Elastic Contact Problems Research Articles

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

"NUMERICAL SIMULATION OF ELASTIC-PLASTIC CONTACT WITH ISOTROPIC HARDENING "

The contacts of mechanical components transmit loads that lead to subsurface stresses developing in the contacting bodies. In an efficient tribological design, these stresses are expected to remain under the yield strength of the softer contacting material. When this condition is not met, plastic flow occurs in the softer body. Under the assumption of isotropic hardening, the yield strength increases with the development of additional plastic strains. As plastic flow processes are dissipative and therefore path dependent, the elastic-plastic problem is unsolvable through analytical endeavours, but can be approached with a numerical algorithm capable of simulating the loading history. The Betti’s reciprocal theorem provides the theoretical framework for the application of superposition principle to elastic-plastic stresses and displacement. An algorithm consisting in three nested loops is assembled from the solutions of simpler problems: (1) the purely elastic rough contact problem, (2) the inclusion problem and (3) the problem of the plastic strain increment. The numerical simulations suggest that the residual stresses decrease the intensity of the total stresses, thus delaying additional plastic flow. With increasing load, the heart-shaped plastic strain volume advances toward the surface, enveloping a plastic core near the initial point of contact. Compared to the purely elastic case, the elastic-plastic pressure shows a flatter distribution, while the contact radius is increased.

Differential variational–hemivariational inequalities with application to contact mechanics

In this paper we study a dynamical system which consists of the Cauchy problem for a nonlinear evolution equation of first order coupled with a nonlinear time-dependent variational–hemivariational inequality with constraint in Banach spaces. The evolution equation is considered in the framework of evolution triple of spaces, and the inequality which involves both the convex and nonconvex potentials. We prove existence of solution by the Kakutani–Ky Fan fixed point theorem combined with the Minty formulation and the theory of hemivariational inequalities. We illustrate our findings by examining a nonlinear quasistatic elastic frictional contact problem for which we provide a result on existence of weak solution.

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 a frictional unilateral contact problem in nonlinear elasticity-existence and smoothing procedure.

This note is devoted to a novel frictional unilateral contact problem in finite-strain elasticity. Here, we adopt the Tresca friction model from linear elasticity. Our analysis relies on the polyconvexity approach to nonlinear elasticity due to J. Ball. We include the delicate case where the elastic body neither is fixed nor has a deformation prescribed along some part of its boundary, but rests on a rigid foundation with a free boundary and is only submitted to forces and loads acting in the interior (like gravity) and at the boundary, respectively. This leads to a loss of coercivity and necessitates an extra condition that prevents the body from escaping by the geometry of the obstacle. This new condition extends a similar condition of Ciarlet and Nečas from the frictionless case to the case of Tresca friction. In addition, as a first step towards a numerical treatment of such nonlinear problems, we present a smoothing procedure that tackles the non-smooth term from Tresca friction and provide a convergence result for the novel smoothing method. This article is part of the theme issue 'Non-smooth variational problems and applications'.

A physics‐based model reduction approach for node‐to‐segment contact problems in linear elasticity

AbstractThe article presents a novel model order reduction method for mechanical problems in linear elasticity with nonlinear contact conditions. Recently, we have proposed an efficient reduction scheme for the node‐to‐node formulation [Manvelyan et al., Comput Mech 68, 1283–1295 (2021)] that leads to linear complementarity problems (LCPs). Here, we enhance the underlying contact problem to a node‐to‐segment formulation, which leads to quadratic inequalities as constraints. The adjoint system corresponds to a nonlinear complementarity problem that describes the Lagrange multiplier. The latter is solved by a Newton‐type iteration based on a LCP solver in each time step. Since the maximal set of potential contact nodes is predefined, an additional substructuring by Craig–Bampton can be performed. This contact treatment turns out to be necessary and allows exclusion of the Lagrange multipliers and the nodal displacements at contact from reduction. The numerical solutions of the reduced contact problem achieve high accuracy and the dynamic contact carries over the behavior of the full order model. Moreover, if the contact area is small compared to the overall structure, the reduction scheme performs very efficiently. The performance of the resulting reduction method is assessed on two 2D computational examples from linear elasticity.

