Unilateral Contact Conditions Research Articles

Error estimates of unilateral piezoelectric contact problem in a curved and smooth boundary domain

We study the linear finite element approximation of piezoelectric unilateral contact problem in a curved and smooth boundary domain. The unilateral contact conditions will be weakly imposed by the penalty method. We derive error estimates which depend on the penalty parameter and the mesh size. In fact, under regularity of the solution, we prove some convergence rate results.

Analysis and Approximation of Hemivariational Inequality for a Frictional Thermo-electro-visco-elastic Contact Problem with Damage

The aim of this paper is to investigate a contact problem involving thermo-electro-visco-elastic body with damage and a rigid foundation. The friction is modelled with a subgradient of a locally Lipschitz mapping, and the contact is described by the Signorini's unilateral contact condition. A parabolic differential inclusion for the damage function is used to include the damaging effect in the model. We establish the model's variational formulation using four systems of three hemivariational inequalities and a parabolic equation, we prove an existence and uniqueness result of this problem. The proof is based on a fixed point argument and a recent finding from hemivariational inequality theory. Finally, by employing the finite element approach, we investigate a fully discrete approximation of the model and we derive error estimates on the approximate solution.

NUMERICAL SIMULATION OF ELASTIC-PLASTIC DEFORMATION OF TORSION SHAFTS OF VEHICLE SPRING SYSTEMS, TAKING INTO ACCOUNT CONTACT INTERACTION

The paper develops a model of stress-strain state of torsion shafts. A variable formulation of the problem on the basis of the theory of variational inequalities was made. Numerical modeling of the stress-head-deformed state was made with the involvement of the finite element method. In the model of the stress-head-deformed state of torsion shafts, its generalized parametric description is integrated. It contains both varied significant design and technological indicators, in particular, the structural design of the torsion head, the galley transition and the diameter of the stem torsion shaft stem, and the parameters of the technological process of double winding: twist angles and features of the shaft material. It is established that the stresses that occur in the torsion shaft are mainly concentrated in the contact interaction zone of the spline shaft head with the spline coupling. The parametric dependencies of the level of equivalent stresses by Mises on the level of double windings and on the ratio of the head diameter and the stem of the torsion shaft stem have been determined. Recommendations have been developed regarding the improvement of strength characteristics of torsion shafts. Contact pressure distribution on the slots of the torsion bar head is also determined. Its characteristic features depend on the type of boundary conditions. If bilateral interaction is enabled on the boundary, the contact pressure may become negative. Furthermore, the contact pressure gets a sharp spike. This non-physical feature disappears if unilateral contact conditions are imposed. Therefore, this version of the boundary conditions is more adequate. It was used to compute the stress-strain state of a torsion bar, taking into account the plastic deformation of the material and the contact interaction with conjugate bodes. Keywords: torsional shaft; stress-strain state; finite element method; theory of variational inequalities; contact interaction

Nonsmooth modal analysis of a varying cross-sectional area bar in unilateral contact

Nonsmooth modes of vibration allow for identification of resonant behaviours and attendant vibratory frequencies in structures prone to unilateral contact conditions on the boundary. The prominent approach for finding nonsmooth modes of vibration entails finding continua of periodic solutions to the system in question. In this paper, nonsmooth modes of a one-dimensional bar of varying cross-sectional area prone to unilateral contact with a rigid obstacle are determined. While numerical and analytical techniques were previously proposed, they were limited to constant cross section bar and could not be applied on the varying area bar for which the classical d’Alembert solution no longer exists. In this article, nonsmooth modal analysis of the varying area bar is performed via a novel treatment of the Signorini conditions within the finite element framework: the nodal boundary method. The nodal boundary method solves the Signorini problem by switching between two sets of shape functions describing either (1) inactive contact motion (motion away from the rigid obstacle) or (2) active contact motion (bar in contact with the rigid obstacle). In the proposed nodal boundary method, the motion of the contacting node does not participate in the resulting governing Ordinary Differential Equation (ODE). Instead, its motion is prescribed by the boundary conditions and is dictated by the motion of internal nodes. The nodal boundary method results in a discontinuous ODE in the internal nodes which can be solved both analytically and via numerical techniques. Solutions obtained by the nodal boundary method exhibit several advantages over existing numerical techniques: no chattering at contact, no penetration of the rigid obstacle, and existence of periodic solutions. Specifically, these periodic solutions are readily detectable via the shooting method with sequential continuation. The nodal boundary method is used successfully for the nonsmooth modal analysis of different models of the varying area bar. Besides, application of the nodal boundary method for nonsmooth modal analysis of the uniform area cantilever bar in Dirichlet or Robin boundary conditions is also demonstrated for comparison with existing literature.

