Equations Of Elastostatics Research Articles

Numerical Analysis of 3D Unilateral Quasi-static Contact: Effect of Coating Thickness and Mechanical Properties

In this study, a numerical model for unilateral quasi-static contact between a rigid sphere and an elastic coating has been developed. The contact problem was solved using a numerical procedure based on the FFT technique, considering various coatings with different thicknesses and mechanical properties, implemented through a Matlab code. The model calculates the contact surface deformation and pressure field through a double iteration process. The first iteration solves the contact problem for a given indenter penetration, while the second iteration refines this penetration by minimizing the difference between the fixed and calculated loads. To achieve this, influence coefficients are derived from the elasto-static equations using the Papkovich-Neuber potentials. The study discusses the influence of coating thickness and friction coefficient value on the tribological behavior of the coating. The results indicate that contact pressure increases (ranging from 1.9 to 2.5) as the coating becomes thicker or more rigid (0.02 mm to 0.2 mm). Additionally, the tribological behavior of the coated surface is affected by the coating's thickness, hardness, and friction coefficient value. Importantly, this model demonstrates versatility by being applicable to both smooth and rough surfaces.

A strong-form meshfree computational method for plane elastostatic equations of anisotropic functionally graded materials via multiple-scale Pascal polynomials

A strong-form meshfree method is proposed for solving plane elastostatic equations of anisotropic functionally graded materials. Any general function may be the grading function and it is changing smoothly from location to location in the material. The proposed method is based on Pascal polynomial basis and multiple-scale technique and it is a genuinely meshfree method since no numerical integrations over domains and meshing processes are required for considered problems. Implementation of the proposed method is straightforward and the method gives very accurate results. Stability of the solutions are examined numerically in occurrence of random noise. Some certain test problems with known exact solutions are solved both on regular and irregular geometries. Acquired solutions by the suggested method are compared with the exact solutions as well as with solutions of some existing numerical techniques in literature, such as boundary element, meshless local Petrov–Galerkin and radial basis function based meshless methods, to show accuracy of the proposed method.

On Representing Strain Gradient Elastic Solutions of Boundary Value Problems by Encompassing the Classical Elastic Solution

The present work aims to primarily provide a general representation of the solution of the simplified elastostatics version of Mindlin’s Form II first-strain gradient elastic theory, which converges to the solution of the corresponding classical elastic boundary value problem as the intrinsic gradient parameters become zero. Through functional theory considerations, a solution representation of the one-intrinsic-parameter strain gradient elastostatic equation that comprises the classical elastic solution of the corresponding boundary value problem is rigorously provided for the first time. Next, that solution representation is employed to give an answer to contradictions arising by two well-known first-strain gradient elastic models proposed in the literature to describe the strain gradient elastostatic bending behavior of Bernoulli–Euler beams.

Revisiting Kirchhoff–Love plate theories for thin laminated configurations and the role of transverse loads

The present article aims to re-derive a (de-)homogenization model for particularly investigating the behavior of thin laminated plates withstanding transverse loads. Instead of starting with Kirchhoff-type of assumptions, we directly apply perturbation analysis, in terms of the small parameter introduced by the thinness of composite plates, to the original three-dimensional governing elastostatic equations. The present article sees its intriguing points in the following three aspects. First, it is shown that transverse loads applied on a thin laminated plate induce an in-plane stress response, which essentially differs from the case of single-layered homogenous plates. A scaling law estimating the magnitude of the in-plane stresses due to transverse loads is then given, and a size effect in such induced in-plane stresses arises. Second, the stress state at any position of interest in the original three-dimensional configuration can be asymptotically estimated following a (de-)homogenization scheme, and the (de-homogenization) accuracy is shown, both theoretically and numerically, to be at a same order of magnitude as the thickness-to-size ratio. Third, the asymptotic analysis here identifies the right order of magnitude for the transverse normal strain, which is often set to vanish, leading to the so-called Poisson’s locking problem in classical thin plate theories.

Simulation of plane elastostatic equations of anisotropic functionally graded materials by integrated radial basis function based on finite difference approach

We present a method based on integrated radial basis function-finite difference for numerical solution of plane elastostatic equations which is a boundary value problem. The two-dimensional version of the governed equation is solved by the proposed method on various geometries such as the rectangular and irregular domains. In the current paper, one of our goals is to present an improved integrated radial basis function method based on the finite difference technique to approximate the second-order mixed partial derivatives with respect to x and y to get more accurate numerical results. Several examples are solved by applying integrated radial basis function based on finite difference method to check its accuracy and validity.

