  We review the theoretical bounds on the effective properties of linear
elastic inhomogeneous solids (including composite materials) in the presence of
constituents having non-positive-definite elastic moduli (so-called
negative-stiffness phases). We show that for statically stable bodies the
classical displacement-based variational principles for Dirichlet and Neumann
boundary problems hold but that the dual variational principle for traction
boundary problems does not apply. We illustrate our findings by the example of
a coated spherical inclusion whose stability conditions are obtained from the
variational principles. We further show that the classical Voigt upper bound on
the linear elastic moduli in multi-phase inhomogeneous bodies and composites
applies and that it imposes a stability condition: overall stability requires
that the effective moduli do not surpass the Voigt upper bound. This
particularly implies that, while the geometric constraints among constituents
in a composite can stabilize negative-stiffness phases, the stabilization is
insufficient to allow for extreme overall static elastic moduli (exceeding
those of the constituents). Stronger bounds on the effective elastic moduli of
isotropic composites can be obtained from the Hashin-Shtrikman variational
inequalities, which are also shown to hold in the presence of negative
stiffness.
