Frictionless Contact Problem Research Articles

Axisymmetric contact problem of piezoelectric coating-substrate system with functionally graded piezoelectric interfacial layer

In this paper, the electromechanical response of piezoelectric coating-functionally graded piezoelectric (FGP) interfacial layer–piezoelectric substrate system under the rigid spherical punch is considered. It is assumed that the piezoelectric coating and piezoelectric substrate are connected by the FGP interfacial layer which is simulated by a multi-layer model. The axisymmetric frictionless contact problem of piezoelectric coating–substrate system with FGP interfacial layer is formulated in terms of singular integral equation with the method of transfer matrix and Hankel integral transform technique. The effect of the gradient of material properties on the stress components and electric displacement components in piezoelectric coating–substrate system is gained by solving the equation numerically. The present results could serve as a useful reference for the design of piezoelectric coating–substrate system.

Optimal control of a two-dimensional contact problem with multiple unilateral constraints

In this article, we are concerned with optimal control of a frictionless contact problem with multiple unilateral constraints for a two-dimensional bar. The existence of an optimal trajectory-control pair is firstly proven under the framework of general cost functional. The Pontryagin maximum principle is then established for the investigational system equipped with many equality and inequality constraints in fixed final horizon case, owing to the Dubovitskii and Milyutin functional analytical approach. A remark concludes the article with the discussion, which address the utilization of obtained necessary optimality condition.

Stable model reduction for linear variational inequalities with parameter-dependent constraints

We consider model reduction for linear variational inequalities with parameter-dependent constraints. We study the stability of the reduced problem in the context of a dualized formulation of the constraints using Lagrange multipliers. Our main result is an algorithm that guarantees inf-sup stability of the reduced problem. The algorithm is computationally effective since it can be performed in the offline phase even for parameter-dependent constraints. Moreover, we also propose a modification of the Cone Projected Greedy algorithm so as to avoid ill-conditioning issues when manipulating the reduced dual basis. Our results are illustrated numerically on the frictionless Hertz contact problem between two half-disks with parameter-dependent radius and on the membrane obstacle problem with parameter-dependent obstacle geometry.

The interpolating element-free Galerkin method for the variational-hemivariational inequality of the frictionless elastic contact problem with normal compliance and unilateral constraint

The interpolating element-free Galerkin method for the variational-hemivariational inequality of the frictionless elastic contact problem with normal compliance and unilateral constraint

Frictionless Multi-field Coupling Contact Problem for a Thermoelectric Layer Loaded by Two Rigid Punches

Frictionless Multi-field Coupling Contact Problem for a Thermoelectric Layer Loaded by Two Rigid Punches

Pointwise A Posteriori Error Control of Discontinuous Galerkin Methods for Unilateral Contact Problems

Abstract In this article, we derive a reliable and efficient a posteriori error estimator in the supremum norm for a class of discontinuous Galerkin (DG) methods for the frictionless unilateral contact problem between two elastic bodies. The proposed error estimator generalizes the basic residual type estimators for the linear problems in linear elasticity taking into account the nonlinearity on a part of the boundary. The analysis hinges on the super- and sub-solutions constructed by modifying the discrete solution appropriately, and it is carried out in a unified manner which holds for several DG methods. The terms arising from the contact stresses in the error estimator vanish on the discrete full contact set. We illustrate the performance of the proposed error estimator via several numerical experiments in two dimensions.

Frictionless discontinuous contact problem of the thermoelectric layer resting on a rigid base

Neglecting the friction between the indenter and the thermoelectric layer (TEL), the discontinuous contact problem between the TEL and the rigid base is studied by using elastic theory and integral transformation technology. The external load is applied to the TEL through the rigid base. When the value of the external load exceeds the critical load (CL) separation occurs along the interface between the TEL and the rigid base. The discontinuous contact problem is transformed into a system of singular integral equations (SIES) by boundary conditions. Numerical results of the contact stress and the separation distance between the TEL and the rigid base are considered by the external load, the specific weight of the TEL and the width of the indenter. The results show that the larger external load and the specific weight of the TEL will cause the separation distance between the TEL and the rigid base to increase. However, the increase in the width of the rigid indenter has the opposite effect on the contact stress between the TEL and the rigid base. The model developed in this article provides guidance for the design and safety and reliability judgment of thermoelectric components in practical applications.