On Harmonic Balance Method-based Lagrangian contact formulations for vibro-impact problems

This article assesses two Harmonic Balance Method-based numerical Lagrangian strategies for the characterization of the periodic response of a mechanical system subject to unilateral contact constraints. The model used for evaluation is an academic rod model facing an obstacle that is firstly considered as equivalent to a stiffness (flexible) and then as a rigid obstacle. Nonlinear frequency response curves are obtained through an arc-length continuation procedure and confronted to the corresponding time integration simulations. An in-depth comparative analysis is conducted on the time signals obtained through both frequency domain strategies: (1) linear complementarity problem (LCP-HBM) and (2) dynamic Lagrangian frequency time (DLFT-HBM). The unilateral contact conditions are thoroughly investigated in the time domain and compared with the reference time integration simulations. Particular attention is paid to the relation between both methods and it is shown that the DLFT-HBM is asymptotically equivalent to the LCP-HBM. This study also shows that the oversampling of time signals inside alternating frequency–time procedures generates non-physical high harmonics. Finally, a convergence analysis is conducted in order to assess the influence of the different numerical parameters on the respect of the unilateral contact laws.

D’Alembert function for exact non-smooth modal analysis of the bar in unilateral contact

Non-smooth modal analysis is an extension of modal analysis to non-smooth systems, prone to unilateral contact conditions for instance. The problem of a one-dimensional bar subject to unilateral contact on its boundary has been previously investigated numerically and the corresponding spectrum of vibration could be partially explored. In the present work, the non-smooth modal analysis of the above system is reformulated as a set of functional equations through the use of both d’Alembert solution to the wave equation and the method of steps for Neutral Delay Differential Equations. The system features a strong internal resonance condition and it is established that irrational and rational periods of vibration should be carefully distinguished. For irrational periods, it was previously proven that the displacement field of the non-smooth modes of vibration is characterized with piecewise-linear functions in space and time and such a motion is unique for a prescribed energy. However, for rational periods, which are the subject of this work, new periodic solutions are found analytically. Findings consist of families of iso-periodic solutions with piecewise-smooth displacement fields in space and time and continua of piecewise-smooth periodic solutions of the same energy and frequency.

Boundary Value Problems in Thermo Viscoplasticity

In this work, we study two uncoupled quasistatic problems for thermo viscoplastic materials. In the model of the equation of generalised thermo viscoplasticity, both the elastic and the plastic rate of deformation depend on a parameter $theta $ which may be interpreted as the absolute temperature. The boundary conditions considered here as displacement-traction conditions as well as unilateral contact conditions. We establish a variational formulation for the model and we prove the existence of a unique weak solution to the problem, reducing the isotherm problem to an ordinary differential equation in a Hilbert space.

First Return Time to the contact hyperplane for $ N $-degree-of-freedom vibro-impact systems

<p style='text-indent:20px;'>The paper deals with the dynamics of conservative <inline-formula><tex-math id="M2">\begin{document}$ N $\end{document}</tex-math></inline-formula>-degree-of-freedom vibro-impact systems involving one unilateral contact condition and a linear free flow. Among all possible trajectories, grazing orbits exhibit a contact occurrence with vanishing incoming velocity which generates mathematical difficulties. Such problems are commonly tackled through the definition of a Poincaré section and the attendant First Return Map. It is known that the First Return Time to the Poincaré section features a square-root singularity near grazing. In this work, a non-orthodox yet natural and intrinsic Poincaré section is chosen to revisit the square-root singularity. It is based on the unilateral condition but is not transverse to the grazing orbits. A detailed investigation of the proposed Poincaré section is provided. Higher-order singularities in the First Return Time are exhibited. Also, activation coefficients of the square-root singularity for the First Return Map are defined. For the linear and periodic grazing orbits from which bifurcate nonlinear modes, one of these coefficients is necessarily non-vanishing. The present work is a step towards the stability analysis of grazing orbits, which still stands as an open problem.</p>

Two-scale thermomechanical damage model for dynamic shear failure in brittle solids

A coupled thermomechanical damage approach for dynamic shear failure in brittle solids is proposed in the present contribution. The model is constructed by asymptotic homogenization from microstructures with dynamically evolving microcracks, in mode II, with unilateral contact and friction conditions on their lips. Crack-tip and frictional heating effects assumed at the small scale give rise to distributed heat sources in the macroscopic temperature equation and specific dissipation terms in the upscaled damage law. The analysis of the effective thermomechanical response of the model reveals strain rate and size effects and the influence of friction and growth of microcracks on the macroscopic thermal evolutions.

