John Murrough Golden

Cluster polylogarithms for scattering amplitudes

Motivated by the cluster structure of two-loop scattering amplitudes in Yang-Mills theory we define cluster polylogarithm functions. We find that all such functions of weight four are made up of a single simple building block associated with the A2 cluster algebra. Adding the requirement of locality on generalized Stasheff polytopes, we find that these A2 building blocks arrange themselves to form a unique function associated with the A3 cluster algebra. This A3 function manifests all of the cluster algebraic structure of the two-loop n-particle MHV amplitudes for all n, and we use it to provide an explicit representation for the most complicated part of the n = 7 amplitude as an example.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ?Cluster algebras in mathematical physics?.

Existence and Uniqueness

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.

A proposal concerning the physical rate of dissipation in materials with memory

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).

THE MINIMUM FREE ENERGY FOR CONTINUOUS SPECTRUM MATERIALS

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].

Thermodynamics of Materials with Memory

We now apply thermodynamic principles to field theories with memory. For general nonlinear, nonisothermal theories, we assume that a free energy is given, this being the fundamental constitutive assumption. Applying a generalization of the approach of Coleman [36], Coleman and Mizel [38], Gurtin and Pipkin [130], we derive the constitutive equations for the theory in Section 5.1. Also, fundamental properties of free energies are derived. Furthermore, some observations are made on the case of periodic histories and in relation to constraints on the nonuniqueness of free energies. In Section 5.2, an expression for the maximum recoverable work is given for general materials, together with an integral equation for the process yielding this maximum. Finally, in Section 5.3, we discuss how free energies can be constructed from combinations of simpler free energies.

Nonisothermal free energies for linear theories with memory

Explicit expressions for minimum, maximum, and intermediate free energies have recently been given for isothermal linear theories. In this paper, these results are generalized to a nonisothermal, linear model. 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 a linear theory. In particular, an integral equation for the continuation associated with the maximum recoverable work from a given 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, generalizing a result previously derived in the isothermal case. A new variational method developed for the isothermal case is extended to nonisothermal conditions. In the time domain, this approach yields integral equations for both the minimum and maximum free energies associated with a given state. In the frequency domain, explicit forms of a family of free energies, associated with a given state of a discrete spectrum material, are derived. This includes both maximum and minimum free energies. These latter developments, given previously for the isothermal scalar case, are generalized (with some restrictions) in this work to the tensor case. The physical relevance of various results are discussed.

A molecular theory of adhesive rubber friction

A theory of adhesive rubber friction is proposed where the final expression for the coefficient of friction obeys the Williams-Landel-Ferry (1955) transform and correlates with the viscoelastic shear loss modulus as shown by Grosch (1963) and others. The approach is based on the bead-spring model for polymer chains of Rouse (1953).

Free energies in a general non-local theory of a material with memory

A general theory of non-local materials, with linear constitutive equations and memory effects, is developed within a thermodynamic framework. Several free energy and dissipation functionals are constructed and explored. These include an expression for the minimum free energy and a functional that is a free energy for important categories of memory kernels and is explicitly a functional of the minimal state. The functionals discussed have a similar general form to the corresponding expressions for simple materials. A number of new results are derived for them, most of which apply equally to both types of material. In particular, detailed formulae are given for these quantities in the case of sinusoidal histories.

Thermomechanics of damage and fatigue by a phase field model

ABSTRACT 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.

Free energies and asymptotic behaviour for incompressible viscoelastic fluids

The existence, uniqueness and asymptotic stability for an incompressible, linear viscoelastic fluid is studied using a new free energy, the representation 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.