becomes feasible choices of finite element subspaces. Finally, we derive optimal a priori error estimates, and provide several numerical results illustrating the performance of the conforming method and confirming the theoretical rates of convergence.

Contact problem for an arbitrarily oriented transversely isotropic half-space

The typical contact problems considered in the past for a transversely isotropic half-space \(z\ge 0\) used the presumption that z \(=\) const. are the planes of isotropy. In 2011, the author has published the first solution to the non-traditional contact problem for a half-space \(x\ge 0\). A more general problem is considered here. We use two systems of Cartesian coordinates: the usual major axes (x, y, z) and the arbitrarily oriented one \((x_{1},x_{2},x_{3})\), which is obtained by rotation of the major axes about axis Ox (axis \(Ox_{1}\) coinciding with major axis Ox) by an arbitrary angle \(\varphi \). The basis for any contact problem is the solution of the Boussinesq problem for the half-space \(x_{3}\ge 0\), related to the axes \((x_{1},x_{2},x_{3})\). This problem is solved by two different methods, to show that the solution is correct. The point force solution leads to the governing integral equation for the contact problem in the case of an arbitrary domain of contact. Several basic contact problems are solved. The complete field of displacements and stresses is presented as contour integrals over a unit circle.

Dual-mixed finite element methods for the stationary Boussinesq problem

We propose and analyze two mixed approaches for numerically solving the stationary Boussinesq model describing heat driven flows. For the fluid equations, the velocity gradient and a Bernoulli stress tensor are introduced as auxiliary unknowns. For the heat equation, we consider primal and mixed-primal formulations; the latter, incorporating additionally the normal component of the temperature gradient on the Dirichlet boundary. Both dual-mixed formulations exhibit the same classical structure of the Navier–Stokes equations. We derive a priori estimates and the existence of continuous and discrete solutions for the formulations. In addition, we prove the uniqueness of solutions and optimal-order error estimates provided the data is sufficiently small. Numerical experiments are given which back up the theoretical results and illustrate the robustness and accuracy of both methods for a classic benchmark problem.

The time viscosity-splitting method for the Boussinesq problem

In this article, we consider the time viscosity-splitting method for the Boussinesq problem. By using this technique, the considered problem is decoupled into two subproblems, and each subproblem is solved more easily than the original one. In the first step, two linear elliptic problems with semi-implicit schemes for the convection terms are solved. The advantage of using such numerical technique is that a linear system with the constant coefficient matrices is obtained and then the computation becomes easy. In the second step, a system consists of a Stokes problem and a linear elliptic problem is solved. The main results of this article include that (1) Establish the stability of numerical solutions in the time viscosity-splitting method; (2) Provide the convergence results of strongly second order for the velocity and temperature and strongly first order for the pressure. Finally, some numerical results are provided to display the performances of the developed numerical algorithms.

An augmented fully-mixed finite element method for the stationary Boussinesq problem

In this paper we propose and analyze a new fully-mixed finite element method for the stationary Boussinesq problem. More precisely, we reformulate a previous primal-mixed scheme for the respective ...

Effect of ring misalignment on the fatigue life of the radial cylindrical roller bearing

An effect of ring misalignment in the radial cylindrical roller bearing on its fatigue life is analysed. In order to predict the fatigue life, a methodology, which allows bearing geometrical parameters, including profiles of rolling generators, bearing radial clearance, angular tilting of rings and complex load to be accounted for, was applied. Stress distributions indispensable to calculate the predicted fatigue life were determined via solving the Boussinesq problem numerically for the elastic half-space and with the FEM. In the calculation of the predicted bearing life fatigue, Lundberg and Palmgren's model was employed. The results of calculations were compared to the recommendations of manufacturers of roller bearings referring to admissible tilt angles of rings.

Elastic Equilibrium of the Half-Space Revisited. Mindlin and Boussinesq Problems

The displacement caused in an isotropic elastic half-space by a point force localized on or beneath its surface is calculated here by a new method. These classical problems are known as Boussinesq and, respectively, Mindlin problems. The motivation for the present work resides in the fact that the original solutions involve some particular procedures, required by the complexity of the boundary conditions, which may limit their general application. The solutions presented here are obtained by including in a generalized Poisson equation the values of the function and its derivatives on the boundary, and by using in-plane Fourier transforms. This method is general and can be extended to other, similar problems.

An Augmented Mixed Finite Element Method for the Navier--Stokes Equations with Variable Viscosity

A new mixed variational formulation for the Navier--Stokes equations with constant density and variable viscosity depending nonlinearly on the gradient of velocity, is proposed and analyzed here. Our approach employs a technique previously applied to the stationary Boussinesq problem and to the Navier--Stokes equations with constant viscosity, which consists firstly of the introduction of a modified pseudostress tensor involving the diffusive and convective terms, and the pressure. Next, by using an equivalent statement suggested by the incompressibility condition, the pressure is eliminated, and in order to handle the nonlinear viscosity, the gradient of velocity is incorporated as an auxiliary unknown. Furthermore, since the convective term forces the velocity to live in a smaller space than usual, we overcome this difficulty by augmenting the variational formulation with suitable Galerkin-type terms arising from the constitutive and equilibrium equations, the aforementioned relation defining the additi...

