Frictionless Elastic Contact Research Articles

A convergence criterion for elliptic variational inequalities

We consider an elliptic variational inequality with unilateral constraints in a Hilbert space X which, under appropriate assumptions on the data, has a unique solution u. We formulate a convergence criterion to the solution u, i.e. we provide necessary and sufficient conditions on a sequence { u n } ⊂ X which guarantee the convergence u n → u in the space X. Then we illustrate the use of this criterion to recover well-known convergence results and well-posedness results in the sense of Tykhonov and Levitin–Polyak. We also provide two applications of our results, in the study of a heat transfer problem and an elastic frictionless contact problem, respectively.

A Self-Adaptive Alternating Direction Multiplier Method for Frictionless Elastic Contact Problems

A Self-Adaptive Alternating Direction Multiplier Method for Frictionless Elastic Contact Problems

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

An Optimization Design of Contact Interface Material Stiffness for Improving the Uniformity of the Contact Pressure

Abstract Material stiffness, a significant parameter of a contact interface, is investigated to improve the uniformity of the contact pressure. A contact interface material stiffness optimization design algorithm is developed based on the modified solid isotropic material with the penalization (SIMP) method. The uniformity of the contact pressure field is represented by its variance and is defined as the optimization objective. A node-to-node frictionless elastic contact theory is adopted to perform the contact analysis. The effectiveness of the interface material stiffness design for improving the uniformity of the contact surface is verified based on two contact cases. Because the relationship between the material stiffness and the hard-and-soft degree of a contact interface is always a positive correlation, the results in this paper could be extended so that the design of the contact interfaces’ hard-and-soft degree will improve the distributing uniformity of the contact surface.

A reduced basis method for parametrized variational inequalities applied to contact mechanics

SummaryWe investigate new developments of the reduced‐basis method for parametrized optimization problems with nonlinear constraints. We propose a reduced‐basis scheme in a saddle‐point form combined with the Empirical Interpolation Method to deal with the nonlinear constraint. In this setting, a primal reduced‐basis is needed for the primal solution and a dual one is needed for the Lagrange multipliers. We suggest to construct the latter using a cone‐projected greedy algorithm that conserves the non‐negativity of the dual basis vectors. The reduction strategy is applied to elastic frictionless contact problems including the possibility of using nonmatching meshes. The numerical examples confirm the efficiency of the reduction strategy.

A class of generalized mixed variational–hemivariational inequalities II: Applications

In this paper we continue the study on generalized mixed variational–hemivariational inequalities which has been recently carried out in Bai et al. (2019). There, several abstract results on existence, uniqueness, and stability of solution are established. To illustrate the applicability of these results, we present two applications. The first application is an elliptic system driven by mixed boundary conditions and a nonhomogeneous partial differential operator. The second one is a frictionless elastic contact model with a unilateral boundary condition and the constitutive law involving the generalized subgradient inclusion.

A robust algorithm for the contact of viscoelastic materials

Existing solutions for the contact problem involving viscoelastic materials often require numerical differentiation and integration, as well as resolution of transcendental equations, which can raise convergence issues. The algorithm advanced in this paper can tackle the contact behaviour of the viscoelastic materials without any convergence problems, for arbitrary contact geometry, arbitrary loading programs and complex constitutive models of linear viscoelasticity. An updated algorithm for the elastic frictionless contact, coupled with a semi-analytical method for the computation of viscoelastic displacement, is employed to solve the viscoelastic contact problem at a series of small time increments. The number of equations in the linear system resulting from the geometrical condition of deformation is set by the number of cells in the contact area, which is a priori unknown. A trial-and-error approach is implemented, resulting in a series of linear systems which are solved on evolving contact areas, until static equilibrium equations and complementarity conditions are fully satisfied for every cell in the computational domain. At any iteration, cells with negative pressure are excluded from the contact area, while cells with negative gap (i.e. cells where the contacting bodies are predicted to overlap) are reincluded. The solution is found when pressure is stabilized in relation to the imposed normal load. This robust algorithm is expected to solve a large variety of contact problems involving viscoelastic materials.

