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Tabulation of Mindlin–Airy stress potentials and the corresponding stress and displacement in polar co-ordinates in couple stress elasticity

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Abstract

The present note tabulates the generalized solution to Mindlin–Airy potential in couple stress elasticity; counterpart to Airy stress potential in classical elasticity. Subsequently stress, couple stress and displacement components are compiled in polar co-ordinates. The utility of the developed tables is demonstrated through solution to two boundary value problems on circular domains in couple stress elasticity.

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Notes

  1. In [12], Eqs. (A9–A12) representing \(\lbrace \sigma _{yy}, \sigma _{xx}, \sigma _{yx}, \sigma _{xy} \rbrace \) should be appended with \(\lbrace Tr_1, -Tr_1, Tr_2, Tr_2 \rbrace \) respectively where \(Tr_1 = \mu b \cos \theta \sin ^2 \theta \Bigl [-8(\ell /r)^2 +\lbrace 4+(r/\ell )^2 \rbrace K_2(r/\ell ) \Bigr ]/(\pi r)\) and \(Tr_2 = \mu b \sin \theta \cos (2\theta ) K_1(r/\ell )/(\pi \ell )\).

References

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Acknowledgements

The author gratefully acknowledges the financial assistance extended by Science and Engineering Research Board (SERB) through the MATRICS Grant MTR/2020/000217. The author also thanks Madhav D. More whose Master’s Thesis Project was the motivation for the presented work.

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Authors and Affiliations

  1. Department of Mechanical Engineering, Indian Institute of Technology Bombay, Mumbai, 400076, India

    Tanmay K Bhandakkar

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  1. Tanmay K Bhandakkar
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Correspondence to Tanmay K Bhandakkar.

Appendix I: Stress and couple stress components

Appendix I: Stress and couple stress components

The terms appearing in tables 2–3 are by-product of the terms in Eqs. (7)–(11) and its involvement in the stress computation through Eqs. (12)–(17). The definition of the terms is given below,

