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The reflection of a three-dimensional plane wave by the stress-free and rigid boundary of a functionally graded triclinic medium

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Abstract

A three-dimensional (3D) plane wave incident at a functionally graded triclinic medium (FGTM) under two different types of boundary (viz. stress free (Case I) and rigid (Case II)) has been modelled. The constitutive equations of the functionally graded triclinic medium are used to deduce the Christoffel matrix of order 3. By using Cardano’s method, we derive the expressions for the phase velocities of the reflected waves (qP, qSV and qSH waves). The expressions of the reflection coefficients (amplitude ratios) are obtained for the present geometry with the help of the generalized Snell’s law and Cramer’s rule. The obtained mathematical expressions are perfectly matched with pre-established results. Further, the slowness surface and energy ratios have been also completed to prove the correctness of the presented problem. The effect of anisotropy and functionally graded parameter on the reflection coefficients has been traced out remarkably through a comparative study by numerical modelling. The comparative analysis reveals that the functionally graded parameter and anisotropy have significant impact on the amplitude ratios and both these attributes contribute simultaneously to simulate a realistic elastodynamic model and predict the response of the medium (geo-medium) with better accuracy.

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Acknowledgements

The authors thank to IIT (ISM), Dhanbad, for providing SRF to Ms. Akanksha Srivastava and also facilitate us with its good facilities for research work.

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Authors and Affiliations

  1. Department of Mathematics and Computing, Indian Institute of Technology (Indian School of Mines), Dhanbad, Jharkhand, 826004, India

    Akanksha Srivastava, Abhishek Kumar Singh & Amares Chattopadhyay

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  1. Akanksha Srivastava
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  2. Abhishek Kumar Singh
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  3. Amares Chattopadhyay
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Correspondence to Akanksha Srivastava.

