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We present in this chapter functionals that are free energies, provided certain assumptions on the relaxation function are valid. In the first section, the general nonisothermal model introduced in Chapter 6 is considered, while in subsequent sections, these functionals are discussed for materials introduced in Chapters 7, 8.

Free energies in electromagnetism, as in mechanics, are not in general uniquely determined by the constitutive equations but form a convex set with a minimum and a maximum element. A formula for the minimum free energy of a dielectric material with a linear memory-dependent constitutive equation was first given in [35], based on results in mechanics [158]. This was generalized in [151, 152], which form the basis of the present chapter.

This chapter is devoted to outlining some ideas involving that part of the theory of inverse problems that is usually referred to as the identification of parameters (numbers, vectors, matrices, functions) appearing in integrodifferential equations describing the evolution of fading memory materials. We recall the celebrated definition by Hadamard of a well-posed problem: it requires the existence and uniqueness of the solution to the problem and its continuous dependence on data.

The classical theory of electromagnetism of continua, without memory effects, is first explored. The case of nonlocal materials is also discussed briefly.

We now develop formulas for the minimum free energy and related quantities applicable to three categories of linear materials (completely linear for solids and heat conductors). The methods differ in detail from those in Chapter 10, though they are equivalent. As in Chapters 7 and 8, we make more use of the abstract terminology and notation introduced in Chapters 3, 4.

Materials with constitutive equations expressed in terms of fractional derivatives [47] are of increasing interest in recent years (see [214, 287]). It is well known that such materials can be considered in the class of materials with memory and may describe elastic, fluid, viscoelastic, and electromagnetic materials, but also other kinds of phenomena, such as heat flux models.

We recall the integrated form of the energy balance relation, given by ( 16.1.29), where \({{\mathcal D}}(t)\) and W(t) are the total dissipation and the work function , respectively, of the material

We now address the problem of finding explicit forms for the free energy of materials with constitutive relations given by linear memory functionals. Such materials are referred to in this work as linear memory materials. As we will see, the equilibrium (or alternatively, the instantaneous) contribution, which is to say the portion of the constitutive equation without memory effects, need not be linear. If the part of the constitutive equation without memory is also linear, we use the description a completely linear material. A linear viscoelastic material is understood to be completely linear, while a viscoelastic material with linear memory need not have this property.

We now focus on the case of materials characterized by memory kernels with only isolated singularities in the frequency domain and derive explicit expressions for a family of free energies, including the minimum free energy discussed in Chap. 11 and the maximum free energy. All of these will be shown to be functionals of the minimal state.

Using a standard representation of a free energy associated with a linear memory constitutive relation, a new condition, involving linear functionals, is derived which, if satisfied, ensures that the free energy is a functional of the minimal state. Using this condition and results on constructing free energy functionals in Chap. 17, it is shown that if the kernel of the rate of dissipation functional is given by sums of products, the associated free energy functional is a FMS. Because this condition is linear rather than a quadratic, it is easier to explore and to apply in new contexts.

The study of differential problems related to materials with fading memory began with the work of Graffi [111, 112]. Later on, these studies were considered by many authors, and in particular, a new important description of such phenomena was given by Dafermos in [47, 46], using semigroup theory, where besides existence and uniqueness of the solution, the interesting problem of asymptotic stability was also examined.

Visco-ferromagnetic materials represented by non-local constitutive equation are considered in the paper. We use fractional derivatives in order to describe memory and spatial effects. Also, thermodynamic principles are formulated and studied.

For helpful comments and input, the authors thank David Abrams, Alma Cohen, Allan Ferrel, Michael Frakes, Mark Lemley, Jonathan Masur, Michael Meurer, Michael Risch, Pam Samuelson, David Schwartz, Ted Sichelman, Charles Silver, Matthew Spitzer, Melissa Wasserman, Heidi Williams, prior anonymous reviewers, and participants in the 2012 Empirical Patent Law Conference sponsored by Cornell Law School and the University of Illinois College of Law, the 2012 Intellectual Property Scholars Conference, the University of Texas School of Law’s Drawing Board workshop, a conference in memory of Ted Eisenberg (Tel Aviv Univ, 2015), and Empirical Legal Studies Workshop (Tel Aviv Univ, 2015). The authors thank Melissa Bernstein, Ross MacDonald, Grace Matthews, and Jane O’Connell for research assistance. The first part of the title is in quotation marks because it was also the first part of the title for a 2005 news story in the journal IP Law & Business (Cohen 2005).

