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Displacement-Based Experimental Stress Analysis of a Circularly-Perforated Asymmetrical Isotropic Structure

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Abstract

This paper demonstrates the ability to determine the individual full-field components of stress, strain and displacement in finite asymmetrical engineering structures from recorded values of a single displacement component while not necessitating differentiation of the measured displacement data; a process which can be ill-conditioned and adversely influenced by data noise and quality. While the present approach of processing recorded displacements with a stress function is not new, previous cases have been restricted to symmetrical scenarios. Full-field displacements are measured here using the digital image correlation (DIC) method. As well as satisfying load equilibrium, the experimental results agree with those predicted using the finite element method (FEM).

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Abbreviations

a0, b0, c0, A0, a1, c1, d1, a ´ 1, c ´ 1, d ´ 1, an, bn, cn, dn, a ´ n, b ´ n, c ´ n, and d ´ n :

Airy coefficients

[A]:

Airy matrix

B:

Load offset distance

{c}:

Airy coefficient column matrix

{d}:

Airy input data column matrix

{d ’}:

Reconstructed {d} vector

E:

Young’s modulus

k :

Number of Airy coefficients

L:

Length

m :

Number of experimental input data

N:

Terminating number

r :

Radial distance (Polar coordinate)

u :

Horizontal displacement

u r :

Radial direction displacement

u θ :

Hoop direction displacement

v :

Vertical displacement

Δ :

Change

ε:

Strain

θ :

Angle (Polar coordinate)

ν:

Poisson’s ratio

σ:

Stress

φ:

Airy stress function

\( \mathit{\nabla} \) 4 :

Biharmonic operator

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Acknowledgments

This research was partially supported by the Hyundai Motor Company, Korea. The technical assistance of A. A. Khaja and I. V. Ilapogu is appreciated.

Author information

Authors and Affiliations

  1. EdXengine, Chennai, India

    S. Paneerselvam

  2. Rochester Institute of Technology, Dubai, United Arab Emirates

    W. A. Samad

  3. University of Ljubljana, Ljubljana, Slovenia

    R. Venkatesh

  4. Hyundai Motor Group, Gyeonggi-do, South Korea

    K. W. Song

  5. University of Wisconsin-Milwaukee, Milwaukee, WI, USA

    R. F. El-Hajjar

  6. University of Wisconsin-Madison, Madison, WI, USA

    R. E. Rowlands

Authors
  1. S. Paneerselvam
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  2. W. A. Samad
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  3. R. Venkatesh
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  4. K. W. Song
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  5. R. F. El-Hajjar
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  6. R. E. Rowlands
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Corresponding author

Correspondence to W. A. Samad.

Appendix

Appendix

From equations. (3) and (4) and Hooke’s Law, one obtains the following radial and tangential components of strain [4],

