  We study gradient field models on an integer lattice with non-convex
interactions. These models emerge in distinct branches of physics and
mathematics under various names. In particular, as zero-mass lattice
(Euclidean) quantum field theory, models of random interfaces, and as
mass-string models of nonlinear elasticity.Our attention is mostly devoted to
the latter with random vector valued fields as displacements for atoms of
crystal structures,where our aim is to prove the strict convexity of the free
energy as a function of affine deformations for low enough temperatures and
small enough deformations. This claim can be interpreted as a form of
verification of the Cauchy-Born rule at small non-vanishing temperatures for a
class of these models. We also show that the scaling limit of the Laplace
transform of the corresponding Gibbs measure (under a proper rescaling)
corresponds to the Gaussian gradient field with a particular covariance.
  The proofs are based on a multi-scale (renormalisation group analysis)
techniques needed in view of strong correlations of studied gradient fields. To
cover sufficiently wide class of models, we extend these techniques from the
standard case with rotationally symmetric nearest neighbour interaction to a
more general situation with finite range interactions without any symmetry. Our
presentation is entirely self-contained covering the details of the needed
renormalisation group methods.