In recent years, the United States Patent and Trademark Office (“PTO”) sought to control results in adjudication by its Patent Trial and Appeals Board (“PTAB”) through a process commonly described as “panel stacking.” In a “strong form” of this practice, the PTO Director or Director’s delegee generated a new panel of administrative judges to conduct rehearing proceedings after an initial panel produced a decision with which the Director or delegee disagreed. This Essay contends that this strong-form practice raises constitutional concerns under the Fifth Amendment’s Due Process Clause. Consequently, the doctrine of constitutional avoidance instructs that courts should understand the Patent Act to preclude strong-form panel stacking. Judges and commentators have repeatedly erred by citing a plurality opinion on panel stacking in In re Alappat as if the plurality opinion authoritatively held that the Patent Act authorizes panel stacking. This Essay seeks to correct that misconception and shows that, once one takes account of constitutional concerns, the Alappat judges’ recognition of statutory ambiguity effectively condemns strong-form panel stacking, rather than “blessing” it.

This chapter:(1) describes the current state of, and normative basis for, the law of reasonable royalties among the leading jurisdictions for patent infringement litigation, as well as the principal arguments for and against various practices relating to the calculation of reasonable royalties; and(2) for each of the major issues discussed, provides one or more recommendations.The chapter’s principal recommendation is that, when applying a “bottom-up” approach to estimating reasonable royalties, courts should replace the Georgia-Pacific factors (and analogous factors used outside the United States) with a smaller list of considerations, specifically:(1) calculating the incremental value of the invention and dividing it appropriately between the parties;(2) assessing market evidence, such as comparable licenses; and (3) where feasible and cost-justified, using each of these first two considerations as a “check” on the accuracy of the other.

It is assumed that any free energy function exhibits strict periodic behavior for histories that have been periodic for all past times. This is not the case for the work function, which, however, has the usual defining properties of a free energy. Forms given in fairly recent years for the minimum and related free energies of linear materials with memory have this property. Materials for which the minimal states are all singletons are those for which at least some of the singularities of the Fourier transform of the relaxation function are not isolated. For such materials, the maximum free energy is the work function, and free energies intermediate between the minimum free energy and the work function should be given by a linear relation involving these two quantities. All such functionals, except the minimum free energy, therefore do not have strict periodic behavior for periodic histories, which contradicts our assumption. A way out of the difficulty is explored which involves approximating the relaxation function by a form for which the minimal states are no longer singletons. A representation can then be given of an arbitrary free energy as a linear combination of the minimum, maximum and intermediate free energies derived in earlier work. This representation obeys our periodicity assumption. Numerical data are presented, supporting the consistency of this approach.

In this paper, we present an isothermal model for describing damage and fatigue by the use of the Ginzburg–Landau (G–L) equation. Fatigue produces progressive damage, which is related with a variation of the internal structure of the material. The G–L equation studies the evolution of the order parameter, which describes the constitutive arrangement of the system and, in this framework, the evolution of damage. The thermodynamic coherence of the model is proved. In the last part of the work, we extend the results of the paper to a nonisothermal system, where fatigue contains thermal effects, which increase the damage of materials.

In this work we consider the thermal convection problem in arbitrary bounded domains of a three-dimensional space for incompressible viscous fluids, with a fading memory constitutive equation for the heat flux. With the help of a recently proposed free energy, expressed in terms of a minimal state functional for such a system, we prove an existence and uniqueness theorem for the linearized problem. Then, assuming some restrictions on the Rayleigh number, we also prove exponential decay of solutions.

This is an Online Appendix to accompany: “Troll” Check? A Proposal for Administrative Review of Patent Litigation (Cohen, Golden, Gurun, and Kominers (2017)), published in the Boston University Law Review.

