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Erschienen in:
Theoretical Analysis Kapitel lesen Erstes Kapitel lesen

2019 | OriginalPaper | Buchkapitel

2. Numerical and Experimental Analyses

verfasst von : Angelo Marcello Tarantino, Luca Lanzoni, Federico Oyedeji Falope

Erschienen in: The Bending Theory of Fully Nonlinear Beams

Verlag: Springer International Publishing

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Abstract

The results provided by the theoretical model proposed in Chap. 1 for the bending of fully nonlinear beams are compared with those given by the numerical and experimental analyses developed in the present Chapter. The numerical model is based on the finite element method (FEM), whereas a test equipment prototype has been designed and manufactured for the experimental analysis. The experimental data have been acquired using the digital image correlation (DIC) instrumentation. The fundamental purpose of these two further analyses for the large bending of slender beams is to justify the hypotheses underlying the theoretical model.

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Vorheriges Kapitel Theoretical Analysis
Nächstes Kapitel Generalization to Variable Bending Moment
Fußnoten
1
For the compression tests, cylindrical squat samples, with 8 mm height and 27 mm diameter, have been realized. Due to symmetry displacement condition, care has been taken in keeping fixed the sample centroid. The upper surface has been subjected to a prescribed displacement in the cylinder axes direction (displacement control test). Even if a lubricating solution has been used for the upper and lower surfaces during the experimental compression test, the observed deformed configuration has highlighted a friction contribute acting at the edges surfaces. For tensile tests, dog-bone rubber specimens with 8 mm width, 4 mm height and 63 mm length have been realized. As for the compression test, the upper edge has been subjected to a prescribed displacement.
 
2
Such a position is based on an observation suggested by Haslach [5]. See also [6‐10].
 
3
Time-dependent effects were not observed [11‐13].
 
4
This rigid constraint reproduces what happens in the experimental tests, where the beam terminal bases are glued to metal plates of the test equipment.
 
5
Imposing the boundary conditions to generate the angle \(2\alpha _{0}\) with the FEM approach reserves some difficulties. In terms of stress, a distribution of stress should be applied in the two end cross sections in order to reproduce along the deformed axis of the beam a diagram of constant bending moments. In terms of displacements, a system of displacements should be applied in the end cross sections such that, retaining the planarity of the cross sections, it is able to deform the beam axis according to a circumferential arc. In both situations, stresses and displacements to be applied depend on the solution of the equilibrium problem and they are not known a priori.
 
6
The irregularities in the isolines are due to the Gauss nodes of the FEM mesh where the solution has been recovered.
 
7
Since the values of the stresses in the central area of the cross section are close to zero, the calculation of the relative error for the stresses becomes meaningless.
 
8
In the experimental tests, the inevitable frictions of the mechanisms run against to the elastic retaining force, therefore, a stiffer behavior for the rubber beam is observed.
 
9
However, since these displacements are very small, the relative errors made in the measurement may be large.
 
10
The last image of Fig. 2.19 indicates the rigid block which, monitored with the DIC instrument, allows to obtain the rotations shown in Fig. 2.12.
 
11
For reasons of representation, the height of the specimen has been doubled.
 
12
The numerical method in the middle section is not affected by the discrepancies generated by the boundary conditions.
 
13
The radius r has been determined locally as the radius of the osculating circle passing through three neighboring points in the deformed configuration.
 
Literatur
1.
L. Lanzoni, A.M. Tarantino, Equilibrium configurations and stability of a damaged body under uniaxial tractions. ZAMP J. Appl. Math. Phys. 66, 171–190 (2015)
2.
ASTM International, Standard Test Methods for Vulcanized Rubber and Thermoplastic Elastomers—Tension Standard, American standard test method, West Conshohocken, PA (2006)
3.
ASTM International, Standard Test Methods for Rubber Property—Compression Set Standard, American standard test method, West Conshohocken, PA (2008)
4.
N.R. Draper, H. Smith, Applied Regression Analysis, vol. 326 (Wiley, New York, 2014)MATH
5.
H.W. Haslach, Constitutive models and singularity types for an elastic biaxially loaded rubber sheet. Math. Mech. Solids 5(1), 41–73 (2000)MathSciNetCrossRef
6.
H. Khajehsaeid, J. Arghavani, R. Naghdabadi, A hyperelastic constitutive law for rubber-like materials. Eur. J. Mech.-A/Solids 38, 144–151 (2013)
7.
L. Meunier et al., Mechanical experimental characterization and numerical modelling of an unfilled silicone rubber. Polym. Test. 27(6), 765–777 (2008)CrossRef
8.
A.M. Tarantino, Equilibrium paths of a hyperelastic body under progressive damage. J. Elast. 114, 225–250 (2014)
9.
L. Lanzoni, A.M. Tarantino, Damaged hyperelastic membranes. Int. J. Nonlinear Mech. 60, 9–22 (2014)CrossRef
10.
L. Lanzoni, A.M. Tarantino, A simple nonlinear model to simulate the localized necking and neck propagation. Int. J. Nonlinear Mech. 84, 94–104 (2016)CrossRef
11.
L. Dezi, G. Menditto, A.M. Tarantino, Homogeneous structures subjected to successive structural system changes. ASCE J. Eng. Mech. 116(8), 1723–1732 (1990)CrossRef
12.
L. Dezi, G. Menditto, A.M. Tarantino, Viscoelastic heterogeneous structures with variable structural system. ASCE J. Eng. Mech. 119(2), 238–250 (1993)CrossRef
13.
L. Dezi, A.M. Tarantino, Creep in continuous composite beams. Part I: theoretical treatment. ASCE J. Struct. Eng. 119(7) 2095–2111 (1993)
Metadaten
Titel
Numerical and Experimental Analyses
verfasst von
Angelo Marcello Tarantino
Luca Lanzoni
Federico Oyedeji Falope
Copyright-Jahr
2019
Buch
The Bending Theory of Fully Nonlinear Beams

Print ISBN: 978-3-030-14675-7

Electronic ISBN: 978-3-030-14676-4

Copyright-Jahr: 2019

DOI
https://doi.org/10.1007/978-3-030-14676-4_2

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