R. Ray Nachlinger

Bone remodeling III: uniqueness and stability in adaptive elasticity theory

A uniqueness theorem for infinitesimal adaptive elasticity is proved. Two theorems establishing sufficient conditions for stability are demonstrated.ZusammenfassungEin Eingigkeitsatz für unendich klein adaptives Elastizitätstheoric ist gebeweisen, Zwei Sätze aussagen hinreichende Bedingingen für Stabilität werden demonstrieren.

On the domain of attraction for steady states in heat conduction

Abstract Energy methods are employed to obtain information on the domain of attraction of steady state solutions to a class of heat conduction problems with nonlinear heat generation. Essential use is made of an inequality of Sobolev type, which may be of interest beyond the present context.

Uniqueness theorems for finite elastodynamics

Described in this paper is a study of the uniqueness of solutions to the boundary-initial value problems of nonlinear dynamical elasticity. Particular consideration is given to the uniqueness implications of certain well-known a priori restrictions on the material response.ZusammenfassungIn diesem Werk wird die Eindeutigkeit der Lösungen von Anfangswert-Randwertproblemen mit nichtlinearer dynamischer Elastizität beschrieben. Besondere Betrachtung ist den Folgerungen der Eindeutigkeit von bestimmten bekannten a priori Beschränkungen auf materielle Reaction gegeben.

On wave propagation and uniqueness in one-dimensional elastic bodies

A notion referred to as the Wave Propagation Property is analyzed in the context of the nonlinear theory of one-dimensional elastic bodies. Roughly speaking, a body possesses this property if mechanical disturbances propagate with bounded speed. A uniqueness theorem is proven with the aid of the results on wave propagation.ZusammenfassungEin Begriff, bezeichnet als Wellenfortpflanzungseigenschaft, wird in Zusammenhang mit der nicht-linearen Theorie des eindimensionalen elastischen Körpers untersucht. Ein Körper besitzt, gross gesprochen, diese Eigenschaft, wenn sich mechanische Störungen mit beschränkter Geschwindigkeit fortpflanzen. Mit Hilfe des Ergebnisses für Wellenfortpflanzung wird ein Eindeutigkeitsatz bewiesen.

Stability of uniform temperature fields in linear heat conductors with memory

Abstract In this paper we consider the asymptotic behavior of temperature fields in linear heat conductors with memory. Using the method of energy integrals, we establish stability for the history problem both in the case when the governing heat equation is parabolic in character and when the heat equation is hyperbolic and predicts finite wave speeds.

On uniqueness and wave propagation in nonlinear heat conductors with memory

Recently, there has been considerable interest in heat conduction in materials with memory. Of particular interest are two theories proposed by Coleman and Gurtin [l] and by Gurtin and Pipkin [2]. The distinguishing feature of both these theories is that they depend on the history of the temperature gradient. Various aspects of these theories have been studies by Nunziato [3, 41, including uniqueness for the linear case, Uniqueness in the linear case was also studied by Finn and Wheeler [5] and by one of the authors [6]. In [5], F inn and Wheeler also established conditions sufficient for disturbances to propagate with finite speed. It is our purpose, here, to study uniqueness and wave propagation in the absence of any linearizing assumptions. The constitutive equations we adopt are a special case of those in [I], since we allow the energy to depend on the current value of the temperature and the summed history of the temperature and temperature gradient, while the heat flux depends also on the current value of the temperature gradient. In Theorem 1, we establish uniqueness of solutions under the conditions that: (1) the conductivity tensor is positive definite, and (2) the heat capacity is positive. For the other results, we consider the case where the heat flux is independent of the current values of the temperature and temperature gradient. Thus, we are then considering a special case of Gurtin and Pipkin’s constitutive equations. For the second case, we first establish in Theorem 2, conditions sufficient for the conductor to possess the Wave Propagation Property (WPP). The WPP (which is made precise in Section 4) is, loosely speaking, a requirement that if a body is disturbed in a bounded set, then for any time t, the disturbed region of the body must be bounded. This property was enunciated in [7], where one of the authors established that if a body possesses the WPP, then solutions to a certain class of boundary-history value problems are unique.

A uniqueness theorem for the general theory of heat conduction with finite wave speeds

Abstract Uniqueness of solutions in nonlinear heat conduction is considered. It is established that for a certain class of heat conductors with memory, at most one solution can exist with perscribed temperature on the boundary.

Wave propagation and uniqueness theorems for elastic materials with internal state variables

Abstract In this paper we consider the general, one-dimensional theory of thermoelastic materials with internal state variables and show that such materials have the wave propagation property, i.e. all smooth structured waves propagate with bounded velocity. We further show that this boundedness property ensures the uniqueness of solutions of unbounded domains.

On the determinacy of motions for the displacement problem in the dynamics of elastic bodies with viscosity

SummaryFor a class of materials in which the stress depends upon the deformation and its rate, we show-without the aid of linearizing assumptions-that the solution to the displacement initial boundary-value problem is unique.ZusammenfassungFür eine Klasse von Materialien, deren Spannung von ihrer Verformung und der Geschwindigkeit derselben abhängt, zeigen wir-ohne die Hilfe von Linearisierungsannahmen- dass die Lösung des Anfangs- und Randwertproblems für die Verschiebungen eindeutig ist.

Ship Motions in Shallow Water

When operating large ships in water depths only slightly greater than the draft, as super tankers at near-shore terminals, it is necessary to know precisely how much clearance is required for safety. Recent developments in hydrodynamics are applied to determine the motions of a ship due to waves and current with draft to depth ratio approaching unity. A computer program written for this purpose is outlined. Computed motions are compared favorably with experimental data.