  Incompressible nonlinearly hyperelastic materials are rarely simulated in
Finite Element numerical experiments as being perfectly incompressible because
of the numerical difficulties associated with globally satisfying this
constraint. Most commercial Finite Element packages therefore assume that the
material is slightly compressible. It is then further assumed that the
corresponding strain-energy function can be decomposed additively into
volumetric and deviatoric parts. We show that this decomposition is not
physically realistic, especially for anisotropic materials, which are of
particular interest for simulating the mechanical response of biological soft
tissue. The most striking illustration of the shortcoming is that with this
decomposition, an anisotropic cube under hydrostatic tension deforms into
another cube instead of a hexahedron with non-parallel faces. Furthermore,
commercial numerical codes require the specification of a `compressibility
parameter' (or `penalty factor'), which arises naturally from the flawed
additive decomposition of the strain-energy function. This parameter is often
linked to a `bulk modulus', although this notion makes no sense for anisotropic
solids; we show that it is essentially an arbitrary parameter and that
infinitesimal changes to it result in significant changes in the predicted
stress response. This is illustrated with numerical simulations for biaxial
tension experiments of arteries, where the magnitude of the stress response is
found to change by several orders of magnitude when infinitesimal changes in
`Poisson's ratio' close to the perfect incompressibility limit of 1/2 are made.
