Biomedical Engineering Reference
In-Depth Information
where w denotes the strain-energy function.
As suggested by physical evidence, S
i
is a strictly increasing function of its
conjugate k
i
. Thus, considering the above relation, S
i
ð
k
i
Þ
, the strain-energy
function is required to be strictly convex (cf. remarks below) and thus must hold
(Marsden and Hughes 1983),
2
w
ok
i
o
[ 0
ð
i
¼
1
;
2
;
3
Þ:
ð
3
:
436
Þ
Condition (
3.434
) and (
3.436
) are referred to as the first and second Baker-
Erickson inequalities.
Remarks:
A function is convex if it satisfies one of the following conditions of convexity:
(i) As outlined in Ciarlet (1989), a function f is convex, or strictly convex if the
inequalities are strict, if the following inequality holds:
f
ð
hu
þð
1
h
Þ
v
Þ
hf
ð
u
Þþð
1
h
Þ
f
ð
v
Þ
for
u
;
v
2
U
;
u
6¼
v
and
0
h
1
ð
3
:
437
Þ
with u and v being two distinct points of convex subset U of a vector space,
i.e. for all u and v in that subset and h in the interval h
2
0
;
1
½
all points
f
ð
hu
þð
1
h
Þ
v
Þ
are situated in that subspace.
(ii) First order condition: a differentiable function f is convex on U, or strictly
convex if the inequalities are strict, if the following inequality holds:
f
ð
v
Þ
f
ð
u
Þþ
f
0
ð
u
Þð
v
u
Þ
for
u
;
v
2
U
;
u
6¼
v
:
ð
3
:
438
Þ
Geometrically speaking (
3.437
) requires the function f to lie above any tan-
gent plane.
(iii) Second order condition: a twice differentiable function f is convex on U,or
strictly convex if the inequalities are strict, if the following inequality holds:
f
00
ð
u
Þð
v
u
;
v
u
Þ
0
for
u
;
v
2
U
;
u
6¼
v
:
ð
3
:
439
Þ
Beside the B
AKER
-E
RICKSEN
inequalities, other classes of inequalities such as the
following exist
• H
ILL
-inequalities (Hill 1970)
• L
EGENDRE
-H
ADAMARD
condition (Hadamard 1903)
• Q
UASI
-convexity condition of M
ORREY
(Morrey 1952)
• C
OLEMAN
-N
OLL
condition (Coleman and Noll 1959)
• Concept of polyconvexity (Ball 1977)