$$ {\in}_{rr}=\frac{1}{E}\left[\begin{array}{l}\left(\left(\frac{1}{r^2}-\frac{3}{2}\frac{r}{R^3}\right)\cdot \left(1+v\right)+\left(\frac{r^3}{2\cdot {R}^5}\right)\cdot \left(1+5\cdot v\right)\right)\cdot {b}_0\\ {}+\left(2\cdot \left(1-v\right)-\frac{3\cdot r}{R}\cdot \left(1+v\right)+\frac{r^3}{R^3}\cdot \left(1+5\cdot v\right)\right)\cdot {c}_0\\ {}+\left(\frac{r\cdot \left(1+v\right)}{2\cdot {R}^3}-\frac{r^3\cdot \left(1+5\cdot v\right)}{2\cdot {R}^5}\right)\cdot \tan \left(3\cdot \theta \right)\cdot {A}_0\\ {}+\left(-\frac{2\cdot {R}^4\cdot \left(1+v\right)}{r^3}+2\cdot r\cdot \left(1-3\cdot v\right)\right)\cdot \sin \left(\theta \right)\cdot {d}_1^{\prime}\\ {}+\left(-\frac{2\cdot {R}^4\cdot \left(1+v\right)}{r^3}+2\cdot r\cdot \left(1-3\cdot v\right)\right)\cdot \cos \left(\theta \right)\cdot {d}_1\\ {}+\left(3\cdot {R}^2\cdot \left(1+v\right)-3\cdot {R}^6\cdot {r}^{-4}\cdot \left(1+v\right)-12\cdot {r}^2\cdot v\right)\cdot \sin \left(2\cdot \theta \right)\cdot {b}_2^{\prime}\\ {}+\left(\left({R}^{-2}+3\cdot {R}^2\cdot {r}^{-4}\right)\cdot \left(1+v\right)-4\cdot {r}^{-2}\right)\cdot \sin \left(2\cdot \theta \right)\cdot {d}_2^{\prime}\\ {}+\left(\frac{24\cdot r\cdot \left(1+v\right)}{R^6}-\frac{12\cdot {r}^3}{R^8}\left(1+5\cdot v\right)-\frac{12}{r^5}\left(1+v\right)\right)\cdot \sin \left(3\cdot \theta \right)\cdot {c}_3^{\prime}\\ {}+\left(\frac{18\cdot r\cdot \left(1+v\right)}{R^4}-\frac{8\cdot {r}^3}{R^6}\left(1+5\cdot v\right)-\frac{2}{r^3}\left(5+v\right)\right)\cdot \sin \left(3\cdot \theta \right)\cdot {d}_3^{\prime}\\ {}+{\displaystyle \sum_{n=4,5,6}^N\left[\left(\begin{array}{l}\left({n}^2-1\right)\cdot {r}^{\left(n-2\right)}\cdot {R}^2\cdot \left(1+v\right)-\left(n+1\right)\cdot {r}^n\cdot \left(n-2+n\cdot v+2\cdot v\right)\\ {}-\left(n+1\right)\cdot {r}^{-\left(n+2\right)}\cdot {R}^{\left(2\cdot n+2\right)}\cdot \left(1+v\right)\end{array}\right)\cdot \sin \left(n\cdot \theta \right)\cdot {b}_n^{\prime}\right]}\\ {}+{\displaystyle \sum_{n=4,5,6}^N\left[\left(\begin{array}{l}\left(n-1\right)\cdot {r}^{\left(n-2\right)}\cdot {R}^{-\left(2\cdot n-2\right)}\cdot \left(1+v\right)-\left(1-{n}^2\right)\cdot {r}^{-\left(n+2\right)}\cdot {R}^2\cdot \left(1+v\right)\\ {}-\left(n-1\right)\cdot {r}^{-(n)}\left(n+2+n\cdot v-2\cdot v\right)\end{array}\right)\cdot \sin \left(n\cdot \theta \right)\cdot {d}_n^{\prime}\right]}\\ {}+{\displaystyle \sum_{n=2,3,4}^N\left[\left(\begin{array}{l}\left({n}^2-1\right)\cdot {r}^{\left(n-2\right)}\cdot {R}^2\cdot \left(1+v\right)-\left(n+1\right)\cdot {r}^n\cdot \left(n-2+n\cdot v+2\cdot v\right)\\ {}-\left(n+1\right)\cdot {r}^{-\left(n+2\right)}\cdot {R}^{\left(2\cdot n+2\right)}\cdot \left(1+v\right)\end{array}\right)\cdot \cos \left(n\cdot \theta \right)\cdot {b}_n\right]}\\ {}+{\displaystyle \sum_{n=2,3,4}^N\left[\left(\begin{array}{l}\left(n-1\right)\cdot {r}^{\left(n-2\right)}\cdot {R}^{-\left(2\cdot n-2\right)}\cdot \left(1+v\right)-\left(1-{n}^2\right)\cdot {r}^{-\left(n+2\right)}\cdot {R}^2\cdot \left(1+v\right)\\ {}-\left(n-1\right)\cdot {r}^{-(n)}\left(n+2+n\cdot v-2\cdot v\right)\end{array}\right)\cdot \cos \left(n\cdot \theta \right)\cdot {d}_n\right]}\end{array}\right] $$
