The Classical Field Theories

Matter is commonly found in the form of materials. Analytical mechanics turned its back upon this fact, creating the centrally useful but abstract concepts of the mass point and the rigid body, in which matter manifests itself only through its inertia, independent of its constitution; “modern” physics likewise turns its back, since it concerns solely the small particles of matter, declining to face the problem of how a specimen made up of such particles will behave in the typical circumstances in which we meet it. Materials, however, continue to furnish the masses of matter we see and use from day to day: air, water, earth, flesh, wood, stone, steel, concrete, glass, rubber, ... All are deformable. A theory aiming to describe their mechanical behavior must take heed of their deformability and represent the definite principles it obeys.

The non-linear field theories of mechanics

We review the development of diffuse-interface models of hydrodynamics and their application to a wide variety of interfacial phenomena. These models have been applied successfully to situations in which the physical phenomena of interest have a length scale commensurate with the thickness of the interfacial region (e.g. near-critical interfacial phenomena or small-scale flows such as those occurring near contact lines) and fluid flows involving large interface deformations and/or topological changes (e.g. breakup and coalescence events associated with fluid jets, droplets, and large-deformation waves). We discuss the issues involved in formulating diffuse-interface models for single-component and binary fluids. Recent applications and computations using these models are discussed in each case. Further, we address issues including sharp-interface analyses that relate these models to the classical free-boundary problem, computational approaches to describe interfacial phenomena, and models of fully miscible fluids.

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Diffuse-interface methods in fluid mechanics

1. Introduction and overview 2. Film stress and substrate curvature 3. Stress in anisotropic and patterned films 4. Delamination and fracture 5. Film buckling, bulging and peeling 6. Dislocation formation in epitaxial systems 7. Dislocation interactions and strain relaxation 8. Equilibrium and stability of surfaces 9. The role of stress in mass transport.

Thin Film Materials: Stress, Defect Formation and Surface Evolution

Solid-solid phase transformations often lead to certain characteristic microstructural features involving fine mixtures of the phases. In martensitic transformations one such feature is a plane interface which separates one homogeneous phase, austenite, from a very fine mixture of twins of the other phase, martensite. In quartz crystals held in a temperature gradient near the α-β transformation temperature, the α-phase breaks up into triangular domains called Dauphine twins which become finer and finer in the direction of increasing temperature. In this paper we explore a theoretical approach to these fine phase mixtures based on the minimization of free energy.

Fine phase mixtures as minimizers of energy

Review of the article ''On the Equilibrium of Heterogeneous Substances'', which was published in ''Transactions of the Connecticut Academy of Arts and Sciences'', vol. 3 (1874-78), pp. 108-248 and 343-524.

J. W. Gibbs

Papers

On the Equilibrium of Heterogeneous Substances

On the Equilibrium of Heterogeneous Substances

Graphical Methods in the Thermodynamics of Fluids

On the equilibrium of heterogeneous substances

The Form of the Teeth of Wheels in Spur Gearing.