Roger Fosdick

Thermodynamics and Stability of Fluids of Third Grade

Today, even though the Clausius-Duhem inequality is widely considered to be of central importance in the subject of continuum thermomechanics, it is also believed to be a somewhat special interpretation of a more fundamental (second) law of thermodynamics. In this work, which is concerned with the relation between thermodynamics and stability for a class of non-Newtonian incompressible fluids of the differential type, we find it essential to introduce the additional thermodynamical restriction that the Helmholtz free energy density be at a minimum value when the fluid is locally at rest. As a background to our main considerations we begin by introducing the general class of Rivlin-Ericksen fluids of complexity n and obtain, for this class, a preliminary set of thermodynamical constitutive restrictions. We then give detailed attention to the special case of fluids of grade 3 and arrive at fundamental inequalities which restrict its (temperature dependent) material moduli. When the moduli are taken to be constant we find that these inequalities require that a body of such a fluid be stable in the sense that its total kinetic energy must tend to zero in time, no matter what its previous mechanical and thermal fields, provided it is both mechanically isolated and immersed in a thermally passive environment at constant temperature from some finite time onward. When the material constants of a fluid of grade 3 are such that the Clausius-Duhem inequality is satisfied but the free energy is not at a minimum in equilibrium, we show that for a broad class of reasonably posed problems the flows are necessarily asymptotically unbounded. Finally, we determine the stability character of non-trivial base flows for fluids of grade 3 with constant material moduli, and establish a uniqueness theorem for the initial-boundary value problem and a uniqueness theorem for problems involving sufficiently slow steady flows.

Uniqueness and drag for fluids of second grade in steady motion

Abstract For incompressible fluids of second grade that are compatible with the Clausius-Duhem inequality, non-uniqueness of steady flows with small Reynolds number (i.e. creeping flows) is possible provided the ‘absorption number’ is also small. We discuss this uniqueness question, generalize a well-known theorem of Tanner concerning how solutions of the Stokes equations may be used to generate solutions of the creeping flow equations for fluids of second grade, and give a new uniqueness theorem appropriate to a class of problems for the steady creeping flow of fluids of second grade. Under the conditions for uniqueness, we obtain a simple formula for the drag force on a fixed body which is immersed in a fluid of second grade which is undergoing uniform creeping flow. For bodies with certain geometric symmetries, the non-Newtonian nature of the fluid has no effect upon the drag.

Rectilinear steady flow of simple fluids

Steady rectilinear motion of in compressible non-Newtonian fluids is, in general, characterized by the two planar fields of pressure and speed, and these are in turn restricted by the three differential equations of motion. Two material functions which depend on the shear rate and which are associated with normal and shear stress effects in viscometric flows enter these equations in such a way that if the material functions are not proportional, the three field equations are independent. For this prevalent case of apparent overdetermination, a well-known and generally accepted conjecture of Ericksen (1956) severely delimits the class of possible flows. In fact, from his conjecture Ericksen drew the remarkable and profound secondary conclusion that rectilinear steady flow through a cylindrical pipe to which the fluid adheres is impossible, except perhaps when the cylinder is composed of portions of planes and right circular cylinders. In this paper, while we show that Ericksen’s conjecture is not fully valid, we also find that in cases of major importance his secondary conclusion is correct in even a sharper form. Our main result (stated in the Introduction) shews that subject to certain technical requirements, rectilinear steady flow with boundary adherence through a fixed straight pipe is possible only when the pipe is either circular or the annulus between two concentric circles. Counter-examples are given which illustrate that the independence condition alone is not sufficient to justify either Ericksen’s conjecture, his secondary conclusion, or our main result. Finally, we show that our main result is valid when the condition of boundary adherence is replaced by a natural slip boundary condition.

Shock induced transitions and phase structures in general media

This volume focuses on the thermodynamics and mechanics of dynamic phase transitions and the consequent issues of rapid solidification, liquification and vaporization. The articles investigate fundamental questions associated with phase stability, metastability, and the reaction kinetics which determine the phase or phases that are attainable. Principal researchers in physics, mathematics, metallurgy, engineering, and molecular dynamics present key experimental observations, realistic modelling criteria, insights gained from large scale computations in molecular dynamics, and mathematical analyses of the resulting models.

Thermodynamics, stability and non-linear oscillations of viscoelastic solids — I. Differential type solids of second grade

Abstract We study the thermodynamics and stability of a viscoelastic second grade solid whose action is characterized by two microstructural coefficients α 1 and α 2 in addition to the Newtonian viscosity μ. We show that it is both necessary and sufficient that μ ⩾ 0, α 1 ⩾ 0 and α 1 + α 2 = 0 if the material model is to be compatible with thermodynamics and its free energy is to be at a local minimum in equilibrium. Then, we construct a stability theorem for second grade solids which undergo mechanically isolated motions wherein it is shown that the motion of the body relative to its center of mass will dissipate away in time. The stability theorem is exemplified by investigating the free oscillation of cylindrical and spherical shells where the equilibrium state is globally stable. When μ = 0, but α 1 ≠ 0, the shells exhibit a larger period than if they were purely elastic in the classical sense.