Implementation of a new strain split to model unilateral contact within the phase field method

SummaryA new orthogonal split of strain tensor into compressive and tensile parts is implemented within the phase field model to mimic unilateral contact condition with which any existing cracks and any crack propagation have to comply. The resulting phase field model offers several advantages as compared to other available schemes. First, it involves rigorous orthogonality between traction and compression parts. Second, it yields remarkably simple, new analytical expressions of the projectors which provide computational saving during the crack propagation simulation. Finally, it can be applied to arbitrary initial elastic anisotropic media, which is not the case of other available strain tensor split operators. A detailed comparison of the fracture responses predicted by the present model and other approaches is provided. It is shown that the present orthogonal decomposition is able to accurately predict experimental results and removes spurious effects found in other schemes for specific loads like compression.

Boundary Element and Augmented Lagrangian Methods for Contact Problem with Coulomb Friction

In this paper, boundary element and augmented Lagrangian methods for Coulomb friction contact problems are presented. Based on the projection technique, both unilateral contact and Coulomb friction conditions are reformulated as fixed point problems. The original problem is deduced to a variational formulation with boundary integral operators. Then, we propose a new augmented Lagrangian method which can be dealt with the semismooth Newton method. Short theoretical results and the algorithm description are given. Numerical simulations show the performance of the method proposed.

Dynamic shear damage with frictional sliding on microcracks

A new damage model for rapid shear failure in brittle solids is constructed in the present contribution using a two-scale approach. The model is obtained by asymptotic homogenization from microstructures with dynamically evolving microcracks, in mode II, with unilateral contact and friction conditions on their lips. The analysis of the local effective response of the model reveals strain-rate and microstructural size effects and the influence of the small-scale friction. Results of numerical simulations for mode II brittle failure of PMMA specimens under impact loading are presented and compared with the experimental data. The failure evolution is found to be in good agreement with that reported in the experiments.

Non-localised contact between beams withcircular and elliptical cross-sections

The key novelty of this contribution is a dedicated technique toefficiently determine the distance (gap) function between parallel or almost parallelbeams with circular and elliptical cross-sections. The technique consists ofparametrizing the surfaces of the two beams in contact, fixing a point on the centroidline of one of the beams and searching for a constrained minimum distance between thesurfaces (two variants are investigated). The resulting unilateral (frictionless)contact condition is then enforced with the Penalty method, which introduces complianceto the, otherwise rigid, beams’ cross-sections. Two contact integration schemes areconsidered: the conventional slave-master approach (which is biased as the contactvirtual work is only integrated over the slave surface) and the so-called two-half-passapproach (which is unbiased as the contact virtual work is integrated over the twocontacting surfaces). Details of the finite element formulation, which is suitablyimplemented using Automatic Differentiation techniques, are presented. A set ofnumerical experiments shows the overall performance of the framework and allows aquantitative comparison of the investigated variants.

A dynamic viscoelastic problem with friction and rate-depending contact interactions

<p style='text-indent:20px;'>The aim of this work is to study a dynamic problem that constitutes a unified approach to describe some rate-depending interactions between the boundaries of two viscoelastic bodies, including relaxed unilateral contact, pointwise friction or adhesion conditions. The classical formulation of the problem is presented and two variational formulations are given as three and four-field evolution implicit equations. Based on some approximation results and an equivalent fixed point problem for a multivalued function, we prove the existence of solutions to these variational evolution problems.

