We describe ellipticity domains for the isochoric elastic energy
F
↦
∥
dev
n
log
U
∥
2
=
∥
log
F
T
F
(
det
F
)
1
/
n
∥
2
=
1
4
∥
log
C
(
det
C
)
1
/
n
∥
2
for
n
=2,3, where
C
=
F
T
F
for
F
∈
GL
+
(
n
). Here,
dev
n
log
U
=
log
U
−
(
1
/
n
)
tr
(
log
U
)
⋅
1
is the deviatoric part of the logarithmic strain tensor
log
U
. For
n
=2, we identify the maximal ellipticity domain, whereas for
n
=3, we show that the energy is Legendre–Hadamard (LH) elliptic in the set
E
3
(
W
H
iso
,
LH
,
U
,
2
3
)
:=
{
U
∈
PSym
(
3
)
|
∥
dev
3
log
U
∥
2
≤
2
3
}
,
which is similar to the von Mises–Huber–Hencky maximum distortion strain energy criterion. Our results complement the characterization of ellipticity domains for the quadratic Hencky energy
W
H
(
F
)
=
μ
∥
dev
3
log
U
∥
2
+
(
κ
/
2
)
[
tr
(
log
U
)
]
2
,
U
=
F
T
F
with
μ
>0 and
κ
>
2
3
μ
, previously obtained by Bruhns
et al.
The exponentiated Hencky energy: anisotropic extension and case studies