On Ericksen's problem for plane deformations with uniform transverse stretch

Abstract Ericksens problem[1] consists of determining static deformation fields which are producible in every isotropic, homogeneous, incompressible, elastic body under the sole action of surface traction. In this paper, we obtain the complete solution of this problem for the ease of plane deformations with uniform transverse stretch. The approach utilizes a novel and convenient parameterization of the deformation gradient field. Aside from the set of homogeneous deformations, the following four families of deformations are found to be the only ones possible: (i) Bending and stretching of a rectangular block, (ii) Straightening and stretching of a sector of a hollow cylinder, (iii) Inflation, bending and extension of an annular wedge. (iv) Inflation, bending, extension and azimuthai shearing of an annular wedge.

Transverse deformations associated with rectilinear shear in elastic solids

The rectilinear deformation of an incompressible, isotropic clastic solid is, in general, characterized by the two planar fields of pressure and displacement magnitude, and these are, in turn, restricted by the three, generally independent, differential equations of equilibrium. The over-determined nature of this situation suggests the possibility that transverse deformations may accompany rectilinear shear—a possibility not supported by the linear theory. Within this context we consider the class of equilibrium non-linear clasticity problems which is associated with cylindrical domains whose various boundaries each are displaced rigidly along their generators. An approximation scheme is developed for determining the cross sectional deformation and a specific example for a cylinder with eccentric circular cross section is given.

The constraint of local injectivity in linear elasticity theory

There are problems in classical linear elasticity theory whose known solutions must be rejected because they predict unacceptable deformation behaviour, such as the interpenetration of material regions. What has been missing is a proper account of the constraint that allowable deformations must be injective. This type of constraint is highly nonlinear and non–convex, even within the classical linear theory, and it is expected to give rise to the existence of an appropriate constraint reaction field. We propose to determine the displacement field u(·) : B →Rn (n = 2, 3) of an elastic body B ⊂Rn such that the potential energy is minimized subject to the constraint that the deformation y = f(x) = x + u(x), x εB, is locally invertible, i.e. det(1 + ▽u) > 0 in B. In linear elasticity theory, the strain energy (assumed positive definite) is a quadratic function of ▽u and, in the context of plane problems where the dimension n= 2, the constraint is properly closed, which allows us to prove, at least in this case, an existence theorem for such minimizers in W1,2(B). We then investigate the form of the corresponding Euler–Lagrange equations in dimension n = 2 or 3 and characterize the associated constraint reaction (Lagrange multiplier) field. Finally, we review an example problem from linear elasticity theory for an aeolotropic disk whose classical solution supports a subregion of material interpenetration, and we give an alternative solution within the constrained theory which avoids this unacceptable behaviour. The subregion of the disk in which the Lagrange multiplier constraint reaction field is active is determined, as is the field itself. We find that the existence of a constraint reaction field is essential if overlapping of the material is to be avoided. Correspondingly, when the constraint of local invertibility is applied we see that the body becomes significantly stiffer in response to the same loads. Though, perhaps, only coincidental, the existence of a constraint reaction field is remindful of the fact that the force between the atoms of a substance becomes strongly repelling when the separation distance between them is made relatively small.

On the impossibility of linear Cauchy and Piola-Kirchhoff constitutive theories for stress in solids

We investigate the possibility of linear elasticity as an infinitesimal theory based on a genuinely linear response function which retains its validity even for finite deformations. Careful consideration of the domain of definition of the stress response function, the definition of linearity and the notion of material frame-indifference leads to our main result that an exact linear constitutive theory for elastic solids is impossible. We then generalize our result to viscoelasticity theory where the stress response is dependent on deformation gradient histories.ZusammenfassungVerf. betrachten die Möglichkeit linearer Elastizität als infinitesimale Theorie begründet auf einer echt linearen Reaktionsfunktion die ihre Gültigkeit sogar für endliche Deformationen behält. Genaue Betrachtung des Definitionsbereiches der Spannungsreaktionsfunktion, der Definition von Linearität, und des Objektivitätsbegriffes führen zum Hauptresultat dass eine echt lineare Theorie für elastische Körper unmöglich ist. Das Resultat wird dann auf viskoelastische Theorie verallgemeinert, wobei die Spannungsreaktion von der Vorgeschichte des Deformationsgradienten abhängt.

Small bending of a circular bar superposed on finite extension or compression

SummarySolutions are presented for the small bending by terminal couples or by a transverse end load of a circular cylinder of incompressible isotropic material which has been initially finitely extended or compressed by axial loading. Numerical results are given for a Mooney material. The results are used to investigate the stability of a circular bar under end thrust. An elementary solution is giver for the small bending by its own weight of a stretched horizontal cylinder.