Abstract:
Many problems in solid dynamics involve moving boundaries, finite elastoplastic
deformations, and strong non-linear waves. Continuum modelling
of such events is difficult on account of these characteristics, and there
exist a number of inadequacies in current numerical algorithms. Furthermore, a
comprehensive understanding of certain underlying processes is yet to be achieved
which places a limit on the derivation of engineering models to simulate these occurrences.
Much needed atomistic studies, capable of revealing much about the governing
physical processes, remain limited by current computational resources. This
thesis is devoted to targeting these difficulties by proposing new continuum numerical
schemes and a means of studying both micro- and macro-scale behaviours via
a dynamic coupling of continuum mechanics and molecular dynamics theory.
Eulerian shock-capturing schemes have advantages for modelling problems involving
complex non-linear wave structures and large deformations in solid media.
Various numerical methods now exist for solving hyperbolic conservation laws that
have yet to be applied to solid dynamics. A three-dimensional finite-volume scheme
on fixed grids is proposed for elastoplastic solids. The scheme is based upon the Godunov
flux method and thus requires solution of the Riemann problem. Both exact
and approximate solutions are proposed for the special case of non-linear elasticity.
An implicit algorithm is developed to allow for resolving rate-dependent inelastic
deformations. The methods are tested against exact solutions in one-dimension, and
symmetrical polar solutions in two- and three-dimensions.
To account for multiple immiscible materials it is necessary to include some
means of tracking material boundaries within a numerical scheme. A moving grid
scheme is a simple means of accommodating transient boundaries. Interface tracking
based on the use of level set functions is an attractive alternative for problems
with sliding interfaces since it allows discontinuous velocity profiles at the material
boundaries whilst employing fixed grids. Both of these methods are explored in
the current context. A series of one-dimensional testcases have been carried out
that demonstrate the ability of the numerical schemes to accurately resolve complex
boundary conditions between interacting free surfaces.
Where singularities occur in a system comprising solid materials, atomistic studies
are invaluable for achieving a fundamental insight into the governing physical
processes. However where non-linear waves are generated, domain size proves to
be a limiting factor in achieving solutions free from numerical artifacts. A domain
decomposition multi-scale modelling strategy is developed that couples the Eulerian
shock capturing scheme with a molecular dynamics solver. The method is
demonstrated for one-dimensional testcases involving strong shear waves and multiple
components. Attention is devoted to resolving transient wave propagation free
from spurious wave reflections through investigation of the numerical parameters.