Receding contact between an inclusion of generalized shape and a remotely-stressed plate

This paper examines the problem of frictionless contact and separation of a rigid inclusion from an infinite elastic plate. The inclusion initially fits perfectly into a hole in the plate, and separates from it upon remote stressing. In a significant generalization of the classical work of Wilson and Goree on receding contacts of elliptical inclusions, the present formulation allows the inclusion to be of quite general shape. In particular, superellipses provide a convenient way to describe a family of such generalized inclusion shapes by varying two parameters. A highly accurate conformal mapping scheme based on rational functions and the Adaptive Antoulas–Anderson (AAA) algorithm is used to map the interior of a unit circle to the exterior of these generalized inclusions. Notably, a rational function representation is far more efficient than a polynomial representation. The governing Singular Integro-Differential Equation (SIDE) of the contact is then derived using the conformal map and analytic continuation, and solved using standard orthogonal polynomial techniques. The contact size and transmitted pressure traction are obtained for a wide range of shapes, including superelliptical, curvilinear hexagonal, and hypotrochoidal square inclusions. The role of the remote stress ratio on the contact response is studied, and a phase diagram demarcating various contact regimes is obtained. A comparison of the pressures for selected inclusion shapes using finite element analysis (FEA) shows good agreement with the present (semi-analytical) SIDE results. For shapes like hypotrochoidal squares, subtle changes in the inclusion geometry can cause significantly different peak contact pressures, pointing to the importance of accurate modeling of geometry in conforming–receding contact problems.

Solvability of Conformable Type Frictionless Contact Problem via Hemivariational Inequalities

In this paper, we study a class of conformable frictionless contact problems with the surface traction driven by the conformable impulsive differential equation. The existence of a mild solution for conformable impulsive hemivariational inequality is obtained by the Rothe method, subjectivity of multivalued pseudomonotone operators and the property of the conformable derivative. Notice that we imply some new fractional viscoelastic constitutive laws.

A Comparison of General Solutions to the Non-Axisymmetric Frictionless Contact Problem with a Circular Area of Contact: When the Symmetry Does Not Matter

The non-axisymmetric problem of frictionless contact between an isotropic elastic half-space and a cylindrical punch with an arbitrarily shaped base is considered. The contact problem is formulated as a two-dimensional Fredholm integral equation of the first type in a fixed circular domain with the right-hand side being representable in the form of a Fourier series. A number of general solutions of the contact problem, which were published in the literature, are discussed. Based on the Galin–Mossakovskii general solution, new formulas are derived for the particular value of the contact pressure at the contact center and the contact stress-intensity factor at the contour of the contact area. Since the named general solution does not employ the operation of differentiation of a double integral with respect to the coordinates that enter it as parameters, the form of the general solution derived by Mossakovskii as a generalization of Galin’s solution for the special case, when the contact pressure beneath the indenter is bounded, is recommended for use as the most simple closed-form general solution of the non-axisymmetric Boussinesq contact problem.

The Problem of Wedge Indenter with Flat-Rounded Bottom Indenting Half-Plane Elastic Body

In this paper, the problem of a wedge indenter with flat-rounded bottom indenting an infinite half-plane elastic body is solved. It can be used to study the penetration force or the surface stress distribution of the cutter when the tool intrudes into an elastic body. First, a first-order continuity method is proposed to eliminate the stress singularity caused by the sharp corners of the wedge indenter with flat bottom. The two-dimensional elastic contact problem is transformed into a Riemann–Hilbert boundary value problem with discontinuous coefficients expressed in the form of complex variables. For the frictionless contact problem, the explicit analytical solution is obtained for contact stress on the half-plane surface. Furthermore, the expression of the stress potential function is also derived to solve the stress field in the half-plane. Verification indicates that the wedge indenter with flat-rounded bottom can degenerate to several well-studied shaped indenters, including the Hertz model and flat-rounded model. The model in this paper is also validated with its numerical counterpart built in the finite element program LS-DYNA. Finally, the sensitivity of different configuration parameters on the normal contact stress and interior stress field in numerical model was investigated.

