Boussinesq Problem Research Articles

Force relaxation response in the poroelastic axisymmetric Boussinesq problem for an indenter of arbitrary profiles I: Permeable and impermeable surface drainage conditions

In this study, we have derived a theoretical solution for the poroelastic axisymmetric Boussinesq problem when an indenter with arbitrary profiles is subjected to step displacement loading. The analysis is conducted within the framework of Biot’s theory, employing the McNamee–Gibson displacement function method. We explore two distinct scenarios for surface drainage boundary conditions: one where the surface allows full permeability and another where it is fully impermeable. The formulation of the mechanical boundary condition at the surface relies on an equation that establishes the relationship between the depth of indentation and the contact radius. Our solution encompasses two aspects: force asymptotes at the undrained and drained limits, and a normalized force relaxation that accounts for transient responses. Specific findings are presented for three indenter shapes: paraboloidal, conical, and cylindrical. Of particular significance, we demonstrate that for these specific indenter shapes, the normalized force relaxation at ω=0 can be expressed in closed form in the Laplace domain. In addition to this analytical breakthrough, we conducted numerical simulations employing actual indenters. Remarkably, the normalized force relaxation curves from actual indenters show negligible dependence on material properties and closely align with our closed-form solution at ω=0. This finding suggests that the closed-form solution could act as a universal master curve capable of characterizing the normalized force relaxation response in instances of a real indenter, irrespective of material properties.

Numerical investigation of convergence of Fourier series, poly-nomials, and method of finite elements

Purpose. To compare the efficiency of using finite elements with variable and averaged mechanical and geometric parameters and to investigate the convergence of results obtained by the semi-analytical finite element method (SAFEM) using Fourier series and polynomials with the results obtained by the finite element method (FEM). The methods. The construction and development of an algorithm for studying the stress-strain state of spatial bodies with variable and averaged mechanical and geometric parameters were carried out based on SAFEM. Findings. Solvability indicators of SAFEM were obtained for calculating nodal reactions and stiffness matrix coefficients of finite elements with variable and averaged mechanical and geometric parameters. Numerical convergence studies of results obtained using SAFEM with Fourier series, polynomials, and the finite element method were conducted for a benchmark example, which was the Boussinesq problem for a half-space subjected to a concentrated force. The results indicate that the convergence of the investigated coordinate functions in the considered problem is of the first order. The originality. The obtained solvability indicators of SAFEM for calculating nodal reactions and stiffness matrix coefficients of finite elements with variable and averaged mechanical and geometric parameters allow for the study of various classes of problems. Numerical convergence studies using Fourier series, polynomials, and the finite element method were conducted for the benchmark example. Practical implementation. The practical significance lies in the development of a methodology for determining the stress-strain state of relevant spatial elements of structures with variable and averaged mechanical and geometric parameters subjected to arbitrarily distributed spatial loads.

On Flamant–Boussinesq problem with dynamical flexoelectric effect and micro-inertia effect in dielectrics subjected to dynamical wave loading

This work investigates the Flamant–Boussinesq problem for a half-space made of a homogeneous and isotropic dielectric material. The dynamical flexoelectric effect and the dynamical flexocoupling between displacement and polarization, due to mechanical and electrical states, are taken in consideration. The mechanical loading is taken as a wave of a decaying behavior in time at the surface of a half-space, while the electric potential is considered in an open circuit with no charge on the terminals. The first strain gradient theory of elasticity is used as a mathematical frame in the problem formulation. The equation of motion for the representative volume element additionally accounts for the micro-inertia effect because of the intrinsic structure of the dielectrics at the nanoscale. The governing equations and the boundary conditions for homogeneous, isotropic dielectric material are presented with reference to previous work, using a variational technique for internal energies and external forces. An analytical harmonic wave solution is obtained for the problem under consideration, involving different coupling parameters arising from the mechanical and electrical loadings. The results are analyzed and discussed. The solutions for the quantities of practical interest are represented graphically with different choices of material parameters and flexocoupling parameters. The solution is finite everywhere. The existing damping phenomenon arises, not only from the various physical parameters in the governing field equations as shown in the figures, but also through the boundary conditions.

