A discrete approach to the calculus

In this contribution, a novel nonlinear micropolar beam model suitable for metamaterials design in a dynamics framework is presented and discussed. The beam model is formulated following a completely discrete approach and it is fully defined by its Lagrangian, i.e., by the kinetic energy and by the potential of conservative forces. Differently from Hencky’s seminal work, which considers only flexibility to compute the buckling load for rectilinear and planar Euler–Bernoulli beams, the proposed model is fully three-dimensional and considers both the extensional and shear deformability contributions to the strain energy and translational and rotational kinetic energy terms. After having introduced the model formulation, some simulations obtained with a numerical integration scheme are presented to show the capabilities of the proposed beam model.

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A Semi-Discrete Approach for the Numerical Simulation of Freestanding Blocks

Computational Modeling of Masonry Structures Using the Discrete Element Method - Advances in Civil and Industrial Engineering ◽

10.4018/978-1-5225-0231-9.ch016 ◽

2016 ◽

pp. 416-439

Author(s):

Fernando Peña

Keyword(s):

Differential Equation ◽

Numerical Simulation ◽

Numerical Modeling ◽

Discrete Model ◽

Mathematical Formulation ◽

Equation Of Motion ◽

Rigid Bodies ◽

The Body ◽

Discrete Approach ◽

Rocking Motion

This chapter addresses the numerical modeling of freestanding rigid blocks by means of a semi-discrete approach. The pure rocking motion of single rigid bodies can be easily studied with the differential equation of motion, which can be solved by numerical integration or by linearization. However, when we deal with sliding and jumping motion of rigid bodies, the mathematical formulation becomes quite complex. In order to overcome this complexity, a Semi-Discrete Model (SMD) is proposed for the study of rocking motion of rigid bodies, in which the rigid body is considered as a mass element supported by springs and dashpots, in the spirit of deformable contacts between rigid blocks. The SMD can detect separation and sliding of the body; however, initial base contacts do not change, keeping a relative continuity between the body and its base. Extensive numerical simulations have been carried out in order to validate the proposed approach.

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