Large deformations of 1D microstructured systems modeled as generalized Timoshenko beams

Abstract

In the present paper we study a natural nonlinear generalization of Timoshenko beam model and show that it can describe the homogenized deformation energy of a 1D continuum with a simple microstructure. We prove the well-posedness of the corresponding variational problem in case of a generic end load, discuss some regularity issues and evaluate the critical load. Moreover, we generalize the model so as to include an additional rotational spring in the microstructure. Finally, some numerical simulations are presented and discussed.

A damaged non-homogeneous Timoshenko beam model for a dam subjected to aging effects

“…First, the aging effects, that here have been modeled by the reduction of the threshold in the neighborhood of the bottom of the dam beam could be substituted by the diffusion of an aging fluid, adding a further kinematical descriptor. In addition, a nonlinear elastic deformation of the beam model could be achieved as in [52, 53]. Moreover, generalization of this diffusion problem to 2D generalization with the inclusion of strain gradient generalization and further dissipative phenomena [54,55] could be achieved.…”

Higher Gradient Materials and Related Generalized Continua

“…This method is very powerful and relatively light from a computational point of view, but just like every energy-related method it is not very suitable to study the multiplicity of arising solutions. For this reason, the numerical technique used here is the same as in Battista et al (2018). Indeed, the boundary value problem for the clamped-free Euler and Timoshenko beams has been solved by means of a shooting technique.…”

“…Interesting generalizations of the Timoshenko beam model have been proposed (see e.g. Romano et al, 1992;Serpieri and Rosati, 2014), while a periodic mechanical system whose homogenized limit is the model (2.5) (in the particular case α ≡ 1) is shown in Battista et al (2018). It is in fact a microstructured 1D system whose unit cell is an articulated parallelogram and equipped with suitably placed rotational springs, and it can be easily obtained by means of 3D printing.…”

“…The most basic one is that it changes the functional set in which the problem is naturally collocated and in particular prevents it from being a vector space, since a strictly positive local axial deformation has to be prescribed. Of course one can obtain this making suitable assumptions on the energy, but in any case the nonautonomous variational problem that arises in the case of a distributed load will present new difficulties with respect to the inextensible case (Battista et al, 2018).…”

Non-Linear Lumped-Parameter Modeling of Planar Multi-Link Manipulators with Highly Flexible Arms

“…Unfortunately, this is the result of a certain compartmentalization of expertise that sometimes occurs [27]. Indeed, it is widely known to those dealing with beam homogenization how to properly assign these discretized constants of stiffness [28][29][30][31][32][33].…”