Signorini Condition Research Articles

From an adhesive to a brittle delamination model in thermo-visco-elasticity

We address the analysis of a model for brittle delamination of two visco-elastic bodies, bonded along a prescribed surface. The model also encompasses thermal effects in the bulk. The related PDE system for the displacements, the absolute temperature, and the delamination variable has a highly nonlinear character. On the contact surface, it features frictionless Signorini conditions and a nonconvex, brittle constraint acting as a transmission condition for the displacements. We prove the existence of (weak/energetic) solutions to the associated Cauchy problem, by approximating it in two steps with suitably regularized problems. We perform the two consecutive passages to the limit via refined variational convergence techniques.

Analysis of dynamic nonlinear thermoviscoelastic beam problems

In this paper, we study a dynamic contact problem between a nonlinear viscoelastic beam and two rigid obstacles. Thermal effects are also taken into account and the contact is modeled using the classical Signorini condition. The existence of solutions is proved by considering approximate problems from a penalization procedure, proving some a priori estimates and passing to the limit. An exponential decay property is also obtained. Then, fully discrete approximations of the approximate problem are provided using the finite element method for the spatial approximation and the implicit Euler scheme for the discretization of the time derivatives. A stability property is obtained and a priori error estimates are proved from which, under some additional regularity conditions, the linear convergence of the algorithm is deduced. Fully discrete approximations of the Signorini problem are then introduced proceeding in a similar way, for which a stability property is obtained. Finally, some numerical simulations are performed to demonstrate the accuracy of the approximation and the behavior of the solution.

A contact problem for electro‐elastic materials

AbstractWe analyze the frictionless unilateral contact between an electro‐elastic body and a rigid electrically conductive foundation. On the potential contact zone, we use the Signorini condition with non‐zero gap and an electric contact condition with a conductivity depending on the Cauchy vector. We provide a weak variationally consistent formulation and show existence, uniqueness and stability of the solution. Our analysis is based on fixed point techniques for weakly sequentially continuous maps. We conclude by a numerical example that illustrates the applicability of the model.

Analysis of a nonlinear beam in contact with a foundation

The paper investigates the contact between a nonlinear dynamic Gao beam and a rigid or reactive foundation. The contact is modeled with the normal compliance condition for the deformable foundation and with the Signorini condition for the rigid foundation. The existence and uniqueness of the weak solution for the problem with normal compliance are obtained. The solution of the Signorini condition for the rigid foundation is obtained by passing to the limit when the normal compliance approaches infinity.

Rapid Contact on a Prestressed Highly Elastic Half-Space: The Sliding Ellipsoid and Rolling Sphere

A neo-Hookean half-space, in equilibrium under uniform Cauchy stress, undergoes contact by a sliding rigid ellipsoid or a rolling rigid sphere. Sliding is resisted by friction, and sliding or rolling speed is subcritical. It is assumed that a dynamic steady state is achieved and that deformation induced by contact is infinitesimal. Transform methods, modified by introduction of quasi-polar coordinates, are used to obtain classical singular integral equations for this deformation. Assumptions of specific contact zone shape are not required. Signorini conditions and the requirement that resultant compressive load is stationary with respect to contact zone stress give an equation for any contact zone span in terms of a reference value and an algebraic formula for the latter. Calculations show that prestress can significantly alter the ratio of spans parallel and normal to the direction of die travel, an effect enhanced by increasing die speed.

Homogenization of the parabolic Signorini boundary-value problem in a thick junction of type 3:2:1

We consider a parabolic Signorini boundary-value problem in a thick junction $\Omega_{\varepsilon}$ which is the union of a domain $\Omega_0$ and a large number of $\varepsilon$-periodically situated thin cylinders. The Signorini conditions are given on the lateral surfaces of the cylinders. The asymptotic analysis of this problem is done as $\varepsilon\to0,$ i.e., when the number of the thin cylinders infinitely increases and their thickness tends to zero. With the help of the integral identity method we prove a convergence theorem and show that the Signorini conditions are transformed (as $\varepsilon\to0)$ in differential inequalities in the region that is filled up by the thin cylinders.