A mechanics model of constant cross-section type disc cutter based on dense core forming mechanism

Based on the solution of Boussinesq problem, this paper studied the stress field beneath contact surface during the Constant Cross-Section (CCS) type disc cutter cutting. The feature of stress field indicated that dense core formed under the contact surface at a depth approximately equal to the half-length of contact surface. And the penetration and cutter thickness significant influence the dense core position. Based on Mohr-Coulomb criterion and the dense core position, a new cutter force model is proposed. And the side force and normal force model is corrected by cutter spacing according to linear correlation assumption. The accuracy and suitability for proposed model was compared with classic CSM model and numerous full-scale linear cutting machine (LCM) database collecting from the others’ researches. The research indicated that the proposed model can accurately predict cutting force in a wide range of rock mechanics parameters, especially on the rock with USC ≥ 40 MPa. In addition, the influence of main parameters on cutting force was discussed. This research offered fundamental insights into dense core forming and provided a theoretical basis for disc cutter design.

Numerical approaches for Boussinesq type equations with its application in Kampar River, Indonesia

In this study, we examine numerous numerical approaches for solving the 1-Dimensional Boussinesq problem. The proposed methods to be discussed include Mohapatra and Chaudhry’s two-four finite difference scheme, the modified two-four finite difference scheme, and the staggered finite volume scheme. The modified two-four finite difference scheme and staggered finite volume scheme has a shorter computational time than the Mohapatra–Chaudhry scheme, which reduces the computational cost. We compare the performance of each numerical scheme against the analytical solutions and approximate solutions using different type of numerical method that is called the MUSCL4 scheme. Furthermore, the calculated results are compared and utilized to assess the contribution of each individual Boussinesq term. Each Boussinesq term was evaluated to examine its affect on the Boussinesq equation’s calculated result. Further, we also implement the numerical schemes that we formulate to investigate the undular bore phenomena in Kampar River. This finding may be useful to those who use the Boussinesq equation to study fluid or wave phenomena.

On Boussinesq's Problem for a Power-Law Graded Elastic Half-Space on Elliptical and General Contact Domains.

The indentation of a power-law graded elastic half-space by a rigid counter body is considered in the framework of linear elasticity. Poisson's ratio is assumed to be constant over the half-space. For indenters with an ellipsoidal power-law shape, an exact contact solution is derived, based on the generalizations of Galin's theorem and Barber's extremal principle for the inhomogeneous half-space. As a special case, the elliptical Hertzian contact is revisited. Generally, elastic grading with a positive grading exponent reduces the contact eccentricity. Fabrikant's approximation for the pressure distribution under a flat punch of arbitrary planform is generalized for power-law graded elastic media and compared with rigorous numerical calculations based on the boundary element method (BEM). Very good agreement between the analytical asymptotic solution and the numerical simulation is obtained for the contact stiffness and the contact pressure distribution. A recently published approximate analytic solution for the indentation of a homogeneous half-space by a counter body, whose shape slightly deviates from axial symmetry but is otherwise arbitrary, is generalized for the power-law graded half-space. The approximate procedure for the elliptical Hertzian contact exhibits the same asymptotic behavior as the exact solution. The approximate analytic solution for the indentation by a pyramid with square planform is in very good agreement with a BEM-based numerical solution of the same problem.

Analytic contact solutions of the Boussinesq and Cattaneo problems for an ellipsoidal power-law indenter

Based on Galin’s theorem for the indentation of an elastic half-space by a polynomial punch and Barber’s theorem for the determination of the contact area in elastic normal contact problems, an exact contact solution is developed for the frictionless indentation of an elastic half-space by an ellipsoidal power-law punch. Within the Cattaneo–Mindlin-approximation of tangential contacts, the tangential contact problem with friction can be reduced to the frictionless normal contact via the Ciavarella–Jäger principle, based on which also the tangential contact solution for the ellipsoidal power-law indenter is given. All known solutions for special cases (axisymmetric contact, elliptical Hertzian contact) are exactly recovered. A comparison of the pressure distribution obtained analytically for the 4-th power ellipsoidal indenter with a corresponding numerical solution based on the boundary element method shows no noticeable differences. The proposed solution procedure can also be used exactly for an arbitrary finite superposition of ellipsoidal power-law profiles, and, in an approximate sense, is applicable for arbitrary monotonous ellipsoidal profiles.

On the complete solution of the three-dimensional solid space problems based on a novel curvilinear elasticity representation

A solution scheme is presented for the three-dimensional elasticity theory in the curvilinear cylindrical coordinate, without introducing any new physically dubious potential function. Exact analytical solutions for the essential 3-D solid full- and half-space problems are developed based on the introduced elasticity approach. The Kelvin's classical problem of concentrated force acting in the interior of an infinite solid space, as well as the Boussinesq's half-space problem of concentrated force normal to the free surface of a semi-infinite 3-D solid are treated. Furthermore, the three-dimensional non-axisymmetric problem of semi-infinite solid space under concentrated tangential force on its surface (the so-called Cerruti's problem) is revisited. It is demonstrated that unlike most of the conventional solutions in the literature for the aforementioned problems, the present approach is capable of obtaining the complete solution without needing to combine with other approaches. Evidently, the completeness issue associated with any new method of potentials is not present for the established direct elasticity approach. At the end, an extension of the approach to a nonhomogeneous elastic media is presented and, as an application, the Boussinesq's problem for a nonhomogeneous half-space is solved analytically. The developed representation may serve as an efficient replacement of the original three-dimensional Navier's elasticity equations for solving different 3-D elastic problems in the future.