(A1)
$$ {\in}_{\theta \theta }=\frac{1}{E}\left[\begin{array}{l}\left(\left(-\frac{1}{r^2}+\frac{3}{2}\frac{r}{R^3}\right)\cdot \left(1+v\right)-\left(\frac{r^3}{2\cdot {R}^5}\right)\cdot \left(5+v\right)\right)\cdot {b}_0\\ {}+\left(2\cdot \left(1-v\right)+\frac{3\cdot r}{R}\cdot \left(1+v\right)-\frac{r^3}{R^3}\cdot \left(5+v\right)\right)\cdot {c}_0\\ {}+\left(-\frac{r\cdot \left(1+v\right)}{2\cdot {R}^3}+\frac{r^3\cdot \left(5+v\right)}{2\cdot {R}^5}\right)\cdot \tan \left(3\cdot \theta \right)\cdot {A}_0\\ {}+\left(\frac{2\cdot {R}^4\cdot \left(1+v\right)}{r^3}+2\cdot r\cdot \left(3-v\right)\right)\cdot \sin \left(\theta \right)\cdot {d_1}^{\prime}\\ {}+\left(\frac{2\cdot {R}^4\cdot \left(1+v\right)}{r^3}+2\cdot r\cdot \left(3-v\right)\right)\cdot \cos \left(\theta \right)\cdot {d}_1\\ {}+\left(-3\cdot {R}^2\cdot \left(1+v\right)+3\cdot {R}^6\cdot {r}^{-4}\cdot \left(1+v\right)+12\cdot {r}^2\right)\cdot \sin \left(2\cdot \theta \right)\cdot {b_2}^{\prime}\\ {}+\left(\left(-{R}^{-2}-3\cdot {R}^2\cdot {r}^{-4}\right)\cdot \left(1+v\right)+4\cdot {r}^{-2}\cdot v\right)\cdot \sin \left(2\cdot \theta \right)\cdot {d_2}^{\prime}\\ {}+\left(-\frac{24\cdot r\cdot \left(1+v\right)}{R^6}+\frac{12\cdot {r}^3}{R^8}\left(5+v\right)+\frac{12}{r^5}\left(1+v\right)\right)\cdot \sin \left(3\cdot \theta \right)\cdot {c_3}^{\prime}\\ {}+\left(-\frac{18\cdot r\cdot \left(1+v\right)}{R^4}+\frac{8\cdot {r}^3}{R^6}\left(5+v\right)+\frac{2}{r^3}\left(1+5\cdot v\right)\right)\cdot \sin \left(3\cdot \theta \right)\cdot {d}_3^{\prime}\\ {}+{\displaystyle \sum_{n=4,5,6}^N\left[\left(\begin{array}{l}-\left({n}^2-1\right)\cdot {r}^{\left(n-2\right)}\cdot {R}^2\cdot \left(1+v\right)+\left(n+1\right)\cdot {r}^n\cdot \left(n+2+n\cdot v-2\cdot v\right)\\ {}+\left(n+1\right)\cdot {r}^{-\left(n+2\right)}\cdot {R}^{\left(2\cdot n+2\right)}\cdot \left(1+v\right)\end{array}\right)\cdot \sin \left(n\cdot \theta \right)\cdot {b_n}^{\prime}\right]}\\ {}+{\displaystyle \sum_{n=4,5,6}^N\left[\left(\begin{array}{l}-\left(n-1\right)\cdot {r}^{\left(n-2\right)}\cdot {R}^{-\left(2\cdot n-2\right)}\cdot \left(1+v\right)+\left(1-{n}^2\right)\cdot {r}^{-\left(n+2\right)}\cdot {R}^2\cdot \left(1+v\right)\\ {}+\left(n-1\right)\cdot {r}^{-(n)}\left(n-2+n\cdot v+2\cdot v\right)\end{array}\right)\cdot \sin \left(n\cdot \theta \right)\cdot {d_n}^{\prime}\right]}\\ {}+{\displaystyle \sum_{n=2,3,4}^N\left[\left(\begin{array}{l}-\left({n}^2-1\right)\cdot {r}^{\left(n-2\right)}\cdot {R}^2\cdot \left(1+v\right)+\left(n+1\right)\cdot {r}^n\cdot \left(n+2+n\cdot v-2\cdot v\right)\\ {}+\left(n+1\right)\cdot {r}^{-\left(n+2\right)}\cdot {R}^{\left(2\cdot n+2\right)}\cdot \left(1+v\right)\end{array}\right)\cdot \cos \left(n\cdot \theta \right)\cdot {b}_n\right]}\\ {}+{\displaystyle \sum_{n=2,3,4}^N\left[\left(\begin{array}{l}-\left(n-1\right)\cdot {r}^{\left(n-2\right)}\cdot {R}^{-\left(2\cdot n-2\right)}\cdot \left(1+v\right)+\left(1-{n}^2\right)\cdot {r}^{-\left(n+2\right)}\cdot {R}^2\cdot \left(1+v\right)\\ {}+\left(n-1\right)\cdot {r}^{-(n)}\left(n-2+n\cdot v+2\cdot v\right)\end{array}\right)\cdot \cos \left(n\cdot \theta \right)\cdot {d}_n\right]}\end{array}\right] $$
(A2)

