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Effect of irregularity and heterogeneity on the stresses produced due to a normal moving load on a rough monoclinic half-space

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Abstract

The present study aims to study the normal and shear stresses produced in a rough irregular heterogeneous monoclinic half-space due to a normal moving load. Closed form expressions of normal and shear stresses have been obtained. It is observed that both normal stress and shear stress are affected not only by depth, the frictional coefficient on a rough surface, and the maximum depth of irregularity but also by the heterogeneity and types of irregularity in the medium. The comparative study has been made to analyze the effect of different types of irregularity on both the stresses. There is a significant effect of depth, frictional coefficient, heterogeneity, maximum depth of irregularity and irregularity factor on the normal and shear stresses in both heterogeneous monoclinic and heterogeneous isotropic medium. A comparison is made to study the effects of the said parameters on the normal and shear stress produced in both heterogeneous medium. These effects are highlighted and depicted by means of graphs. As a special case of the problem, the stress produced due to normal moving load in an isotropic half-space with and without heterogeneity, irregularity has been discussed.

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Acknowledgments

The authors convey their sincere thanks to Indian School of Mines, Dhanbad for providing JRF to Mr. Santan Kumar and also facilitating us with its best facility for research.

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Authors and Affiliations

  1. Department of Applied Mathematics, Indian School of Mines, Dhanbad, 826004, Jharkhand, India

    Abhishek Kumar Singh, Santan Kumar & Amares Chattopadhyay

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  1. Abhishek Kumar Singh
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  2. Santan Kumar
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  3. Amares Chattopadhyay
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Correspondence to Abhishek Kumar Singh.

Appendices

Appendix I

$$ \begin{aligned} a = C_{44}^{0} - C_{22}^{0} q^{2} - \rho^{0} V^{2} ,\quad b = \left( {C_{42}^{0} + C_{24}^{0} } \right)q, \hfill \\ c = \left( {C_{24}^{0} q^{2} - C_{43}^{0} } \right),\quad d = \left( {C_{23}^{0} + C_{44}^{0} } \right)q, \hfill \\ \end{aligned} $$