Accuracy of a DtN Finite Element Approach for the Elastic Half-Space

In this work, the accuracy of a DtN finite element approach is numerically assessed. This procedure allows an efficient and accurate numerical solution of boundary-value problems of axisymmetric elasticity in semi-infinite domains. For this kind of problems, no explicit closed-form expression for the associated DtN map exists, so a suitable semi-analytical approximation of it, in series form, is used to impose exact boundary conditions on a semi-spherical artificial boundary. The procedure is tested by solving the classical Boussinesq problem, whose exact solution is known analytically. The computed numerical solution is compared to the analytical solution, achieving an excellent agreement between both solutions, both for displacements and stresses. The convergence of the analytical solution to the numerical solution is numerically demonstrated, in terms of artificial boundary location, series truncation order and finite element mesh size.

Quasi-variational inequality for the nonlinear indentation problem: a power-law hardening model.

The Boussinesq problem, which describes quasi-static indentation of a rigid punch into a deformable body, is studied within the context of nonlinear constitutive equations. By this, the material response expresses the linearized strain in terms of the stress and cannot be inverted in general. A contact area between the punch and the body is unknown a priori, whereas the total contact force is prescribed and yields a non-local integral condition. Consequently, the unilateral indentation problem is stated as a quasi-variational inequality for unknown variables of displacement, stress and indentation depth. The Lagrange multiplier approach is applied in order to establish well-posedness to the underlying physically and geometrically nonlinear problem based on augmented penalty regularization and applying the minimax theorem of Ekeland and Témam. A sufficient solvability condition implies response functions that are bounded, hemi-continuous, coercive and obey a convex potential. A typical example is power-law hardening models for titanium alloys, Norton-Hoff and Ramberg-Osgood materials. This article is part of the theme issue 'Non-smooth variational problems and applications'.

Dynamical behavior of the long waves on the surface of the water with a small amplitude in none–dimensional nonlinear lattices

Analytic and approximation solutions to the Ill–posed Boussinesq (IPB) problem are constructed using two modern computational and numerical approaches. In shallow water, this equation represents the propagation of waves under the effect of gravity. It also examines the propagation of nonlinear strings and lattices in one-dimensional nonlinear systems. Additionally, the nonlinear strings and none-dimensional nonlinear lattices have lengthy waves on the surface with modest amplitudes described by the IPB model. The primary aims of our work are to obtain new analytical solutions and utilize those answers to assess beginning and boundary conditions to apply numerical methods to the same problem. In order to determine how accurate a solution is, the absolute value of the error between numerical and analytical results is calculated. Additionally, several diagrams are provided to describe the physical features of these waves in greater detail.

A NUMERICAL APPROACH TO THE CONTACT OF NOMINALLY FLAT SURFACES

Machined surfaces can be described by heights and wavelengths of the surface asperities that show a statistical variation. Considering that a regular wavy surface with a sinusoidal profile is the crudest model for a rough surface, studying the contact of regular wavy surfaces is a good approximation for the contact of nominally flat surfaces. Such contact problems exhibit periodicity that can be simulated with the aid of computational techniques derived for contact mechanics in the frequency domain. The displacement calculation, which is a necessary step in the resolution of the contact problem, is mathematically a convolution product that can be calculated in the frequency domain with increased computational efficiency. The displacement induced by a unit surface load can be expressed in the frequency domain by the frequency response functions, which are counterparts of the space domain solutions to half-space fundamental problems such as the Boussinesq problem. The displacement induced by a periodic pressure distribution can be computed by executing the convolution product between the frequency response function and pressure on a single period. It should be noted that the convolution calculation in the spectral domain implies that the contributions of all neighbouring pressure periods are accounted for. The need to treat numerically only a single period results in remarkable computational efficiency, allowing for high density meshes that can capture the essential features of any textured real surface. The displacement calculation promotes the solution of the contact problem by an iterative approach. The advanced method is benchmarked against existing analytical solutions for the 3D contact of surfaces possessing two-dimensional waviness. This essentially deterministic model, supported by a direct numerical solution that can be obtained for samples of real rough surfaces, presents itself as a worthy alternative to the existing statistical models for rough contact interaction.

