Signorini Condition Research Articles

IDENTIFICATION OF POSITIONS OF SEPARATED RIGID INCLUSIONS BY BOUNDARY MEASUREMENTS FOR A SIGNORINI PROBLEM

A variational inequality describing an elastic body with a finite set of rigid inclusions is considered. The Signorini condition is imposed on a part of the boundary of the body. On the other part a homogeneous Dirichlet boundary condition is specified. The inclusions are arranged such that distance between any two inclusions is not less than a given positive number. All inclusions are located at a nonzero distance from the outer boundary. An inverse problem is investigated, which consists in identification of positions of the rigid inclusions from the measurement of displacements on an observation boundary. Continuous dependency of the solution of the forward problem on parameters of inclusions’ location and rotation is established. This provides existence of a solution for the inverse identification problem.

Numerical analysis of a history-dependent mixed hemivariational-variational inequality in contact problems

This paper is devoted to a study of a history-dependent mixed hemivariational-variational inequality arising in contact mechanics. The contact problem concerns the deformation of a viscoelastic body with long memory, subject to a general friction law on one part of the boundary and a frictionless Signorini condition on another part of the boundary. The solution existence and uniqueness of the history-dependent mixed hemivariational-variational inequality based on a Lagrange multiplier approach are proved. Then, a fully discrete scheme is introduced and studied. The trapezoidal rule is used to approximate the integral in the history-dependent operator. For the spatial discretization, the linear finite elements are used to discretize the displacement field, and dual basis functions are used in the approximation of the Lagrange multiplier. Optimal order error estimates are derived for the displacement and the Lagrange multiplier under appropriate solution regularity assumptions. Numerical results are presented to illustrate the theoretical prediction of the convergence orders.

Analysis and Numerical Simulation of Time-Fractional Derivative Contact Problem with Friction in Thermo-Viscoelasticity

Abstract The objective of this study is to analyze a quasistatic frictional contact problem involving the interaction between a thermo-viscoelastic body and a thermally conductive foundation. The constitutive relation in our investigation is constructed using a fractional Kelvin–Voigt model to describe displacement behavior. Additionally, the heat conduction aspect is governed by a time-fractional derivative parameter that is associated with temperature. The contact is modeled using the Signorini condition, which is a version of Coulomb’s law for dry friction. We develop a variational formulation for the problem and establish the existence of its weak solution using a combination of techniques, including the theory of monotone operators, Caputo derivative, Galerkin method, and the Banach fixed point theorem. To demonstrate the effectiveness of our approach, we include several numerical simulations that showcase the performance of the method.

Strange Operator in Homogenization of the Diffusion Equation in a Domain Perforated Along of a Manifold with Dynamic Signorini Condition on Perforation Boundary. Critical Case

Strange Operator in Homogenization of the Diffusion Equation in a Domain Perforated Along of a Manifold with Dynamic Signorini Condition on Perforation Boundary. Critical Case

Variational and Numerical Analysis for an Electro-Viscoelastic Unilateral Contact Problem with Adhesion

In this paper we investigate a mathematical model for the unilateral contact between an electro-viscoelastic body and a conductive foundation. The Signorini conditions are used to model the contact, and adhesion between the contact surfaces is also taken into account. We establish the existence and uniqueness of a weak solution to this problem. Furthermore, we propose a fully discrete numerical scheme to approximate the solution, and we present and prove the main result concerning the estimation of errors.

Variational approach for nonsmooth elasto-plastic dynamics with contact and impacts

The objective of this article is the modeling and the numerical simulation of the response of elastoplastic structures to impacts. To this end, a numerical method is proposed that takes into account one-sided contact (Signorini condition) and impact phenomena together with plasticity in a monolithic solver, while accounting for the non-smooth character of the dynamics. The formulation of the plasticity and the contact laws are based on inclusions into normal cones of convex sets, or equivalently, variational inequalities following the pioneering work of Moreau (1974) and Halphen and Nguyen (1975), who introduced the assumptions of normal dissipation and of generalized standard materials (GSM) in the framework of associated plasticity with strain hardening. The proposed time–stepping method is an extension of the Jean and Moreau (1987) scheme for nonsmooth dynamics. The discrete energy balance shows that spurious numerical damping can be suppressed and that the scheme is in practice unconditionally stable. Furthermore, the finite-dimensional variational inequality at each time–step is well–posed and can be solved by optimization methods for convex quadratic programs, providing an interesting alternative to the return mapping algorithm coupled with a dedicated frictional contact method. The paper is completed by illustrative numerical examples of impacts on steel structures consisting of beams.