$$\begin{aligned} sT_{1r\theta }=-\frac{1}{\ell r} \bigl [ G_{01}I_1(r/\ell ) -H_{01}K_1(r/\ell )\bigr ], \end{aligned}$$
(A1)
$$\begin{aligned} sT_{1\theta r}&=\frac{1}{\ell ^2}\Bigl [ G_{01}\lbrace I_0(r/\ell ) -(\ell /r)I_1(r/\ell )\rbrace \\ &\quad +H_{01}\lbrace K_0(r/\ell )+(\ell /r)K_1(r/\ell ) \rbrace \Bigr ],\end{aligned}$$
(A2)
$$\begin{aligned} csT_{1r}=uT_{1\theta }=\frac{1}{\ell } \bigl [ G_{01}I_1(r/\ell ) -H_{01}K_1(r/\ell )\bigr ], \end{aligned}$$
(A3)
$$\begin{aligned} sT_{2r}&= \frac{1}{r^2}\Bigl [ F_1 \lbrace 2 K_1 (r/\ell ) + (r/\ell ) K_0(r/\ell )\rbrace \\ &\quad +G_1 \lbrace 2 I_1 (r/\ell ) -(r/\ell )I_0(r/\ell ) \rbrace \Bigr ]\cos \theta ,\end{aligned}$$
(A4)
$$\begin{aligned} sT_{2\theta }&=\frac{1}{r^2} \Bigl [ -F_1 \lbrace 2 K_1 (r/\ell ) + (r/\ell ) K_0(r/\ell )\rbrace \\ &\quad -G_1 \lbrace 2 I_1 (r/\ell ) -(r/\ell )I_0(r/\ell ) \rbrace \Bigr ] \cos \theta ,\end{aligned}$$
(A5)
$$\begin{aligned} sT_{2r\theta }&= \frac{1}{r^2} \Bigl [ F_1 \lbrace 2 K_1 (r/\ell ) + (r/\ell ) K_0(r/\ell )\rbrace \\ &\quad + G_1 \lbrace 2 I_1 (r/\ell ) -(r/\ell )I_0(r/\ell ) \rbrace \Bigr ] \sin \theta ,\end{aligned}$$
(A6)
$$\begin{aligned} sT_{2\theta r}&= \frac{1}{r^2} \Bigl [ F_1 \lbrace 2 K_1 (r/\ell ) + (r/\ell ) K_0(r/\ell ) \\ &\quad + (r/\ell )^2 K_1(r/\ell ) \rbrace +G_1 \lbrace 2 I_1 (r/\ell ) - (r/\ell )I_0(r/\ell ) \\ &\quad +(r/\ell )^2 I_1(r/\ell ) \rbrace \Bigr ] \sin \theta ,\end{aligned}$$
(A7)
$$\begin{aligned} csT_{2r}&= \frac{1}{\ell } \Bigl \lbrace -F_1 \bigl [ (\ell /r) K_1(r/\ell ) + K_0(r/\ell )\bigr ] \\ &+\quad G_1 \bigl [ I_0(r/\ell )-(\ell /r)I_1(r/\ell )\bigr ] \Bigr \rbrace \sin \theta ,\end{aligned}$$
(A8)
$$\begin{aligned} csT_{2\theta }= \frac{1}{r}\bigl [G_1 I_1(r/\ell )+F_1 K_1(r/\ell )\bigr ] \cos \theta , \end{aligned}$$
(A9)
$$\begin{aligned} uT_{2r}= -\frac{1}{r}\bigl [G_1 I_1(r/\ell )+F_1 K_1(r/\ell )\bigr ]\cos \theta , \end{aligned}$$
(A10)
$$\begin{aligned} uT_{2\theta }&= -\frac{1}{r}\Bigl [G_1 \lbrace I_1(r/\ell )-(r/\ell )I_0(r/\ell )\rbrace \\ &\quad +F_1\lbrace K_1(r/\ell )+(r/\ell )K_0(r/\ell )\rbrace \Bigr ]\sin \theta ,\end{aligned}$$
(A11)
$$\begin{aligned} sT_{3r}&= {} \frac{1}{r^2}\Bigl [E_1 \lbrace (r/\ell )I_0(r/\ell )-2I_1(r/\ell )\rbrace \\ &\quad -H_1 \lbrace (r/\ell )K_0(r/\ell )+2K_1(r/\ell ) \rbrace \Bigr ] \sin \theta ,\end{aligned}$$
(A12)
$$\begin{aligned} sT_{3\theta }&= {} \frac{1}{r^2}\Bigl [E_1 \lbrace 2I_1(r/\ell )-(r/\ell )I_0 (r/\ell )\rbrace \\ &\quad +H_1 \lbrace (r/\ell )K_0 (r/\ell )+2K_1 (r/\ell ) \rbrace \Bigr ] \sin \theta ,\end{aligned}$$
(A13)
$$\begin{aligned} sT_{3r\theta }&= \frac{1}{r^2} \Bigl [E_1 \lbrace 2I_1(r/\ell )-(r/\ell )I_0(r/\ell ) \rbrace \\ &\quad +H_1 \lbrace 2K_1(r/\ell )+(r/\ell )K_0(r/\ell ) \rbrace \Bigr ] \cos \theta ,\end{aligned}$$
(A14)
$$\begin{aligned} sT_{3\theta r}&= \frac{1}{r^2} \Bigl [E_1 \lbrace (2+(r/\ell )^2)I_1(r/\ell )-(r/\ell )I_0(r/\ell ) \rbrace \\ &\quad + H_1 \lbrace (2+(r/\ell )^2)K_1(r/\ell )+(r/\ell )K_0(r/\ell ) \rbrace \Bigr ] \cos \theta ,\end{aligned}$$
(A15)
$$\begin{aligned} csT_{3r}&= uT_{3\theta } = \frac{1}{\ell }\Bigl [E_1 \lbrace I_0(r/\ell )-(\ell /r)I_1(r/\ell ) \rbrace \\ &\quad -H_1 \lbrace K_0(r/\ell )+(\ell /r)K_1(r/\ell )\rbrace \Bigr ] \cos \theta ,\end{aligned}$$
