This paper describes two folded metamaterials based on the Miura-ori fold pattern. The structural mechanics of these metamaterials are dominated by the kinematics of the folding, which only depends on the geometry and therefore is scale-independent. First, a folded shell structure is introduced, where the fold pattern provides a negative Poisson's ratio for in-plane deformations and a positive Poisson's ratio for out-of-plane bending. Second, a cellular metamaterial is described based on a stacking of individual folded layers, where the folding kinematics are compatible between layers. Additional freedom in the design of the metamaterial can be achieved by varying the fold pattern within each layer.I n this paper, we describe the use of origami for mechanical metamaterials, where the fold patterns introduce kinematic deformation modes that dominate the overall structural response. The geometry and kinematics of two types of folded metamaterial are described: a folded shell structure and a folded cellular metamaterial. The examples presented here are both based on a particular fold geometry: the classic Miura-ori pattern. This pattern has previously been considered for applications, such as deployable solar panels (1), and was observed in the biaxial compression of stiff thin membranes on a soft elastic substrate (2, 3).In recent years, origami has seen a surge in research interest from engineers and physicists. Developments include folded sandwich panel cores (4, 5), origami-inspired stents (6), selffolding membranes (7), and cellular materials made from folded cylinders (8). An important concept is rigid origami, where the fold pattern is modeled as rigid panels connected through frictionless hinges. These assumptions make the study of origami folding a matter of kinematics. Of particular interest here are fold patterns where four fold lines meet at each vertex (so-called degree-4 vertices). Each such vertex has one degree of freedom, a tessellated fold pattern is overconstrained, and folding is only possible under strict geometric conditions. In a landmark paper, Huffman (9) studied rigid folding using spherical geometry; recent work includes the modeling of crease patterns using quaternions (10) and an increased understanding of the foldability conditions for partly folded quadrilateral surfaces (11,12).In describing the properties of the folded metamaterials, we are here primarily concerned with the deformation kinematics. If required, these models can straightforwardly be extended to include simple constitutive behavior at the fold lines [for instance, elastic (13) or plastic (14) behavior].The paper is structured as follows. First, the Miura-ori unit cell is introduced, because its geometry plays a key role in the mechanical properties of the folded metamaterials. The first such metamaterial is based on a single planar Miura-ori sheet: a folded shell structure. Of particular interest are the shell's outof-plane kinematics. Second, a bulk metamaterial is proposed based on the stacking of individual Miura-or...

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Thin cylindrical shell structures can show interesting bistable behaviour. If made unstressed from isotropic materials they are only stable in the initial configuration, but if made from fibre-reinforced composites they may also have a second, stable configuration. If the layup of the composite is antisymmetric, this alternative stable configuration forms a tight coil; if the layup is symmetric the alternative stable configuration is helical. A simple two-parameter model for these structure is presented that is able to distinguish between these different behaviours.

A simple derivation of the tangent stiffness matrix for a prestressed pin-jointed structure is given, and is used to compare the diverse formulations that can be found in the literature for finding the structural response of prestressed structures.

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We determine upper bounds for the maximum order of an element of a finite almost simple group with socle T in terms of the minimum index m(T ) of a maximal subgroup of T : for T not an alternating group we prove that, with finitely many exceptions, the maximum element order is at most m(T ). Moreover, apart from an explicit list of groups, the bound can be reduced to m(T )/4. These results are applied to determine all primitive permutation groups on a set of size n that contain permutations of order greater than or equal to n/4. 2000 Mathematics Subject Classification. 20B15, 20H30.

a b s t r a c tMaxwell's rule from 1864 gives a necessary condition for a framework to be isostatic in 2D or in 3D. Given a framework with point group symmetry, group representation theory is exploited to provide further necessary conditions. This paper shows how, for an isostatic framework, these conditions imply very simply stated restrictions on the numbers of those structural components that are unshifted by the symmetry operations of the framework. In particular, it turns out that an isostatic framework in 2D can belong to one of only six point groups. Some conjectures and initial results are presented that would give sufficient conditions (in both 2D and 3D) for a framework that is realized generically for a given symmetry group to be an isostatic framework.