Fundamentals - Preventive Biomechanics - page 143

On the basis of H ILL ' S inequality for instance (Ogden 1972a), deduced for
incompressible materials, using a strain-energy function of the form (see ( 3.203 )in
Sect. 3.2.6.2 )

w ¼ X

N

l j
a j

k a 1 þ k a j 2 þ k a j 3 3 ð 3 : 440 Þ

j ¼ 1

the following restriction on the parameters

l j a j [ 0 ð j ¼ 1 ; ... ; N Þ:

ð 3 : 441 Þ

Equation ( 3.441 ) is required to hold for each term j of the series expansion of
( 3.440 ) with no summation over j.
The latter class of inequalities of the above listed, the polyconvexity require-
ment (polyconvexity guarantees the existence of real wave speeds in the material
for all possible deformations), was shown by B ALL to be crucial for a strain-energy
function to clear out the conflicts with physical requirements (e.g. buckling phe-
nomena, ( 3.393 ) 1 and (3.393) 2 . In Ball (1977), it is outlined that a strain-energy
function is polyconvex if it is convex in its respective arguments, namely F, Cof
F and det F (see Sect. 3.2.6.2 Eq. ( 3.206 ))

w ðÞ¼ w F ; Cof F ; det F ð Þ ð 3 : 442 Þ

where the cofactor of the deformation gradient F according to ( 3.207 ) 4 is defined
as Cof F : ¼ð det F Þ F T .

The criterion of polyconvexity leads, according to Ciarlet (1988), p.174-181 et
seq. and Dhondt (2004), p.189, to the definition of a general class of strain-energy
functions of the form (without the normalizing constant 3 ¼ tr I)

w ð k 1 ; k 2 ; k 3 Þ¼ X a i ð k c 1 þ k c 2 þ k c 3 Þþ X

m n

b j J d j ð k d 1 þ k d 2 þ k d 3 Þþ f ð J Þ

i ¼ 1 j ¼ 1

ð 3 : 443 Þ

with the following requirements on the coefficients and exponents respectively,
and the function f ð J Þ ,

a i [ 0 and c i 1 ; 1 i m ;
b j [ 0 and d j 1 ; 1 j n ;
f ð J Þ is convex for 0 J þ1:

ð 3 : 444 Þ

Comparison of the O GDEN -H ILL model, ( 3.209 ), with ( 3.443 ) with b j ¼ 0 leads
with the polyconvexity requirements ( 3.444 ), to the following restrictions on l j
and a j of ( 3.209 )

l j [ 0 and a j 1 for j ¼ 1 ; ... ; N : ð 3 : 445 Þ