$$ \begin{aligned} a_{1} = C_{42}^{0} q^{2} - C_{34}^{0} ,\quad b_{1} = \left( {C_{44}^{0} + C_{32}^{0} } \right)q, \hfill \\ c_{1} = \left( { - \rho^{0} V^{2} + C_{33}^{0} - C_{44}^{0} q^{2} } \right),\quad d_{1} = \left( {C_{43}^{0} + C_{34}^{0} } \right)q, \hfill \\ \end{aligned} $$
$$ \begin{aligned} A^{'} = \left( {C_{44}^{0} - \rho V^{2} } \right),\quad B^{'} = \left( {C_{23}^{0} + C_{44}^{0} } \right),\quad C^{'} = \left( {C_{42}^{0} + C_{24}^{0} } \right),\, \hfill \\ D^{'} = \left( {A^{'} B^{'} - C^{'} C_{43}^{0} } \right),\quad E^{'} = \left( {C^{'} C_{24}^{0} - C_{22}^{0} B^{'} } \right),\quad F^{'} = \left( {C_{34}^{0} + C_{43}^{0} } \right), \hfill \\ G^{'} = \left( {C_{44}^{0} + C_{32}^{0} } \right),\quad H^{'} = \left( {C_{33}^{0} - \rho V^{2} } \right),\quad I^{'} = \left( {C_{42}^{0} F^{'} - G^{'} C_{44}^{0} } \right), \hfill \\ J^{'} = \left( {G^{'} H^{'} - C_{34}^{0} F^{'} } \right),\quad K^{'} = \left( {F^{'2} - 2C_{44}^{0} H^{'} } \right),\quad Y = \left( {B^{'2} - 2C_{24}^{0} C_{43}^{0} } \right), \hfill \\ \end{aligned} $$
$$ \begin{aligned} a_{0} & = E^{'} C_{44}^{02} + I^{'} C_{24}^{02} ,\quad 3b_{0} = \left[ {D^{'} C_{44}^{02} + E^{'} K^{'} + I^{'} Y + J^{'} C_{24}^{02} } \right], \\ 3c_{0} & = \left[ {D^{'} K^{'} + E^{'} H^{'2} + I^{'} C_{43}^{02} + J^{'} Y} \right],\quad d_{0} = D^{'} H^{'2} + J^{'} C_{43}^{02} , \\ \end{aligned} $$
$$ Z = a_{0} x + b_{0} ,\quad H = a_{0} c_{0} - b_{0}^{2} ,\quad G = a_{0}^{2} d_{0} - 3a_{0} b_{0} c_{0} + 2b_{0}^{3} , $$
$$ s_{1} = \left\{ {\frac{1}{2}\left[ { - G + \sqrt {G^{2} + 4H^{3} } } \right]} \right\}^{{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-0pt} 3}}} ,\quad n_{1} = \left\{ {\frac{1}{2}\left[ { - G - \sqrt {G^{2} + 4H^{3} } } \right]} \right\}^{{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-0pt} 3}}} ,\quad x = q^{2} , $$
$$ \begin{aligned} t_{1} = C_{22}^{0} q_{1} + m_{1} C_{23}^{0} ,\quad t_{2} = C_{22}^{0} q_{2} + m_{2} C_{23}^{0} ,\quad t_{3} = \left( {q_{1} m_{1} - 1} \right)C_{24}^{0} ,\quad t_{4} = \left( {q_{2} m_{2} - 1} \right)C_{24}^{0} , \hfill \\ t_{5} = C_{42}^{0} q_{1} + m_{1} C_{43}^{0} ,\quad t_{6} = C_{42}^{0} q_{2} + m_{2} C_{43}^{0} ,\quad t_{7} = \left( {m_{1} q_{1} - 1} \right)C_{44}^{0} ,\quad t_{8} = \left( {m_{2} q_{2} - 1} \right)C_{44}^{0} , \hfill \\ \end{aligned} $$
$$ D_{1} = \left| {\begin{array}{*{20}c} {t_{1} } & {t_{2} } & {t_{3} } & {t_{4} } \\ { - t_{3} } & { - t_{4} } & {t_{1} } & {t_{2} } \\ {t_{5} } & {t_{6} } & {t_{7} } & {t_{8} } \\ { - t_{7} } & { - t_{8} } & {t_{5} } & {t_{6} } \\ \end{array} } \right|. $$

