# AMS: The theory of peridynamics

The theory of peridynamics is a fairly new idea, having only been first suggested by Stewart Silling of Sandia National Laboratories in 2000. However, it has been garnering more and more interest from those in both mathematics and material science as of late for its apparent ability to do something that has never been done before: Provide an accurate model for the phenomena of fracture dynamics. In her presentation at this week’s installment of the AMS Department’s colloquia series, Prof. Petronela Radu of the University of Nebraska-Lincoln explained in further detail what peridynamics are, how this theory works, and what its further applications could possibly be.

So what exactly is peridynamics? According to Radu, it is simply a means of defining material properties under conditions of deformity and discontinuity (specifically, fracture). The purpose for this theory being developed by Silling was that basic continuum mechanics laws could not be applied easily, if at all, to these particular fractured situations. Under continuum mechanics, Radu explained, formulations are done based upon the use of partial differential equations, meaning that computations of partial derivatives must be possible for the given material and its conditions. However, the necessary derivatives simply cannot be found on fracture surfaces, cracks, or other material deformities, meaning that another method besides continuum mechanics had to be found in order to solve these problems. This is where Silling’s peridynamic theory was introduced, according to Radu.

She pointed out that because of the lack of partial derivatives in deformity situations, Silling sought to devise a method that did not have to rely on derivatives at all. What he determined would work as an acceptable substitute were integral equations, as they required no derivation calculations at all and could be used at virtually all points of a given situation. To accomplish this, Radu explained that in basic terms, one needs to identify the differential operators within a continuum mechanics model and replace them with uniquely singular integral operators.

When numerically tested with numerous deformation situations, such as fracturing, material tearing, and bursting (as in the case of a popped balloon, for example), it was found that making the change to integral operators and the peridynamic theory produced remarkably close representations of the true physical situations. Projected crack paths were similar to the experimental peridynamic model results, including data regarding the material’s damaged zone and the crack’s branching patterns. The model also accurately portrayed the correct series of angles and structures created from tearing a material, as well as modeling with great precision the size, structure, and movement characteristics of material fragments from a burst balloon example.

Despite all of its benefits, Radu explained that an issue many in the mathematics field have with peridynamics is that there’s very little true application of rationale and logic. These opponents of the theory claim that it is basically a fortuitous result of simply “playing around” with certain parameters and that there is very little actual theory behind it. However, as Radu often cited, no other method to date has been able to model fracture dynamics and material deformities as accurately as peridynamics has, so there is most certainly good reason behind further study of this method and its possible applications.

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