Homogenization of Friction in a 2D Linearly Elastic Contact Problem

Contact problems with Coulomb friction in linear elasticity are notoriously difficult, and their mathematical analysis is still largely incomplete. In this paper, a model problem with heterogeneous friction coefficient is considered in two-dimensional elasticity. For this model problem, an existence and uniqueness result is proved, relying heavily on harmonic analysis. A complete and rigorous homogenization analysis can be performed in the case of a highly oscillating friction coefficient, being the first result in that direction. The Coulomb law is found to hold in the limit, and an explicit formula is provided to calculate the effective friction coefficient. This effective friction coefficient is found to differ from the spatial average, showing an influence of the coupling between friction and elasticity on the homogenized limit.

Penalty method for a class of differential nonlinear system arising in contact mechanics

The main goal of this paper is to study a class of differential nonlinear system involving parabolic variational and history-dependent hemivariational inequalities in Banach spaces by using the penalty method. We first construct a penalized problem for such a nonlinear system and then derive the existence and uniqueness of its solution to obtain an approximating sequence for the nonlinear system. Moreover, we prove the strong convergence of the obtained approximating sequence to the solution of the original nonlinear system when the penalty parameter converges to zero. Finally, we apply the obtained convergence result to a long-memory elastic frictional contact problem with wear and damage in mechanics. First part title: IntroductionSecond part title: PreliminariesThird part title: Convergence result for (1.1)Fourth part title: An application

A new finite element paradigm to solve contact problems with roughness

This article’s main scope is the presentation of a computational method for the simulation of contact problems within the finite element method involving complex and rough surfaces. The approach relies on the MPJR (eMbedded Profile for Joint Roughness) interface finite element proposed in [Paggi, M., Reinoso, J., 2020. Mech. Adv. Mater. Struct. 27:1731–1747], which is nominally flat but can embed at the nodal level any arbitrary height to reconstruct the displacement field due to contact in the presence of roughness. Here, the formulation is generalized to handle 3D surface height fields and any arbitrary nonlinear interface constitutive relation, including friction and adhesion. The methodology is herein validated with BEM solutions for linear elastic contact problems. Then, a selection of nonlinear contact problems prohibitive to be simulated by BEM and by standard contact algorithms in FEM are detailed, to highlight the promising aspects of the proposed method for tribology.

Fractal Contact Mechanics Model for the Rough Surface of a Beveloid Gear with Elliptical Asperities

Understanding the contact mechanics of rough tooth surfaces is critical in order to understand phenomena such as tooth surface flash temperature, tooth surface wear, and gear vibration. In this paper, the contact behavior between the meshing tooth surfaces of beveloid gear pairs with elliptical asperities is the focus. The contact area distribution function of the elliptical asperity was proposed for the point contact of curved surfaces by transforming the elastic contact problem between gear meshing surfaces into the contact between elastic curved surfaces with an arbitrary radius of curvature. In addition, a fractal contact mechanics model for the rough surface of a beveloid gear with elliptical asperities was established. The influence of tooth surface topography on the contact load and contact stiffness under different fractal parameters was investigated, and the results demonstrated that the real contact load and the contact stiffness of curved surfaces increase with the increase in the fractal dimension D and the contact coefficient λ. Conversely, the real contact load and normal contact stiffness decrease with the increase in the fractal roughness G and eccentricity e.

