Multi-field approach in mechanics of structural solids

Abstract

a b s t r a c tWe overview the basic concepts, models, and methods related to the multi-field continuum theory of solids with complex structures. The multi-field theory is formulated for structural solids by introducing a macrocell consisting of several primitive cells and, accordingly, by increasing the number of vector fields describing the response of the body to external factors. Using this approach, we obtain several continuum models and explore their essential properties by comparison with the original structural models. Static and dynamical problems as well as the stability problems for structural solids are considered. We demonstrate that the multi-field approach gives a way to obtain families of models that generalize classical ones and are valid not only for long-, but also for short-wavelength deformations of the structural solids. Some examples of application of the multi-field theory and directions for its further development are also discussed.

Gradient elasticity in statics and dynamics: An overview of formulations, length scale identification procedures, finite element implementations and new results

“…(23) have been derived on many occasions. The exact format of the stress-strain relations depends on the types of particle interactions that are taken into account, but a general representation of such models can be written as (Chang and Gao, 1995;Mühlhaus and Oka, 1996;Suiker and de Borst, 2001;Askes and Metrikine, 2005;Vasiliev et al, 2010) r ij ¼ C ijkl e kl þ ' 2 e kl;mm…”

“…It is also mentioned that with the dominance of stable strain gradients over unstable strain gradients unbounded phase velocities are found for the higher wave numbers, see Section 2.5.1. Vasiliev et al (2010) distinguish multiple displacement fields which are related hierarchically. In their continualisation approach, each displacement field is associated with a distinct range of wave numbers, by which an improved correspondence with discrete lattice models can be obtained.…”

Lattice dynamics from a continuum viewpoint

“…As one example, we can mention a static multi-field theory (Charlotte and Truskinovsky, 2008) allowing one to reduce the discrete problem to a set of coupled second order partial differential equations. Another example is a dynamic multi-field theory (Il'iushina, 1969;Kunin, 1982;Vasiliev et al, 2010) introducing internal variables with nontrivial inertia. The problem with this general approach is that the new continuum structures are accounted for not only where they are essential, but also where the simpler CC theory could be used as well.…”

A hyperboloid structure as a mechanical model of the carbon bond

“…The use of a pairwise potential leads to the maximizing of a packing density that in turn causes the collapsing of a model. A possible solution is to take the rotational degrees of freedom for the carbon atoms into account [22,23,24] A C C E P T E D M A N U S C R I P T and develop a generalized moment (torque) potentials [25,26,27,28] for simulations. In these works the atoms are assumed to be the solid bodies, not just the material points, and interaction between them is described both by the forces and the torques.…”