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We show that a flat two dimensional network of connected vertices, whenstretched, may deform plastically by producing `pleats'; system spanning linearstructures with width comparable to the lattice spacing, where the networkoverlaps on itself. To understand the pleating process, we introduce anexternal field that couples to local {\em non-affine} displacements, i.e. thosedisplacements of neighbouring vertices that cannot be represented as a localaffine strain. We obtain both zero and finite temperature phase diagrams in thestrain -- field plane. Pleats occur here as a result of an equilibriumfirst-order transition from the homogeneous network to a heterogeneous phasewhere stress is localised within pleats and eliminated elsewhere. We show thatin the thermodynamic limit the un-pleated state is always metastable atvanishing field for infinitesimal strain. Plastic deformation of the initiallyhomogeneous network is akin to the decay of a metastable phase via a dynamicaltransition. We make predictions concerning local stress distributions andthermal effects associated with pleats which may be observable in suitableexperimental systems.

Plastic deformation of a permanently bonded network: stress relaxation by pleats