Equilibrium paths of Hencky pantographic beams in a three-point bending problem

Abstract

We investigate the mechanical behavior of so-called pantographic beams undergoing large deformations. To this aim, an exact-kinematics Hencky pantographic beam model has been employed in a three-point bending test. Given the occurrence of local snap-through instabilities and limit points, said Hencky model has been solved by means of a step-by-step strategy based on Riks's arclength method. Such a method has been particularly adapted for the case of problems with prescribed displacements, as opposed to those with prescribed forces. Numerical simulations performed by varying the stiffness parameters are discussed, aimed at getting an insight into the different behaviors which can be exhibited by pantographic beams. Numerical simulations performed by varying the quantity of unit cells for fixed total length allow instead to understand whether the observed features are inherent to the pantographic beam structure or size-dependent. Therefore, beyond being interesting for possible future engineering exploitation, we believe this phenomenological evidence to be useful in guiding the formulation of conjectures regarding observed microscale local snap-through instability phenomena in the framework of a previously proposed macroscale continuum model for pantographic beams obtained by asymptotic homogenization.

Effective strain gradient continuum model of metamaterials and size effects analysis

“…The size effects can also be examined by making use of the homogenization method for different metamaterials under different boundary conditions, for example, metamaterials of D4-invariant, D6invariant, Z2-invariant, etc., as shown in [9] under stretching, shearing, transverse bending [86]. In a specific case, the pantographic metamaterial which has been largely studied by dell'Isola and his colleagues [7,29,31,32,68,70,72,73,78,79] is a so-called higher gradient material [2][3][4]16,27,28]. It is also possible to determine the effective material parameters of a pantographic metamaterial by using this method.…”

Size effects of mechanical metamaterials: a computational study based on a second-order asymptotic homogenization method

“…It will be interesting to investigate differences of the identified parameters by means of these two methods. -Hencky-type discrete models are the fundamental building block for continuum models [57,58]. Physical systems studied in the present paper can be also modeled by the so-called Hencky-type models.…”

Stepwise analysis of pantographic beams subjected to impulsive loads

“…In the case of the pantographic beam in the large displacement regime, even when neglecting inertia contributions, the equilibrium path is rather complicated, presenting jump phenomena, see, e.g., [28, 43]. Therefore, it is mandatory to construct a robust and, if possible, simple mechanical model along with a reliable algorithm for the solution of the nonlinear equilibrium equations.…”

“…We consider a planar pantographic beam, such as that represented in the top of Figure 1, constructed from two orthogonal families of extensible Euler-Bernoulli beams connected in their intersecting midpoints by torsional links. 3 As done in [16,[25][26][27] for pantographic lattices and in [28] for pantographic beams, we adopt the description sketched in the bottom of Figure 1, where interactions, using the laws reported next, consist of extensional and rotational springs. Obviously, this description could be enhanced in a h-refinement fashion, as shown in [28].…”