Existence of solution and uniform decay for a contact problem in laminated beam

In this paper, we study Signorini’s problem for a laminated Timoshenko beam with interfacial slip. We assume that the transverse displacements at the end of the beam are in the presence of two rigid obstacles (Signorini conditions). We prove the existence of solution and analyze the asymptotic behavior of the system. We use the hybrid-penalized method to show the global existence of at least one solution to Signorini’s problem and finally, numerical experiments verify the theoretical results.

Saturated‐unsaturated seepage simulation and hydrological dam models for quantifying the heterogeneity to hydrological state variables: Impacts of auxiliary structures and leakage anomalies

AbstractSeepage analysis is the central issue in solving the unconfined seepage problems from complex earth‐rock dams. The precision and efficiency of seepage simulation are the major technical barriers to exploring the heterogeneities of hydrological state variables in dams. This paper addresses the heterogeneity of the saturated‐unsaturated zones of dams and develops the smoothed finite element method (S‐FEM) capable of solving the seepage problem in domains with arbitrary geometry and continuously varied permeability. The gradient smoothing technique, in which the area integrals are transformed into the line integrals around edges of smoothing cells, is used to obtain the element matrices, which is used to simplify the solution of seepage problem. To be specific, we employ Van Genuchten method to modify the hydraulic conductivity of unsaturated zones medium and Signorini condition to convert the outflow boundary into the water head boundaries. The precision of the proposed method together with the importance of considering the unsaturated zones are first deliberated by the verification tests. Then, to quantify the impacts on the hydrological state variables, we focus on the heterogeneity of the entire zones through exploratory tests. The results show that the presence of leakage anomalies and auxiliary dam structures has a significant effect on the redistribution of hydrological state variables, which convinces that the findings may lay the foundation for the development of multi‐parameter seepage inversion and the derivation of coupled hydro‐geophysical inversion using the most appropriate starting model.

Model-based simulations of pulsed laser ablation using an embedded finite element method

A model of thermal ablation with application to multi-pulsed laser lithotripsy is presented. The approach is based on a one-sided Stefan–Signorini model for thermal ablation, and relies on a level-set function to represent the moving interface between the solid phase and a fictitious gas phase (representing the ablated material). The model is discretized with an embedded finite element method, wherein the interface geometry can be arbitrarily located relative to the background mesh. Nitsche’s method is adopted to impose the Signorini condition on the moving interface. A bound constraint is also imposed to deal with thermal shocks that can arise during representative simulations of pulsed ablation with high-power lasers. We report simulation results based on experiments for pulsed laser ablation of wet BegoStone samples treated in air, where Begostone has been used as a phantom material for kidney stone. The model is calibrated against experimental measurements by adjusting the percentage of incoming laser energy absorbed at the surface of the stone sample. Simulation results are then validated against experimental observations for the crater area, volume, and geometry as a function of laser pulse energy and duration. Our studies illustrate how the spreading of the laser beam from the laser fiber tip with concomitantly reduced incident laser irradiance on the damaged crater surface explains trends in both the experimental observations and the model-based simulation results.

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.

Analysis of a Frictionless Electro Viscoelastic Contact Problem with Signorini Conditions

This study considered a mathematical model to describe the process of a quasi-static contact between a piezoelectric body and an electrically conductive foundation. The behavior of the material was modeled with a nonlinear electro-viscoelastic constitutive law, the contact was frictionless, and the result was described with the Signorini condition. A variational formulation was derived for the problem, proving the existence and uniqueness of a weak solution of the model. The proof was based on arguments for nonlinear equations with maximal monotone operators.