Explicit expressions for the minimum free energy of a linear viscoelastic material and Noll’s definition of state are used here to explore spatial energy decay estimates for viscoelastic bodies, in the full dynamical case and in the quasi-static approximation. In the inertial case, Chirita et al. obtained a certain spatial decay inequality for a space–time integral over a portion of the body and over a finite time interval of the total mechanical energy. This involves the work done on histories, which is not a function of state in general. Here it is shown that for free energies which are functions of state and obey a certain reasonable property, the spatial decay of the corresponding space–time integral is stronger than the one involving the work done on the past history. It turns out that the bound obtained is optimal for the minimal free energy. Two cases are discussed for the quasi-static approximation. The first case deals with general states, so that general histories belonging to the equivalence class of any given state can be considered. The continuity of the stress functional with respect to the norm based on the minimal free energy is proved, and the energy measure based on the minimal free energy turns out to obey the decay inequality derived Chirita et al. for the quasi-static case. The second case explores a crucial point for viscoelastic materials, namely that the response is influenced by the rate of application of loads. Quite surprisingly, the analysis of this phenomenon in the context of Saint-Venant principles has never been carried out explicitly before, even in the linear case. This effect is explored by considering states, the related histories of which are sinusoidal. The spatial decay parameter is shown to be frequency-dependent, i.e. it depends on the rate of load application, and it is proved that of those considered, the most conservative estimate of the frequency-dependent decay is associated with the minimal free energy. A comparison is made of the results for sinusoidal histories at low frequencies and general histories.

The free energy for most materials with memory is not unique. There is a convex set of free energy functionals with a minimum and a maximum element. Various functionals have been shown to have the properties of a free energy for materials with particular types of relaxation behaviour. Also, over the last decade or more, forms have been given for the minimum and related free energies. These are all quadratic functionals which yield linear memory terms in the constitutive equations for the stress. &nbsp A difficulty in constructing free energy functionals arises in making choices that ensure a non-negative quadratic form both for the free energy and for the rate of dissipation. We propose a technique which renders this task more straightforward. Instead of constructing the free energy and determining from this the rate of dissipation, which may not have the required non-negativity, the procedure is reversed, which guarantees a satisfactory free energy functional. &nbsp Certain results for quadratic functionals in the time and frequency domains are derived, providing a platform for this alternative approach, which produces new free energies, including a family of functionals that are generalizations of the minimum and related free energies.

The classical expressions for the free energy, introduced for viscoelastic solids, can be adapted to rigid heat conductors with memory, with a few changes. This includes an expression for the maximum free energy. For such materials, a free energy can also be given in terms of the minimal state descriptor, which corresponds to a new form recently derived for viscoelastic solids. Moreover, an expression for the internal dissipation associated with each of these free energies is derived. New forms of the free energy, with associated rates of dissipation, relating to the internal energy, are also derived.

The existence, uniqueness and asymptotic stability for an incompressible, linear viscoelastic fluid is studied using a new free energy, the represen- tation of which is based on the concept of a minimal state. A restriction imposed by thermodynamics is also used. Furthermore, an expression for the minimum free energy in the time domain is derived, which shows explicitly its dependence on the minimal state.

We now examine how the formulas emerging from the methodology developed in Chapter 10 apply to materials other than those exhibiting a discrete-spectrum response, in particular for materials with a branch-cut-type singularity. We confine our considerations to the case that the cut is on the imaginary axis. Such a material is said to have a continuous-spectrum response, i.e., thosematerials for which the relaxation function is given by an integral of a density function multiplying a strictly decaying exponential. The results reported in this chapter were first presented in [61].

The concept of a minimal state was introduced in recent decades, based on earlier work by Noll. The property that a given quantity is a functional of the minimal state is of central interest in the present work. Using a standard representation of a free energy associated with a linear memory constitutive relation, a new condition, involving linear functionals, is derived which, if satisfied, ensures that the free energy is a functional of the minimal state. Using this result and recent work on constructing free energy functionals, it is shown that if the kernel of the rate of dissipation functional is given by sums of products, the associated free energy functional is a functional of the minimal state.

