Theoretical and computational peridynamics for fracture of materials and structures.

The classical theory of continuum mechanics has been in use by engineers for more than
a century. The birth and development of di erential calculus gave Navier and Cauchy a
powerful tool, with which to build its foundations. In time, technology also progressed,
which, together with the development of associated numerical methods, such as the -
nite element (FE) method, allowed engineers and scientists to tackle many practical and
theoretical problems of increasing complexity, within the framework of this theory.
However, with this increasing demand, many problems emerged which could not be
adequately addressed within the existing theoretical framework. Explaining the failure
of materials, modelling materials with internal length scale, such as brous or composite
materials, or problems involving di erent length scales fall into this category.
In the eld of biology, there are many phenomena which, when modelled from an
engineering point of view, lead to the considerations mentioned above. In
cell migration, for example, which occurs in many cases, such as wound healing or, less fortunately, tumour
spreading,long-range coordinated movement of cells has been observed, moving like a
family. As another example, many tissues in the human body, such as the skin or arteries, are
partly comprised of a network of collagen bers and it is these bers that are responsible
for the strength of these tissues. [8, 9]. These ber networks, on the other hand, have
mechanical properties which are long-range correlated , up to a distance of approximately 0=2, where L0 is the ber length and these correlations are dependent on the scale ofobservation. These long-range correlations, introduced by the nite length of the bers, is not adequately captured in classical continuum mechanics since the derivatives
which are employed by it, by their de nition as limits, assume this length scale to vanish.
In an attempt to address such issues but, at the same time, building on the existing
foundations, scientists and engineers created
ad hoc additions to the classical theory. The
linear elastic fracture mechanics attempted to model failure from a theoretical point o
view, while specialised numerical techniques, such as the extended nite element methods,
attempted to complement this form a numerical point of view, in the same way that nite
elements complement the classical theory.
On the other hand, as we will see, several approaches emerged to account, to some
degree, for length scale and nonlocality, either in the constitutive model or balance laws.
But these methods seemed somewhat \arti cial", because they attempted to build on
pre-existing foundations, to which these limitations were inherent.
i
It was to this end that Silling in 2000 introduced the
peridynamic formulation, which addresses many of the shortcoming of the classical theory and is in a natural way, using
integration, rather than di erentiation. It is, therefore, suited for applications in these
areas.
The formulation di ers from classical continuum mechanics in that it is based on dis-
placements and forces, rather than strains and stresses. Therefore, constitutive models in
peridynamics should, also, be based on the former primal elds and not the latter, which is
their derivatives. This is because relative displacements and the corresponding forces can
be formulated for nite distances, taking into account long-range interactions mentioned
above, while strains and stresses, by their de nition, limit these interactions to in nitesimal
distances.