The weak solution of an antiplane contact problem for electro-viscoelastic materials with long-term memory

We study a mathematical model which describes the antiplane shear deformation of a cylinder in frictionless contact with a rigid foundation. The material is assumed to be electro-viscoelastic with long-term memory, and the friction is modeled with Tresca’s law and the foundation is assumed to be electrically conductive. First we derive the classical variational formulation of the model which is given by a system coupling an evolutionary variational equality for the displacement field with a time-dependent variational equation for the potential field. Then we prove the existence of a unique weak solution to the model. Moreover, the proof is based on arguments of evolution equations and on the Banach fixedpoint theorem.     

Quasistatic Thermoviscoelastic Frictional Contact Problem with Damped Response

We analyze a problem which describes the frictional contact between a thermoviscoelastic body and a rigid foundation. The process is assumed to be quasistatic and the contact is modeled by a general normal damped response condition with friction law and heat exchange. Then we present a variational formulation of the problem, which is set in an abstract form as a system of evolution equations for the displacements and temperature. We establish the existence and uniqueness of the weak solution, using general results on evolution equations with monotone operators and fixed point arguments. Finally, we study the continuous dependence of the solution with respect to the initial data and contact conditions.     

A frictional contact problem with wear and damage for electro-viscoelastic materials

We consider a quasistatic contact problem for an electro-viscoelastic body. The contact is frictional and bilateral with a moving rigid foundation which results in the wear of the contacting surface. The damage of the material caused by elastic deformation is taken into account, its evolution is described by an inclusion of parabolic type. We present a weak formulation for the model and establish existence and uniqueness results. The proofs are based on classical results for elliptic variational inequalities, parabolic inequalities and fixed point arguments.     

Collisions d'un point et d'un plan

A predictive theory of rigid bodies collision is developed that is mathematical and mecanical coherent. We investigate the existence of a solution of the evolution problem of a point above a plane including friction and general constitutive laws during collisions.     

Exact Null Controllability of a Nonlinear Thermoelastic Contact Problem

We study the controllability properties of a nonlinear parabolic system that models the temperature evolution of a one-dimensional thermoelastic rod that may come into contact with a rigid obstacle. Basically the system dynamics is described by a one-dimensional nonlocal heat equation with a nonlinear and nonlocal boundary condition of Newmann type. We focus on the control problem and treat the case when the control is distributed over the whole space domain. In this case the system is proved to be exactly null controllable provided the parameters of the system are smooth. The proof is based on changing the control variable and using Aubins Compactness Lemma to obtain an invariant set for the linearized controllability map. Then, by proving that the found solution is sufficiently smooth, we get the null controllability for the original system.     

Existence of solutions for the equations

We introduce a concept of weak solution for a boundary value problem modelling the interactive motion of a coupled system consisting in a rigid body immersed in a viscous fluid. The fluid, and the solid are contained in a fixed open bounded set of R3. The motion of the fluid is governed by the incompresible Navier-Stokes equations and the standard conservation's laws of linear, and angular momentum rules the dynamics of the rigid body. The time variation of the fluid's domain (due to the motion of the rigid body) is not known apriori, so we deal with a free boundary value problem. Our main theorem asserts the existence of at least one weak solution for this problem. The result is global in time provided that the rigid body does not touch the boundary     

Exact controllability of a fluid–rigid body system

This paper is devoted to the controllability of a 2D fluid–structure system. The fluid is viscous and incompressible and its motion is modelled by the Navier–Stokes equations whereas the structure is a rigid ball which satisfies Newton's laws. We prove the local null controllability for the velocities of the fluid and of the rigid body and the exact controllability for the position of the rigid body. An important part of the proof relies on a new Carleman inequality for an auxiliary linear system coupling the Stokes equations with some ordinary differential equations.     

Motion of a rigid body in a viscous fluid

We introduce a concept of weak solution for a boundary value problem modelling the motion of a rigid body immersed in a viscous fluid. The time variation of the fluid's domain (due to the motion of the rigid body) is not known a priori, so we deal with a free boundary value problem. Our main theorem asserts the existence of at least one weak solution for this problem. The result is global in time provided that the rigid body does not touch the boundary.     