An Isogeometric analysis based method for frictional elastic contact problems with randomly rough surfaces

The problem addressed in this paper concerns the frictional contact between an elastomer and a rigid body with randomly rough surfaces. The work is accomplished on the basis of two crucial elements: a framework for generating random geometries and a robust frictional contact algorithm, both of which are realised in the approach of Isogeometric analysis (IGA) for its high accuracy and robustness. For the former, a new Isogeometric framework for random geometry modelling is proposed, which combines the random field generation based on Karhunen–Loève expansion theory with Non-Uniform Rational B-Spline (NURBS) interpolation method. For the latter, a mortar-based frictional contact algorithm in 2D large deformation regime is adopted incorporating a modified closest point projection method for detection of contact. Numerical experiments are conducted with several settings such as ‘rough–smooth’, ‘smooth–rough’ and ‘rough–rough’ contact, depending on which side of the contact pair the randomly rough surface belongs to. The ratio of the global coefficient of friction to the prescribed local one and the ratio of true contact area to the nominal contact area are characterised under these settings, and factors like the root mean square roughness and correlation length of the random surface and the external traction are discovered to have a significant influence on the two ratios.

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.

Topology optimization of elastic contact problems with maximum contact pressure constraint

Topology optimization of elastic contact problems with maximum contact pressure constraint

"The penalty method for generalized mixed variational-hemivariational inequality problems"

"It is well known that many popular variational, quasi-variational, hemivariational inequalities and variational inclusions involving constraints in a Banach space to convert a fixed point problems for finding the solution of such problems. This paper is to infuse a sequence of penalized problems without constraints and we show under the few reasonable assumptions to the Kuratowski upper limit with respect to the weak topology of the sets of solutions to penalized problems is nonempty. As an application, we explore two complicated partial differential systems of elliptic mixed boundary value problem involving a nonlinear nonhomogeneous differential operator with an obstacle effect, and a nonlinear elastic contact problem in mechanics with unilateral constraints."

Stress-based topology optimization of continuum structures for the elastic contact problems with friction.

Structural problems have various nonlinearities in the real world and these nonlinearities should be accommodated in structural topology optimization. This work proposes a topology optimization method for minimizing the maximum von Mises stress of elastic continuum structures with frictional contact under material usage constraint, using an extended Bi-directional Evolutionary Structural Optimization (BESO) method. Stresses are treated as global performance (objective) function, the global von Mises stress is measured by the p-norm stress aggregation approach, and the friction behavior is governed by the Coulomb friction law regularized in analogy with the perfect elasto-plastic theory. BESO method based on discrete variables which can avoid the well-known stress singularity and the numerical instability issue in frictional contact problems. The adjoint sensitivity analysis method is adopted to derive the sensitivity numbers. The effectiveness of the proposed method is validated through a series of comparison studies including elastic-rigid and elastic-elastic contact problems. The influence of varying friction coefficient on the optimized results and the stress distributions are investigated in comparison with the maximum stiffness design. The effect of different parameters including p-norm, volume fraction and mesh density on the optimized results are discussed. The optimized results, for elastic-rigid contact, indicate that the maximum stress can be reduced compared with elastic-elastic contact. The optimized stress decreases as the friction coefficient increases because the friction behavior resists the tangential deformation at the contact interface. The results also show that the proposed approach can achieve a reasonable design that effectively controls the stress level and reduces the stress concentration effect at the critical stress areas.

Effects of Near-Surface Composites on Frictional Rolling Contact Solved by a Semi-Analytical Model

Abstract A semi-analytical model (SAM) to tackle the steady-state elastic frictional rolling contact problem involving composites is presented. Specifically, the frictional rolling contact is categorized into two subtypes, namely, normal and tangential problems, and the conjugate gradient method (CGM) is used to figure out the normal pressure and tangential traction. In SAM, the equivalent inclusion method (EIM) is applied to analyze the influence of composites on the matrix, and the displacement disturbance resulting from such composites is added to the total surface displacement, which implements the coupling between surface contact and composites. The accuracy of the proposed model is verified by the finite element (FE) model. The effects of composites on the frictional rolling contact behavior are investigated. The results indicate that Young’s modulus, as well as the size and location of the composites, are correlated with the distributions of tangential traction, subsurface stresses, and the sizes of stick and sliding zones.

