241 research outputs found
The configurational and standard force balances are not always statements of a single law
By studying the asympototic connection between phase-field and sharp-interface theories for transitions between two phases distinguished only by their constant free-energy densities, we show that the configurational force balance of the sharpinterface theory is completely unrelated to standard force balance. This demonstrates the fallacy of a recently asserted view that the balances for configurational and standard forces are never independent.published or submitted for publicationis not peer reviewe
Theory for atomic diffusion on fixed and deformable crystal lattices
We develop a theoretical framework for the diffusion of a single
unconstrained species of atoms on a crystal lattice that provides a
generalization of the classical theories of atomic diffusion and
diffusion-induced phase separation to account for constitutive nonlinearities,
external forces, and the deformation of the lattice. In this framework, we
regard atomic diffusion as a microscopic process described by two independent
kinematic variables: (i) the atomic flux, which reckons the local motion of
atoms relative to the motion of the underlying lattice, and (ii) the time-rate
of the atomic density, which encompasses nonlocal interactions between
migrating atoms and characterizes the kinematics of phase separation. We
introduce generalized forces power-conjugate to each of these rates and require
that these forces satisfy ancillary microbalances distinct from the
conventional balance involving the forces that expend power over the rate at
which the lattice deforms. A mechanical version of the second law, which takes
the form of an energy imbalance accounting for all power expenditures
(including those due to the atomic diffusion and phase separation), is used to
derive restrictions on the constitutive equations. With these restrictions, the
microbalance involving the forces conjugate to the atomic flux provides a
generalization of the usual constitutive relation between the atomic flux and
the gradient of the diffusion potential, a relation that in conjunction with
the atomic balance yields a generalized Cahn-Hilliard equation.Comment: To appear in Journal of Elasticity, 18 pages, requires kluwer macr
Microforces and the Theory of Solute Transport
A generalized continuum framework for the theory of solute transport in
fluids is proposed and systematically developed. This framework rests on the
introduction of a generic force balance for the solute, a balance distinct from
the macroscopic momentum balance associated with the mixture. Special forms of
such a force balance have been proposed and used going back at least as far as
Nernst's 1888 theory of diffusion. Under certain circumstances, this force
balance yields a Fickian constitutive relation for the diffusive solute flux,
and, in conjunction with the solute mass balance, provides a generalized
Smoluchowski equation for the mass fraction. Our format furnishes a systematic
procedure for generalizing convection-diffusion models of solute transport,
allowing for constitutive nonlinearities, external forces acting on the
diffusing constituents, and coupling between convection and diffusion.Comment: 17 page
Importance and effectiveness of representing the shapes of Cosserat rods and framed curves as paths in the special Euclidean algebra
We discuss how the shape of a special Cosserat rod can be represented as a
path in the special Euclidean algebra. By shape we mean all those geometric
features that are invariant under isometries of the three-dimensional ambient
space. The representation of the shape as a path in the special Euclidean
algebra is intrinsic to the description of the mechanical properties of a rod,
since it is given directly in terms of the strain fields that stimulate the
elastic response of special Cosserat rods. Moreover, such a representation
leads naturally to discretization schemes that avoid the need for the expensive
reconstruction of the strains from the discretized placement and for
interpolation procedures which introduce some arbitrariness in popular
numerical schemes. Given the shape of a rod and the positioning of one of its
cross sections, the full placement in the ambient space can be uniquely
reconstructed and described by means of a base curve endowed with a material
frame. By viewing a geometric curve as a rod with degenerate point-like cross
sections, we highlight the essential difference between rods and framed curves,
and clarify why the family of relatively parallel adapted frames is not
suitable for describing the mechanics of rods but is the appropriate tool for
dealing with the geometry of curves.Comment: Revised version; 25 pages; 7 figure
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