Abstract:
|
Elastomers
are viscoelastic polymers with low Young's modulus and high failure strain
that are used in many
c
ivil
e
ngineering applications, including bridge bearings, seismic isolators for
buil
dings and resilient rail wheels.
Their constitutive behaviour
i
s characterized by a nonlinear stress
-
strain relation with
an
extensibility limit
. This contrasts
with materials that have
instead
a limit on the tensile stresses, such
as mild steel.
This
MSc
thesis
is concerned with the
numerical modeling
of elastomers.
T
his involves
dealing with a medium
with two phases:
a constrained region, wher
e
the particles have reached their
maxi
mum allowable deformation, and a
free region, where the
inextensibility constraint
is still
inactive.
Moreover, one can think of an interf
ace splitting the two phases of the medium. If the focus is
put in obtaining methods to
locate and evolve such interface, then a two phase medium with a
moving interface
is considered.
From the mathematical point of view, this is a constrained minimization
problem. One of the
strategies to solve it is to turn the minimization problem into a shape equilibrium one. This
approach has b
een successfully employed for an
interface location problem
in small strains
and
serves as the starting point of this work.
T
hu
s, t
he main purpose of this thesis is to extend this formulation to a
large strains
interface
locating and evolving scenario. A first analysis of the problem reveals two sources of
nonlinearity
:
the inextensibility constraint and the kinematics in large st
rains.
A simple
but
thorough
one
-
dimensional
study of the problem is
then
developed
to find methods
to sort out
both
nonlinearities.
Following this,
explicit iterative schemes to locate and evolve one or
multiple interfaces
are straightforwardly obtained i
n 1D linear elasticity.
However, the same ideas applied to a simple
St.Venant
-
Kirchhoff
hyperelasticity
model
,
evidences that even very simple 1D problems become rather complex and cannot be solved as
directly and explicit as before.
Numerical examples are
provided throughout this analysis and they are also useful to conclude
that both locating and evolving the interface can be
essentially
seen as the same problem, but with
different driving effects.
After that,
an extension of the one
-
dimensional schemes t
o two or more dimensions
is explored
.
Although the same ideas can be applied, more sophisticated modeling tools are required, namely,
the
X
-
FEM and Level set
methods, the
shape sensitivity
analysis and the
Arbitrary Lagrangian
-
Eulerian methods
.
A
complemen
tary
numerical implementation of the pr
oposed strategy is to show its
computational benefits. In particular, a combination of the three previous techniques shall make
unnecessary a stepwise update of the Level set.
T
he work presented here
may not be limite
d to this particular case and
be r
elevant to
other
engineering problems involving
moving interfaces
and boundaries
, such as plasticity analysis or
the saturation of a porous medium. |