Two- and three-dimensional elastic networks with rigid junctions: modeling within the theory of micropolar shells and solids

Abstract

For two-and three-dimensional elastic structures made of families of flexible elastic fibers undergoing finite deformations, we propose homogenized models within the micropolar elasticity. Here we restrict ourselves to networks with rigid connections between fibers. In other words, we assume that the fibers keep their orthogonality during deformation. Starting from a fiber as the basic structured element modeled by the Cosserat curve beam model, we get 2D and 3D semi-discrete models. These models consist of systems of ordinary differential equations describing the statics of a collection of fibers with certain geometrical constraints. Using a specific homogenization technique, we introduce two-and three-dimensional equivalent continuum models which correspond to the six-parameter shell model and the micropolar continuum, respectively. We call two models equivalent if their approximations coincide with each other up to certain accuracy. The twoand three-dimensional constitutive equations of the networks are derived and discussed within the micropolar continua theory. The author acknowledges the support of the Government of the Russian Federation (Contract No. 14.Z50.31.0046).

Equilibrium of Two-Dimensional Cycloidal Pantographic Metamaterials in Three-Dimensional Deformations

“…Herein, a superimposed tilde denotes non-dimensional quantities. Therefore, the material parameters appearing in Equations (26) and (27) were normalized in the following way:…”

“…From a geometric point of view, a pantographic surface resembles very closely an elastic network structure. Therefore, many of the achievements in modeling these systems can be fruitfully employed also for the pantographic sheets (see, e.g., [23][24][25][26][27][28][29]). According to these results, the pantographic structure is identified with a surface composed of two families of continuously distributed fibers in the framework of continuum theory.…”

Stiffness optimization in nonlinear pantographic structures

“…In this paper, we will design novel metamaterials using the method of structural optimization: as a demonstrative example, the microconstitutive parameters for planar pantographic structures will be optimized, with a focus on the torsional stiffness of the pivots that interconnect the two families of beam fibres. Note that the shear stiffness effect and rigid junctions, respectively, were studied [13, 14] and [15]. As such exemplary metamaterials have proven to be extremely tough in extension and shearing, see [16, 17], we aim to increase their already described resistance and resilience to large elongations.…”

Effective strain gradient continuum model of metamaterials and size effects analysis

“…Higher-order theories, including strain gradient theory, couple stress theory, micropolar theory, micromorphic theory, and so on, cover the microstructural information, and they are able to mimic the size effects. They have been studied by some researchers extensively, for example, see [5,30,35,36,38,52,59,62,76]. The main challenge for the higher-order continuum approach is to determine the corresponding additional constitutive parameters, which are difficult to measure experimentally.…”