Adjoint slip inversion under a constrained optimization framework: revisiting the 2006 Guerrero slow slip event

SUMMARYTo shed light on the prevalently slow, aseismic slip interaction between tectonic plates, we developed a new static slip inversion strategy, the ELADIN (ELastostatic ADjoint INversion) method, that uses the adjoint elastostatic equations to compute the gradient of the cost function. ELADIN is a 2-step inversion algorithm to efficiently handle plausible slip constraints. First it finds the slip that best explains the data without any constraint, and then refines the solution by imposing the constraints through a Gradient Projection Method. To obtain a self-similar, physically consistent slip distribution that accounts for sparsity and uncertainty in the data, ELADIN reduces the model space by using a von Karman regularization function that controls the wavenumber content of the solution, and weights the observations according to their covariance using the data precision matrix. Since crustal deformation is the result of different concomitant interactions at the plate interface, ELADIN simultaneously determines the regions of the interface subject to both stressing (i.e. coupling) and relaxing slip regimes. For estimating the resolution, we introduce a mobile checkerboard analysis that allows to determine lower-bound fault resolution zones for an expected slip-patch size and a given stations array. We systematically test ELADIN with synthetic inversions along the whole Mexican subduction zone and use it to invert the 2006 Guerrero Slow Slip Event (SSE), which is one of the most studied SSEs in Mexico. Since only 12 GPS stations recorded the event, careful regularization is thus required to achieve reliable solutions. We compared our preferred slip solution with two previously published models and found that our solution retains their most reliable features. In addition, although all three SSE models predict an upward slip penetration invading the seismogenic zone of the Guerrero seismic gap, our resolution analysis indicates that this penetration might not be a reliable feature of the 2006 SSE.

General-purpose kernel regularization of boundary integral equations via density interpolation

This paper presents a general high-order kernel regularization technique applicable to all four integral operators of Calderón calculus associated with linear elliptic PDEs in two and three spatial dimensions. Like previous density interpolation methods, the proposed technique relies on interpolating the density function around the kernel singularity in terms of solutions of the underlying homogeneous PDE, so as to recast singular and nearly singular integrals in terms of bounded (or more regular) integrands. We present here a simple interpolation strategy which, unlike previous approaches, does not entail explicit computation of high-order derivatives of the density function along the surface. Furthermore, the proposed approach is kernel- and dimension-independent in the sense that the sought density interpolant is constructed as a linear combination of point-source fields, given by the same Green’s function used in the integral equation formulation, thus making the procedure applicable, in principle, to any PDE with known Green’s function. For the sake of definiteness, we focus here on Nystr‘̀om methods for the (scalar) Laplace and Helmholtz equations and the (vector) elastostatic and time-harmonic elastodynamic equations. The method’s accuracy, flexibility, efficiency, and compatibility with fast solvers are demonstrated by means of a variety of large-scale three-dimensional numerical examples.

A Computational Model for Simulation of Shallow Water Waves by Elastic Deformations in the Topography

We propose a coupled model to simulate shallow water waves induced by elastic deformations in the bed topography. The governing equations consist of the depth-averaged shallow water equations including friction terms for the water freesurface and the well-known second-order elastostatics formulation for the bed deformation. The perturbation on the free-surface is assumed to be caused by a sudden change in the bottom beds. At the interface between the water flow and the bed topography, transfer conditions are implemented. Here, the hydrostatic pressure and friction forces are considered for the elastostatic equations whereas bathymetric forces are accounted for in the shallow water equations. The focus in the present study is on the development of a simple and accurate representation of the interaction between water waves and bed deformations in order to simulate practical shallow water flows without relying on complex partial differential equations with free boundary conditions. The effects of location and magnitude of the deformation on the flow fields and free-surface waves are investigated in details. Numerical simulations are carried out for several test examples on shallow water waves induced by sudden changes in the bed. The proposed computational model has been found to be feasible and satisfactory.

