  We show that a flat two dimensional network of connected vertices, when
stretched, may deform plastically by producing `pleats'; system spanning linear
structures with width comparable to the lattice spacing, where the network
overlaps on itself. To understand the pleating process, we introduce an
external field that couples to local {\em non-affine} displacements, i.e. those
displacements of neighbouring vertices that cannot be represented as a local
affine strain. We obtain both zero and finite temperature phase diagrams in the
strain -- field plane. Pleats occur here as a result of an equilibrium
first-order transition from the homogeneous network to a heterogeneous phase
where stress is localised within pleats and eliminated elsewhere. We show that
in the thermodynamic limit the un-pleated state is always metastable at
vanishing field for infinitesimal strain. Plastic deformation of the initially
homogeneous network is akin to the decay of a metastable phase via a dynamical
transition. We make predictions concerning local stress distributions and
thermal effects associated with pleats which may be observable in suitable
experimental systems.
