Myocardium cells are tiny cylindrical structures arranged in a 
repetitive way.  Recent biological measurements using optical tweezers 
or a gel stretching method made it possible to obtain some information 
on their mechanical behavior. This was the impetus for trying to derive 
a macroscopic constitutive law for the cardiac muscle from 
microscopical data and, possibly, to perform a comparison with 
phenomenological laws that are commonly used in biomechanics. As 
opposed to classical laws, the homogenized law is implicit and 
knowledge of the stress vectors associated with a deformation gradient 
can only be obtained by solving a  nonlinear set of  equations with 
vectorial unknowns.

In a totally different field, the equilibrium of  the  hexagonal 
network of  atoms that make a graphene (an unrolled nanotube) is a 
minimizer of an elastic energy that contains both interatomic and 
interbond terms -- at least for simple models.  The Euler-Lagrange 
equation of this minimization problem is similar in form to the 
equilibrium system of cadiomyocytes. Therefore, the same homogenization 
procedure applies. A nonlinear membrane model of continuum mechanics is 
derived for the large deformations of the lattice.  For R^3-valued 
deformations, linearization only makes sense around a pre-stressed 
configuration. An explicit form of the linearized constituttive law can 
be given, which interestingly keeps track of the antisymmetric part of 
the deformation gradient. When deformations are restricted to be 
planar, linearizing aroud rest makes sense. A Hooke's law is recovered 
and we  perform comparisons between the mechanical constants we compute 
from mechanical nano data with macroscopical mechanical constants 
provided by the experimental literature.