Finite Elements in Solid Mechanics - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 292

(2) For so-called neo-Hookean materials ,

1
2 µ [trace (C − I) +

2

W(x,C) =

β { ( det C) − β/ 2

− 1 } ] , ( 2 . 7 )

2 ν

2 ν .
We note that (2.6) is restricted to strains which are not too large. Indeed, we
expect that

where β =

1 −

W(x,F) −→ ∞

as det F → 0 , ( 2 . 8 )

since det F → 0 means that the density of the deformed material becomes very

large. The condition (2.8) implies that W is not a convex function of F . Indeed,
the set of matrices

3

B ={ F ∈ M ; det F> 0 } ( 2 . 9 )

=
0 which are the convex combination of two matrices F 1 and F 2 with positive
determinants. By the continuity of W at F 1 and F 2 , we would get the boundedness
in a neighborhood of F 0 whenever W is assumed to be convex.

is not a convex set; see Problem 2.4. There are many matrices F 0 with det F 0

Problems

2.4 Show that (2.9) does not define a convex set by considering the convex
combinations of the matrices
2 −

1

2 and − 4 .

2 1

2.5 Consider a St. Venant-Kirchhoff material with the energy function (2.6), and
show that there would exist negative energy states if µ< 0 were to hold.

2.6 Consider a neo-Hookean material for small strains, and establish Hooke's
law with the same parameters µ and ν .

2.7 Often the energy functional depends on J : = det F . Show that the derivate

is given by

1

D F J : δF = J trace (F − 1 δF ),

2 J trace (C − 1 δC).

D C J : δC =

Hint. For F = I we have obviously D I JδF = trace (δF ) and det (F + δF) =

det F det (I + F − 1 δF) .