A hybrid finite volume/finite element method for shallow water waves by static deformation on seabeds

Purpose The purpose of this study is twofold: first, to derive a consistent model free-surface runup flow problems over deformable beds. The authors couple the nonlinear one-dimensional shallow water equations, including friction terms for the water free-surface and the two-dimensional second-order solid elastostatic equations for the bed deformation. Second, to develop a robust hybrid finite element/finite volume method for solving free-surface runup flow problems over deformable beds. The authors combine the finite volume for free-surface flows and the finite element method for bed elasticity. Design/methodology/approach The authors propose a new model for wave runup by static deformation on seabeds. The model consists of the depth-averaged shallow water system for the water free-surface coupled to the second-order elastostatic formulation for the bed deformation. At the interface between the water flow and the seabed, transfer conditions are implemented. Here, hydrostatic pressure and friction forces are considered for the elastostatic equations, whereas bathymetric forces are accounted for in the shallow water equations. As numerical solvers, the authors propose a well-balanced finite volume method for the flow system and a stabilized finite element method for elastostatics. Findings The developed coupled depth-averaged shallow water system and second-order solid elastostatic system is well suited for modeling wave runup by deformation on seabeds. The derived coupling conditions at the interface between the water flow and the bed topography resolve well the condition transfer between the two systems. The proposed hybrid finite volume element method is accurate and efficient for this class of models. The novel technique used for wet/dry treatment accurately captures the moving fronts in the computational domain without generating nonphysical oscillations. The presented numerical results demonstrate the high performance of the proposed methods. Originality/value Enhancing modeling and computations for wave runup problems is at an early stage in the literature, and it is a new and exciting area of research. To the best of our knowledge, solving wave runup problems by static deformation on seabeds using a hybrid finite volume element method is presented for the first time. The results of this research study, and the research methodologies, will have an important influence on a range of other scientists carrying out research in related fields.

Harmonic balance-boundary element and continuation methods for steady-state wave scattering by interior and surface-breaking cracks with contact acoustic nonlinearity

A numerical method for nonlinear steady-state wave scattering due to a crack with contact acoustic nonlinearity (CAN) is developed in this article. The in-plane scattering problem is addressed here. The two systems studied are composed of an unbounded elastic solid that includes an interior crack and an elastic half plane that includes a surface-breaking crack. A time-harmonic plane wave is incident, and clapping and rubbing motions on the crack faces are induced as nonlinear phenomena. A harmonic balance-boundary element method (HB-BEM) is utilized to solve the nonlinear steady-state problem. The system of equations obtained by the HB-BEM consists of an elastostatic boundary integral equation (BIE) and frequency-domain elastodynamic BIEs of several different frequencies, which are coupled by nonlinear boundary conditions. To solve the system numerically, the nonlinear boundary conditions are regularized. In the regularized boundary conditions, traction is expressed by a smooth single-valued function with respect to crack-opening displacement. The solution of the steady-state system is tracked with a parameter continuation method to investigate bifurcations. The parameters that vary in the continuation method are the incident frequency and static field variables. To determine the stability of the steady-state solutions, a stability analysis method based on Hill’s infinite determinant is proposed.The stable solutions obtained by the proposed method agree with the conventional time-domain solutions after enough time has elapsed. A 1/2-order subharmonic resonance is observed with bifurcations for particular parameters. The interaction between the local resonant properties due to the crack and the CAN causes subharmonic resonance.

Direct meshless local Petrov-Galerkin method to investigate anisotropic potential and plane elastostatic equations of anisotropic functionally graded materials problems

The meshless local technique is employed to simulate potential problems involving material anisotropy and also plane elastostatic equations of anisotropic functionally graded materials (PEE-AFGM). We utilize direct meshless local Petrov Galerkin (DMLPG) technique to discrete the spatial directions. The DMLPG5 (based on weak form) is applied to solve potential problems involving material anisotropy and the DMLPG2 (based on strong form) is employed to simulate PEE-AFGM. The coefficients of the anisotropic conductivity require the gradient term to be modified accordingly. The grading function can be any general function that varies smoothly from point to point in the material. The developed procedures are tested on various examples on non-rectangular domains to check their accuracy.