A dynamic electroviscoelastic problem with thermal effects

We consider a mathematical model which describes the dynamic pro- cess of contact between a piezoelectric body and an electrically conductive foun- dation. We model the material's behavior with a nonlinear electro-viscoelastic constitutive law with thermal e ects. Contact is described with the Signorini condition, a version of Coulomb's law of dry friction. A variational formulation of the model is derived, and the existence of a unique weak solution is proved. The proofs are based on the classical result of nonlinear rst order evolution inequali- ties, the equations with monotone operators, and the xed point arguments.

Nonsmooth modal analysis via the boundary element method for one-dimensional bar systems

A numerical scheme grounded on the Boundary Element Method expressed in the Frequency Domain is proposed to perform Nonsmooth Modal Analysis of one-dimensional bar systems. The latter aims at finding continuous families of periodic orbits of mechanical components featuring unilateral contact constraints. The proposed formulation does not assume a semi-discretization in space of the governing Partial Differential Equations, as achieved in the Finite Element Method, and so mitigates a few associated numerical difficulties, such as chattering at the contact interface, or the questionable approximation of internal resonance conditions. The nonsmooth Signorini condition stemming from the unilateral contact constraint is enforced in a weighted residual sense via the Harmonic Balance Method. Periodic responses are investigated in the form of energy-frequency backbone curves along with the associated displacement fields. It is found that for the one-bar systems, the results compare well with existing works and the proposed methodology stands as a viable option in the field of interest. The two-bar system, for which no known results are reported in the literature, exhibits very rich nonsmooth modal dynamics with entangled nonsmooth modal motions combining hardening and softening effects via the intricate interaction of various, possibly subharmonic and internally resonant, nonsmooth modes of the two bars.

Modelling and simulation of a dynamic contact problem in thermo-piezoelectricity

In this work, we numerically study a dynamic frictional contact problem between a thermo-piezoelectric body and a conductive foundation. The linear thermo-electro-elastic constitutive law is employed to model the thermo-piezoelectric material. The contact is modelled by the Signorini condition and the friction by the Coulomb law. A frictional heat generation and heat transfer across the contact surface are assumed. The heat exchange coefficient is assumed to depend on contact pressure. Hybrid formulation is introduced, it is a coupled system for the displacement field, the electric potential, the temperature and two Lagrange multipliers. The discrete scheme of the coupled system is introduced based on a finite element method to approximate the spatial variable and an Euler scheme to discretize the time derivate. The thermo-mechanical contact is treated by using an augmented Lagrangian approach. A solution algorithm is discussed and implemented. Numerical simulation results are reported, illustrating the mechanical behavior related to the contact condition.

Hybrid high-order discretizations combined with Nitsche’s method for Dirichlet and Signorini boundary conditions

Abstract We present two primal methods to weakly discretize (linear) Dirichlet and (nonlinear) Signorini boundary conditions in elliptic model problems. Both methods support polyhedral meshes with nonmatching interfaces and are based on a combination of the hybrid high-order (HHO) method and Nitsche’s method. Since HHO methods involve both cell unknowns and face unknowns, this leads to different formulations of Nitsche’s consistency and penalty terms, either using the trace of the cell unknowns (cell version) or using directly the face unknowns (face version). The face version uses equal-order polynomials for cell and face unknowns, whereas the cell version uses cell unknowns of one order higher than the face unknowns. For Dirichlet conditions, optimal error estimates are established for both versions. For Signorini conditions, optimal error estimates are proven only for the cell version. Numerical experiments confirm the theoretical results and also reveal optimal convergence for the face version applied to Signorini conditions.