Appendix II

$$ A = \left| {\begin{array}{*{20}c} {\frac{1}{\pi }} & {t_{2} } & {t_{3} } & {t_{4} } \\ 0 & { - t_{4} } & {t_{1} } & {t_{2} } \\ {\frac{1}{\pi }} & {t_{6} } & {t_{7} } & {t_{8} } \\ 0 & { - t_{8} } & {t_{5} } & {t_{6} } \\ \end{array} } \right|,\,\,B = \left| {\begin{array}{*{20}c} {t_{1} } & {\frac{1}{\pi }} & {t_{3} } & {t_{4} } \\ { - t_{3} } & 0 & {t_{1} } & {t_{2} } \\ {t_{5} } & {\frac{R}{\pi }} & {t_{7} } & {t_{8} } \\ { - t_{7} } & 0 & {t_{5} } & {t_{6} } \\ \end{array} } \right|,\,\,C = \left| {\begin{array}{*{20}c} {t_{1} } & {t_{2} } & {\frac{1}{\pi }} & {t_{4} } \\ { - t_{3} } & { - t_{4} } & 0 & {t_{2} } \\ {t_{5} } & {t_{6} } & {\frac{R}{\pi }} & {t_{8} } \\ { - t_{7} } & { - t_{8} } & 0 & {t_{6} } \\ \end{array} } \right|,\,\,D = \left| {\begin{array}{*{20}c} {t_{1} } & {t_{2} } & {t_{3} } & {\frac{1}{\pi }} \\ { - t_{3} } & { - t_{4} } & {t_{1} } & 0 \\ {t_{5} } & {t_{6} } & {t_{7} } & {\frac{R}{\pi }} \\ { - t_{7} } & { - t_{8} } & {t_{5} } & 0 \\ \end{array} } \right| $$
$$ E_{1} = \left| {\begin{array}{*{20}c} { - S_{1} } & {t_{2} } & {t_{3} } & {t_{4} } \\ { - S_{3} } & { - t_{4} } & {t_{1} } & {t_{2} } \\ { - S_{5} } & {t_{6} } & {t_{7} } & {t_{8} } \\ { - S_{7} } & { - t_{8} } & {t_{5} } & {t_{6} } \\ \end{array} } \right|,\,\,E_{2} = \left| {\begin{array}{*{20}c} { - S_{2} } & {t_{2} } & {t_{3} } & {t_{4} } \\ { - S_{4} } & { - t_{4} } & {t_{1} } & {t_{2} } \\ { - S_{6} } & {t_{6} } & {t_{7} } & {t_{8} } \\ { - S_{8} } & { - t_{8} } & {t_{5} } & {t_{6} } \\ \end{array} } \right|,\,\,F_{1} = \left| {\begin{array}{*{20}c} {t_{1} } & { - S_{1} } & {t_{3} } & {t_{4} } \\ { - t_{3} } & { - S_{3} } & {t_{1} } & {t_{2} } \\ {t_{5} } & { - S_{5} } & {t_{7} } & {t_{8} } \\ { - t_{7} } & { - S_{7} } & {t_{5} } & {t_{6} } \\ \end{array} } \right|,\,\, $$
$$ F_{2} = \left| {\begin{array}{*{20}c} {t_{1} } & { - S_{2} } & {t_{3} } & {t_{4} } \\ { - t_{3} } & { - S_{4} } & {t_{1} } & {t_{2} } \\ {t_{5} } & { - S_{6} } & {t_{7} } & {t_{8} } \\ { - t_{7} } & { - S_{8} } & {t_{5} } & {t_{6} } \\ \end{array} } \right|,\,\,G_{1} = \left| {\begin{array}{*{20}c} {t_{1} } & {t_{2} } & { - S_{1} } & {t_{4} } \\ { - t_{3} } & { - t_{4} } & { - S_{3} } & {t_{2} } \\ {t_{5} } & {t_{6} } & { - S_{5} } & {t_{8} } \\ { - t_{7} } & { - t_{8} } & { - S_{7} } & {t_{6} } \\ \end{array} } \right|,\,\,G_{2} = \left| {\begin{array}{*{20}c} {t_{1} } & {t_{2} } & { - S_{2} } & {t_{4} } \\ { - t_{3} } & { - t_{4} } & { - S_{4} } & {t_{2} } \\ {t_{5} } & {t_{6} } & { - S_{6} } & {t_{8} } \\ { - t_{7} } & { - t_{8} } & { - S_{8} } & {t_{6} } \\ \end{array} } \right|,\,\, $$
$$ H_{1} = \left| {\begin{array}{*{20}c} {t_{1} } & {t_{2} } & {t_{3} } & { - S_{1} } \\ { - t_{3} } & { - t_{4} } & {t_{1} } & { - S_{3} } \\ {t_{5} } & {t_{6} } & {t_{7} } & { - S_{5} } \\ { - t_{7} } & { - t_{8} } & {t_{5} } & { - S_{7} } \\ \end{array} } \right|,\,\,H_{2} = \left| {\begin{array}{*{20}c} {t_{1} } & {t_{2} } & {t_{3} } & { - S_{2} } \\ { - t_{3} } & { - t_{4} } & {t_{1} } & { - S_{4} } \\ {t_{5} } & {t_{6} } & {t_{7} } & { - S_{6} } \\ { - t_{7} } & { - t_{8} } & {t_{5} } & { - S_{8} } \\ \end{array} } \right|,\,\, $$
