  We derive a general theory of effective properties in metasolids based on
phononic crystals with low frequency resonances. We demonstrate that in general
these structures need to be described by means of a frequency-dependent and
non-local anisotropic mass density, stiffness tensor and a third- rank coupling
tensor, which shows that they behave like a non-local Willis medium. The effect
of non-locality and coupling tensor manifest themselves for some particular
resonances whereas they become negligible for other resonances. Considering the
example of a two-dimensional phononic crystal, consisting of triangular
arrangements of cylindrical shells in an elastic matrix, we show that its mass
density tensor is strongly resonant and anisotropic presenting both positive
and negative divergent values, while becoming scalar in the quasi-static limit.
Moreover, it is found that the negative value of transverse component of the
mass density is induced by a dipolar resonance, while that of the vertical
component is induced by a monopolar one. Finally, the dispersion relation
obtained by the effective parameters of the crystal is compared with the band
structure, showing a good agreement for the low-wave number region, although
the non-local effects are important given the existence of some resonant values
of the wave number.