It has been known for several decades that the free energy and entropy of a material with memory is not in general uniquely determined, nor are the total dissipation in the material over a given time period and the rate of dissipation. The dissipation in a material element would in particular seem to be a quantity that has immediate physical objectivity. It must be seen therefore as a significant weakness in the thermodynamics of materials exhibiting memory effects, that a quantity as basic as the rate of dissipation cannot be predicted in terms of the constitutive parameters. The objective of the present work is to propose a formula for the physical free energy of a linear scalar viscoelastic material in terms of a family of free energies, each of which can be regarded as an estimate of the physical quantity. This formula follows from a new physical hypothesis of Maximum Parametric Symmetries, which states that the physical free energy and dissipation have the closest possible level of symmetry among the parameters of the theory to that of the work function. This results in the assignment of explicit weights to all members of the family of free energies, each of these being associated with a particular factorization of a quantity closely related to the loss modulus of the material. It is interesting that the final formula proposed for the physical free energy can be expressed in simple, closed form. Once the free energy is known, the corresponding physical rate of dissipation can also be determined without difficulty. It is shown that non-trivial equivalence classes of states, in the sense of Noll, exist only if the material has a relaxation function derivative, the Fourier transform of which has only isolated singularities in the complex frequency plane. The members of the family of free energies used to determine the physical free energy are all functions of such an equivalence class. The derivation of their form is a generalization of work reported in Maximum and minimum free energies for a linear viscoelastic material, Quart. Appl. Math. 60, 341 - 381 (2002).

Certain results about free energies of materials with memory are proved, using the abstract formulation of thermodynamics, both in the general case and as applied within the theory of linear viscoelasticity. In particular, an integral equation for the strain continuation associated with the maximum recoverable work from a given linear viscoelastic state is shown to have a unique solution and is solved directly, using the Wiener-Hopf technique. This leads to an expression for the minimum free energy, previously derived by means of a variational technique in the frequency domain. A new variational method is developed in both the time and frequency domains. In the former case, this approach yields integral equations for both the minimum and maximum free energies associated with a given viscoelastic state. In the latter case, explicit forms of a family of free energies, associated with a given state of a discrete spectrum viscoelastic material, are derived. This includes both maximum and minimum free energies.

Explicit forms of the minimum free energy and the corresponding rate of dissipation, for general histories of strain, are derived and discussed within the context of fractional derivative models of materials with memory. Simple formulae are also given for sinusoidal and exponential histories.

The aim of this work is to determine what free energy functionals are expressible as quadratic forms of the state functional \(I^t\) which is discussed in earlier papers. The single integral form is shown to include the functional \(\psi _F\) proposed a few years ago, and also a further category of functionals which are easily described but more complicated to construct. These latter examples exist only for certain types of materials. The double integral case is examined in detail, against the background of a new systematic approach developed recently for double integral quadratic forms in terms of strain history, which was used to uncover new free energy functionals. However, while, in principle, the same method should apply to free energies which can be given by quadratic forms in terms of \(I^t\), it emerges that this requirement is very restrictive; indeed, only the minimum free energy can be expressed in such a manner.

Some expressions for the free energy in the case of incompressible viscoelastic fluids are given. These are derived from free energies already introduced for other viscoelastic materials, adapted to incompressible fluids. A new free energy is given in terms of the minimal state descriptor. The internal dissipations related to these different functionals are also derived. Two equivalent expressions for the minimum free energy are given, one in terms of the history of strain and the other in terms of the minimal state variable. This latter quantity is also used to prove a theorem of existence and uniqueness of solutions to initial boundary value problems for incompressible fluids. Finally, the evolution of the system is described in terms of a strongly continuous semigroup of linear contraction operators on a suitable Hilbert space. Thus, a theorem of existence and uniqueness of solutions admitted by such an evolution problem is proved.

The problem of a viscoelastic material with an infinite cylindrical cavity occupied by a cylinder of another viscoelastic material and subject to varying stresses at infinity, is considered in the non-inertial approximation. The case where a constant compressive stress is applied along one axis and a varying stress along the other, is solved in detail. A condition determining the contact region is given. This region depends on material parameters in a more complex way than in the elastic case. Integral equations for the pressure and gap functions are derived and numerically solved.¶Detailed numerical results are presented for materials modelled as standard linear solids. One result of interest is that during the unloading phase, the pressure distribution is quite different from the elastic circular shape. It becomes more like a uniform distribution with a sharp fall-off near the end of the contact region. For materials with large viscoelastic losses, there is a small peak near the edge of the interval and the maximum pressure is out at the edges rather than at the centre.

