Meir Shillor

A contact problem for bingham fluid with friction

We consider a mathematical model which describes the stationary flow of a Bingham fluid in a domain which is partially bounded by a deformable obstacle, such as a die. The contact between the fluid and the die is modeled by a nonlocal viscoplastic friction law. We present the classical formulation of the problem and derive a variational formulation for the velocity field. We establish the existence of a weak solution and, under additional assumption, its uniqueness. The proofs are based on classical results for elliptic variational inequalities and fixed point arguments. We also establish the continuous dependence of the solution on the yield limit.

Dynamic Gao Beam in Contact with a Reactive or Rigid Foundation

This chapter constructs and analyzes a model for the dynamic behavior of nonlinear viscoelastic beam, which is acted upon by a horizontal traction, that may come in contact with a rigid or reactive foundation underneath it. We use a model, first developed and studied by D.Y. Gao, that allows for the buckling of the beam when the horizontal traction is sufficiently large. In contrast with the behavior of the standard Euler–Bernoulli linear beam, it can have three steady states, two of which are buckled. Moreover, the Gao beam can vibrate about such buckled states, which makes it important in engineering applications. We describe the contact process with either the normal compliance condition when the foundation is reactive, or with the Signorini condition when the foundation is perfectly rigid. We use various tools from the theory of pseudomonotone operators and variational inequalities to establish the existence and uniqueness of the weak or variational solution to the dynamic problem with the normal compliance contact condition. The main step is in the truncation of the nonlinear term and then establishing the necessary a priori estimates. Then, we show that when the viscosity of the material approaches zero and the stiffness of the foundation approaches infinity, making it perfectly rigid, the associated solutions of the problem with normal compliance converge to a solution of the elastic problem with the Signorini condition.

Dynamic Evolution of an Elastic Beam in Frictional Contact with an Obstacle

Problems involving contact and friction phenomena have received a great deal of attention in recent years and by now there is a considerable body of engineering literature devoted to this subject. In contrast, there are relatively few general mathematical results available in this area, due to the substantial difficulties encountered in establishing existence results for initial-boundary value problems that model these phenomena. Moreover, in both cases, most of the existing literature deals with static situations, or, occasionally, with a sequence of static problems, which arise from the time discretization of an evolution problem. Modeling and mathematical analysis of such problems can be found in Duvaut and Lions [DL], Moreau et al. [MPS], Kikuchi and Oden [KO], and Telega [Tel], and the references therein (see also Curnier [Cu]). There are, however, some recent results on quasistatic and dynamic behavior in Andersson [An], Telega [Te2], Klarbring et al. [KMS2] and Oden and Martins [OM].

A Beam in Adhesive Contact

A quasistatic process of contact with adhesion between an elastic beam and a foundation is considered. The contact is modeled with the Signorini condition when the foundation is rigid, and with normal compliance when it is deformable. The adhesion is modeled by introducing the bonding function β, the evolution of which is described by an ordinary differential equation. The existence and uniqueness of the weak solution for each of the problems is established. A fully-discrete scheme for numerical solutions of the problem with normal compliance is described.

Nonsmooth dynamic frictional contact of a thermoviscoelastic body

Abstract This paper studies a system of two hemivariational inequalities modeling a dynamic thermoviscoelastic contact problem with general nonmonotone and multivalued subdifferential boundary conditions. Thermal effects are included in the Kelvin–Voigt thermoviscoelastic constitutive law and in the boundary conditions, and so in frictional heat generation, which takes place on the boundary and enters the condition for the temperature. The existence of a weak solution to the problem is established using a recent surjectivity result for differential inclusions associated with pseudomonotone operators.

Models of Debonding Caused by Vibrations, Heat and Humidity

This paper describes a 3D model for the process of debonding of two adhesively bonded rectangular components or solids caused by mechanical vibrations, temperature variations and moisture or humidity. These issues are very common in many parts of the world in which systems, such as automotive systems that have parts that are glued together, have to operate in adverse conditions of large variations in temperature and very high humidity. The model consists of a coupled system of dynamic equations for the displacements of the two components, the evolution equations for the temperature in each body and the evolution inclusions for the bonding field and the moisture field in the thin layer of glue on the contact surface. The solids may be either thermoelastic or thermoviscoelastic, and the viscosity may be of the short-memory or long-memory types. Then, this work presents 1D variations of the model. In particular, a model in which each of the components is described both as a rod and a beam so as to capture the tangential and vertical motions, which affect the strength of the bonding field. These models raise many interesting questions: theoretical, computational and experimental that are described in some detail. In particular, the 1D models may be used for parameter identification purposes, especially of the debonding source function.

