Macroscopic limits of microscopic models

Many physical systems are comprised of several discrete elements, the equations of motion of each element being known. If the system has a large number of degrees of freedom, it may be possible to treat it as a continuous system. In this event, one might wish to derive the equations of motion of the continuous (macroscopic) system by taking a suitable limit of the equations governing the discrete (microscopic) system. The classical example of this involves a row of particles with each particle connected to its nearest neighbor by a linear spring, its continuum counterpart being a linearly elastic bar; see Fig. 1. In a typical undergraduate engineering subject on, for example, Dynamics, the transition from a discrete system to a continuous system is usually carried out through a formal Taylor expansion of the terms of the discrete model about some reference configuration. The aim of this paper is to draw attention to the fact that a macroscopic model derived in this way should be examined critically in order to confirm that it provides a faithful representation of the underlying microscopic model. We use a specific (striking) example to make this point. In this example, a simple solution of the discrete model can be stable or unstable depending on the state of the system. However, the corresponding solution of the continuous system is always unstable! We go on to show how the dispersion relations of the two models can be used to identify the source of the discrepancy and to suggest how one might modify the continuous model

A One-Dimensional Peridynamic Model of Defect Propagation and its Relation to Certain Other Continuum Models

The peridynamic model of a solid does not involve spatial gradients of the displacement field and is therefore well suited for studying defect propagation. Here, bond-based peridynamic theory is used to study the equilibrium and steady propagation of a lattice defect -- a kink -- in one dimension. The material transforms locally, from one state to another, as the kink passes through. The kink is in equilibrium if the applied force is less than a certain critical value that is calculated, and propagates if it exceeds that value. The kinetic relation giving the propagation speed as a function of the applied force is also derived. In addition, it is shown that the dynamical solutions of certain differential-equation-based models of a continuum are the same as those of the peridynamic model provided the micromodulus function is chosen suitably. A formula for calculating the micromodulus function of the equivalent peridynamic model is derived and illustrated. This ability to replace a differential-equation-based model with a peridynamic one may prove useful when numerically studying more complicated problems such as those involving multiple and interacting defects

On a shock-induced martensitic phase transition

A recently developed continuum-mechanical model for stress-induced phase transitions in solids is applied to a transition generated by impact. The role of transition kinetics in determining the macroscopic response to impact is discussed; in addition, the special way that “overdriven” phase boundaries emerge in this model is described. The predictions of the model are compared with experiments involving shock-induced graphite-to-diamond phase transitions

A continuum model for the growth of dendritic actin networks

Polymerization of dendritic actin networks underlies important mechanical processes in cell biology such as the protrusion of lamellipodia, propulsion of growth cones in dendrites of neurons, intracellular transport of organelles and pathogens, among others. The forces required for these mechanical functions have been deduced from mechano-chemical models of actin polymerization; most models are focused on single growing filaments, and only a few address polymerization of filament networks through simulations. Here we propose a continuum model of surface growth and filament nucleation to describe polymerization of dendritic actin networks. The model describes growth and elasticity in terms of macroscopic stresses, strains and filament density rather than focusing on individual filaments. The microscopic processes underlying polymerization are subsumed into kinetic laws characterizing the change of filament density and the propagation of growing surfaces. This continuum model can predict the evolution of actin networks in disparate experiments. A key conclusion of the analysis is that existing laws relating force to polymerization speed of single filaments cannot predict the response of growing networks. Therefore a new kinetic law, consistent with the dissipation inequality, is proposed to capture the evolution of dendritic actin networks under different loading conditions. This model may be extended to other settings involving a more complex interplay between mechanical stresses and polymerization kinetics, such as the growth of networks of microtubules, collagen filaments, intermediate filaments and carbon nanotubes

Discontinuous Deformation Gradients in Plane Finite Elastostatics of Incompressible Materials. (I) General Considerations. (II) An Example

This investigation is concerned with the possibility of the change of type of the differential equations governing finite plane elastostatics for incompressible elastic materials, and the related is sue of the existence of equilibrium fields with discontinuous deformation gradients. Explicit necessary and sufficient conditions on the deformation invariants and the material for the ellipticity of the plane displacement equations of equilibrium are established. The issue of the existence, locally, of "elastostatic shocks" -- elastostatic fields with continuous displacements and discontinuous deformation gradients -- is then investigated. It is shown that an elastostatic shock exists only if the governing field equations suffer a loss of ellipticity at some deformation. Conversely, if the governing field equations have lost ellipticity at a given deformation at some point, an elastostatic shock can exist, locally, at that point. The results obtained are valid for an arbitrary homogeneous, isotropic, incompressible, elastic material. In order to illustrate the occurrence of elastostatic shocks in a physical problem, a specific displacement boundary value problem is studied. Here, a particular class of isotropic, incompressible, elastic materials which allow for a loss of ellipticity is considered. It is shown that no solution which is smooth in the classical sense exists to this problem for certain ranges of the applied loading. Next, we admit solutions involving elastostatic shocks into the discussion and find that the problem may then be solved completely. When this is done, however, there results a lack of uniqueness of solutions to the boundary value problem. In order to resolve this non-uniqueness, dissipativity and stability are investigated.</p