Well-posedness of constrained evolutionary differential variational–hemivariational inequalities with applications

A system of a first order history-dependent evolutionary variational– hemivariational inequality with unilateral constraints coupled with a nonlinear ordinary differential equation in a Banach space is studied. Based on a fixed point theorem for history dependent operators, results on the well-posedness of the system are proved. Existence, uniqueness, continuous dependence of the solution on the data, and the solution regularity are established. Two applications of dynamic problems from contact mechanics illustrate the abstract results. First application is a unilateral viscoplastic frictionless contact problem which leads to a hemivariational inequality for the velocity field, and the second one deals with a viscoelastic frictional contact problem which is described by a variational inequality.

Sliding frictional contact problem of a layer indented by a rigid punch in couple stress elasticity

This paper investigates the frictional contact problem of a layer indented by a rigid punch within the framework of the couple stress elasticity. It is assumed that the layer is homogeneous, isotropic, and fully bonded to a rigid substrate. The mixed-boundary value problem is converted using Fourier transform into a singular integral equation in which the unknown is the contact pressure between the layer and the punch. The integral equation is further derived for the flat and cylindrical punch case profiles, normalized and then solved numerically using the Gauss–Jacobi integration formula. The obtained results are first validated based on those published for the case of a frictionless contact problem of a half-plane indented by a rigid punch and solved within the context of couple stress theory. An extensive parametric study is then conducted to investigate the effect of several parameters on the contact stresses for the both the flat and cylindrical punch profiles. These parameters include the characteristic material length, the layer height, the friction coefficient, the indentation load, and the shear modulus.

A singularity analysis in Keer’s elastic indentation problem

A non-axisymmetric three-dimensional frictionless contact problem for an elastic half-space and a cylindrical indenter with the face in a wedge form is considered under the assumption of a circular region of contact. Explicit formulas are derived for the critical indenter displacement and contact force that are needed to ensure full contact over the entire contact region. It is shown that the contact pressure beneath the indenter maintains a logarithmic singularity, which is produced by the wedge edge and is similar to the singularity of the contact pressure in the corresponding two-dimensional contact problem for a wedge rigid indenter.

History-dependent fractional hemivariational inequality with time-delay system for a class of new frictionless quasistatic contact problems

We study a new frictionless quasistatic contact problem for viscoelastic materials, in which contact conditions are described by the fractional Clarke generalized gradient of nonconvex and nonsmooth functions and a time-delay system. In addition, our constitutive relation is modeled using the fractional Kelvin–Voigt law with long memory. The existence of mild solutions for new history-dependent fractional differential hemivariational inequalities with a time-delay system are obtained by the Rothe method, properties of the Clarke generalized gradient, and a fixed-point theorem.

Convergence results for a class of multivalued variational–hemivariational inequality

In the present paper, we focus on a class of multivalued variational–hemivariational inequalities with pseudomonotone operator and constraint set in Hilbert spaces. By employing the penalty method and the Moreau–Yosida approximation technique, we construct an approximating problem for the original multivalued variational–hemivariational inequality under consideration. The main result in the paper shows that every weak cluster of the solution sequence for the approximating problem is always a solution of the original problem. Moreover, based on the obtained weak convergence result, another two strong convergence results are obtained when the condition of pseudomonotonicity is reinforced. Finally, we illustrate the application of our abstract results in the study of a frictionless contact problem in mechanics with unilateral constraint.