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Appendix

Appendix

$$\begin{aligned} B_{11}= & {} \overline{M_{11}} {p_1^{\left( n \right) }}^2+\overline{M_{66}} {p_2^{\left( n \right) }}^2+\overline{M_{55}} {p_3^{\left( n \right) }}^2+2\overline{M_{16}}p_1^{\left( n \right) }p_2^{\left( n \right) }+2\overline{M_{15}}p_1^{\left( n \right) }p_3^{\left( n \right) }+2\overline{M_{56}}p_2^{\left( n \right) }p_3^{\left( n \right) }-\frac{il}{k_{n}}\overline{M_{16}}p_{1}^{(n)}\\&-\,\frac{il}{k_{n}}\overline{M_{66}}p_{2}^{(n)}-\frac{il}{k_{n}}\overline{M_{56}}p_{3}^{(n)},\\ B_{12}= & {} \overline{M_{16}} {p_1^{\left( n \right) }}^2+\overline{M_{26}} {p_2^{\left( n \right) }}^2+\overline{M_{45}} {p_3^{\left( n \right) }}^2+(\overline{M_{12}}+\overline{M_{66}})p_1^{\left( n \right) }p_2^{\left( n \right) }+(\overline{M_{14}}+\overline{M_{56}})p_1^{\left( n \right) }p_3^{\left( n \right) }\\&+\,(\overline{M_{46}}+\overline{M_{25}})p_2^{\left( n \right) }p_3^{\left( n \right) }-\frac{il}{k_{n}}\overline{M_{66}}p_{1}^{(n)}-\frac{il}{k_{n}}\overline{M_{26}}p_{2}^{(n)}-\frac{il}{k_{n}}\overline{M_{46}}p_{3}^{(n)},\\ B_{13}= & {} \overline{M_{15}} {p_1^{\left( n \right) }}^2+\overline{M_{46}} {p_2^{\left( n \right) }}^2+\overline{M_{35}} {p_3^{\left( n \right) }}^2+(\overline{M_{13}}+\overline{M_{55}})p_1^{\left( n \right) }p_3^{\left( n \right) }+(\overline{M_{14}}+\overline{M_{56}})p_1^{\left( n \right) }p_2^{\left( n \right) }\\&+\,(\overline{M_{36}}+\overline{M_{45}})p_2^{\left( n \right) }p_3^{\left( n \right) }-\frac{il}{k_{n}}\overline{M_{56}}p_{1}^{(n)}-\frac{il}{k_{n}}\overline{M_{46}}p_{2}^{(n)}-\frac{il}{k_{n}}\overline{M_{36}}p_{3}^{(n)},\\ B_{21}= & {} \overline{M_{16}} {p_1^{\left( n \right) }}^2+\overline{M_{26}} {p_2^{\left( n \right) }}^2+\overline{M_{45}} {p_3^{\left( n \right) }}^2+(\overline{M_{12}}+\overline{M_{66}})p_1^{\left( n \right) }p_2^{\left( n \right) }+(\overline{M_{14}}+\overline{M_{56}})p_1^{\left( n \right) }p_3^{\left( n \right) }\\&+\,(\overline{M_{46}}+\overline{M_{25}})p_2^{\left( n \right) }p_3^{\left( n \right) }-\frac{il}{k_{n}}\overline{M_{12}}p_{1}^{(n)}-\frac{il}{k_{n}}\overline{M_{26}}p_{2}^{(n)}-\frac{il}{k_{n}}\overline{M_{25}}p_{3}^{(n)},\\ B_{22}= & {} \overline{M_{66}} {p_1^{\left( n \right) }}^2+\overline{M_{22}} {p_2^{\left( n \right) }}^2+{p_3^{\left( n \right) }}^2+2\overline{M_{26}}p_1^{\left( n \right) }p_2^{\left( n \right) }+2\overline{M_{46}}p_1^{\left( n \right) }p_3^{\left( n \right) }+2\overline{M_{24}}p_2^{\left( n \right) }p_3^{\left( n \right) }-\frac{il}{k_{n}}\overline{M_{26}}p_{1}^{(n)}\\&-\,\frac{il}{k_{n}}\overline{M_{22}}p_{2}^{(n)}-\frac{il}{k_{n}}\overline{M_{24}}p_{3}^{(n)}, \end{aligned}$$