NUMERICAL SIMULATION OF TRI-LAYERED MATERIALS UNDER CONTACT LOAD

Various biomedical components, such as dental crowns and hip prostheses, data processing devices, and other numerous mechanical components that transmit load through a mechanical contact, may benefit from a tri-layer design. The coating may be optimized for wear protection and corrosion prevention, whereas the intermediate layer provides increased adhesion between the outer layer and the substrate, and confines the crack propagation. The solution to the contact problem involving tri-layered materials can be pursued numerically with the finite element or the boundary element methods, but semi-analytical techniques benefitting from the efficiency of the fast Fourier transform (FFT) technique have also been successfully applied. At the heart of the FFT-assisted approach lie the frequency response functions (FRFs), which are analytical solutions for fundamental problems of elasticity such as the Boussinesq and Cerruti problems, but expressed in the frequency domain. Considering recent efforts and results in application of FFT to convolution calculations in contact problems, the displacement arising in a tri-layer configuration is computed in the frequency domain, and the contact problem is subsequently solved in the space domain using a state-of-the-art algorithm based on the conjugate gradient method. The method relies on the FRFs derived in the literature for tri-layered materials, and the efficiency and accuracy of computations in the frequency domain is assured by using the Discrete Convolution Fast Fourier Technique (DCFFT) with influence coefficients derived from the FRFs. The computer program reproduces well-known results for bi-layered materials. Numerical simulations are performed for various configurations in which the elastic properties of the layers, as well as the frictional coefficient, are varied. By using the newly advanced simulation technique, design recommendations may be advanced for the optimal configuration of tri-layered materials under contact load.

An analysis of weak Galerkin finite element method for a steady state Boussinesq problem

In this article, we present and analyze a weak Galerkin finite element method (WG-FEM) for the coupled Navier–Stokes/temperature (or Boussinesq) problems. In this WG-FEM, discontinuous functions are applied to approximate the velocity, temperature, and the normal derivative of temperature on the boundary while piecewise constants are used to approximate the pressure. The stability, existence and uniqueness of solution of the associated WG-FEM are proved in detail. An optimal a priori error estimate is then derived for velocity in the discrete H1 and L2 norms, pressure in the L2 norm, temperature in the discrete H1 and L2 norms, and the normal derivative of temperature in H−1/2 norm. Finally, to complete this study some numerical tests are presented which illustrate that the numerical errors are consistent with theoretical results.

Normal modes with boundary dynamics in geophysical fluids

Three-dimensional geophysical fluids support both internal and boundary-trapped waves. To obtain the normal modes in such fluids, we must solve a differential eigenvalue problem for the vertical structure (for simplicity, we only consider horizontally periodic domains). If the boundaries are dynamically inert (e.g., rigid boundaries in the Boussinesq internal wave problem and flat boundaries in the quasigeostrophic Rossby wave problem), the resulting eigenvalue problem typically has a Sturm–Liouville form and the properties of such problems are well-known. However, when restoring forces are also present at the boundaries, then the equations of motion contain a time-derivative in the boundary conditions, and this leads to an eigenvalue problem where the eigenvalue correspondingly appears in the boundary conditions. In certain cases, the eigenvalue problem can be formulated as an eigenvalue problem in the Hilbert space L2⊕C and this theory is well-developed. Less explored is the case when the eigenvalue problem takes place in a Pontryagin space, as in the Rossby wave problem over sloping topography. This article develops the theory of such problems and explores the properties of wave problems with dynamically active boundaries. The theory allows us to solve the initial value problem for quasigeostrophic Rossby waves in a region with sloping bottom (we also apply the theory to two Boussinesq problems with a free-surface). For a step-function perturbation at a dynamically active boundary, we find that the resulting time-evolution consists of waves present in proportion to their projection onto the dynamically active boundary.

Pressure profile definition over the metal orthotropic deck surface of the decking superstructure ballast pocket from the local load action

The article discusses the interaction issue between the track and the rolling stock to determine the forces working on the track during the passage of a rolling load, and their further distribution through the ballast bed. The study aims to determine the vertical pressure profile working on the orthotropic metal ballast bed slab of the railway superstructure. To determine the load share from the rolling stock transferred from the rails to the cross-sleepers, the rail is represented as a beam of infinite length with one or several concentrated forces. To determine the cross-sleeper stress-strain state, it is considered as a short beam on a cushion course. The article presents the substantiation of the flow mechanics formulas application possibility to the ballast bed. Based on the Boussinesq problem solution, calculation formulas are obtained for determining the pressure along with the ballast bed plate. The results obtained by the proposed method were compared with other authors’ experimental data and the finite element modeling results in the Midas FEA software package. The analysis took into account the design features of the ballast bed plates and the track superstructure on the existing superstructures. A theoretical study of the influence of the sheet of the covering bending stiffness on the pressure profile northing is carried out. The proposed method is applicable for various superstructure and the ballast bed slab, and also takes into account the uneven pressure distribution on the inner and outer rails from the trainload, changes in eccentricity along the length of the structure, different thickness of the ballast bed under the cross-sleepers ends, as well as the effect of centrifugal force.