A numerical solution to the cattaneo-mindlin problem for viscoelastic materials

The problem of the frictional mechanical contact with slip and stick, also referred to as the Cattaneo-Mindlin problem, is an important topic in engineering, with applications in the modeling of particle-flow simulations or in the study of contact between rough surfaces. In the frame of Linear Theory of Elasticity, accurate description of the slip-stick contact can only be achieved numerically, due to mutual interaction between normal and shear contact tractions. Additional difficulties arise when considering a viscoelastic constitutive law, as the mechanical response of the contacting materials depends explicitly on time. To overcome this obstacle, an existing algorithm for the purely elastic slip-stick contact is coupled with a semi-analytical method for viscoelastic displacement computation. The main advantage of this approach is that the contact model can be divided in subunits having the same structure as that of the purely elastic frictionless contact model, for which a well-established solution is readily available. In each time step, the contact solver assesses the contact area, the pressure distribution, the stick area and the shear tractions that satisfy the contact compatibility conditions and the static force equilibrium in both normal and tangential directions. A temporal discretization of the simulation windows assures that the memory effect, specific to both viscoelasticity and friction as a path-dependent processes, is properly replicated.

A majorized Newton-CG augmented Lagrangian-based finite element method for 3D restoration of geological models

In this paper, we propose a majorized Newton-CG augmented Lagrangian-based finite element method for 3D elastic frictionless contact problems. In this scheme, we discretize the restoration problem via the finite element method and reformulate it to a constrained optimization problem. Then we apply the majorized Newton-CG augmented Lagrangian method to solve the optimization problem, which is very suitable for the ill-conditioned case. Numerical results demonstrate that the proposed method is a very efficient algorithm for various large-scale 3D restorations of geological models, especially for the restoration of geological models with complicated faults.

A linear elastic-brittle interface model: application for the onset and propagation of a fibre-matrix interface crack under biaxial transverse loads

A new linear elastic and perfectly brittle interface model for mixed mode is presented and analysed. In this model, the interface is represented by a continuous distribution of springs which simulates the presence of a thin elastic layer. The constitutive law for the continuous distribution of normal and tangential initially-linear-elastic springs takes into account possible frictionless elastic contact between adherents once a portion of the interface is broken. A perfectly brittle failure criterion is employed for the springs, which enables the study of crack onset and propagation. This interface failure criterion takes into account the variation of the interface fracture toughness with the fracture mode mixity. A unified way to represent several phenomenological both energy and stress based failure criteria is introduced. A proof relating the energy release rate and tractions at an interface point (not necessarily a crack tip point) is introduced for this interface model by adapting Irwin’s crack closure technique for the first time. The main advantages of the present interface model are its simplicity, robustness and computational efficiency, even in the presence of snap-back and snap-through instabilities, when the so-called sequentially linear (elastic) analysis is applied. This model is applied here in order to study crack onset and propagation at the fibre-matrix interface in a composite under tensile/compressive remote biaxial transverse loads. Firstly, this model is used to obtain analytical predictions about interface crack onset, while investigating a single fibre embedded in a matrix which is subjected to uniform remote transverse loads. Then, numerical results provided by a 2D boundary element analysis show that a fibre-matrix interface failure is initiated by the onset of a finite debond in the neighbourhood of the interface point where the failure criterion is first reached (under increasing proportional load); this debond further propagates along the interface in mixed mode or even, in some configurations, with the crack tip under compression. The analytical predictions of the debond onset position and associated critical load are used for several parametric studies of the influence of load biaxiality, fracture-mode sensitivity and brittleness number, and for checking the computational procedure implemented.

Frictionless elastic contact of crowned roller: Approximate analytical calculation of compliance and contact area

Formulas for Hertzian frictionless elastic contact are well-known and have been used for a long time. However, they are not directly applicable for a crowned roller, a cylindrical roller with symmetrically rounded corners, deflecting a half-space. For this punch shape, the approximate analytical method of Barber and Billings ( Int J Mech Sci 1990; 32(12): 991–997) was adapted to compute the load–deflection curve and the contact boundary avoiding calculation of pressure distribution. Comparisons were performed with finite element method (FEM) simulations, an adaptive Hertzian quadratic punch surface approximation (AHA) and other published results. Good agreement of analytical load–deflection curves with FEM results was noted but the AHA results deviated significantly from the former two. The FEM contact area resembled a dog-bone shape, the analytical a bow-tie shape and the AHA, by design, an ellipse. Though the AHA contact area was closer to the FEM result numerically, the contact boundary aspect ratio of the analytical model was closer to the FEM. The analytical results simulating conditions from other publications agreed quantitatively for the load–deflection curves and qualitatively for the contact boundary. The non-linear spring characteristic of the crowned roller was quantitatively evaluated for a diverse range of geometry.

