J Elasticity (2007) 86:235–243
DOI 10.1007/s10659-006-9091-z
On the Rank 1 Convexity of Stored Energy Functions
of Physically Linear Stress-Strain Relations
Albrecht Bertram · Thomas Böhlke · Miroslav Šilhavý
Received: 1 June 2005 / Accepted: 27 September 2006 /
Published online: 16 November 2006
© Springer Science + Business Media B.V. 2006
Abstract The rank 1 convexity of stored energy functions corresponding to isotropic
and physically linear elastic constitutive relations formulated in terms of generalized
stress and strain measures [Hill, R.: J. Mech. Phys. Solids 16, 229–242 (1968)] is
analyzed. This class of elastic materials contains as special cases the stress-strain
relationships based on Seth strain measures [Seth, B.: Generalized strain measure
with application to physical problems. In: Reiner, M., Abir, D. (eds.) Second-order
Effects in Elasticity, Plasticity, and Fluid Dynamics, pp. 162–172. Pergamon, Oxford,
New York (1964)] such as the St.Venant–Kirchhoff law or the Hencky law. The
stored energy function of such materials has the form
˜
W( F ) = W(α) :=
1
2
3
i=1
f (α
i
)
2
+ β
1≤i< j≤3
f (α
i
) f (α
j
),
where f : (0, ∞) → R is a function satisfying f (1) = 0, f
′
(1) = 1, β ∈ R, and α
1
, α
2
,
α
3
are the singular values of the deformation gradient F. Two general situations are
determined under which
˜
W is not rank 1 convex: (a) if (simultaneously) the Hessian
of W at α = (1, 1, 1) is positive definite, β = 0, and f is strictly monotonic, and/or (b)
A. Bertram
Institute of Mechanics, Department of Engineering Mechanics,
Magdeburg University, Germany
e-mail: bertram@mb.uni-magdeburg.de
T. Böhlke (B )
Institute of Engineering Mechanics, Department of Mechanical Engineering,
University of Karlsruhe, P.O. BOX 6980, Germany
e-mail: boehlke@itm.uni-karlsruhe.de
M. Šilhavý
Mathematical Institute, Academy of Sciences of the CR, Prague, Czech Republic
e-mail: silhavy@math.cas.cz