Liftings and stresses for planar periodic frameworks

Emergence of zero modes in disordered solids under periodic tiling

In computational models of particle packings with periodic boundary conditions, it is assumed that the packing is attached to exact copies of itself in all possible directions. The periodicity of the boundary then requires that all of the particles' images move together. An infinitely repeated structure, on the other hand, does not necessarily have this constraint. As a consequence, a jammed packing (or a rigid elastic network) under periodic boundary conditions may have a corresponding infinitely repeated lattice representation that is not rigid or indeed may not even be at a local energy minimum. In this manuscript, we prove this claim and discuss ways in which periodic boundary conditions succeed in capturing the physics of repeated structures and where they fall short.

Infinitesimal Periodic Deformations and Quadrics

We describe a correspondence between the infinitesimal deformations of a periodic bar-and-joint framework and periodic arrangements of quadrics. This intrinsic correlation provides useful geometric characteristics. A direct consequence is a method for detecting auxetic deformations, identified by a pattern consisting of homothetic ellipsoids. Examples include frameworks with higher crystallographic symmetry.

Periodic Auxetics: Structure and Design

Materials science has adopted the term of auxetic behavior for structural deformations where stretching in some direction entails lateral widening, rather than lateral shrinking. Most studies, in the last three decades, have explored repetitive or cellular structures and used the notion of negative Poisson's ratio as the hallmark of auxetic behavior. However, no general auxetic principle has been established from this perspective. In the present article, we show that a purely geometric approach to periodic auxetics is apt to identify essential characteristics of frameworks with auxetic deformations and can generate a systematic and endless series of periodic auxetic designs. The critical features refer to convexity properties expressed through families of homothetic ellipsoids.

Auxetic deformations and elliptic curves

In materials science and engineering, auxetic behavior refers to deformations of flexible structures where stretching in some direction involves lateral widening, rather than lateral shrinking. We address the problem of detecting auxetic behavior for flexible periodic bar-and-joint frameworks. Currently, the only known algorithmic solution is based on the rather heavy machinery of fixed-dimension semi-definite programming. In this paper we present a new, simpler algorithmic approach which is applicable to a natural family of three-dimensional periodic bar-and-joint frameworks with three degrees of freedom. This class includes most zeolite structures, which are important for applications in computational materials science. We show that the existence of auxetic deformations is related to properties of an associated elliptic curve. A fast algorithm for recognizing auxetic capabilities is obtained via the classical Aronhold invariants of the cubic form defining the curve. A related alternative is also considered.

New principles for auxetic periodic design

We show that, for any given dimension d ≥ 2, the range of distinct possible designs for periodic frameworks with auxetic capabilities is infinite. We rely on a purely geometric approach to auxetic trajectories developed within our general theory of deformations of periodic frameworks.

Mechanisms and states of self-stress of planar trusses using graphic statics, part I: The fundamental theorem of linear algebra and the Airy stress function

The fundamental theorem of linear algebra establishes a duality between the statics of a pin-jointed truss structure and its kinematics. Graphic statics visualizes the forces in a truss as a reciprocal diagram that is dual to the truss geometry. In this article, we combine these two dualities to provide insights not available from a graphical or algebraic approach alone. We begin by observing that the force diagram of a statically indeterminate truss, although itself typically a kinematically loose structure, must support a self-stress state of its own. Such an “extra” self-stress state is described by the fundamental theorem of linear algebra. We show that the self-stress states of a truss are in a one-to-one correspondence with linkage-mechanism displacements of its reciprocal, and the relative centers of rotation of these mechanism displacements correspond to centers of perspective of a projection of a plane-faced three-dimensional polyhedral mesh. We prove that this polyhedral mesh is exactly the continuum Airy stress function, restricted to describe equilibrium of a truss structure. We use the Airy function to prove James Clerk Maxwell’s conjecture that a two-dimensional truss structure of arbitrary topology has a self-stress state if and only if its geometry is given by the projection of a three-dimensional plane-faced polyhedron. Although a very limited number of engineers have been aware of the relationship between trusses and a polyhedral Airy function, the authors believe that this is the first truly rigorous elucidation. We summarize the properties of this “dual duality,” which has the Airy function at its core, and conclude by showing applications to design of tensegrities, planar panelization of architectural surfaces, and optimization of trusses.

Geometric auxetics

We formulate a mathematical theory of auxetic behaviour based on one-parameter deformations of periodic frameworks. Our approach is purely geome- tric, relies on the evolution of the periodicity lattice and works in any dimension. We demonstrate its usefulness by predicting or recognizing, without experiment, computer simulations or numerical approximations, the auxetic capabilities of several well-known structures available in the literature. We propose new principles of auxetic design and rely on the stronger notion of expansive behaviour to provide an infinite supply of planar auxetic mechanisms and several new three-dimensional structures.