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Asymptotic analysis of an anti-plane shear dispersion of an elastic five-layered structure amidst contrasting properties

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Abstract

The present paper studies the anti-plane shear motion of an inhomogeneous elastic five-layered plate amidst the four contrasting material setups. The asymptotic analysis method in regard to the various material parameters is adopted for the study. The respective displacements, stresses and the Rayleigh-Lamb dispersion relation corresponding to the antisymmetric anti-plane motion with perfect interlayer and traction-free (on the outer faces) boundary conditions are determined. Furthermore, in order to analyze the said dispersion relation in the presence of these contrasts, a unification of parameters was proposed. The overall cutoff frequencies and the low-frequency estimates are determined for both the generalized and unified settings. A comparative analysis between the unified Rayleigh-Lamb dispersion relation and the optimal shortened polynomial dispersion relation is carried for each contrast. We also established some asymptotic formulae for the related unified displacements and stresses.

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Acknowledgements

The first author, Rahmatullah Ibrahim Nuruddeen, sincerely acknowledges the 2017 CIIT-TWAS Full-time Postgraduate Fellowship Award (FR Number: 3240299480).

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Authors and Affiliations

  1. Department of Mathematics, COMSATS University Islamabad, Park Road, Chak Shahzad, Islamabad, 44000, Pakistan

    Rahmatullah Ibrahim Nuruddeen, R. Nawaz & Q. M. Zaigham Zia

  2. Department of Mathematics, Faculty of Science, Federal University Dutse, Dutse, Jigawa State, Nigeria

    Rahmatullah Ibrahim Nuruddeen

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  2. R. Nawaz
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Appendices

Appendix A

The dispersion matrix posed by the problem in Sect. 4 is as follows:

$$\begin{aligned} \left( \begin{array}{ccccc} \sinh \left( p_1\right) &{} -\cosh \left( p_2\right) &{} -\sinh \left( p_2\right) &{} 0 &{} 0 \\ 0 &{} \cosh \left( p_3\right) &{} \sinh \left( p_3\right) &{} -\cosh \left( p_4\right) &{} -\sinh \left( p_4\right) \\ \cosh \left( p_1\right) R_{\text {ic}} \mu _{\text {ic}} &{} -\sinh \left( p_2\right) R_{\text {oc}} \mu _{\text {oc}} &{} -\cosh \left( p_2\right) R_{\text {oc}} \mu _{\text {oc}} &{} 0 &{} 0 \\ 0 &{} \sinh \left( p_3\right) R_{\text {oc}} \mu _{\text {oc}} &{} \cosh \left( p_3\right) R_{\text {oc}} \mu _{\text {oc}} &{} -\sinh \left( p_4\right) R_s \mu _s &{} -\cosh \left( p_4\right) R_s \mu _s \\ 0 &{} 0 &{} 0 &{} \sinh \left( p_5\right) &{} \cosh \left( p_5\right) \\ \end{array} \right) , \end{aligned}$$

with the following shortend terms in the matrix above

$$\begin{aligned} p_1=h_1 R_{\text {ic}}, \ \ p_2=h_1 R_{\text {oc}}, \ \ p_3=\left( h_1+h_2\right) R_{\text {oc}}, \ \ p_4=\left( h_1+h_2\right) R_s, \ \ p_5=\left( h_1+h_2+h_3\right) R_s, \end{aligned}$$

where

$$\begin{aligned} R_s=\sqrt{k^2-\frac{\omega ^2}{c_s^2}}, \ \ \ \ R_{\text {ic}}=\sqrt{k^2-\frac{\omega ^2}{c_{\text {ic}}^2}}, \ \ \ \ R_{\text {oc}}=\sqrt{k^2-\frac{\omega ^2}{c_{\text {oc}}^2}}. \end{aligned}$$

Note that the dimensionless form of the above formulae is used in the main text via Eqs. (9)–(11).