Hybridizable discontinuous Galerkin projection methods for Navier–Stokes and Boussinesq equations

Schemes for the incompressible Navier–Stokes and Boussinesq equations are formulated and derived combining the novel Hybridizable Discontinuous Galerkin (HDG) method, a projection method, and Implicit–Explicit Runge–Kutta (IMEX-RK) time-integration schemes. We employ an incremental pressure correction and develop the corresponding HDG finite element discretization including consistent edge-space fluxes for the velocity predictor and pressure correction. We then derive the proper forms of the element-local and HDG edge-space final corrections for both velocity and pressure, including the HDG rotational correction. We also find and explain a consistency relation between the HDG stability parameters of the pressure correction and velocity predictor. We discuss and illustrate the effects of the time-splitting error. We then detail how to incorporate the HDG projection method time-split within standard IMEX-RK time-stepping schemes. Our high-order HDG projection schemes are implemented for arbitrary, mixed-element unstructured grids, with both straight-sided and curved meshes. In particular, we provide a quadrature-free integration method for a nodal basis that is consistent with the HDG method. To prevent numerical oscillations, we develop a selective nodal limiting approach. Its applications show that it can stabilize high-order schemes while retaining high-order accuracy in regions where the solution is sufficiently smooth. We perform spatial and temporal convergence studies to evaluate the properties of our integration and selective limiting schemes and to verify that our solvers are properly formulated and implemented. To complete these studies and to illustrate a range of properties for our new schemes, we employ an unsteady tracer advection benchmark, a manufactured solution for the steady diffusion and Stokes equations, and a standard lock-exchange Boussinesq problem.

Fixed point strategies for mixed variational formulations of the stationary Boussinesq problem

In this paper, we report on the main results concerning the solvability analysis of two new mixed variational formulations for the stationary Boussinesq problem. More precisely, we introduce mixed-primal and fully-mixed approaches, both of them suitably augmented with Galerkin-type equations, and show that the resulting schemes can be rewritten, equivalently, as fixed-point operator equations. Then, classical arguments from linear and nonlinear functional analysis are employed to conclude that they are well-posed.

Point force and point electric charge applied to the boundary of three-dimensional anisotropic piezoelectric solid

The Green's functions for a three-dimensional semi-infinite fully anisotropic piezoelectric material are derived using the plane wave theory method. The solution gives the complete set of electromechanical fields due to an arbitrarily oriented point force and a point electric charge applied to the boundary of the half-space. The solution constitutes generalization of Boussinesq's and Cerruti's problems of elastic isotropy for the anisotropic piezoelectric materials. On the example of piezoceramics PZT-6B, the present results are compared with the previously obtained solution for the special case of transversely isotropic piezoelectric solid subjected to the same boundary condition.

Analysis of an augmented mixed‐primal formulation for the stationary Boussinesq problem

In this article, we propose and analyze a new mixed variational formulation for the stationary Boussinesq problem. Our method, which uses a technique previously applied to the Navier–Stokes equations, is based first on the introduction of a modified pseudostress tensor depending nonlinearly on the velocity through the respective convective term. Next, the pressure is eliminated, and an augmented approach for the fluid flow, which incorporates Galerkin‐type terms arising from the constitutive and equilibrium equations, and from the Dirichlet boundary condition, is coupled with a primal‐mixed scheme for the main equation modeling the temperature. In this way, the only unknowns of the resulting formulation are given by the aforementioned nonlinear pseudostress, the velocity, the temperature, and the normal derivative of the latter on the boundary. An equivalent fixed‐point setting is then introduced and the corresponding classical Banach Theorem, combined with the Lax–Milgram Theorem and the Babuška–Brezzi theory, are applied to prove the unique solvability of the continuous problem. In turn, the Brouwer and the Banach fixed‐point theorems are used to establish existence and uniqueness of solution, respectively, of the associated Galerkin scheme. In particular, Raviart–Thomas spaces of order k for the pseudostress, continuous piecewise polynomials of degree ≤ k+1 for the velocity and the temperature, and piecewise polynomials of degree ≤ k for the boundary unknown become feasible choices. Finally, we derive optimal a priori error estimates, and provide several numerical results illustrating the good performance of the augmented mixed‐primal finite element method and confirming the theoretical rates of convergence. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 445–478, 2016

Verification of the hemispherical finite element model of elastic space

The present paper outlines the verification process of the finite element model of elastic space. The axially-symmetrical model in the shape of a hemisphere made of linear-elastic material of one sort, i.e. homogeneous and isotropic, was considered. The solution of the generalised Boussinesq problem was used for verification. The suggested model enables the reduction in number of finite elements and consequently shortens calculating times in comparison to universally implemented shapes of models. It forms the basis for more complicated and comprehensive models. Results of calculations were compared with their equivalents related to the cylindrical model. Basic information on the application of the Finite Element Method (FEM) to issues of pavement modelling were featured. Likewise, commonly applied shapes of pavement models were described.