On the applicability of the Boussinesq influence function in modelling the frictionless elastic contact between a rectangular indenter with rounded edges and a half-plane

The applicability of the Boussinesq influence function in modelling the frictionless elastic contact between a rectangular indenter with rounded edges and a half-plane is numerically explored. The potential of the asymptotic matching method combined with classical fracture mechanics results is investigated. Manageable design formulae for evaluating the maximum equivalent stress are analytically derived with the aid of the asymptotic matching method.

A COMPRESSIBLE LAGRANGIAN FRAMEWORK FOR MODELING THE FLUID–STRUCTURE INTERACTION IN THE UNDERWATER IMPLOSION OF AN ALUMINUM CYLINDER

We propose a fully Lagrangian monolithic system for the simulation of the underwater implosion of cylindrical aluminum containers. A variationally stabilized form of the Lagrangian shock hydrodynamics is exploited to deal with the ultrahigh compression shock waves that travel in both air and water domains. The aluminum cylinder, which separates the internal atmospheric-pressure air from the external high-pressure water, is modeled by a three-node rotation-free shell element. The cylinder undergoes fast transient deformations, large enough to produce self-contact along it. A novel elastic frictionless contact model is used to detect contact and compute the non-penetrating forces in the discretized domain between the mid-planes of the shell. Mesh quality in the vicinity of the cylinder is guaranteed by regenerating the mesh in the air and water domains when large displacements occur. A monolithic fluid–structure interaction (FSI) system is then solved. Two schemes are tested, implicit using the predictor/multi-corrector Bossak scheme, and explicit, using the forward Euler scheme. The results of the two simulations are compared with experimental data.

Frictionless elastic contact analysis of a functionally graded vitreous enameled low carbon steel plate and a rigid spherical indenter

Frictionless elastic contact of a functionally graded vitreous enameled low carbon steel plate and a rigid spherical punch is studied in this paper. The graded modulus of elasticity of the plate is assumed to vary through the thickness and is found experimentally by means of nano-indentation technique and also estimated mathematically by a mathematical model originally proposed by Tamura, Tomota and Ozawa (TTO model). The contact problem of the plate and the rigid sphere is studied both analytically and numerically by means of ABAQUS finite element package. In the analytical approach, a higher order displacement field plate theory along with an exact Green’s function is used to find the contact parameters. The acquired results are then used to investigate the effect of material distribution, punch radius and plate span on the contact parameters. The results show that increasing the metal phase volume fraction of graded medium increases both the contact force and stress in the graded plate. In addition, in the vitreous enameled graded plates, if a polynomial function is fitted to the contact force–indentation relation, the power is equal to 2.0 regardless of the material distribution and plate geometry.

Effective macroscopic adhesive contact behavior induced by small surface roughness

In this paper we study a model contact problem involving adhesive elastic frictionless contact between rough surfaces. The problem's most notable feature is that it captures the phenomenon of depth-dependent-hysteresis (DDH) (e.g., see Kesari et al., 2010), which refers to the observation of different contact forces during the loading and unloading stages of a contact experiment. We specifically study contact between a rigid axi-symmetric punch and an elastic half-space. The roughness is represented as arbitrary periodic undulations in the punch's radial profile. These undulations induce multiple equilibrium contact regions between the bodies at each indentation-depth. Assuming that the system evolves so as to minimize its potential energy, we show that different equilibrium contact regions are selected during the loading and unloading stages at each indentation-depth, giving rise to DDH. When the period and amplitude of our model's roughness is reduced, we show that the evolution of the contact force and radius with the indentation-depth can be approximated with simpler curves, the effective macroscopic behavior, which we compute. Remarkably, the effective behavior depends solely on the amplitude and period of the model's roughness. The effective behavior is useful for estimating material properties from contact experiments displaying DDH. We show one such example here. Using the effective behavior for a particular roughness model (sinusoidal) we analyze the energy loss during a loading/unloading cycle, finding that roughness can toughen the interface. We also estimate the energy barriers between the different equilibrium contact regions at a fixed indentation-depth, which can be used to assess the importance of ambient energy fluctuations on DDH.