Vibrations of a beam between stops: Collision events and energy balance properties

<p style='text-indent:20px;'>We consider the model problem of an elastic beam vibrating between two stops. More precisely the beam is clamped at its left end while its right end may undergo contact and collision events with two stops. We model the interaction between the beam and the stops either with Signorini complementarity conditions when the stops are perfectly rigid or with a normal compliance contact law allowing some penetration within the stops and given by a linear relationship between the shear stress and the penetration at some positive power <inline-formula><tex-math id="M1">\begin{document}$ \beta $\end{document}</tex-math></inline-formula> when contact occurs. <p style='text-indent:20px;'>Motivated by computational issues we study the evolution of the energy functional defined as the sum of the kinetic energy and the potential energy of elastic deformation of the beam. When contact is modelled with a normal compliance law we prove an energy conservation property. Then we interpret the relationship between the shear stress and the penetration in case of contact as a penalization of the non-penetration condition. We show that the solutions of the penalized problems converge to a <i>strong solution</i> of the problem with Signorini conditions as defined in [<xref ref-type="bibr" rid="b26">26</xref>] and we prove that the limit satisfies an energy conservation property through instantaneaous collision events.

Numerical approximation of a frictional contact problem in elasto‐plasticity based on the penalty approach

AbstractA numerical model based on the penalty method for the unilateral contact problem with friction between an elasto‐plastic body and a rigid foundation is presented in this paper. The process is supposed to be static, the material's behavior is described by the nonlinear elastic constitutive Hencky's law, the contact and friction are modeled, respectively, with Signorini's condition and a version of Coulomb's law in which the coefficient of friction depends on the slip. Next, the existence of a unique weak solution to the penalized problem is proved, and its convergence towards that of the variational problem is confirmed. Then, the finite element method is a numerical method that can be successfully used to generate an approximate solution for a problem of this kind. Afterward, a successive iteration technique to solve the problem numerically is proposed, and a convergence result is established. Finally, to prove the reliability and the performance of this approach, numerical experiments of two‐dimensional test problems are carried out.

On the Signorini’s Contact Problem with Non-local Coulomb’s Friction in Thermo-Piezoelectricity

This paper deals with the quasi-static frictional contact problem between a thermo-piezoelectric body and a rigid foundation. The body’s material is modelled with a linear thermo–electro-elastic constitutive laws. The contact is described with a temperature depends Signorini conditions and a version of Coulomb’s friction law with slip dependent coefficient of friction. We derive a variational formulation for the model, it is a coupled system for the displacements, the electric potential and the temperature. We then prove, under some smallness assumptions of the friction coefficient, existence of weak solution to the problem. Finally, we study the continuous dependence of solutions on friction coefficient and on displacement and temperature initial data.

Analysis and Numerical Approximation of a Contact Problem Involving Nonlinear Hencky-Type Materials with Nonlocal Coulomb’s Friction Law

A static frictional contact problem between an elasto-plastic body and a rigid foundation is considered. The material’s behavior is described by the nonlinear elastic constitutive Hencky’s law. The contact is modeled with the Signorini condition and a version of Coulomb’s law in which the coefficient of friction depends on the slip. The existence of a weak solution is proved by using Schauder’s fixed-point theorem combined with arguments of abstract variational inequalities. Afterward, a successive iteration technique, based on the Kačanov method, to solve the problem numerically is proposed, and its convergence is established. Then, to improve the conditioning of the iterative problem, an appropriate Augmented Lagrangian formulation is used and that will lead us to Uzawa block relaxation method in every iteration. Finally, numerical experiments of two-dimensional test problems are carried out to illustrate the performance of the proposed algorithm.

Optimal location of a rigid inclusion in equilibrium problems for inhomogeneous two‐dimensional bodies with a crack

AbstractA two‐dimensional model describing equilibrium of a cracked inhomogeneous body with a rigid inclusion is studied. We assume that the Signorini condition, ensuring non‐penetration of the crack faces, is satisfied. For a family of corresponding variational problems, we analyze the dependence of their solutions on the location of the rigid inclusion. The existence of a solution of the optimal control problem is proven. For this problem, a cost functional is defined by an arbitrary continuous functional on a suitable Sobolev space, with the location parameter of the inclusion is chosen as a control parameter.