The problem of a smooth rigid indentor under variable loading moving across a viscoelastic half-space in one direction with variable speed is considered. The motion is assumed to be frictionless and the standard linear model is adopted to describe the viscoelastic material response. An integral equation is derived and a numerical algorithm for its solution subject to appropriate subsidiary conditions is constructed. The contact interval length, pressure, and coefficient of hysteretic friction are presented and the results discussed.

A general closed expression is given for the isothermal minimum free energy of a linear viscoelastic material in terms of Fourier-transformed quantities. A one-parameter family of free energies is constructed, ranging continuously from the maximum to the minimum free energies.

At first glance, the Court’s compromise seems Solomonic, but the wisdom of its decision has yet to be seen.

A general integral equation is derived for the problem of a rigid punch moving across a viscoelastic half-space with inertial effects included. When the half-space is modelled as a standard linear solid, it is shown that the problem is formally equivalent to a non-inertial problem with the half-space response described by a continuous-spectrum viscoelastic function. The resulting integral equation is solved numerically. The pressure function and the coefficient of hysteretic friction are plotted for various materials. The discussion is restricted to punch velocities less than the lowest speed of Rayleigh waves in the material. The theory predicts that internal frictional losses, and therefore hysteretic friction, are low for large and small viscoelastic decay times. In some cases, this gives rise to a hump-shaped curve when hysteretic friction is plotted against velocity, just as for the non-inertial theory. However, because hysteretic friction always increases sharply as the lowest Rayleigh speed is approached, its behaviour as a function of velocity, for some material densities, may be monotonic.

The problem of several indentors moving on a viscoelastic half-plane is considered in the noninertial approximation. The solution of this mixed boundary value problem is formulated in terms of a coupled system of integral equations in space and time. These are solved numerically in the steady-state limit for the case of two indentors. The phenomena of hysteretic friction and interaction between the two indentors are explored.

The solution of the problem of a crack in a viscoelastic medium subjected to a sinusoidal bending moment is formulated in terms of non-singular integral equations in space and time. These are solved numerically in the steady-state limit. The partial closing of the crack is smooth, though generally rapid, rather than sudden, in contrast to the behaviour of an elastic medium. An approximate method of solution, much simpler than the exact method, is also proposed. The integral equation method used to solve this problem is one of considerable generality. It was previously used to solve problems with transversely moving indentors on viscoelastic media. Potentially, it can be adapted to solve any non-inertial viscoelastic boundary value problem that cannot be treated by easier techniques. However, it will generally be the case that numerical methods must be used to solve the integral equations.

The steady-state problem is considered for periodic normal loading by a smooth, rigid indentor on a half-space exhibiting general viscoelastic behaviour. A technique is developed for summing the infinite series that arise. The method is applicable to the case in which the viscoelastic behaviour is described by a discrete spectrum model, in other words, by a finite sum of decaying exponentials. Numerical results are presented for two decay times. The extension to any number of decay times is straightforward.

We now focus mainly on the case ofmaterials characterized by memory kernels with only isolated singularities and derive explicit expressions for a family of free energies, including the minimum free energy discussed in Chapter 10 and the maximum free energy, which, in this context, will turn out to be less than the work function. All of these will be shown to be functionals of the minimal state.

In this initial chapter, we introduce various fundamentals: description of deformation, definition and interpretation of the strain and stress tensors, balance laws, and general restrictions on constitutive equations. These provide the foundation for later developments.

We consider in this chapter, in a mathematically abstract way, the evolution equations of the kind discussed in Chapters 18 and 21, comparing the new state formulation with the traditional history approach. Relevant background to this discussion is the concept of a minimal state discussed in Part III from Section 6.4 onward, in particular, certain conclusions of Section 15.2.

The evolution of any material system is described by means of partial differential equations.With a suitable choice of controls, which may be source terms or boundary conditions, we can act on a given state of the material.

In this chapter, we discuss various fundamental concepts and results in continuum thermodynamics. Some examples are given in terms of the materials discussed in Part I, generalized to a nonisothermal context.