$$ \begin{aligned} \alpha_{1} = \frac{\det A}{{\det D_{1} }},\quad \alpha_{2} = \frac{\det B}{{\det D_{1} }},\quad \alpha_{3} = \frac{\det C}{{\det D_{1} }},\quad \alpha_{4} = \frac{\det D}{{\det D_{1} }},\quad \alpha_{5} = \frac{{\det E_{1} }}{{\det D_{1} }},\quad \alpha_{6} = \frac{{\det E_{2} }}{{\det D_{1} }}, \hfill \\ \alpha_{7} = \frac{{\det F_{1} }}{{\det D_{1} }},\quad \alpha_{8} = \frac{{\det F_{2} }}{{\det D_{1} }},\quad \alpha_{9} = \frac{{\det G_{1} }}{{\det D_{1} }},\quad \alpha_{10} = \frac{{\det G_{2} }}{{\det D_{1} }},\quad \alpha_{11} = \frac{{\det H_{1} }}{{\det D_{1} }},\quad\alpha_{12} = \frac{{\det H_{2} }}{{\det D_{1} }}, \hfill \\ \end{aligned} $$
$$S_{1} = \alpha_{1} t_{1} + \alpha_{2} t_{2} + \alpha_{3} t_{3} + \alpha_{4} t_{4} ,\quad S_{2} = - \left( {\alpha_{1} t_{1} q_{1} + \alpha_{2} t_{2} q_{2} + \alpha_{3} t_{3} q_{1} + \alpha_{4} t_{4} q_{2} } \right),\quad S_{3} = - \alpha_{1} t_{3} - \alpha_{2} t_{4} + \alpha_{3} t_{1} + \alpha_{4} t_{2} ,\quad S_{4} = \alpha_{1} t_{3} q_{1} + \alpha_{2} t_{4} q_{2} - \alpha_{3} t_{1} q_{1} - \alpha_{4} t_{2} q_{2} ,\quad S_{5} = \alpha_{1} t_{5} + \alpha_{2} t_{6} + \alpha_{3} t_{7} + \alpha_{4} t_{8} ,\quad S_{6}= - \left( {\alpha_{1} t_{5} q_{1} + \alpha_{2} t_{6} q_{2} + \alpha_{3} t_{1} q_{1} + \alpha_{4} t_{8} q_{2} } \right),\quad S_{7} = - \alpha_{1} t_{7} - \alpha_{2} t_{8} + \alpha_{3} t_{5} + \alpha_{4} t_{6} ,\quad S_{8} = \alpha_{1} t_{7} q_{1} + \alpha_{2} t_{8} q_{2} - \alpha_{3} t_{5} q_{1} - \alpha_{4} t_{6} q_{2} , $$
$$ \begin{aligned} Q_{1} = \alpha_{1} + \varepsilon h\upsilon \alpha_{5} ,\quad Q_{2} = \varepsilon h\alpha_{6} ,\quad Q_{3} = \alpha_{3} + \varepsilon h\upsilon \alpha_{9} ,\quad Q_{4} = \varepsilon h\alpha_{10} ,\, \hfill \\ Q_{5} = \alpha_{2} + \varepsilon h\upsilon \alpha_{7} ,\quad Q_{6} = \varepsilon h\alpha_{8} ,\quad Q_{7} = \alpha_{4} + \varepsilon h\upsilon \alpha_{11} ,\quad Q_{8} = \varepsilon h\alpha_{12} , \hfill \\ \end{aligned} $$
$$ \begin{aligned} \phi_{1} = q_{1}^{2} y^{2} + \left( {z - Vt} \right)^{2} ,\quad \phi_{2} = q_{2}^{2} y^{2} + \left( {z - Vt} \right)^{2} , \hfill \\ \phi_{3} = q_{1}^{2} y^{2} - \left( {z - Vt} \right)^{2} ,\quad \phi_{4} = q_{2}^{2} y^{2} - \left( {z - Vt} \right)^{2} . \hfill \\ \end{aligned} $$

Appendix III

$$ \begin{aligned} t_{1}^{'} &= \left( {\lambda + 2\mu } \right)q_{1}^{'} + m_{1}^{'} \lambda , \hfill \\ \,t_{2}^{'} &= \left( {\lambda + 2\mu } \right)q_{2}^{'} + m_{2}^{'} \lambda , \hfill \\ \,t_{3}^{'} &= t_{4}^{'} = t_{5}^{'} = t_{6}^{'} = 0, \hfill \\ \,t_{7}^{'} & = \left( {m_{1}^{'} q_{1}^{'} - 1} \right)\mu , \hfill \\ \,t_{8}^{'} & = \left( {q_{2}^{'} m_{2}^{'} - 1} \right)\mu . \hfill \\ \end{aligned} $$

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Singh, A.K., Kumar, S. & Chattopadhyay, A. Effect of irregularity and heterogeneity on the stresses produced due to a normal moving load on a rough monoclinic half-space. Meccanica 49, 2861–2878 (2014). https://doi.org/10.1007/s11012-014-0033-8

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