Interior-stress fields produced by a general axisymmetric punch

This work is a supplement to the work of Sneddon on axisymmetric Boussinesq problem in 1965 in which the distributions of interior-stress fields are derived here for a punch with general profile. A novel set of mathematical procedures is introduced to process the basic elastic solutions (obtained by the method of Hankel transform, which was pioneered by Sneddon) and the solution of the dual integral equations. These processes then enable us to not only derive the general relationship of indentation depth D and total load P that acts on the punch but also explicitly obtain the general analytical expressions of the stress fields beneath the surface of an isotropic elastic half-space. The usually known cases of punch profiles are reconsidered according to the general formulas derived in this study, and the deduced results are verified by comparing them with the classical results. Finally, these general formulas are also applied to evaluate the von Mises stresses for several new punch profiles.

Use of an Efficient Proxy Solution for the Hillslope‐Storage Boussinesq Problem in Upscaling of Subsurface Stormflow

AbstractThe hillslope‐storage Boussinesq (hsB) equation is a modification of the one‐dimensional Boussinesq representation of saturated subsurface flow that accommodates the impact of converging and diverging flows resulting from a hillslope with nonconstant width. Here, a surrogate model, the hsB Proxy, is developed which closely emulates the results of high‐quality numerical solutions of the hsB to reproduce the drainage response of a wide range of wedge‐shaped hillslopes in response to a recharge time series at a fraction of the computational cost of a numerical solution and with minimal loss of fidelity. The efficacy of the Proxy in emulating the hsB response of a single hillslope and the aggregate response of a network of hillslopes is demonstrated. The Proxy applies to wedge‐shaped hillslopes with homogenous hydraulic conductivity under uniform recharge, with the expectation that these hillslopes may inform future investigations into the upscaling behavior that results in the known power‐law relationship between subsurface storage and outflow in a basin. The utility of the hsB as a tool for evaluating upscaling problems is demonstrated in two applications. First, the Proxy response is compared to the known basin‐scale recession behavior in the Panola Mountain Research Watershed, under both transient and steady‐state conditions. Second, the Proxy is applied to derive the single best effective hillslope in a basin. The Proxy is expected to be a useful tool for rapid simulation of subsurface flow from hillslopes for use in hydrological models and land surface schemes.

The stationary Boussinesq problem under singular forcing

In Lipschitz two- and three-dimensional domains, we study the existence for the so-called Boussinesq model of thermally driven convection under singular forcing. By singular we mean that the heat source is allowed to belong to [Formula: see text], where [Formula: see text] is a weight in the Muckenhoupt class [Formula: see text] that is regular near the boundary. We propose a finite element scheme and, under the assumption that the domain is convex and [Formula: see text], show its convergence. In the case that the thermal diffusion and viscosity are constants, we propose an a posteriori error estimator and show its reliability. We also explore efficiency estimates.

Lagrange multiplier approach to unilateral indentation problems: Well-posedness and application to linearized viscoelasticity with non-invertible constitutive response

The Boussinesq problem describing indentation of a rigid punch of arbitrary shape into a deformable solid body is studied within the context of a linear viscoelastic model. Due to the presence of a non-local integral constraint prescribing the total contact force, the unilateral indentation problem is formulated in the general form as a quasi-variational inequality with unknown indentation depth, and the Lagrange multiplier approach is applied to establish its well-posedness. The linear viscoelastic model that is considered assumes that the linearized strain is expressed by a material response function of the stress involving a Volterra convolution operator, thus the constitutive relation is not invertible. Since viscoelastic indentation problems may not be solvable in general, under the assumption of monotonically non-increasing contact area, the solution for linear viscoelasticity is constructed using the convolution for an increment of solutions from linearized elasticity. For the axisymmetric indentation of the viscoelastic half-space by a cone, based on the Papkovich–Neuber representation and Fourier–Bessel transform, a closed form analytical solution is constructed, which describes indentation testing within the holding-unloading phase.