Abstract
In this work, an extension of the strain energy for fibrous metamaterials composed of two families of parallel fibers lying on parallel planes and joined by connective elements is proposed. The suggested extension concerns the possibility that the constituent fibers come into contact and eventually scroll one with respect to the other with consequent dissipation due to friction. The fibers interact with each other in at least three different ways: indirectly, through microstructural connections that could allow a relative sliding between the two families of fibers; directly, as the fibers of a family can touch each other and can scroll introducing dissipation. From a mathematical point of view, these effects are modeled first by introducing two placement fields for the two fiber families and adding a coupling term to the strain energy and secondly by adding two other terms that take into account the interdistance between the parallel fibers and the Rayleigh dissipation potential (to account for friction).
Highlights
An important modeling issue in the design of metamaterials micro-architecture concerns the need of describing those phenomena occurring when parts of the afore-mentioned microstructure, initially distant, enter in contact because of deformation
In order to prove that the synthesis of second gradient metamaterials with non-negligible second gradient energetic terms is physically conceivable, but can be approached mathematically, it has been proven that [1,2]: i. pantographic architectures provide for the synthesis of Casal-type beams [69,70]; ii. it is possible to synthesize second gradient plates with two families of pantographic substructures: i.e., plates having strain energy depending on the in-plane second gradients of displacement [71]; iii. when perfect hinges are provided, the macro-strain energy of short beam pantographic structures does not incorporate at all the shear first gradient terms, and in the pantographic fabrics the so-called floppy-modes can be observed: i.e., homogeneous deformations that correspond to a null strain energy
A possibility is to employ the so-called Hamilton-Rayleigh Principle [48]. It can be summarized as follows: i. we separate the problem in two parts, one conservative and the other dissipative, and we describe only the conservative part using an action functional; ii. we calculate the first variation of the action functional; iii. we add suitable linear functionals to model dissipative phenomena (Rayleigh dissipation functionals); iv. the Hamilton-Rayleigh Principle can be formulated by summing the Lagrangian and the dissipative virtual works and postulating this sum to be vanishing for every admissible variation of motion
Summary
An important modeling issue in the design of metamaterials micro-architecture concerns the need of describing those phenomena occurring when parts of the afore-mentioned microstructure, initially distant, enter in contact because of deformation. Starting from the basics outlined in [1,2,3], we want to present a first approach that intends to give some research perspectives in the study of contact phenomena occurring in the micro-architecture of fibrous metamaterials To this end, we will follow in the presentation of the results already available in the literature a criterion of simplicity so that in the following we can introduce more the new concepts that are discussed here for the first time. The mathematical implication, as we will see below, is that the governing equations can be obtained considering a higher gradient continuum, a second gradient one In this context, the role of the pivots, we will see, is merely to ensure that the two fiber layers that make up the pantographic metamaterial can exhibit relative rotations at their points of contact. This has been attempted in [6, 7]
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