Analysis of a unilateral contact problem taking into account adhesion and friction

In this paper, we investigate a contact problem between a viscoelastic body and a rigid foundation, when both the effects of the (irreversible) adhesion and of the friction are taken into account. We describe the adhesion phenomenon in terms of a damage surface parameter according to Frémondʼs theory, and we model unilateral contact by Signorini conditions, and friction by a nonlocal Coulomb law. All the constraints on the internal variables as well as the contact and the friction conditions are rendered by means of subdifferential operators, whence the highly nonlinear character of the resulting PDE system. Our main result states the existence of a global-in-time solution (to a suitable variational formulation) of the related Cauchy problem. It is proved by an approximation procedure combined with time discretization.

Homogenization of a parabolic signorini boundary value problem in a thick plane junction

We consider a parabolic Signorini boundary value problem in a thick plane junction Ω ε which is the union of a domain Ω0 and a large number of ε−periodically situated thin rods. The Signorini conditions are given on the vertical sides of the thin rods. The asymptotic analysis of this problem is done as ε → 0, i.e., when the number of the thin rods infinitely increases and their thickness tends to zero. With the help of the integral identity method we prove a convergence theorem and show that the Signorini conditions are transformed (as ε → 0) in differential inequalities in the region that is filled up by the thin rods in the limit passage. Bibliography: 31 titles. Illustrations: 1 figure.

Location of Bifurcation Points for a Reaction-Diffusion System with Neumann-Signorini Conditions

Abstract We consider a reaction-diffusion system of activator-inhibitor or substrate-depletion type in one space dimension which is subject to diffusion-driven instability. We determine the change of bifurcation when a pure Neumann condition is supplemented with a Signorini condition. We show that this change differs essentially from the known case when also Dirichlet conditions are assumed.

A new parabolic variational inequality formulation of Signorini's condition for non‐steady seepage problems with complex seepage control systems

AbstractBy extending Darcy's law to the dry domain above the free surface and specifying the boundary condition on the potential seepage surfaces as Signorini's type, a partial differential equation (PDE) defined in the entire domain of interest is formulated for non‐steady seepage flow problems with free surfaces. A new parabolic variational inequality (PVI) formulation equivalent to the PDE formulation is then proposed, in which the flux part of the complementary condition of Signorini's type in the PDE formulation is transformed into natural boundary condition. Consequently, the singularity at the seepage points is eliminated and the difficulty in selecting the trial functions is significantly reduced. By introducing an adaptive penalized Heaviside function in the finite element analysis, the numerical stability of the discrete PVI formulation is well guaranteed. The proposed approach is validated by the existing laboratory tests with sudden rise and dropdown of water heads, and then applied to capture the non‐steady seepage flow behaviors in a homogeneous rectangular dam with five drainage tunnels during a linear dropdown of upstream water head. The non‐steady seepage flow in the surrounding rocks of the underground powerhouse in the Shuibuya Hydropower Project is further modeled, in which a complex seepage control system is involved. Comparisons with the in situ monitoring data show that the calculation results well illustrate the non‐steady seepage flow process during impounding and the operation of the reservoir as well as the seepage control effects of the drainage hole arrays and drainage tunnels. Copyright © 2010 John Wiley & Sons, Ltd.

History-dependent quasi-variational inequalities arising in contact mechanics

We consider a class of quasi-variational inequalities arising in a large number of mathematical models, which describe quasi-static processes of contact between a deformable body and an obstacle, the so-called foundation. The novelty lies in the special structure of these inequalities that involve a history-dependent term as well as in the fact that the inequalities are formulated on the unbounded interval of time [0, +∞). We prove an existence and uniqueness result of the solution, then we complete it with a regularity result. The proofs are based on arguments of monotonicity and convexity, combined with a fixed point result obtained in [22]. We also describe a number of quasi-static frictional contact problems in which we model the material's behaviour with an elastic or viscoelastic constitutive law. The contact is modelled with normal compliance, with normal damped response or with the Signorini condition, as well, associated to versions of Coulomb's law of dry friction or to the frictionless condition. We prove that all these models cast in the abstract setting of history-dependent quasi-variational inequalities, with a convenient choice of spaces and operators. Then, we apply the abstract results in order to prove the unique weak solvability of each contact problem.