Magnetic resonance in layered ferrite-piezoelectric structures

The aim of this work is to study the electric field induced shift and broadening of ferromagnetic resonance (FMR) line in layered structures based on ferrite and piezoelectric. Theoretical modeling is based on the solution of equations of motion for magnetization, elastostatic and constitutive equations. The paper shows that in asymmetric structures, the applied electric field causes a broadening of the FMR lines. In symmetric layered structures, the applied electric field gives rise to a shift of lines and no broadening of resonance lines.

A boundary element approach for solving plane elastostatic equations of anisotropic functionally graded materials

A boundary element method is proposed for the numerical solution of an important class of boundary value problems governed by plane elastostatic equations of anisotropic functionally graded materials. The grading function of the material properties may be any general function that varies smoothly from point to point in the material. The proposed boundary element method is applied to solve some specific problems to check its validity and accuracy.

On the Fredholm Property of the Trace Operators Associated with the Elastic Layer Potentials

We deal with the system of equations of linear elastostatics, governing the equilibrium configurations of a linearly elastic body. We recall the basics of the theory of the elastic layer potentials and we extend the trace operators associated with the layer potentials to suitable sets of singular densities. We prove that the trace operators defined, for example, on W 1 − k − 1 / q , q ( ∂ Ω ) (with k ≥ 2 , q ∈ ( 1 , + ∞ ) and Ω an open connected set of R 3 of class C k ), satisfy the Fredholm property.

Formulation and analysis of fully-mixed methods for stress-assisted diffusion problems

This paper is devoted to the mathematical and numerical analysis of a mixed-mixed PDE system describing the stress-assisted diffusion of a solute into an elastic material. The equations of elastostatics are written in mixed form using stress, rotation and displacements, whereas the diffusion equation is also set in a mixed three-field form, solving for the solute concentration, for its gradient, and for the diffusive flux. This setting simplifies the treatment of the nonlinearity in the stress-assisted diffusion term. The analysis of existence and uniqueness of weak solutions to the coupled problem follows as combination of Schauder and Banach fixed-point theorems together with the Babuška–Brezzi and Lax–Milgram theories. Concerning numerical discretization, we propose two families of finite element methods, based on either PEERS or Arnold–Falk–Winther elements for elasticity, and a Raviart–Thomas and piecewise polynomial triplet approximating the mixed diffusion equation. We prove the well-posedness of the discrete problems, and derive optimal error bounds using a Strang inequality. We further confirm the accuracy and performance of our methods through computational tests.

Spectral permanence in a space with two norms

A generalization of a classical argument of Mark G. Krein leads us to the conclusion that the Neumann–Poincar´e operator associated to the Lamé system of linear elastostatics equations in two dimensions has the same spectrum on the Lebesgue space of the boundary as the more natural energy space. A similar result for the Neumann–Poincaré operator associated to the Laplace equation was stated by Poincaré and was proved rigorously a century ago by means of a symmetrization principle for non-selfadjoint operators. We develop the necessary theoretical framework underlying the spectral analysis of the Neumann–Poincaré operator, including also a discussion of spectral asymptotics of a Galerkin type approximation. Several examples from function theory of a complex variable and harmonic analysis are included.

Fast and accurate two-field reduced basis approximation for parametrized thermoelasticity problems

This paper concerns a two-field reduced basis algorithm for the metamodelling of parametrized one-way coupled thermoelasticity problems based on the constitutive relation error (CRE) estimation. The coupled system consists of parametrized thermal diffusion and elastostatic equations which are explicitly coupled in a one-way manner. The former can be solved in advance independently and the latter can be solved afterwards using the solution of the former. For the fast and accurate analysis of the coupled system, we developed an algorithm that can choose adaptively the number of reduced basis functions of the temperature field to approximate the CRE equality of the mechanical field. We compute approximately the upper bound for the true errors of displacement and stress fields in energy norms. To enable this, a two-field greedy sampling strategy is adopted to construct the displacement and stress fields in an efficient manner. The computational efficiency of the proposed approach is demonstrated with computing the effective coefficient of thermal expansion of heterogeneous materials.