Boundary element method for the elastic contact problem with hydrostatic load at the contact interface

When a seal is applied to prevent leakage of a pressurized fluid it must be designed to sustain the hydrostatic load it is subjected to. One important aspect that the design must take into consideration is the possibility that the hydrostatic fluid load may deform the sealing surfaces to such an extent that leakage occurs. In this paper, we describe the development and implementation of a novel boundary element method (BEM) that can be used to study the effects of the pressurized fluid presents at the interface between the sealing surfaces. The proposed method utilizes a connected region labeling algorithm to identify the fluid-filled regions and it solves the coupled solid-solid and fluid-solid contact problem with an FFT-based boundary element method (DC-FFT). When benchmarked against a FEM implementation of the same problem, it is found that the results are in good agreement. In particular, it is shown that the proposed method can capture the deformation of the sealing surfaces, and how the fluid front advances when the hydrostatic pressure increases.

Numerical investigation into pressure-assisted sintering using fully coupled mechano-diffusional phase-field model

To investigate the effect of contact stress between particles on the evolution of microstructure in pressure-assisted sintering, an improved fully coupled mechano-diffusional phase-field model (PFM) was presented, in which the design of the interpolation method for elastic modulus ensured that the normalised contact stress distribution was approximated to that in conventional contact problems. The immediate advantage was that the consequent strain energy in the contact and neck area could be directly characterised without additional approximate treatment.The simulation results showed that the sintering neck evolved under the joint action of the mechanical mechanism of elastic contact and the mass diffusion mechanism for pressure-assisted sintering. Elastic contact played a dominant role before significant neck growth in the early stage of sintering. As the neck grew, mass diffusion became the dominant factor, but the normalised contact stress distribution in the contact area always satisfied that of the elastic contact problem. Owing to the additional contribution of strain energy, the greater the pressure, the faster the sintering neck growth. Further study showed that the diffusion-induced stress hindered the growth of the sintering neck according to the presented fully coupled model presented, and that the strain energy stored within the contact area will accelerate the sintering process.

Evaluation of Young’s modulus of construction materials by instrumented indentation using a ball indenter

Methods for evaluation of Young’s modulus (Em) of structural materials by instrumented indentation using ball indenter have been considered. All these techniques are based on the solution of elastic contact problems performed by H. Hertz. It has been shown that registration of the initial elastic region in the «load – displacement» indentation diagram provides the Em determination for metals and alloys. However, it is necessary to evaluate accurately the elastic compliance of a device, to use an indenter with a large radius R, and ensure a high surface quality of the test material in advance. Methods for Em determation, when indentation diagrams are recorded in the elastoplastic indentation region, should include the effect of plastic deformation on the elastic displacement calculated by H. Hertz expression. However, it appeared essential to determine the relation between the elastic αel and plastic h components of the total elastoplastic displacement α and the elastic displacement α0 estimated by H. Hertz expression for a definite indentation load. A close correlation between α0 and αel is revealed for steels, aluminum, magnesium, and titanium alloys when using indenters with a radius of R = 0.2 – 5 mm (diameter D = 0.4 – 10 mm) and maximum indentation load Fmax = 47 – 29430 N (4.8 – 3000 kgf). It is also shown that a gradual decrease in Em is observed with an increase in R(D) at the same degree of loading F/D2 for the same material. This fact was explained by the scale factor effect.

A Numerical Study on Plane Contact Problem Involving Inclusions

Hertz’s contact theory is widely used in elastic solid contact problems. In this paper, the applicability of Hertz’s contact theory in conformal contact problems was examined. Taking deep groove ball bearing and the outer raceway model as an example, both the FEM analyses and the theoretical studies were conducted. It is found that when the radius coefficient of groove curvature is greater than 0.55, Hertz’s theory can be practically applied for conformal contact. Contact problems of rolling bearings are generally solved by Hertz’s contact theory without considering inclusions and inhomogeneities in the material. In this paper, inclusions are presented in the contact analyses, and the corresponding elastic field distribution of the contact-inclusion model is investigated by using the finite element method.