Appendix B

Some of the polynomial coefficients of Eq. (15) of Sect. 4 are as follows:

$$\begin{aligned} \gamma _1= & {} \frac{1}{2} \left( h^*\right) ^2 \mu _* \mu ^*+h h^* \mu \mu _*+h h_* \mu \mu ^*+h \mu \mu _* \mu ^*+\frac{\mu _* \mu ^*}{2}, \nonumber \\ \gamma _2= & {} \frac{1}{6} h^3 \mu \mu _* \mu ^*+\frac{1}{6} h_* \left( h^*\right) ^2 h \mu \mu ^*+\frac{1}{2} \left( h^*\right) ^2 h \mu \mu _* \mu ^*+\frac{1}{4} \left( h^*\right) ^2 \mu _* \mu ^* +\frac{1}{6} \left( h^*\right) ^3 h \mu \mu _* \nonumber \\&+ \frac{1}{2} h^* h \mu \mu _* + \frac{1}{6} h^* h^3 \mu \mu _* + \frac{1}{6} h_* h \mu \mu ^* +\frac{1}{6} h \mu \mu _* \mu ^*, \nonumber \\ \gamma _3= & {} -\frac{1}{6} h^3 \mu \mu _* \mu ^*-\frac{h_* h^3 \mu ^2 \mu ^*}{2 \rho }-\frac{h^3 \mu ^2 \mu _* \mu ^*}{6 \rho }-\frac{h^2 \mu \mu _* \mu ^*}{4 \rho }-\frac{\left( h^*\right) ^2 h \mu \mu _* \left( \mu ^*\right) ^2}{2 \rho ^*} \nonumber \\&-\frac{h^* h \mu \mu _* \mu ^*}{2 \rho ^*}-\frac{\left( h^*\right) ^3 h \mu \mu _* \mu ^*}{3 \rho ^*} -\frac{h_* \left( h^*\right) ^2 h \mu \left( \mu ^*\right) ^2}{6 \rho ^*}-\frac{\left( h^*\right) ^2 \mu _* \left( \mu ^*\right) ^2}{4 \rho ^*} \nonumber \\&-\frac{1}{6} h_* \left( h^*\right) ^2 h \mu \mu ^*-\frac{1}{2} \left( h^*\right) ^2 h \mu \mu _* \mu ^*-\frac{1}{4} \left( h^*\right) ^2 \mu _* \mu ^*-\frac{1}{2} h^* h \mu \mu _*, \nonumber \\ \gamma _4= & {} -\frac{h^2 \mu \mu _* \mu ^*}{2 \rho }-\frac{h^* h \mu \mu _* \mu ^*}{\rho ^*}-\frac{\left( h^*\right) ^2 \mu _* \left( \mu ^*\right) ^2}{2 \rho ^*}-h \mu \mu _* \mu ^*-h h_* \mu \mu ^*-\frac{\mu _* \mu ^*}{2},\nonumber \\ \gamma _5= & {} \frac{h_* h^3 \mu ^2 \mu ^*}{2 \rho }+\frac{h^3 \mu ^2 \mu _* \mu ^*}{6 \rho }+\frac{h^2 \mu \mu _* \mu ^*}{4 \rho }+\frac{\left( h^*\right) ^2 h \mu \mu _* \left( \mu ^*\right) ^2}{2 \rho ^*}+\frac{h^* h \mu \mu _* \mu ^*}{2 \rho ^*} \nonumber \\&+\frac{h_* \left( h^*\right) ^2 h \mu \left( \mu ^*\right) ^2}{6 \rho ^*} + \frac{\left( h^*\right) ^3 h \mu \mu _* \left( \mu ^*\right) ^2}{6 \left( \rho ^*\right) ^2}+\frac{\left( h^*\right) ^2 \mu _* \left( \mu ^*\right) ^2}{4 \rho ^*} \nonumber \\&+\frac{h^* h^3 \mu ^2 \mu _* \mu ^*}{6 \rho \rho ^*}+\frac{\left( h^*\right) ^2 h^2 \mu \mu _* \left( \mu ^*\right) ^2}{4 \rho \rho ^*}+\frac{1}{6} h_* h \mu \mu ^*+\frac{1}{6} h \mu \mu _* \mu ^*, \nonumber \\ \gamma _6= & {} \frac{1}{36} h^3 \mu \mu _* \mu ^*+\frac{h_* h^3 \mu ^2 \mu ^*}{6 \rho }+\frac{h^3 \mu ^2 \mu _* \mu ^*}{18 \rho }+\frac{\left( h^*\right) ^3 h \mu \mu _* \mu ^*}{6 \rho ^*}+\frac{h_* \left( h^*\right) ^2 h \mu \left( \mu ^*\right) ^2}{18 \rho ^*} \nonumber \\&+\frac{\left( h^*\right) ^3 h \mu \mu _* \left( \mu ^*\right) ^2}{12 \left( \rho ^*\right) ^2} +\frac{1}{36} h_* \left( h^*\right) ^2 h \mu \mu ^*+\frac{1}{12} \left( h^*\right) ^2 h \mu \mu _* \mu ^* +\frac{h^* h^3 \mu ^2 \mu _*}{12 \rho } \nonumber \\&+\frac{h^* h^3 \mu \mu _* \mu ^*}{12 \rho ^*}+\frac{\left( h^*\right) ^3 h^3 \mu \mu _* \left( \mu ^*\right) ^2}{36 \left( \rho ^*\right) ^2}+\frac{h_* \left( h^*\right) ^2 h^3 \mu ^2 \left( \mu ^*\right) ^2}{12 \rho \rho ^*} +\frac{h^* h^3 \mu ^2 \mu _* \mu ^*}{12 \rho \rho ^*} \nonumber \\&+\frac{\left( h^*\right) ^3 h^3 \mu ^2 \mu _* \mu ^*}{18 \rho \rho ^*}+\frac{h_* \left( h^*\right) ^2 h^3 \mu ^2 \mu ^*}{12 \rho }+\frac{\left( h^*\right) ^2 h^3 \mu ^2 \mu _* \mu ^*}{12 \rho }+\frac{\left( h^*\right) ^2 h^2 \mu \mu _* \mu ^*}{8 \rho }, \nonumber \\ \gamma _7= & {} -\frac{1}{18} h^3 \mu \mu _* \mu ^*-\frac{h_* h^3 \mu ^2 \mu ^*}{12 \rho }-\frac{h^3 \mu ^2 \mu _* \mu ^*}{36 \rho }-\frac{\left( h^*\right) ^3 h \mu \mu _* \mu ^*}{6 \rho ^*}-\frac{\left( h^*\right) ^2 h \mu \mu _* \left( \mu ^*\right) ^2}{12 \rho ^*} \nonumber \\&-\frac{h_* \left( h^*\right) ^2 h \mu \left( \mu ^*\right) ^2}{36 \rho ^*} -\frac{1}{18} h_* \left( h^*\right) ^2 h \mu \mu ^*-\frac{1}{6} \left( h^*\right) ^2 h \mu \mu _* \mu ^*-\frac{1}{12} \left( h^*\right) ^3 h \mu \mu _* \nonumber \\&-\frac{h^* h^3 \mu ^2 \mu _*}{12 \rho } -\frac{\left( h^*\right) ^3 h^3 \mu ^2 \mu _*}{36 \rho }-\frac{\left( h^*\right) ^2 h^3 \mu \mu _* \left( \mu ^*\right) ^2}{12 \rho ^*} -\frac{h^* h^3 \mu \mu _* \mu ^*}{12 \rho ^*} \nonumber \\&-\frac{\left( h^*\right) ^3 h^3 \mu \mu _* \mu ^*}{18 \rho ^*}-\frac{1}{12} \left( h^*\right) ^2 h^3 \mu \mu _* \mu ^* -\frac{h_* \left( h^*\right) ^2 h^3 \mu ^2 \mu ^*}{12 \rho }-\frac{\left( h^*\right) ^2 h^3 \mu ^2 \mu _* \mu ^*}{12 \rho }, \nonumber \\&\vdots \end{aligned}$$

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Nuruddeen, R.I., Nawaz, R. & Zia, Q.M.Z. Asymptotic analysis of an anti-plane shear dispersion of an elastic five-layered structure amidst contrasting properties. Arch Appl Mech 90, 1875–1892 (2020). https://doi.org/10.1007/s00419-020-01702-6

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