Viscoplasticity with frictional contact and rapid growth

We consider a problem in the inelastic deformation theory with a quasistatic deformation process of the gradient‐monotone type. We assume that the body has contact with a rigid foundation: the body moves on the foundation with friction. The frictional contact is modelled by a velocity‐dependent dissipation functional. This makes an evolution problem with two nonlinear monotone operators. We consider the gradient‐monotone inelastic constitutive function with a rapid growth at infinity. This leads us to a nonreflexive Orlicz space as an operational base. The frictional dissipation potential brings about a minimalization problem in this nonreflexive Orlicz space. Copyright © 2016 John Wiley & Sons, Ltd.     

Constant mean curvature tori as stationary solutions to the Davey–Stewartson equation

A well known result of Da Rios and Levi-Civita says that a closed planar curve is elastic if and only if it is stationary under the localized induction (or smoke ring) equation, where stationary means that the evolution under the localized induction equation is by rigid motions. We prove an analogous result for surfaces: an immersion of a torus into the conformal 3-sphere has constant mean curvature with respect to a space form subgeometry if and only if it is stationary under the Davey–Stewartson flow.     

Optimal Control of Rigid Body Angular Velocity with Quadratic Cost

In this paper, we consider the problem of obtaining optimal controllers which minimize a quadratic cost function for the rotational motion of a rigid body. We are not concerned with the attitude of the body and consider only the evolution of the angular velocity as described by the Euler equations. We obtain conditions which guarantee the existence of linear stabilizing optimal and suboptimal controllers. These controllers have a very simple structure.     

Evolution of elastic thin films with curvature regularization via minimizing movements

The evolution equation with curvature regularization that models the motion of a two-dimensional thin film by evaporation-condensation on a rigid substrate is considered. The film is strained due to the mismatch between the crystalline lattices of the two materials. Here, short time existence, uniqueness and regularity of the solution are established using De Giorgi’s minimizing movements to exploit the $L^{2}$ -gradient flow structure of the equation. This seems to be the first analytical result for the evaporation-condensation case in the presence of elasticity.     

Computer simulation of underthrusting and subduction due to collision of slabs

We come up with a mathematical simulation of a collision between lithospheric slabs (plates) where one slab is forced into the mantle beneath another. Problems of the Earth’s crust and mantle deformation are solved numerically: for spatial discretization of equations of deformable solid mechanics, a finite-element method is used, and for evolution of the collision process, a stepwise integration of quasistatic deformation equations is applied. Problems of plate motion are solved within a geometrically nonlinear setting in a two-dimensional approximation (plane deformation) with due regard for large deformations of bodies and contact interactions of slabs with the mantle. A numerical solution is obtained via a MSC.Marc 2005 code, encompassing formulations of equations with required types of nonlinearities. A part of the Earth’s crust that has no tendency to delving into the mantle is simulated by a prescribed motion of a rigid body. A part of the Earth’s crust that should sink by virtue of properties of initial geometry is simulated as a deformable solid made up of elastoplastic strain-hardening material. The mantle is simulated by an ideal elastoplastic material with a low yield stress value. We are concerned with parts of the Earth’s crust that have different geometric parameters. Computer simulation of plate collision shows that under standard conditions, underthrusting of one slab beneath another occurs; at sites of initial thickening of a slab in a contact zone, subduction (deep sinking) of the slab into the mantle is expected. In the latter case account should be taken of a well-known experimental fact, that of material compaction of the sunken piece of a slab.     

New cases of integrability in dynamics of a rigid body with the cone form of its shape interacting with a medium

The purpose of the activity is to elaborate the qualitative methods for studying the dynamics of rigid bodies interacting with a resisting medium under quasistationarity conditions. This material refers equally to the qualitative theory of ordinary differential equations and the dynamics of rigid bodies. We use the properties of body's motion in a medium under conditions of the jet flow past this body. We study the plane model problems of the motion of a body with the cone form of its shape in a resisting medium. The new families of phase portraits of variable dissipation systems are obtained, their absolute or relative roughness is demonstrated. The integrable cases of equations of motion of rigid bodies are found. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A qualitative analysis of solutions to microelectromechanical systems with curvature and nonlinear permittivity profile

An investigation of qualitative properties of solutions to microelectromechanical systems with nonlinear permittivity profile is established. The considered problem is described by a quasilinear evolution equation for the displacement of a membrane, coupled with an elliptic moving boundary problem for the electrostatic potential between the moving membrane and a rigid ground plate. The system is shown to be well-posed locally in time for arbitrary values λ of the applied voltage. For small values of λ the solution exists even globally in time such that the membrane never touches down on the ground plate. In addition to those fundamental results concerning existence and uniqueness of solutions, sufficient conditions are specified which guarantee non-positivity of the membrane's displacement on the one hand and the occurrence of a singularity after a finite time T on the other hand.     

