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The Quasiconvex Envelope of Conformally Invariant Planar Energy Functions in Isotropic Hyperelasticity
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The Quasiconvex Envelope of Conformally Invariant Planar Energy Functions in Isotropic Hyperelasticity
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Journal Article
The Quasiconvex Envelope of Conformally Invariant Planar Energy Functions in Isotropic Hyperelasticity
2020
Overview
We consider
conformally invariant
energies
W
on the group
GL
+
(
2
)
of
2
×
2
-matrices with positive determinant, i.e.,
W
:
GL
+
(
2
)
→
R
such that
W
(
A
F
B
)
=
W
(
F
)
for all
A
,
B
∈
{
a
R
∈
GL
+
(
2
)
|
a
∈
(
0
,
∞
)
,
R
∈
SO
(
2
)
}
,
where
SO
(
2
)
denotes the special orthogonal group and provides an explicit formula for the (notoriously difficult to compute)
quasiconvex envelope
of these functions. Our results, which are based on the representation
W
(
F
)
=
h
(
λ
1
λ
2
)
of
W
in terms of the singular values
λ
1
,
λ
2
of
F
, are applied to a number of example energies in order to demonstrate the convenience of the singular-value-based expression compared to the more common representation in terms of the distortion
K
:
=
1
2
‖
F
‖
2
det
F
. Applying our results, we answer a conjecture by Adamowicz (in: Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Serie IX. Matematica e Applicazioni, vol 18(2), pp 163, 2007) and discuss a connection between polyconvexity and the Grötzsch free boundary value problem. Special cases of our results can also be obtained from earlier works by Astala et al. (Elliptic partial differential equations and quasiconformal mappings in the plane, Princeton University Press, Princeton, 2008) and Yan (Trans Am Math Soc 355(12):4755–4765, 2003). Since the restricted domain of the energy functions in question poses additional difficulties with respect to the notion of quasiconvexity compared to the case of globally defined real-valued functions, we also discuss more general properties related to the
W
1
,
p
-quasiconvex envelope on the domain
GL
+
(
n
)
which, in particular, ensure that a stricter version of
Dacorogna’s formula
is applicable to conformally invariant energies on
GL
+
(
2
)
.
Publisher
Springer US,Springer Nature B.V