Partial slip contact problem between a transversely isotropic half-space of multi-ferroic composite medium and a spherical indenter

This article concerns partial slip contact problem of a composite medium half-space punched by a spherical indenter. The indenter is subjected to the normal indentation force, electric and magnetic loads, and then exerted by a progressively increasing tangential load. The tangential load, which is not sufficient to cause complete sliding, results in the contact region split into the slip region and the stick region. Coulomb friction law is adopted in the slip region and four possible physical cases are considered in view of the magneto-electric properties of indenter. Three-dimensional (3D) magneto-electro-elastic coupling fields associated with four cases are obtained in terms of the elementary functions by use of the potential theory and ignoring effect of the tangential load on the generalized normal stresses in the contact region. The contact radius and the stick radius are determined by the equilibrium equations. The stress intensity factors are introduced to characterize the stress singularity. The obtained solutions are verified by comparing with results related to the frictionless contact problem and the fundamental solutions. Influence of the frictional coefficient and the magneto-electric properties of indenter on the physical quantities are analyzed. It is worth noting that the frictional coefficients have obvious effect on the von Mises stress and the maximum shear stress. The suitable electric and magnetic loads may eliminate the stress concentration. These results not only are the theoretical fundamental to the forthcoming indentation experiments and the fretting contact, but also serve as guiding design of the smart devices.

Boundary Element Calculations for Normal Contact of Soft Materials With Tensed Surface Membrane

This work considers the non-adhesive frictionless contact problem of soft materials with surface being tensed by equi-biaxial tension. The boundary element method based on Fast Fourier Transform and conjugate gradient algorithm is extended to deal with this problem. By comparing with existing analytical solutions for the axisymmetric contact between a rigid parabolic indenter and an elastic half space with constant membrane/surface tension, our numerical simulations are validated having great accuracy and efficiency. Then, with this numerical tool, we further calculate the contact responses of a soft substrate indented by a smooth indenter with general quadric profile and a rough indenter with self-affine fractal surface, respectively. Some nontrivial contact behavior resulted from the presence of membrane tension are demonstrated and discussed.

Depth-sensing indentation of spherical particles on corrugated substrates — An asymptotic model

An AFM-based nanoindentation of a spherical particle absorbed on a corrugated surface of rigid substrate is considered in the framework of continuum mechanics. The corresponding problem of unilateral frictionless contact with multiple contact zones is formulated using isotropic infinitesimal elasticity. Both the AFM probe surface and the surfaces of rigid supports, which are exposed to contact with the elastic sphere, are assumed to be spherical. The method of matched asymptotic expansions is utilized for constructing the leading order asymptotic model of multiple contact. Explicit formulas are derived for the force-displacement relationship and the incremental indentation stiffness. It is shown that the difference between the one-point contact and the multi-point contact (e.g., with two and three local supports) is found to be significant. The case of tight hexagonal packing of the support spheres is considered in detail and illustrated by a comparison with the Hertzian theory of local contact.

Incomplete contact between a coated elastic substrate and rigid foundation perturbed by a rigid disc

This paper investigates an axisymmetric and frictionless contact problem of a coated elastic substrate pressed onto a rigid foundation with a rigid disc by remoted uniform compressive stress. Such an incomplete contact problem is formulated as a three-parts mixed boundary values problem (MBVP). By means of Hankel transform technique and a new solution derived for a multilayered elastic solid subjected to surface displacement (dislocation), the MBVP is converted to a system of triple integral equations. A series expansion method characterized by expressing the unknown surface displacement as Bessel function series is adopted to solve the triple integral equations. Explicit analytical expressions for contact stress, resultant contact force and Mode-I stress intensity factor are obtained. The accuracy of the present solutions is verified by comparing to the results available in the literatures for the simple cases of a single layer and a homogeneous half space. Numerical studies are also performed to investigate the contact mechanics of coated material under incomplete contact. Results from this study are valuable for the design of coating to protect engineering machine components against third-body abrasive particles or solid contaminants induced contact damage.