Theoretical and computational results are presented clarifying the role of long-ranged strain interactions in determining the charge and orbital ordering in colossal magnetoresistance manganites. The strain energy contribution is found to be of order 20 − 30 meV /M n and in particular stabilizes the anomalous 'zig-zag chain' order observed in many half-doped manganites.Perovskite manganites of chemical formula Ln 1−x Ak x MnO 3 (Ln is a lanthanide rare earth such as La or Pr; Ak is a divalent alkali such as Sr or Ca) present a complex phase diagram associated with the interplay of charge, spin, orbital and lattice degrees of freedom.1 Charge and orbitally ordered phases appear in wide parameter ranges, for example in La 1−x Ca x MnO 3 for x ≥ 0.5, and in Pr 1−x Ca x MnO 3 for 0.3 ≤ x ≤ 0.7. The physical mechanism underlying the ordering remains the subject of debate. Particular attention has focussed on the half-doped (x = 0.5) materials, including La 0.5 Ca 0.5 MnO 3 , Pr 0.5 Ca 0.5 MnO 3 and Nd 0.5 Sr 0.5 MnO 3 , which possess a strongly insulating ground state with a particularly interesting ('zig-zag chain') spin, charge and orbital ordering pattern shown in the inset to Fig. 1a. Stabilizing this state requires 'second neighbor' interactions, which as noted by Khomskii and Kugel 2 and by Ahn and Millis 3 may reasonably be expected to arise from electron-lattice interactions. However, experiments have also indicated that long ranged strain plays an important role. Insulating ordered phases appearing in bulk La 0.5 Ca 0.5 MnO 3 are suppressed in thin films 4 ; it is argued that the difference arises from the lattice mismatch with the substrate, which prevents the occurrence of lattice distortions necessary to accommodate orbital ordering. Similar conclusions have been drawn from recent work on Pr 0.6 Ca 0.4 MnO 3 films subjected both to compressive and tensile strain.5 Other sets of experiments in polycrystalline La 0.5 Ca 0.5 MnO 3 show that the charge/orbital ordering transition temperature is progressively suppressed as grain size is decreased; the interpretation again is that boundary effects inhibit the formation, in small grains, of the strains imposed by the orbital ordering.6 Perhaps most significantly, polarized optical microscopy studies on Bi 0.2 Ca 0.8 MnO 3 single crystals 7 reveal that as temperature is decreased to just below the charge/orbital ordering transition, lenticular shaped domains appear and grow slowly with time. This behavior is well known in martensitic systems 8 , where it is attributed to the interplay between a tendency of the system to favor a state with large strain, and a boundary condition which prevents the existence of a truly uniform strain.Despite the considerable experimental evidence that long ranged strain is important, strain physics has not received much theoretical attention. The original analyses 9,10,11 and many subsequent works 12,13,14 focused on local interactions, including magnetic, Coulomb and Jahn-Teller electron-phonon coupling. In this paper we study lon...