A porous layer of negative value of Poisson’s ratio under a flat-ended and rigid cylinder indenter

Abstract

Materials whose Poisson’s ratio is negative (e.g., so-called auxetic materials) have a number of unusual mechanical and functional properties. This paper solves an orthotropic and elastic layer under a rigid cylinder indenter. The intensity factor of the stress field is adopted to characterize the concentration of the stresses at the edge of the indenter. Closed form solutions can be obtained if the diameter of the indenter is considerably small than the thickness of the elastic layer. Otherwise, the field intensity factors can be found numerically based on the dual integral equation technique. The effects of the layer thickness and Poisson’s ratio of the material on stress concentration at the indenter tip are discussed. The results show that materials with negative Poisson’s ratio can substantially reduce the stress concentration at the edge of the indenter and increase the energy absorbing. This research is helpful for understanding the possible failure behavior and to optimize the mechanical properties of the materials (i.e., Poisson’s ratio and elastic modulus) in order to achieve a maximum indentation resistance for auxetic materials.

Appendix A

Displacements and stresses on the surface of the elastic layer.

$$u(r,0) = \frac{2}{\pi }\frac{{W_{0} }}{{M_{11} }}\frac{1}{r}\sum\limits_{\alpha = 1}^{2} {b_{\alpha } A_{1\alpha } } \left[ {a - \sqrt {a^{2} - r^{2} } } \right],\quad r < a$$

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$$u(r,0) = \frac{2}{\pi }\frac{{W_{0} }}{{M_{11} }}\frac{a}{r}\sum\limits_{\alpha = 1}^{2} {b_{\alpha } A_{1\alpha } } ,\quad r \ge a$$

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$$w(r,0) = \left\{ {\begin{array}{*{20}l} {W_{0} ,\quad r < a} \hfill \\ {\frac{{2W_{0} }}{\pi }\arcsin \left( \frac{a}{r} \right),\quad r \ge a} \hfill \\ \end{array} } \right.$$

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$$\sigma_{zz} (r,0) = \frac{2}{\pi }\frac{{W_{0} }}{{M_{11} }}\frac{1}{{\sqrt {a^{2} - r^{2} } }},r \le a$$

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$$\sigma_{rr} (r,0) = \frac{2}{\pi }\frac{{W_{0} a}}{{M_{11} }}\frac{{c_{12} - c_{11} }}{{r^{2} }}\sum\limits_{\alpha = 1}^{2} {b_{\alpha } A_{1\alpha } } ,r > a$$

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$$\sigma_{rr} (r,0) = \frac{2}{\pi }\frac{{W_{0} }}{{M_{11} }}\sum\limits_{\alpha = 1}^{2} {b_{\alpha } C_{4\alpha } } \frac{1}{{\sqrt {a^{2} - r^{2} } }} + \frac{2}{\pi }\frac{{W_{0} }}{{M_{11} }}\frac{{c_{12} - c_{11} }}{{r^{2} }}\sum\limits_{\alpha = 1}^{2} {b_{\alpha } A_{1\alpha } } [a - \sqrt {a^{2} - r^{2} } ],r \le a$$

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$$\sigma_{\theta \theta } (r,0) = \frac{2}{\pi }\frac{{W_{0} a}}{{M_{11} }}\frac{{c_{11} - c_{12} }}{{r^{2} }}\sum\limits_{\alpha = 1}^{2} {b_{\alpha } A_{1\alpha } } ,r > a$$

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$$\sigma_{\theta \theta } (r,0) = \frac{2}{\pi }\frac{{W_{0} }}{{M_{11} }}\sum\limits_{\alpha = 1}^{2} {b_{\alpha } C_{5\alpha } } \frac{1}{{\sqrt {a^{2} - r^{2} } }} + \frac{2}{\pi }\frac{{W_{0} }}{{M_{11} }}\frac{{c_{11} - c_{12} }}{{r^{2} }}\sum\limits_{\alpha = 1}^{2} {b_{\alpha } A_{1\alpha } } [a - \sqrt {a^{2} - r^{2} } ],r \le a$$

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