Analytical Solutions of 2-dimensional Second Gradient Linear Elasticity for Continua with Cubic-D4 Microstructure

Placidi, Luca; Rosi, Giuseppe; Barchiesi, Emilio

doi:10.1007/978-3-030-13307-8_26

Cited by 12 publications

(7 citation statements)

References 64 publications

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“…To determine these newly appeared parameters, experiments have to be performed. [13][14][15][16] In this research, uniaxial tensile tests were conducted on 3D-printed pantographic metamaterials made out of PETG. Three different variations of pantographics with differently sized substructures are investigated and discussed.…”

Section: Introductionmentioning

confidence: 99%

Investigation of deformation behavior of PETG-FDM-printed metamaterials with pantographic substructures based on different slicing strategies

Özen

Ganzosch

Barchiesi³

et al. 2021

Composites and Advanced Materials

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Based on the progress and advances of additive manufacturing technologies, design and production of complex structures became cheaper and therefore rather possible in the recent past. A promising example of such complex structure is a so-called pantographic structure, which can be described as a metamaterial consisting of repeated substructure. In this substructure, two planes, which consist of two arrays of beams being orthogonally aligned to each other, are interconnected by cylinders/pivots. Different inner geometries were taken into account and additively manufactured by means of fused deposition modeling technique using polyethylene terephthalate glycol (PETG) as filament material. To further understand the effect of different manufacturing parameters on the mechanical deformation behavior, three types of specimens have been investigated by means of displacement-controlled extension tests. Different slicing approaches were implemented to eliminate process-related problems. Small and large deformations are investigated separately. Furthermore, 2D digital image correlation was used to calculate strains on the outer surface of the metamaterial. Two finite-element simulations based on linear elastic isotropic model and linear elastic transverse isotropic model have been carried out for small deformations. Standardized extension tests have been performed on 3D-printed PETG according to ISO 527-2. Results obtained from finite-element method have been validated by experimental results of small deformations. These results are in good agreement with linear elastic transverse isotropic model (up to about [Formula: see text] of axial elongation), though the response of large deformations indicates a nonlinear inelastic material behavior. Nevertheless, all samples are able to withstand outer loading conditions after the first rupture, resulting in resilience against ultimate failure.

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Section: Introductionmentioning

confidence: 99%

Investigation of deformation behavior of PETG-FDM-printed metamaterials with pantographic substructures based on different slicing strategies

Özen

Ganzosch

Barchiesi³

et al. 2021

Composites and Advanced Materials

Self Cite

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“…From a general viewpoint, for verification purposes, a numerical solution have to be compared to analytical results [79]. Some analytical derivations were presented for 2D cases [22,69,74,75,95]. Obtaining results of this kind for 3D cases is rather challenging.…”

Section: Introductionmentioning

confidence: 99%

Verification of strain gradient elasticity computation by analytical solutions

et al. 2021

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As there are different computational methods for simulating problems in generalized mechanics, we present simple applications and their closed-form solutions for verifying a numerical implementation. For such a benchmark, we utilize these analytical solutions and examine three-dimensional numerical simulations by the finite element method (FEM) using IsoGeometric Analysis (IGA) with the aid of open source codes, called tIGAr, developed within the FEniCS platform. A study for the so-called wedge forces and double tractions help to comprehend their roles in the displacement solution as well as examine the significance by comparing to the closed form solutions for given boundary conditions. It is found that numerical results are in a good agreement with the analytical solutions if wedge forces and double tractions are considered. It is also presented how the wedge forces become necessary in order to maintain equilibrium in strain gradient materials.

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“…So far we tried to describe the deformation of a Continuum medium obtaining a useful tool for complex micro-structures not easily analyzed by Cauchy Continuum theory generating and generating big quantity of experimental data. It is known as Classical Cauchy continua are not able to give accuracy prediction in highly non-homogeneous microstructure; to this aim generalizations have to be introduced, either considering additional degrees of freedom to account for the kinematics at the level of the microstructure, [8][9][10][11][12][13][14][15][16][17][18][19][20], or including in the deformation energy density higher gradients of the displacement than the first one [21][22][23][24][25][26][27][28][29][30][31][32][33]. The latter is a particularly relevant topic considering the technological interest in developing exotic mechanical metamaterials able to perform targeted tasks [34][35][36][37][38][39][40][41], and therefore the investigation of new and efficient algorithms is of great interest at the moment.…”

Section: Introductionmentioning

confidence: 99%

A Tool to Describe Particle System Evolution from Swarm Robotics Behavior

Dell'Erba¹

2020

Developments and Novel Approaches in Nonlinear Solid Body Mechanics

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In previous works we have described time evolution of a two-dimensional particles lattice, subject to deformation, without to use Newton's laws. Our experience, in evolution and control of robotic swarm, has suggested as the new position of a particle could be determined by the spatial position of its neighbours; therefore we have used an interaction law based on the spatial position between the particles themselves without defining forces. The tool that we have realized reproduced some behaviour of deformable bodies both according to the standard Cauchy model and second gradient theory. It gives a plausible simulation of continuum deformation and fracture. It can be useful to describe final and sometime intermediate, configuration of a continuum material under assigned strain of some of its points; the advantages are in saving computational time, with respect to solving classical differential equation. Many aspects have to be still investigated, like the relationships describing the interaction rules between particles and its physical meaning and some results does not sound very good. In this paper we try to focus the job done and what is coming over.

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Analytical Solutions of 2-dimensional Second Gradient Linear Elasticity for Continua with Cubic-D4 Microstructure

Cited by 12 publications

References 64 publications

Investigation of deformation behavior of PETG-FDM-printed metamaterials with pantographic substructures based on different slicing strategies

Investigation of deformation behavior of PETG-FDM-printed metamaterials with pantographic substructures based on different slicing strategies

Verification of strain gradient elasticity computation by analytical solutions

A Tool to Describe Particle System Evolution from Swarm Robotics Behavior

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