Phase field simulation of early-age fracture in cement-based materials

In this paper, a new computational approach, which is based on a multi-physics coupling of the chemo-thermo-mechanical process, is developed for modeling failure mechanism in cement-based materials at early-age hydration. A new constitutive law describing mechanical behavior of young concrete is thus derived, taking into account the effects of several factors including thermal expansion, shrinkage, damage, and both basic and transient thermal creeps. A strongly coupled model describing the interaction between fracture and multi-physics problems of cement hydration is obtained. The present phase field model in terms of smeared crack approach is highly suitable for simulating crack, especially when considering the unilateral contact conditions at crack surfaces. The capability of the proposed model in modeling complex crack initiation and propagation under arbitrary boundary conditions is highlighted. Fracture phenomena at both macroscopic and mesoscopic scales are considered and analyzed. The effects of creep and microstructural heterogeneity in different length scale are investigated, showing the great potential of the developed approach.

Alternating direction method of multiplier for the unilateral contact problem with an automatic penalty parameter selection

We propose an alternating direction method of multiplier (ADMM) for the unilateral (frictionless) contact problem with an optimal parameter selection. We first introduce an auxiliary unknown to seprate the linear elasticity subproblem from the unilateral contact condition. Then an alternating direction is applied to the corresponding augmented Lagrangian. By eliminating the primal and auxiliary unknowns, at the discrete level, we derive a pure dual algorithm, starting point for the convergence analysis and the optimal parameter approximation. Numerical experiments are proposed to illustrate the efficiency of the proposed (optimal) penalty parameter selection method.

Phase field modeling of interfacial damage in heterogeneous media with stiff and soft interphases

A new interfacial cracking model in the phase field framework is proposed. The developed method is able to capture the effects of both stiff and soft interphases on the fracture behavior of heterogeneous materials. A dimensional-reduced model based on a rigorous asymptotic analysis is adapted to derive the zero thickness imperfect interface models from an original configuration containing thin interphase. Then, the energetic approach is used to describe the material degradation both on the interface and in bulk within the context of the phase field method for fracture. This technique allows to effectively model the competition between the interface and bulk cracking. Furthermore, a unilateral contact condition is also proposed to ensure the physical crack propagation patterns in the case of spring imperfect interface. The complex cracking phenomena on interfaces such as initiation, delamination, coalescence, deflection are successfully predicted by the present method. The numerical implementation using a staggered algorithm provides an extremely robust approach. The performance of the proposed framework is demonstrated through numerical examples involving complex microcracking of both stiff and soft interfaces in complex microstructures.

Exploiting Lagrange duality for topology optimizationwith frictionless unilateral contact

This paper presents tractable reformulations of topology optimization problems of structures subject to frictionless unilateral contact conditions. Specifically, weconsider stiffness maximization problems of trusses and continua. Based on the Lagrange duality theory, we derive formulations that do not involve complementarity constraints. It is often that a structural optimization problem with contact conditions is formulated as a mathematical programming problem with complementarityconstraints (MPCC problem). However, MPCC usually requires special treatment for numerical solution, because it does not satisfy standard constraint qualifications. In contrast, to the formulation presented in this paper, we can apply standard optimization approaches. Numerical experiments on trusses and continua are performed to examine efficiency of the proposed approach.

Optimal convergence analysis of the unilateral contact problem with and without Tresca friction conditions by the penalty method

We study the linear finite element approximation of the elasticity equations with and without unilateral friction contact (of Tresca type) conditions in a polygonal or polyhedral domain. The unilateral contact condition is weakly imposed by the penalty method. We derive error estimates which depend on the penalty parameter ε and the mesh size h . In fact, under the H 3 2 + ν ( Ω ) , 0 < ν ≤ 1 2 , regularity of the solution of the contact problems (with and without friction) and with the requirement ε > h , we prove a convergence rate of O ( h 1 2 + ν + ε 1 2 + ν ) in the energy norm. Therefore, if the penalty parameter is taken as ε : = c h θ where 0 < θ ≤ 1 , the convergence rate of O ( h θ ( 1 2 + ν ) ) is obtained. In particular, we obtain an optimal linear convergence when ε behaves like h (i.e. θ = 1 ) and ν = 1 2 .

Derivation of imperfect interface models coupling damage and temperature

In this paper we introduce a model describing a layered structure composed by two thermoelastic adherents and a thin adhesive subject to a degradation process. By an asymptotic expansion method, we derive a model of imperfect interface coupling damage and temperature evolution. Moreover, assuming that the behaviour of the adhesive is ruled by two different regimes, one in traction and one in compression, we derive a second limit model where unilateral contact conditions on the interface are also included.