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On Poisson’s functions of compressible elastomeric materials under compression tests in the framework of linear elasticity theory

Abstract

We analyze the compression of a right cylinder made of an elastomeric material, sliding on a rigid plate in the framework of linear elasticity theory. Lubrication seems to reduce the heterogeneous effects of lateral bulging so that the deformations can be considered as “quasi-homogeneous.” As a result, we show that the ratio between the transversal deformation and axial strain is not a Poisson’s ratio but a Poisson’s function.

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Acknowledgements

We would like to thank the reviewers for their careful reading of the original manuscript. Their comments and suggestions have been incorporated into the improved final version. Thanks also due Mr. N. Ghezzou, English Teacher in the Department of Mechanical Engineering, University of Bejaia, to refine the English of this paper.

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Appendix A

Appendix A

We consider the following boundary value problem:

$$\begin{aligned}&\bar{{m}}^{4}S^{4}\bar{{\phi }}-2 \bar{{m}}^{2}S^{2}\left( {{\bar{{\phi }}}''+\frac{{\bar{{\phi }}}'}{\bar{{r}}}} \right) +\frac{1}{\bar{{r}}}\left[ {\bar{{r}} \left( {{\bar{{\phi }}}''+\frac{{\bar{{\phi }}}'}{\bar{{r}}}} \right) ^{\prime }} \right] ^{\prime }=0 \end{aligned}$$
(A.1)

where

$$\begin{aligned}&\bar{{\phi }}\left( 0 \right) =\bar{{\phi }}\left( 1 \right) = 1\hbox { and }{\bar{{\phi }}}'\left( 0 \right) =0. \end{aligned}$$
(A.2)

We pose: \(\forall r \in ] {-R, R} [ , \bar{{\phi }}\left( {\bar{{r}}} \right) =\sum \nolimits _{n\ge 0} {a_n } \bar{{r}}^{n}\).

So, we have then

$$\begin{aligned} \bar{{\phi }}\left( 0 \right) =1\hbox { and }{\bar{{\phi }}}'\left( 0 \right) =0\Rightarrow a_0 =1\hbox { and }a_1 =0. \end{aligned}$$
(A.3.1-4)

We substitute Eq. (A.3.1) in (A.1), and by a changing of the indices, we get the following equation:

$$\begin{aligned} \left\{ {\begin{array}{ll} a_{n+1} =- \frac{2 \bar{{m}}^{2}S^{2}\left( {n-1} \right) ^{2}a_{n-1} -\bar{{m}}^{4}S^{4}a_{n-3} }{\left( {n+1} \right) ^{2}\left( {n-1} \right) ^{2}}, &{}\quad \hbox {if }n\ge 3; \\ a_n =0, &{}\quad \hbox {if }n=2p+1, \forall p\ge 0 \\ \end{array}} \right. \end{aligned}$$

In order to obtain the coefficients \(a_{n}\), we vary the value of \(a_{2}\) until to satisfy the boundary condition \(\bar{{\phi }}\left( 1 \right) =1\).

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Bechir, H., Djema, A. & Bouzidi, S. On Poisson’s functions of compressible elastomeric materials under compression tests in the framework of linear elasticity theory. Acta Mech 230, 2491–2504 (2019). https://doi.org/10.1007/s00707-019-02407-w

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  • DOI: https://doi.org/10.1007/s00707-019-02407-w