Volume 9 issue 1 Nuclear Future


Conversely, a PhD or EngD route is well suited to this kind of work and provides an ideal way for the nuclear industry to work with universities on problems directly relevant to their business. The expectation is that this work will continue to develop and completing students will begin to be embedded into industry.


Peridynamics modelling


The fuel-performance model described above is based on FE methods. Although already extensively deployed in industry, the model has difficulties describing the behaviour of discontinuities such as cracks, bubbles and splinters. By comparison, peridynamics is a relatively recent method that overcomes these issues. We will now go on to give a brief description of the method before continuing to describe some of the work being carried out within the Centre in applying peridynamics to nuclear engineering problems. Peridynamics is a non-local modelling technique capable


of describing a wide range of length scales and material phenomena. It has been developed, primarily over the last decade, to model ballistic impact effects in brittle solids. The non-local approach behind it has been around for many more years, however. The name peridynamics was coined by Silling in 19984,5


; it means ‘all around force’ and is derived from the


integral approach to solving the constitutive equations. Rather than assuming a point within a material is influenced by other points that are confined to its immediate locality, or are an infinitesimal distance away (as in classical continuum theories), the point is influenced by points within a finite cut-off radius known as the horizon distance. Figure 2 illustrates the peridynamics horizon in two


dimensions: it shows the bonds between material points and the other points within their horizon. For clarity, the bonds associated with an individual material point have been highlighted in red within the figure. In order to calculate the forces for this central material point, one now only needs to integrate the bond responses within the horizon (the dashed red- line within the figure). Most of the work published so far has been concerned with


the development and extension of the peridynamics theory, with only a few papers written on its application to real engineering problems6


. While it has mostly been developed for stress, strain


and ultimately fracture of brittle solids, it can be used to model many other problems including heat transfer and diffusion7,8 phase changes9


, and plasticity10 . Often it can do this without the


numerical problems associated with more conventional methods such as FE. Because peridynamics is an integral approach it does not have problems dealing with the formation of discontinuities in the material, such as cracks, that cannot be defined by the partial differential equation approach of classical FE, for surfaces or crack tips. Peridynamics can also accurately describe material behaviour from the nano-scale11


(where local approaches can


fail due to non-local interactions) up to the macro-scale; and unlike many approaches, including FE, it does not force the user to pre-impose a particular damage path. At the micro-scale and larger, the pairwise force function has a direct relationship to the bulk material properties and so is easily defined. Further, for many problems such as brittle fracture, the FE approach is highly mesh-dependent; by comparison, this is not a problem in peridynamics12


. Even the extended finite element method (XFEM), being


developed to combat the shortcomings of the FE approach to damage problems, has disadvantages compared with peridynamics. For instance, in the case of dynamic cracks in brittle solids (e.g. glass) XFEM can have difficulty in describing the speed of crack growth, which is not problematic for peridynamics13


. The only significant disadvantages of


peridynamics in comparison with more established methods such as XFEM, is that it has not been widely deployed in industry and consequently does not have the same kind of support in commercial modelling packages as FE approaches (although this will surely follow as the technique gains acceptance). A number of discontinuities within nuclear fuel can limit its


in-reactor performance. These take the form of bubbles, broken fuel slivers that are bonded to the cladding, and both radial and circumferential fuel cracks. Within the Centre an implicit peridynamics framework has been developed to look at these


Materials modelling at Imperial 45


Copyright © Imperial College London / Tanya Lloyd


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