Wood Frogs Population in a Changing Environment

We present new results for a mathematical model for the dynamics of a population of Wood Frogs, which continue the investigation begun in [1]. Thexa0 model is in the form of a system of nonlinear impulsive differentialxa0equations forxa0xa0each developmental stage (larvae, juvenile, and mature).xa0 Itxa0 alsoxa0xa0takes into account the differences in the growthxa0 of the early, middle, and latexa0xa0juvenile stages.xa0We describe numerical simulations for the study of the environmentalxa0xa0impact on the population, in particular we investigate three issues: the existence of periodic solutionsxa0for the model; the recovery of the population from 1-3 dry years in which no larvaexa0xa0hatch; and the dependence of the model on the system parameters.xa0It is seen that the results agree qualitatively with the observed data, which allowsxa0us to use the model for a tentative prediction of next years development. We alsoxa0present some additional mathematical and numerical issues for future study. [1]xa0N. Al-Asuoad, R. Anguelov, K. Berven, M. Shillor, Model and Simulations of a Wood Frog Population, Biomath 1 (2012), 1209032, http://dx.doi.org/10.11145/j.biomath.2012.09.032

Mathematical Models for Chagas Disease

We present mathematical and computational results for a model for the dynamics of Chagas disease. It is caused by the parasite T. cruzi that is transported by the vectors Triatoma infestans, and affects millions of humans and domestic mammalsxa0 throughout rural areas in Centralxa0 and South America.xa0 The chronic disease causes mortality and severe morbidity.xa0 To control the disease spread periodic insecticide spraying of thexa0xa0village houses is used and also bank blood screening. The basic model for the disease dynamics consists of four nonlinearxa0xa0ordinary differential equations for the populations of the vectors and of infected vectors,xa0xa0humans, and domestic animals. It has time-dependent periodic coefficients to accountxa0for seasonality, and was developed in [1]. The mainxa0motivation for the model was to optimize the insecticide spraying schedules. The model was extended to take into account congenital transmission in both humans and domestic mammals as well as oral transmission in domestic mammals [2].xa0In particular, oral transmission provides an alternative to vector biting as an infection xa0route for the domestic mammals, who are key to the infection cycle.xa0 This may lead toxa0xa0high infection rates in domestic mammals even when the vectors have a low preferencexa0xa0for biting them, and ultimately results in high infection levels in humans.xa0xa0Another extension was to allow for random coefficients, reflectingxa0 the uncertainty in their values.xa0xa0The simulations show that the variations in some of the model parameters lead toxa0xa0considerable variations in the numbers of infected humans and domestic mammals. [1] A.M. Spagnuolo, M. Shillor, G.A. Stryker, A model for Chagas disease with controlled spraying , J. Biological Dynamics 5 (4)(2010) 299--317. [2] D.J. Coffield Jr., E. Mema, B. Pell, A. Pruzinsky, M. Shillor, A.M. Spagnuolo, and A. Zetye, A Model for Chagas Disease with vector consumption and transplacental transmission, to appear in PLOS.

A Survey of 1-D Problems of Dynamic Contact or Damage

We survey four of our recent results on one-dimensional dynamic contact with or without friction or with damage. We present the classical models, the weak or variational formulations and state our results. Тhe purpose of considering one-dimensional problems is to gain insight into the behavior of dynamic models for contact, without having to address the considerable mathematical complications that arise in two or three dimensions.

Rock’s Interface Problem Including Adhesion

A rock’s dynamic contact model taking into account friction and adhesion phenomena is discussed. It consists of a hemivariational inequality because of the adhesion process. A weak solution is obtained as a limit of a sequence of solutions to some regularized problems after establishing the necessary estimates.