Existence of weak solutions for a Bingham fluid-rigid body system

We consider the motion of a rigid body in a viscoplastic material. This material is modeled by the 3D Bingham equations, and the Newton laws govern the displacement of the rigid body. Our main result is the existence of a weak solution for the corresponding system. The weak formulation is an inequality (due to the plasticity of the fluid), and it involves a free boundary (due to the motion of the rigid body). We approximate it by regularizing the convex terms in the Bingham fluid and by using a penalty method to take into account the presence of the rigid body.     

A note on the rigid core of affine surfaces

During recent decades, $$\mathbb {G}_a$$-actions, especially certain invariants of $$\mathbb {G}_a$$-actions, have been important tools in the study of affine varieties. The $$\mathbb {G}_a$$-actions are usually studied through locally nilpotent derivations in characteristic zero and exponential maps (see Definition 1.1) in arbitrary characteristic. The “Makar-Limanov invariant” of locally nilpotent derivations played a pivotal role in solving the linearization conjecture in the 1990s, while invariants of exponential maps were central to N. Gupta’s resolution of the Zariski cancellation problem in positive characteristic. In the study of locally nilpotent derivations on commutative algebras containing $$\mathbb {Q}$$, Freudenburg and Moser-Jauslin (Mich Math J 62:227–258, (2013), Theorem 6.1) have introduced a new invariant called “rigid core” and used it to formulate an alternative version of Mason’s theorem and to prove a well-known analogue of Fermat’s last theorem for rational functions (Freudenburg and Moser-Jauslin (2013), Corollary 6.1). In this note, we consider the concept of the rigid core in the framework of exponential maps on commutative algebras over an algebraically closed field k of arbitrary characteristic. We observe that for any factorial k-domain B with $${\text {tr.deg}}_k(B)=2$$, the concept of rigid core coincides with the Makar-Limanov invariant. We also show that over any affine two-dimensional normal k-domain B, its rigid core is a stable invariant.     

On the stability of uniformly rotating liquid in a weak magnetic field

We analyze the simplest free boundary problem of magnetohydrodynamics governing the evolution of an isolated mass of a viscous incompressible liquid in the presence of the magnetic field. The motion of the liquid is governed by the Navier–Stokes equations, and for the magnetic field we have the Maxwell equations with an excluded displacement current. The magnetic field should be determined not only in the domain filled with the liquid, but also in the surrounding vacuum region. On the free boundary of the liquid standard jump conditions for the magnetic field are prescribed, as well as kinematic and dynamic boundary conditions, where the magnetic stress tensor is taken into account. We prove that the solution corresponding to a rigid rotation of the fluid and to zero magnetic field is stable if the functional of potential energy has a positive second variation. Bibliography: 11 titles.     

Understanding rigid geometric transformations: Jeff's learning path for translation

This article describes the development of knowledge and understanding of translations of Jeff, a prospective elementary teacher, during a teaching experiment that also included other rigid transformations. His initial conceptions of translations and other rigid transformations were characterized as undefined motions of a single object. He conceived of transformations as movement and showed no indication about what defines a transformation. The results of the study indicate that the development of his thinking about translations and other rigid transformations followed an order of (1) transformations as undefined motions of a single object, (2) transformations as defined motions of a single object, and (3) transformations as defined motions of all points on the plane. The case of Jeff is part of a bigger study that included four prospective teachers and analyzed their development in understanding of rigid transformations. The other participants also showed a similar evolution.     

On the problem of a thin rigid inclusion embedded in a Maxwell material

We consider a plane viscoelastic body, composed of Maxwell material, with a crack and a thin rigid inclusion. The statement of the problem includes boundary conditions in the form of inequalities, together with an integral condition describing the equilibrium conditions of the inclusion. An equivalent variational statement is provided and used to prove the uniqueness of the problem’s solution. The analysis is carried out in respect of perfect and non-perfect bonding of the rigid inclusion. Additional smoothness properties of the solutions, namely the existence of the time derivative, are also established.