Thin Auxetic Plates and Shells

Abstract

This chapter opens with a discussion on flexural rigidity of auxetic plates vis-à-vis conventional ones, followed by an analysis of circular auxetic plates. Bending moment result from uniform loading of circular plates suggests that the optimal Poisson’s ratio is −1/3 if the plate is simply-supported at the edge. Based on bending and twisting moment minimization on a rectangular plate under sinusoidal load, the optimal Poisson’s ratio for a square plate is 0, and this value reduces until −1 for a rectangular plate with aspect ratio 1 + √2. Auxetic materials are not suitable for uniformly loaded and simply supported square plates , as moment minimization study suggests an optimal Poisson’s ratio of 0.115, but are highly suitable for central point loaded and simply supported square plates, as moment minimization study suggests an optimal Poisson’s ratio of −1. In the study of auxetic plates on auxetic foundation, the plotted results suggest that, in addition to selecting materials of sufficient strength and mechanical designing of plate for reduced stressed concentration, the use of plate and/or foundation materials with negative Poisson’s ratio is useful for designing against failure. The investigations on width-constrained plates under uniaxial in-plane pressure by Strek et al. (J Non-Cryst Solids 354(35–39):4475–4480, 2008) and Pozniak et al. (Rev Adv Mat Sci 23(2):169–174, 2010) exhibit a remarkable and surprising result—at extreme negative Poisson’s ratios the displacement vector has components which are anti-parallel to the direction of loading. In the study of spherical shells under uniform load, the use of auxetic material reduces the ratio of maximum bending stress to the membrane stress , thereby implying that if the shell material possesses a Poisson’s ratio that is sufficiently negative, such as −1, and the boundary condition permits free rotation and lateral displacement, then the use of membrane theory of shell is sufficient even though the shell thickness is significant. Results also recommend the use of auxetic material for spherical shells with simple supports because the bending stress is significantly reduced. However the use of auxetic material as spherical shells, with built-in edge, is not recommended due to the sharp increase in the bending stress as the Poisson’s ratio of the shell material becomes more negative.

Appendices

Appendix A

Writing the LHS of Eq. (8.4.37) as

$$\sum\limits_{m = 1}^{\infty } {\sum\limits_{n = 1}^{\infty } {\frac{{m^{2} }}{{\left( {m^{2} + n^{2} } \right)^{2} }}} } = \sum\limits_{m = 1}^{\infty } {\sum\limits_{n = 1}^{\infty } {\frac{1}{{\left( {m + \frac{{n^{2} }}{m}} \right)^{2} }}} } = \sum\limits_{m = 1}^{\infty } {\sum\limits_{f = 1}^{\infty } {\frac{1}{{\left( {m + f} \right)^{2} }}} } + \sum\limits_{m = 1}^{\infty } {\sum\limits_{g}^{\infty } {\frac{1}{{\left( {m + g} \right)^{2} }}} }$$

(A.1)

where f and g are the positive integer and positive non-integer of n ²/m, respectively, we can see that

$$\sum\limits_{m = 1}^{\infty } {\sum\limits_{n = 1}^{\infty } {\frac{{m^{2} }}{{\left( {m^{2} + n^{2} } \right)^{2} }}} } > \sum\limits_{m = 1}^{\infty } {\sum\limits_{f = 1}^{\infty } {\frac{1}{{\left( {m + f} \right)^{2} }}} } .$$

(A.2)

Since the RHS of Eq. (A.2) is divergent (Ghorpade and Limaye 2010), it follows that the LHS is also divergent by comparison.

Appendix B

The relations between the moments and stresses are given as (Ventsel and Krauthammer 2001)

$$\left\{ {\begin{array}{*{20}c} {M_{x} } \\ {M_{y} } \\ {M_{xy} } \\ \end{array} } \right\} = - D\left[ {\begin{array}{*{20}c} 1 & v & 0 \\ v & 1 & 0 \\ 0 & 0 & {v - 1} \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\frac{{\partial^{2} w}}{{\partial x^{2} }}} \\ {\frac{{\partial^{2} w}}{{\partial y^{2} }}} \\ {\frac{{\partial^{2} w}}{\partial x\partial y}} \\ \end{array} } \right\}$$

(B.1)

$$\left\{ {\begin{array}{*{20}c} {\upsigma_{x} } \\ {\upsigma_{y} } \\ {\uptau_{xy} } \\ \end{array} } \right\}_{\hbox{max} } = \pm \frac{6}{{h^{2} }}\left\{ {\begin{array}{*{20}c} {M_{x} } \\ {M_{y} } \\ {M_{xy} } \\ \end{array} } \right\}_{\hbox{max} }$$

(B.2)

$$\left\{ {\begin{array}{*{20}c} {{\tau }_{xz} } \\ {{\tau }_{yz} } \\ \end{array} } \right\} = - \left\{ {\begin{array}{*{20}c} {\int\limits_{- h/2}^{+ h/2}{\left( {\frac{{\partial {\sigma }_{x} }}{\partial x} + \frac{{\partial{\tau }_{xy} }}{\partial y}} \right)dz} } \\ {\int\limits_{- h/2}^{+ h/2}{\left( {\frac{{\partial {\sigma }_{y} }}{\partial y} + \frac{{\partial {\tau }_{yx} }}{\partial x}} \right)dz} } \\ \end{array} } \right\} = \frac{{E\left( {z^{2} - \frac{{h^{2} }}{4}} \right)}}{{2\left( {1 - v^{2} } \right)}}\left\{ {\begin{array}{*{20}c} {\frac{\partial }{\partial x}{\nabla }^{2} w} \\ {\frac{\partial }{\partial y}{\nabla }^{2} w} \\ \end{array} } \right\}$$

(B.3)