Mesh refitting approach: a simple method to model mixed-mode crack propagation in nonlinear elastic solids

We devise a finite element methodology to trace quasi-static through-thickness crack paths in nonlinear elastic solids. The main feature of the proposed method is that it can be directly implemented into existing large scale finite element solvers with minimal effort. The mesh topology modifications that are essential in propagating a crack through the finite element mesh are accomplished by utilizing a combination of a mesh refitting procedure and a nodal releasing approach. The mesh refitting procedure consists of two steps: in the first step, the nodes are moved by solving the elastostatic equations without touching the connectivity between the elements; in the next step, if necessary, quadrilateral elements attached to crack tip nodes are split into triangular elements. This splitting of elements allows the straightforward modification of element connectivity locally, and is a key step to preserve the quality of the mesh throughout the simulation. All the geometry related operations required for crack propagation are addressed in detail with full emphasis on computer implementation. Solving several examples involving single and multiple cracks, and comparing them with experimental or other numerical approaches indicate that the proposed method captures crack paths accurately.

Learning about Hydrothermal Volcanic Activity by Modeling Induced Geophysical Changes

Motivated by ongoing efforts to understand the nature and the energy potential of geothermal resources, we devise a coupled numerical model (hydrological, thermal, mechanical), which may help in the characterization and monitoring of hydrothermal systems through computational experiments. Hydrothermal areas in volcanic regions arise from a unique combination of geological and hydrological features which regulate the movement of fluids in the vicinity of magmatic sources capable of generating large quantities of steam and hot water. Numerical simulations help in understanding and characterizing rock-fluid interaction processes and the geophysical observations associated with them. Our aim is the quantification of the response of different geophysical observables (i.e. deformation, gravity and magnetic field) to hydrothermal activity on the basis of a sound geological framework (e.g. distribution and pathways of the flows, the presence of fractured zones, caprock). A detailed comprehension and quantification of the evolution and dynamics of the geothermal systems and the definition of their internal state through a geophysical modeling approach are essential to identify the key parameters for which the geothermal system may fulfill the requirements to be exploited as a source of energy. For the sake of illustration only, the numerical computations are focused on a conceptual model of the hydrothermal system of Vulcano Island by simulating a generic 1-year unrest and estimating different geophysical changes. We solved (i) the mass and energy balance equations of flow in porous media for temperature, pressure and density changes, (ii) the elastostatic equation for the deformation field and (iii) the Poisson’s equations for gravity and magnetic potential fields. Under the model assumptions, a generic unrest of 1-year engenders on the ground surface low amplitude changes in the investigated geophysical observables, that are, however, above the accuracies of the modern state-of-the-art instruments. Devising multidisciplinary and easy-to-use computational experiments enable us to learn how the hydrothermal system responds to un unrest and which fingerprints it may leave in the geophysical signals.

Conformal invariance and new exact solutions of the elastostatics equations

We fulfilled a group foliation of the system of n-dimensional (n ≥ 2) Lame equations of the classical static theory of elasticity with respect to the infinite subgroup contained in normal subgroup of main group of this system. It permitted us to move from the Lame equations to the equivalent unification of two first-order systems: automorphic and resolving. We obtained a general solution of the automorphic system. This solution is an n-dimensional analogue of the Kolosov-Muskhelishvili formula. We found the main Lie group of transformations of the resolving system of this group foliation. It turned out that in the two-dimensional and three-dimensional cases, which have a physical meaning, this system is conformally invariant, while the Lame equations admit only a group of similarities of the Euclidean space. This is a big success, since in the method of group foliation, resolving equations usually inherit Lie symmetries subgroup of the full symmetry group that was not used for the foliation. In the three-dimensional case for the solutions of the resolving system, we found the general form of the transformations similar to the Kelvin transformation. These transformations are the consequence of the conformal invariance of the resolving system. In the three-dimensional case with a help of the complex dependent and independent variables, the resolving system is written as a simple complex system. This allowed us to find non-trivial exact solutions of the Lame equations, which direct for the Lame equations practically impossible to obtain. For this complex system, all the essentially distinct invariant solutions of the maximal rank we have found in explicit form, or we reduced the finding of those solutions to the solving of the classical one-dimensional equations of the mathematical physics: the heat equation, the telegraph equation, the Tricomi equation, the generalized Darboux equation, and other equations. For the resolving system, we obtained double wave of a special type and double waves of the shear deformations in an elastic medium. By the obtained formulas for generating new solutions, these solutions give an infinite set of exact solutions of this complex system. By the three-dimensional analog of the Kolosov-Muskhelishvili formula, the found solutions produce an infinite-dimensional vector space of exact solutions of the Lame equations.