(A16)
$$\begin{aligned} csT_{3\theta }&= -uT_{3r}= -\frac{1}{r}\bigl [E_1 I_1(r/\ell )+H_1 K_1(r/\ell )\bigr ] \sin \theta ,\end{aligned}$$
(A17)
$$\begin{aligned} sT_{4r}&= \frac{1}{r^2}\Bigl [G_n \lbrace (n-n^2)I_n(r/\ell )-n(r/\ell )I_{n+1}(r/\ell )\rbrace \\ &\quad +F_n \lbrace (n-n^2)K_n(r/\ell )+n(r/\ell )K_{n+1}(r/\ell )\rbrace \Bigr ] \cos (n\theta ),\end{aligned}$$
(A18)
$$\begin{aligned} sT_{4\theta }&= \frac{1}{r^2}\Bigl [G_n \lbrace (n^2-n)I_n(r/\ell )+n(r/\ell )I_{n+1}(r/\ell )\rbrace \\ &\quad +F_n \lbrace (n^2-n)K_n(r/\ell )-n(r/\ell )K_{n+1}(r/\ell )\rbrace \Bigr ] \cos (n\theta ),\end{aligned}$$
(A19)
$$\begin{aligned} sT_{4r\theta }&= \frac{1}{r^2}\Bigl [G_n \lbrace (n^2-n)I_n(r/\ell )-(r/\ell )I_{n+1}(r/\ell )\rbrace \\&\quad +F_n \lbrace (n^2-n)K_n(r/\ell )+(r/\ell )K_{n+1}(r/\ell )\rbrace \Bigr ] \sin (n\theta ),\end{aligned}$$
(A20)
$$\begin{aligned} sT_{4\theta r}&= \frac{1}{r^2}\Bigl [G_n \lbrace (n^2-n+(r/\ell )^2)I_n(r/\ell )-(r/\ell )I_{n+1}(r/\ell )\rbrace \\ &\quad +F_n \lbrace (n^2-n+(r/\ell )^2)K_n(r/\ell )+(r/\ell )K_{n+1}(r/\ell )\rbrace \Bigr ] \sin (n\theta ),\end{aligned}$$
(A21)
$$\begin{aligned} csT_{4r}&= uT_{4\theta }=\frac{1}{\ell }\Bigl [G_n \lbrace I_{n+1}(r/\ell )+n(\ell /r)I_n(r/\ell ) \rbrace \\ &\quad +F_n \lbrace -K_{n+1}(r/\ell )+n(\ell /r)K_n(r/\ell ) \rbrace \Bigr ] \sin (n\theta )\end{aligned}$$
(A22)
$$\begin{aligned} csT_{4\theta }&=-uT_{4r} = \frac{n}{r}\bigl [G_nI_n(r/\ell )+F_n K_n(r/\ell )\bigr ]\cos (n\theta ),\end{aligned}$$
(A23)
$$\begin{aligned} sT_{5r}&= \frac{1}{r^2}\Bigl [E_n \lbrace (n^2-n)I_n(r/\ell )+n(r/\ell )I_{n+1}(r/\ell )\rbrace \\ &\quad +H_n \lbrace (n^2-n)K_n(r/\ell )-n(r/\ell )K_{n+1}(r/\ell )\rbrace \Bigr ] \sin (n\theta ),\end{aligned}$$
(A24)
$$\begin{aligned} sT_{5\theta }&= \frac{1}{r^2}\Bigl [E_n \lbrace (n-n^2)I_n(r/\ell )-n(r/\ell )I_{n+1}(r/\ell )\rbrace \\ &\quad +H_n \lbrace (n-n^2)K_n(r/\ell )+n(r/\ell )K_{n+1}(r/\ell )\rbrace \Bigr ] \sin (n\theta ),\end{aligned}$$
(A25)
$$\begin{aligned} sT_{5r\theta }&= \frac{1}{r^2}\Bigl [E_n \lbrace (n^2-n)I_n(r/\ell )-(r/\ell )I_{n+1}(r/\ell )\rbrace \\ &\quad +H_n \lbrace (n^2-n)K_n(r/\ell )+(r/\ell )K_{n+1}(r/\ell )\rbrace \Bigr ] \cos (n\theta ),\end{aligned}$$
(A26)
$$\begin{aligned} sT_{5\theta r}&= \frac{1}{r^2}\Bigl [E_n \lbrace (n^2-n+(r/\ell )^2)I_n(r/\ell )-(r/\ell )I_{n+1}(r/\ell )\rbrace \\ &\quad + H_n \lbrace (n^2-n+(r/\ell )^2)K_n(r/\ell )+(r/\ell )K_{n+1}(r/\ell )\rbrace \Bigr ] \cos (n\theta ),\end{aligned}$$
(A27)
$$\begin{aligned} csT_{5r}&= uT_{5\theta } = \frac{1}{\ell }\Bigl [E_n \lbrace I_{n+1}(r/\ell )+n(\ell /r)I_n(r/\ell ) \rbrace \\ &\quad +H_n \lbrace -K_{n+1}(r/\ell )+n(\ell /r)K_n(r/\ell ) \rbrace \Bigr ] \cos (n\theta )\end{aligned}$$
(A28)
$$\begin{aligned} csT_{5\theta }&= -uT_{5r}= -\frac{n}{r}\bigl [E_nI_n(r/\ell )+H_n K_n(r/\ell )\bigr ]\sin (n\theta ). \end{aligned}$$
(A29)

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Bhandakkar, T.K. Tabulation of Mindlin–Airy stress potentials and the corresponding stress and displacement in polar co-ordinates in couple stress elasticity. Sādhanā 48, 24 (2023). https://doi.org/10.1007/s12046-023-02076-5

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