Quasistatic Elastic Contact with Adhesion

The aim of this paper is the variational study of the contact with adhesion between an elastic material and a rigid foundation in the quasistatic process where the deformations are supposed to be small. The behavior of this material is modelled by a nonlinear elastic law and the contact is modelled with Signorini′s conditions and adhesion. The evolution of bonding field is described by a nonlinear differential equation. We derive a variational formulation of the mechanical problem, and we prove the existence and uniqueness of the weak solution using a theorem on variational inequalities, the theorem of Cauchy‐Lipschitz, a lemma of Gronwall, as well as the fixed point of Banach.

Some energy conservative schemes for vibro-impacts of a beam on rigid obstacles

Caused by the problem of unilateral contact during vibrations of satellite solar arrays, the aim of this paper is to better understand such a phenomenon. Therefore, it is studied here a simplified model composed by a beam moving between rigid obstacles. Our purpose is to describe and compare some families of fully discretized approximations and their properties, in the case of non-penetration Signorini's conditions. For this, starting from the works of Dumont and Paoli, we adapt to our beam model the singular dynamic method introduced by Renard. A particular emphasis is given in the use of a restitution coefficient in the impact law. Finally, various numerical results are presented and energy conservation capabilities of the schemes are investigated.

Homogenization of the Signorini boundary-value problem in a thick junction and boundary integral equations for the homogenized problem

We consider a mixed boundary-value problem for the Poisson equation in a thick junction Ωε which is the union of a domain Ω0 and a large number of ε—periodically situated thin cylinders. The non-uniform Signorini conditions are given on the lateral surfaces of the cylinders. The asymptotic analysis of this problem is done as ε→0, i.e. when the number of the thin cylinders infinitely increases and their thickness tends to zero. We prove a convergence theorem and show that the non-uniform Signorini boundary conditions are transformed in the limiting variational inequalities in the region that is filled up by the thin cylinders as ε→0. The convergence of the energy integrals is proved as well. The existence and uniqueness of the solution to this non-standard limit problem is established. This solution can be constructed by using a penalty formulation and successive iteration. For some subclass, these problems can be reduced to an obstacle problem in Ω0 and an appropriate postprocessing. The equations in Ω0 finally are also treated with boundary integral equations. Copyright © 2010 John Wiley & Sons, Ltd.

Consistency results on Newmark methods for dynamical contact problems

The paper considers the time integration of frictionless dynamical contact problems between viscoelastic bodies in the frame of the Signorini condition. Among the numerical integrators, interest focuses on the classical Newmark method, the improved energy dissipative version due to Kane et al., and the contact-stabilized Newmark method recently suggested by Deuflhard et al. In the absence of contact, any such variant is equivalent to the Stormer–Verlet scheme, which is well-known to have consistency order 2. In the presence of contact, however, the classical approach to discretization errors would not show consistency at all because of the discontinuity at the contact. Surprisingly, the question of consistency in the constrained situation has not been solved yet. The present paper fills this gap by means of a novel proof technique using specific norms based on earlier perturbation results due to the authors. The corresponding estimation of the local discretization error requires the bounded total variation of the solution. The results have consequences for the construction of an adaptive timestep control, which will be worked out subsequently in a forthcoming paper.