$$\begin{aligned} B_{23}= & {} \overline{M_{56}} {p_1^{\left( n \right) }}^2+\overline{M_{24}} {p_2^{\left( n \right) }}^2+\overline{M_{34}} {p_3^{\left( n \right) }}^2+(\overline{M_{46}}+\overline{M_{25}})p_1^{\left( n \right) }p_2^{\left( n \right) }+(\overline{M_{36}}+\overline{M_{45}})p_1^{\left( n \right) }p_3^{\left( n \right) }\\&+\,(\overline{M_{23}})p_2^{\left( n \right) }p_3^{\left( n \right) }-\frac{il}{k_{n}}\overline{M_{25}}p_{1}^{(n)}-\frac{il}{k_{n}}\overline{M_{24}}p_{2}^{(n)}-\frac{il}{k_{n}}\overline{M_{23}}p_{3}^{(n)},\\ B_{31}= & {} \overline{M_{15}} {p_1^{\left( n \right) }}^2+\overline{M_{46}} {p_2^{\left( n \right) }}^2+\overline{M_{35}} {p_3^{\left( n \right) }}^2+(\overline{M_{13}}+\overline{M_{55}})p_1^{\left( n \right) }p_3^{\left( n \right) }+(\overline{M_{14}}+\overline{M_{56}})p_1^{\left( n \right) }p_2^{\left( n \right) }\\&+\,(\overline{M_{36}}+\overline{M_{45}})p_2^{\left( n \right) }p_3^{\left( n \right) }-\frac{il}{k_{n}}\overline{M_{14}}p_{1}^{(n)}-\frac{il}{k_{n}}\overline{M_{46}}p_{2}^{(n)}-\frac{il}{k_{n}}\overline{M_{45}}p_{3}^{(n)},\\ B_{32}= & {} \overline{M_{56}} {p_1^{\left( n \right) }}^2+\overline{M_{24}} {p_2^{\left( n \right) }}^2+\overline{M_{34}} {p_3^{\left( n \right) }}^2+(\overline{M_{46}}+\overline{M_{25}})p_1^{\left( n \right) }p_2^{\left( n \right) }+(\overline{M_{36}}+\overline{M_{45}})p_1^{\left( n \right) }p_3^{\left( n \right) }\\&+\,(\overline{M_{23}})p_2^{\left( n \right) }p_3^{\left( n \right) }-\frac{il}{k_{n}}\overline{M_{46}}p_{1}^{(n)}-\frac{il}{k_{n}}\overline{M_{24}}p_{2}^{(n)}-\frac{il}{k_{n}}p_{3}^{(n)},\\ B_{33}= & {} \overline{M_{55}} {p_1^{\left( n \right) }}^2+ {p_2^{\left( n \right) }}^2+\overline{M_{33}} {p_3^{\left( n \right) }}^2+2\overline{M_{45}}p_1^{\left( n \right) }p_2^{\left( n \right) }+2\overline{M_{35}}p_1^{\left( n \right) }p_3^{\left( n \right) }+2\overline{M_{34}}p_2^{\left( n \right) }p_3^{\left( n \right) }-\frac{il}{k_{n}}\overline{M_{45}}p_{1}^{(n)}\\&-\,\frac{il}{k_{n}}p_{2}^{(n)}-\frac{i\gamma }{k_{n}}\overline{M_{34}}p_{3}^{(n)},\\ \overline{M_{ij}}= & {} \frac{M_{ij}}{M_{44}},\\ P_{n}= & {} {\overline{M_{11}}}\,{p_{1}^{(n)}}{d_{1}^{(n)}} + {\overline{M_{12}}}\,{p_{2}^{(n)}}{d_{2}^{(n)}} + {\overline{M_{13}}}\,{p_{3}^{(n)}}{d_{3}^{(n)}} + {\overline{M_{14}}}\,({p_{3}^{(n)}}{d_{2}^{(n)}}+{p_{2}^{(n)}}{d_{3}^{(n)}})\\&+ \,{\overline{M_{15}}}\,({p_{3}^{(n)}}{d_{1}^{(n)}}+{p_{1}^{(n)}}{d_{3}^{(n)}}) +{\overline{M_{16}}}\,({p_{2}^{(n)}}{d_{1}^{(n)}}+{p_{1}^{(n)}}{d_{2}^{(n)}}), \end{aligned}$$
$$\begin{aligned} Q_{n}= & {} {\overline{M_{12}}}\,{p_{1}^{(n)}}{d_{1}^{(n)}} + {\overline{M_{22}}}\,{p_{2}^{(n)}}{d_{2}^{(n)}} + {\overline{M_{23}}}\,{p_{3}^{(n)}}{d_{3}^{(n)}} + {\overline{M_{24}}}\,({p_{3}^{(n)}}{d_{2}^{(n)}}+{p_{2}^{(n)}}{d_{3}^{(n)}})\\&+\, {\overline{M_{25}}}\,({p_{3}^{(n)}}{d_{1}^{(n)}}+{p_{1}^{(n)}}{d_{3}^{(n)}}) + {\overline{M_{26}}}\,({p_{2}^{(n)}}{d_{1}^{(n)}}+{p_{1}^{(n)}}{d_{2}^{(n)}}),\\ R_{n}= & {} {\overline{M_{13}}}\,{p_{1}^{(n)}}{d_{1}^{(n)}} + {\overline{M_{23}}}\,{p_{2}^{(n)}}{d_{2}^{(n)}} + {\overline{M_{33}}}\,{p_{3}^{(n)}}{d_{3}^{(n)}} + {\overline{M_{34}}}\,({p_{3}^{(n)}}{d_{2}^{(n)}}+{p_{2}^{(n)}}{d_{3}^{(n)}})\\&+\,{\overline{M_{35}}}\,({p_{3}^{(n)}}{d_{1}^{(n)}}+{p_{1}^{(n)}}{d_{3}^{(n)}}) +{\overline{M_{36}}}\,({p_{2}^{(n)}}{d_{1}^{(n)}}+{p_{1}^{(n)}}{d_{2}^{(n)}}),\\ S_{n}= & {} {\overline{M_{14}}}\,{p_{1}^{(n)}}{d_{1}^{(n)}} + {\overline{M_{24}}}\,{p_{2}^{(n)}}{d_{2}^{(n)}} + {\overline{M_{34}}}\,{p_{3}^{(n)}}{d_{3}^{(n)}} + {\overline{M_{44}}}\,({p_{3}^{(n)}}{d_{2}^{(n)}}+{p_{2}^{(n)}}{d_{3}^{(n)}})\\&+\, {\overline{M_{45}}}\,({p_{3}^{(n)}}{d_{1}^{(n)}}+{p_{1}^{(n)}}{d_{3}^{(n)}}) +{\overline{M_{46}}}\,({p_{2}^{(n)}}{d_{1}^{(n)}}+{p_{1}^{(n)}}{d_{2}^{(n)}}),\\ T_{n}= & {} {\overline{M_{15}}}\,{p_{1}^{(n)}}{d_{1}^{(n)}} + {\overline{M_{25}}}\,{p_{2}^{(n)}}{d_{2}^{(n)}} + {\overline{M_{35}}}\,{p_{3}^{(n)}}{d_{3}^{(n)}} + {\overline{M_{45}}}\,({p_{3}^{(n)}}{d_{2}^{(n)}}+{p_{2}^{(n)}}{d_{3}^{(n)}})\\&+\, {\overline{M_{55}}}\,({p_{3}^{(n)}}{d_{1}^{(n)}}+{p_{1}^{(n)}}{d_{3}^{(n)}}) + {\overline{M_{56}}}\,({p_{2}^{(n)}}{d_{1}^{(n)}}+{p_{1}^{(n)}}{d_{2}^{(n)}}),\\ Z_{n}= & {} {\overline{M_{16}}}\,{p_{1}^{(n)}}{d_{1}^{(n)}} + {\overline{M_{26}}}\,{p_{2}^{(n)}}{d_{2}^{(n)}} + {\overline{M_{36}}}\,{p_{3}^{(n)}}{d_{3}^{(n)}} + {\overline{M_{46}}}\,({p_{3}^{(n)}}{d_{2}^{(n)}}+{p_{2}^{(n)}}{d_{3}^{(n)}})\\&+\,{\overline{M_{56}}}\,({p_{3}^{(n)}}{d_{1}^{(n)}}+{p_{1}^{(n)}}{d_{3}^{(n)}}) + {\overline{M_{66}}}\,({p_{2}^{(n)}}{d_{1}^{(n)}}+{p_{1}^{(n)}}{d_{2}^{(n)}}),\\ {Z_{n}}^{'}= & {} {C_{16}}\,{p_{1}^{(n)}}{d_{1}^{(n)}} +{C_{26}}\,{p_{2}^{(n)}}{d_{2}^{(n)}} + {C_{36}}\,{p_{3}^{(n)}}{d_{3}^{(n)}} + {C_{46}}\,({p_{3}^{(n)}}{d_{2}^{(n)}}+{p_{2}^{(n)}}{d_{3}^{(n)}})\\&+\, {C_{56}}\,({p_{3}^{(n)}}{d_{1}^{(n)}}+{p_{1}^{(n)}}{d_{3}^{(n)}}) + {C_{66}}\,({p_{2}^{(n)}}{d_{1}^{(n)}}+{p_{1}^{(n)}}{d_{2}^{(n)}}),\\ {Q_{n}}^{'}= & {} {C_{12}}\,{p_{1}^{(n)}}{d_{1}^{(n)}} +{C_{22}}\,{p_{2}^{(n)}}{d_{2}^{(n)}} + {C_{23}}\,{p_{3}^{(n)}}{d_{3}^{(n)}} + {C_{24}}\,({p_{3}^{(n)}}{d_{2}^{(n)}}+{p_{2}^{(n)}}{d_{3}^{(n)}})\\&+\, {C_{25}}\,({p_{3}^{(n)}}{d_{1}^{(n)}}+{p_{1}^{(n)}}{d_{3}^{(n)}}) + {C_{26}}\,({p_{2}^{(n)}}{d_{1}^{(n)}}+{p_{1}^{(n)}}{d_{2}^{(n)}}),\\ {S_{n}}^{'}= & {} {C_{14}}\,{p_{1}^{(n)}}{d_{1}^{(n)}} + {C_{24}}\,{p_{2}^{(n)}}{d_{2}^{(n)}} + {C_{34}}\,{p_{3}^{(n)}}{d_{3}^{(n)}} + {C_{44}}\,({p_{3}^{(n)}}{d_{2}^{(n)}}+{p_{2}^{(n)}}{d_{3}^{(n)}})\\&+\, {C_{45}}\,({p_{3}^{(n)}}{d_{1}^{(n)}}+{p_{1}^{(n)}}{d_{3}^{(n)}}) +{C_{46}}\,({p_{2}^{(n)}}{d_{1}^{(n)}}+{p_{1}^{(n)}}{d_{2}^{(n)}}).\\ \end{aligned}$$

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Srivastava, A., Singh, A.K. & Chattopadhyay, A. The reflection of a three-dimensional plane wave by the stress-free and rigid boundary of a functionally graded triclinic medium. Acta Mech 233, 4523–4534 (2022). https://doi.org/10.1007/s00707-022-03328-x

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