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2019 | OriginalPaper | Chapter

1. Theoretical Analysis

Authors : Angelo Marcello Tarantino, Luca Lanzoni, Federico Oyedeji Falope

Published in: The Bending Theory of Fully Nonlinear Beams

Publisher: Springer International Publishing

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Abstract

This chapter deals with the equilibrium problem of fully nonlinear beams in bending by extending the model for the anticlastic flexion of solids recently proposed in the context of finite elasticity by Lanzoni and Tarantino (J Elast 131:137–170, 2018, [1]). Initially, kinematics is reformulated and, subsequently, a nonlinear theory for the bending of slender beams has been developed. In detail, no hypothesis of smallness is introduced for the deformation and displacement fields, the constitutive law is considered nonlinear and the equilibrium is imposed in the deformed configuration. Explicit formulas are obtained which describe the displacement field, stretches and stresses for each point of the beam using both the Lagrangian and Eulerian descriptions.

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next chapter Numerical and Experimental Analyses
Footnotes
1
Recent contributions on the nonlinear damage theory can be found in [33‐36].
 
2
\(\mathcal {V}\) is the vector space associated with \(\mathcal {E}\).
 
3
Lin is the set of all (second order) tensors whereas \(Lin^{+}\) is the subset of tensors with positive determinant.
 
4
For slender beams, the kinematic model contains only one unknown geometrical parameter: the anticlastic radius r, while for a block the geometrical parameters are three [1].
 
5
The existence of at least one arc, for which \(\lambda _{Z}=1\), is ensured by the continuity of deformation. Uniqueness is guaranteed by the hypothesis 2 of conservation of the planarity of the cross section.
 
6
In the case of a block, the two arcs, for which \(\lambda _{Z}=1\) and \(\lambda _{Y}=1\), are distinct and do not pass through the point \(O'\). Instead when a slender beam is considered, the distance between the two arcs vanishes [1].
 
7
In the sequel, it will be found the relationship between this angle \(\alpha _{0}\) and the pair of self-balanced bending moment to apply to the end faces of the beam.
 
8
It is assumed that the isotropy property is preserved in the deformed configuration.
 
9
The other components of tensor C are zero.
 
10
Or equivalently by the principal invariants of the right Cauchy-Green strain tensor \(\mathbf {C}\).
 
11
The following notations: \(\parallel \mathbf {A}\parallel =\left( \mathrm {tr}\mathbf {A}^{\mathrm {T}}\mathbf {A}\right) ^{1/2}\) for the tensor norm in the linear tensor space Lin and \(\mathbf {A}^{\star }=(\mathrm {det}\mathbf {A})\mathbf {A}^{\mathrm {-T}}\) for the cofactor of the tensor A (if A is invertible) are used.
 
12
This function is polyconvex and satisfies the growth conditions: \(\omega \rightarrow \infty \) as \(\lambda \rightarrow 0^{+}\) or \(\lambda \rightarrow +\infty \). It was used, for example, in [57‐60].
 
13
Similar positions can be found in [61‐64].
 
14
The first equation of system (1.22) is also verified for all points of the vertical plane \(X=0\).
 
15
The same symbols are used for normalized and non-normalized constants.
 
16
Using (1.25), it can be promptly verified that, in the absence of deformation, \(\lambda =\lambda _{Z}=1\), is \(S_{Z}=N=0\).
 
17
From (1.32)\(_{3}\) the quantity in square brackets is attained and then replaced into (1.32)\(_{2}\), obtaining (1.33)\(_{3}\). Similarly, from (1.32)\(_{1}\), \(r\,e^{-\frac{Y}{r}}\) is evaluated and then substituted into (1.32)\(_{2}\), obtaining (1.33)\(_{1}\). Expression (1.33)\(_{2}\) is evaluated directly from (1.32)\(_{2}\) using (1.33)\(_{1}\) and (1.33)\(_{3}\).
 
18
It can be see that, by taking \(B=2\beta _{0}r\), (1.41) reduces to BH as \(r\rightarrow \infty \). Similarly, by taking \(L=2\alpha _{0}R_{0}\), (1.43) becomes \(V'=BHL\) as \((R_{0},\,r)\rightarrow \infty \).
 
19
The Landau symbols are used.
 
20
Using the Taylor series expansions, the following approximations are employed:
$$ e^{-\frac{Y}{r}}\simeq 1-\frac{Y}{r}+\frac{Y^{2}}{2\,r^{2}}+o(r^{-2}),\quad \sin \frac{X}{r}\simeq \frac{X}{r}+o(r^{-2}),\quad \cos \frac{X}{r}\simeq 1-\frac{X^{2}}{2\,r^{2}}+o(r^{-3}), $$
$$ \sin \frac{Z}{R_{0}}\simeq \frac{Z}{R_{0}}+o(R_{0}^{-2}),\quad \cos \frac{Z}{R_{0}}\simeq 1-\frac{Z^{2}}{2\,R_{0}^{2}}+o(R_{0}^{-3}). $$
 
21
In the sequel, the infinitesimal terms of higher order are omitted definitively.
 
22
After linearization, the following relationships hold: \(\mathrm {\mathbf {R}}=\mathbf {I}+\mathbf {W}\), \(\mathbf {U}=\mathbf {I}+\mathbf {E}\).
 
23
Using the Taylor series expansions, the following approximation is employed:
$$ \sinh \frac{H}{2r}\simeq \frac{H}{2r}+o(r^{-2}), $$
as well as similar expressions for different arguments of hyperbolic sine function.
 
24
Using the Taylor series expansions, the following approximation is employed:
$$ \frac{1}{\lambda }\simeq 1+\frac{Y}{r}+o(r^{-1}), $$
and the relationship among the constitutive constants (1.25) has been used to obtain (1.63).
 
25
Using the Taylor series expansions, the following approximation is employed:
$$ \frac{1}{\lambda ^{2}\lambda _{Z}}\simeq 1+\frac{2Y}{r}-\frac{Y}{R_{0}}+o(r^{-1})+o(R_{0}^{-1}). $$
.
 
26
Of course, the same result can be achieved for a compressible Mooney-Rivlin material that satisfies the conditions (1.69). In effect, replacing (1.70) into (1.63), it is found
$$ S=\left[ -(4a+12b+8c)+(4b+4c)\,\frac{1}{\nu }\right] \frac{Y}{r}=0, $$
$$ S_{Z}=\left[ -(8b+8c)\,\nu +(4a+8b+4c)\right] \,\frac{Y}{R_{0}}=E\,\varepsilon _{z}. $$
.
 
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Metadata
Title
Theoretical Analysis
Authors
Angelo Marcello Tarantino
Luca Lanzoni
Federico Oyedeji Falope
Copyright Year
2019
Book
The Bending Theory of Fully Nonlinear Beams

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

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

Copyright Year: 2019

DOI
https://doi.org/10.1007/978-3-030-14676-4_1

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