Whirl Mappings on Generalised Annuli and the Incompressible Symmetric Equilibria of the Dirichlet Energy
Whirl Mappings on Generalised Annuli and the Incompressible Symmetric Equilibria of the Dirichlet...
Morris, Charles; Taheri, Ali
2018-05-29 00:00:00
In this paper we show a striking contrast in the symmetries of equilibria and extremisers of the total elastic energy of a hyperelastic incompressible annulus subject to pure displacement boundary conditions. Indeed upon considering the equilibrium equations, here, the nonlinear second order elliptic system formulated for the deformation
u
=
(
u
1
,
…
,
u
N
)
$u=(u_{1}, \ldots, u_{N})$
:
E
L
[
u
,
X
]
=
{
Δ
u
=
div
(
P
(
x
)
cof
∇
u
)
in
X
,
det
∇
u
=
1
in
X
,
u
≡
φ
on
∂
X
,
$$ {\mathbb{E}} {\mathbb{L}}[u, {\mathbf {X}}] = \left \{ \textstyle\begin{array}{l@{\quad}l} \Delta u = \operatorname{div}(\mathscr{P} (x) \operatorname{cof} \nabla u) & \textrm{in }{\mathbf {X}},\\ \det\nabla u = 1 & \textrm{in }{\mathbf {X}},\\ u \equiv\varphi& \textrm{on }\partial{\mathbf {X}}, \end{array}\displaystyle \right . $$
where
X
${\mathbf {X}}$
is a finite, open, symmetric
N
$N$
-annulus (with
N
≥
2
$N \ge2$
),
P
=
P
(
x
)
$\mathscr{P}=\mathscr{P}(x)$
is an unknown hydrostatic pressure field and
φ
$\varphi$
is the identity mapping, we prove that, despite the inherent rotational symmetry in the system, when
N
=
3
$N=3$
, the problem possesses no non-trivial symmetric equilibria whereas in sharp contrast, when
N
=
2
$N=2$
, the problem possesses an infinite family of symmetric and topologically distinct equilibria. We extend and prove the counterparts of these results in higher dimensions by way of showing that a similar dichotomy persists between all odd vs. even dimensions
N
≥
4
$N \ge4$
and discuss a number of closely related issues.
http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.pngJournal of ElasticitySpringer Journalshttp://www.deepdyve.com/lp/springer-journals/whirl-mappings-on-generalised-annuli-and-the-incompressible-symmetric-XzuEbDAkB8
Whirl Mappings on Generalised Annuli and the Incompressible Symmetric Equilibria of the Dirichlet Energy
In this paper we show a striking contrast in the symmetries of equilibria and extremisers of the total elastic energy of a hyperelastic incompressible annulus subject to pure displacement boundary conditions. Indeed upon considering the equilibrium equations, here, the nonlinear second order elliptic system formulated for the deformation
u
=
(
u
1
,
…
,
u
N
)
$u=(u_{1}, \ldots, u_{N})$
:
E
L
[
u
,
X
]
=
{
Δ
u
=
div
(
P
(
x
)
cof
∇
u
)
in
X
,
det
∇
u
=
1
in
X
,
u
≡
φ
on
∂
X
,
$$ {\mathbb{E}} {\mathbb{L}}[u, {\mathbf {X}}] = \left \{ \textstyle\begin{array}{l@{\quad}l} \Delta u = \operatorname{div}(\mathscr{P} (x) \operatorname{cof} \nabla u) & \textrm{in }{\mathbf {X}},\\ \det\nabla u = 1 & \textrm{in }{\mathbf {X}},\\ u \equiv\varphi& \textrm{on }\partial{\mathbf {X}}, \end{array}\displaystyle \right . $$
where
X
${\mathbf {X}}$
is a finite, open, symmetric
N
$N$
-annulus (with
N
≥
2
$N \ge2$
),
P
=
P
(
x
)
$\mathscr{P}=\mathscr{P}(x)$
is an unknown hydrostatic pressure field and
φ
$\varphi$
is the identity mapping, we prove that, despite the inherent rotational symmetry in the system, when
N
=
3
$N=3$
, the problem possesses no non-trivial symmetric equilibria whereas in sharp contrast, when
N
=
2
$N=2$
, the problem possesses an infinite family of symmetric and topologically distinct equilibria. We extend and prove the counterparts of these results in higher dimensions by way of showing that a similar dichotomy persists between all odd vs. even dimensions
N
≥
4
$N \ge4$
and discuss a number of closely related issues.
Journal
Journal of Elasticity
– Springer Journals
Published: May 29, 2018
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