A multiscale finite-element method for solving rough-surface elastic-contact problems

A novel multiscale finite-element method to investigate the elastic contact of two-dimensional rough surfaces is presented. The aim of the method is to find the microscopic curve that describes the deformed shape of a solid with a smooth boundary surface in frictionless contact with a rigid rough surface. In addition, the real contact area is studied through the surface deformation. The contact traction on the contact surface and the maximum shear stress around the contact region are analyzed. This method is based on the variational inequality approach for solving the elastic frictionless contact problem. The strategy is to separate a small slice within the contact region and solve it as an independent system. Then the contact traction is obtained through iterations between the solution of the independent small slice and the solution of the total solid body. We observe a much higher pressure than the result of Hertz theory around the asperities. The main conclusions are (1)~the actual contact area and surface traction are dependent on the wavelength and amplitude of the surface roughness, (2) there is a much higher pressure around the asperities than predicted by Hertz theory, and (3) the location of maximum shear stress tends to be shifted toward the surface as compared with the case of smooth-surface contact. The method has the potential to be extended to solve three-dimensional rough contact problems.PACS Nos.: 03.40.D, 46.30.P, 62.20.P

Micromechanics of Crenulated Fibers

The stresses at the interface of fully bonded and fully disbonded crenulated carbon fibers in a carbon matrix are presented. Interface stresses for two loading cases, transverse tensile loading and a negative temperature change, are discussed. A crenulated fiber is one which has a wavy or scalloped outer radius, the amplitude and frequency of the waviness resulting from the manufacturing process. A square-packed array of fibers is assumed and the results are obtained using the finite-element method. For comparison to the crenulated fiber, the stresses for similarly loaded bonded and disbanded circular fibers are also presented. The level of disbond in each case is measured by the size of the radial gap between the fiber and the matrix. When fully disbanded, the fiber and matrix are assumed to interact only through frictionless elastic contact. Results from the fully disbonded case show that mechanical interference occurs between the fiber and the matrix. In general, the results of the study show that the stress levels present with the crenulated fiber are significantly higher than for the circular fiber. In particular, large stress concentations arise in the matrix tangential stress component at high crenulation amplitudes and high crenulation frequencies. Overall, the work described provides new insight into the complex interactions between fiber and matrix, specifically interactions that reflect on the reality of disbond, and innovations regarding fiber architecture.

A full numerical solution for the elastic contact of three-dimensional real rough surfaces

A full numerical method for the dry frictionless elastic contact of three-dimensional real rough surfaces is presented. The partial flexibility matrix store method, successive under-relaxation iteration method and contact domain contraction method are used to obtain the full numerical solution in the whole nominal contact domain. The pressure distribution, surface deformation, real contact area and the distribution of real contact area can be obtained. The calculated results of two contact problems of three-dimensional real rough surfaces are given. The area-load relationship obtained by this numerical method is compared with that obtained by experiment in which the total reflection technique and the image process technique are adopted. The effects of the interaction of asperities on the contact parameters are examined.

Boundary variational formulations and numerical solution techniques for unilateral contact problems

In this paper the numerical solution of the elastic frictionless contact problem is obtained by means of boundary discretization techniques. Variational formulations in terms of boundary tractions are given in presence of both bilateral and unilateral constraints. The discretization of the boundary functional is examined from the point of view of the theory of approximation and it is proved that the coerciveness (but not the symmetry) of the continuum problem is preserved when standard B.E.Ms are employed. As a consequence, the contact problem can be cast as a L.C.P. having, as coefficient matrix, a generally non symmetric P matrix. A simple, but meaningful example is discussed in some detail.

Frictionless elastic contact problem for a curved rigid punch of arbitrary shape

A new method is proposed for the analysis of elastic contact problems for a curved punch of non-elliptic planform under the action of a normal force. The punch base is assumed to be a quadratic surface. The method is based on an integral representation for the reciprocal distance between two points obtained by the author earlier. Some general relationships are established between the applied force and the punch settlement. Specific formulae are derived for a punch whose planform has a shape of a polygon, a rectangle, a rhombus and a cross. An example of a finite rigid cylinder lying on its generator and pressed against an elastic half-space is considered in detail. The method allows to have singular stresses at the cylinder edges and zero stresses at the rest of the boundary of the contact domain. The last condition serves for defining the width of the domain of contact. All the formulae are checked against the solutions known in the literature, and a good accuracy is confirmed in a sufficiently wide range of the aspect ratio.