Analysis of admissible steady-state fracture processes in discrete lattice structures

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Student thesis: Doctoral ThesisDoctor of Philosophy

Original languageEnglish
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Award date2017
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Abstract

The purpose of this work is to study physically possible crack propagation
at constant velocity inside a discrete solid by means of theoretical analysis
supported by numerical simulations.

Analytical solutions are delivered for fracture problems in one dimensional
chains, a double chain and square lattices. Evaluation of obtained solutions required implementation of numerical algorithms for computation of integral transforms. Consideration of one-dimensional cases, namely a simple chain of oscillators and a chain of masses with non-local interactions, allowed to examine the validity of derived formulae by a complementary computer
simulation of a corresponding dynamic system. Starting from simple models, the analysis of physically admissible and forbidden fracture regimes has been performed. The analytical predictions of possible steady states found a good agreement with a purely numerical scheme.

The work discusses the advantages of different approaches to study
steady-state failure processes: either with energetic or load characteristics.
These attributes of fracture mechanics are shown to be effient for quantifying
global predictions, e.g. a choice a particular loading condition for achieving a certain value of a crack speed. However, it was demonstrated that derivation of these characteristics is not enough and consideration of the displacement or stress fields should be performed.

The results on chains with non-local interactions between the oscillators
illustrated the features of failure at micro-level. Namely, different combinations of microscopic parameters, that result in the same bulk quantities, reflect different patterns of crack propagation in discrete solids.

A problem of a separation a double chain compounded by two chains
with different properties shows the peculiarities of parameters mismatch.
Particularly, it was established that, contrary to quasi-static problems, a
steady-state separation is necessarily caused by forces, applied to each chain,
of different values. Furthermore, distinct material parameters of chains give
a chance for the observation of the supersonic fracture of the structure.

Increasing the problem dimension from chains to lattices, several new features
emerged. For instance, the behaviour of displacements along a crack
path changes. Moreover, the admissibility analysis is expanded to the consideration of possible fracture behind a crack tip. The outcomes predict
crack propagation regimes with high energy release rates be accompanied
by snapping of the springs on the faces of the original moving crack. The
evaluation of displacement eld in the direction orthogonal to a crack path
is also presented. The contrast in material properties in anisotropic lattices
and mismatch of material properties in dissimilar lattices unveiled different
scenarios of admissible regimes.

Furthermore, the question of the choice of a particular fracture criterion
is addressed. Two history-dependent criteria are compared to the classical
one of threshold elongation for linear bonds. The results show that steadystate
regimes can be reached in the low subsonic crack speed range which
can not be according to the classical criterion. Repercussions in terms of
load and crack opening versus velocity are explained in details. Once known
the steady-state regimes of fracture propagation, a procedure for applying
history-dependent criteria emerges as not restricted to the two examined
ones and opens the way to dierent and more complex problems.