Peridynamics

Q:So what exactly is peridynamics?
A: Well, it's a new formulation of solid mechanics, which in turn, is part of continuum mechanics. Continuum mechanics represents those parts of the macroscopic world which can be idealised as continuous, extended entities. If you've got a gas or a liquid, you can represent it using fluid mechanics. Fluids, however, don't have strength, whereas solids do. To represent a solid, you need to use solid mechanics.

Q: So why the need for a new formulation?
A: Well, it's basically all about fracture. The trouble with fracture is that, by definition, it constitutes a discontinuity in a solid, and given that solid mechanics is predicated upon the continuity of things, the conventional formulation struggles to deal with fracture.

Q: And what does peridynamics postulate to resolve the problem?
A: Cauchy's momentum equation, the governing equation of continuum mechanics, defines the force at a point by the divergence of the stress tensor. The divergence is, of course, a differential operator, and if your equations are based upon derivatives, then your equations will fail in the presence of a discontinuity. Peridynamics attempts to get around this by replacing the spatial derivatives of the stress tensor at each point with the integral of a force density function centred at that point. This, then, is a radical approach, which attempts to generalise from Cauchy's conception of the internal stresses in a solid. The field equations in this formulation, it is claimed, can be applied to discontinuities such as cracks.

Q: Are there any philosophical implications?
A: Definitely, yes. On smaller length scales, where fluids and solids are discrete, people use something called Molecular Dynamics to represent substances. And the equations of Molecular Dynamics are intrinsically non-local; the net force on each particle is determined by the joint effect of all the inter-atomic forces due to other particles, not just those immediately adjacent to the particle in question. Finding the force on a particle by adding all the contributions from particles in a neighbourhood of that particle, is a discrete version of an integral. Conventional solid mechanics, however, is distinctly local. This means that the inter-theoretic relationship between Molecular Dynamics and conventional solid mechanics is very unsatisfactory. However, by using the non-local reformulation provided by peridynamics, the inter-theoretic relationship is far more satisfactory.

It's an interesting case, which demonstrates that macroscopic theories sometimes need to be reformulated using concepts and structures taken from the microscopic theory.