Controllability of a string submitted to unilateral constraint

This article studies the controllability property of a homogeneous linear string of length one, submitted to a time dependent obstacle (described by the function {ψ(t)}0⩽t⩽T) located below the extremity x=1. The Dirichlet control acts on the other extremity x=0. The string is modelled by the wave equation y″−yxx=0 in (t,x)∈(0,T)×(0,1), while the obstacle is represented by the Signorini's conditions y(t,1)⩾ψ(t), yx(t,1)⩾0, yx(t,1)(y(t,1)−ψ(t))=0 in (0,T). The characteristic method and a fixed point argument allow to reduce the problem to the analysis of the solutions at x=1. We prove that, for any T>2 and initial data (y0,y1)∈H1(0,1)×L2(0,1) with ψ(0)⩽y0(1), the system is null controllable with controls in H1(0,T). Two distinct approaches are used. We first introduce a penalized system in yϵ, transforming the Signorini's condition into the simpler one yϵ,x(t,1)=ϵ−1[yϵ(t,1)−ψ(t)]−, ϵ being a small positive parameter. We construct explicitly a family of controls of the penalized problem, uniformly bounded with respect to ϵ in H1(0,T). This enables us to pass to the limit and to obtain a control for the initial equation. A more direct approach, based on differential inequalities theory, leads to a similar positive conclusion. Numerical experiments complete the study.

63975 MODELING AND ANALYSIS OF CONTACT PHENOMENA IN MULTIBODY SYSTEMS USING A LINEAR COMPLEMENTARITY FORMULATION(Contact, Impact, and Friction)

The dynamic modeling and analysis of rigid multibody systems that experience contact-impact events is presented and discussed in this study. The methodology is based on the nonsmooth dynamics approach, in which the interaction of the colliding bodies is modeled with multiple frictional unilateral constraints. The generalized contact kinematics is formulated in terms of gap functions and normal and tangential relative velocities. The dynamics of rigid multibody systems are stated as an equality of measures, which are formulated at the velocity-impulse level. The equations of motion are complemented with constitutive laws for the normal and tangential directions. In the present work, the unilateral constraints are described by a set-valued force law of the type of Signorini's condition, while the frictional contacts are characterized by a set-valued force law of the type of Coulomb's law for dry friction. Then, the resulting contact-impact problem is formulated and solved as a linear complementarity problem, which is embedded in the Moreau's time-stepping method. This method is considered here mainly due to its simplicity and robustness. The effectiveness of the methodologies presented in this study is demonstrated throughout the dynamic simulation of a planar multibody system.

Analysis and Numerical Approximation of an Electro-elastic Frictional Contact Problem

We consider the problem of frictional contact between an piezoelectric body and a conductive foundation. The electro-elastic constitutive law is assumed to be nonlinear and the contact is modelled with the Signorini condition, nonlocal Coulomb friction law and a regularized electrical conductivity condition. The existence of a unique weak solution of the model is established. The finite elements approximation for the problem is presented, and error estimates on the solutions are derived

A Perturbation Result for Dynamical Contact Problems

This paper is intended to be a first step towards the continuous dependence of dynamical contact problems on the initial data as well as the uniqueness of a solution. Moreover, it provides the basis for a proof of the convergence of popular time integration schemes as the Newmark method. We study a frictionless dynamical contact problem between both linearly elastic and viscoelastic bodies which is formulated via the Signorini contact conditions. For viscoelastic materials fulfilling the Kelvin-Voigt constitutive law, we find a characterization of the class of problems which satisfy a perturbation result in a non-trivial mix of norms in function space. This characterization is given in the form of a stability condition on the contact stresses at the contact boundaries. Furthermore, we present perturbation results for two well-established approximations of the classical Signorini condition: The Signorini condition formulated in velocities and the model of normal compliance, both satisfying even a sharper version of our stability condition.

Modeling and analysis of the unilateral contact of a piezoelectric body with a conductive support

We consider a mathematical model which describes the quasistatic process of contact between a piezoelectric body and an electrically conductive support, the so-called foundation. We model the material's behavior with a nonlinear electro-viscoelastic constitutive law; the contact is frictionless and is described with the Signorini condition and a regularized electrical conductivity condition. We derive a variational formulation for the problem and then we prove the existence of a unique weak solution to the model. The proof is based on arguments of nonlinear equations with multivalued maximal monotone operators and fixed point. Then we introduce a fully discrete scheme, based on the finite element method to approximate the spatial variable and the backward Euler scheme to discretize the time derivatives. We treat the unilateral contact conditions by using an augmented Lagrangian approach. We implement this scheme in a numerical code then we present numerical simulations in the study of two-dimensional test problems, together with various comments and interpretations.