with N being the terminating limit for finite series. Combing eqns (A1) through (A4), and integrating, the individual components of displacement in polar co-ordinates of eqns (A5) and (A6) are obtained

$$ \frac{\partial {u}_r}{\kern0.5em \partial \mathrm{r}} = {\upepsilon}_{rr} = \frac{1}{\mathrm{E}}\ \left({\sigma}_{rr}\hbox{-}\ \upupsilon\ {\upsigma}_{\uptheta \uptheta}\right) $$
(A3)
$$ \frac{\partial {u}_{\theta }}{\partial \theta } = r.{\varepsilon}_{\theta \theta } - {u}_r $$
(A4)
$$ {u}_r=\frac{1}{E}\left[\begin{array}{l}\left(\left(-\frac{1}{r}-\frac{3}{4}\frac{r^2}{R^3}\right)\cdot \left(1+v\right)+\left(\frac{r^4}{8\cdot {R}^5}\right)\cdot \left(1+5\cdot v\right)\right)\cdot {b}_0\\ {}+\left(2\cdot r\cdot \left(1-v\right)-\frac{3\cdot {r}^2}{2\cdot R}\cdot \left(1+v\right)+\frac{r^4}{4\cdot {R}^3}\cdot \left(1+5\cdot v\right)\right)\cdot {c}_0\\ {}+\left(\frac{r^2\cdot \left(1+v\right)}{4\cdot {R}^3}-\frac{r^4\cdot \left(1+5\cdot v\right)}{8\cdot {R}^5}\right)\cdot \tan \left(3\cdot \theta \right)\cdot {A}_0\\ {}+\left(\frac{R^4\cdot \left(1+v\right)}{r^2}+{r}^2\cdot \left(1-3\cdot v\right)\right)\cdot \sin \left(\theta \right)\cdot {d}_1^{\prime}\\ {}+\left(\frac{R^4\cdot \left(1+v\right)}{r^2}+{r}^2\cdot \left(1-3\cdot v\right)\right)\cdot \cos \left(\theta \right)\cdot {d}_1\\ {}+\left(3\cdot r\cdot {R}^2\cdot \left(1+v\right)+{R}^6\cdot {r}^{-3}\cdot \left(1+v\right)-4\cdot {r}^3\cdot v\right)\cdot \sin \left(2\cdot \theta \right)\cdot {b}_2^{\prime}\\ {}+\left(\left({R}^{-2}\cdot r-{R}^2\cdot {r}^{-3}\right)\cdot \left(1+v\right)+4\cdot {r}^{-1}\right)\cdot \sin \left(2\cdot \theta \right)\cdot {d}_2^{\prime}\\ {}+\left(\frac{12\cdot {r}^2\cdot \left(1+v\right)}{R^6}-\frac{3\cdot {r}^4}{R^8}\left(1+5\cdot v\right)+\frac{3}{r^4}\left(1+v\right)\right)\cdot \sin \left(3\cdot \theta \right)\cdot {c}_3^{\prime}\\ {}+\left(\frac{9\cdot {r}^2\cdot \left(1+v\right)}{R^4}-\frac{2\cdot {r}^4}{R^6}\left(1+5\cdot v\right)+\frac{1}{r^2}\left(5+v\right)\right)\cdot \sin \left(3\cdot \theta \right)\cdot {d}_3^{\prime}\\ {}+{\displaystyle \sum_{n=4,5,6}^N\left[\left(\begin{array}{l}\left(n+1\right)\cdot {r}^{\left(n-1\right)}\cdot {R}^2\cdot \left(1+v\right)-{r}^{n+1}\cdot \left(n-2+n\cdot v+2\cdot v\right)\\ {}+{r}^{-\left(n+1\right)}\cdot {R}^{\left(2\cdot n+2\right)}\cdot \left(1+v\right)\end{array}\right)\cdot \sin \left(n\cdot \theta \right)\cdot {b}_n^{\prime}\right]}\\ {}+{\displaystyle \sum_{n=4,5,6}^N\left[\left(\begin{array}{l}{r}^{\left(n-1\right)}\cdot {R}^{-\left(2\cdot n-2\right)}\cdot \left(1+v\right)+\left(1-n\right)\cdot {r}^{-\left(n+1\right)}\cdot {R}^2\cdot \left(1+v\right)\\ {}+{r}^{\left(-n+1\right)}\left(n+2+n\cdot v-2\cdot v\right)\end{array}\right)\cdot \sin \left(n\cdot \theta \right)\cdot {d}_n^{\prime}\right]}\\ {}+{\displaystyle \sum_{n=2,3,4}^N\left[\left(\begin{array}{l}\left(n+1\right)\cdot {r}^{\left(n-1\right)}\cdot {R}^2\cdot \left(1+v\right)-{r}^{n+1}\cdot \left(n-2+n\cdot v+2\cdot v\right)\\ {}+{r}^{-\left(n+1\right)}\cdot {R}^{\left(2\cdot n+2\right)}\cdot \left(1+v\right)\end{array}\right)\cdot \cos \left(n\cdot \theta \right)\cdot {b}_n\right]}\\ {}+{\displaystyle \sum_{n=2,3,4}^N\left[\left(\begin{array}{l}{r}^{\left(n-1\right)}\cdot {R}^{-\left(2\cdot n-2\right)}\cdot \left(1+v\right)+\left(1-n\right)\cdot {r}^{-\left(n+1\right)}\cdot {R}^2\cdot \left(1+v\right)\\ {}+{r}^{\left(-n+1\right)}\left(n+2+n\cdot v-2\cdot v\right)\end{array}\right)\cdot \cos \left(n\cdot \theta \right)\cdot {d}_n\right]}\\ {}+g\left(\theta \right)\end{array}\right] $$
(A5)
$$ {u}_{\theta }=\frac{1}{E}\left[\begin{array}{l}\left(\left(\frac{9}{4}\frac{r^2}{R^3}\right)\cdot \left(1+v\right)-\left(\frac{r^4}{8\cdot {R}^5}\right)\cdot \left(21+9\cdot v\right)\right)\cdot \theta \cdot {b}_0\\ {}+\left(\frac{9\cdot {r}^2}{2\cdot R}\cdot \left(1+v\right)-\frac{r^4}{4\cdot {R}^3}\cdot \left(21+9\cdot v\right)\right)\cdot \theta \cdot {c}_0\\ {}+\left(\frac{r^2\cdot \left(1+v\right)}{4\cdot {R}^3}-\frac{r^4\cdot \left(7+3\cdot v\right)}{8\cdot {R}^5}\right)\cdot \ln \left( \cos \left(3\cdot \theta \right)\right)\cdot {A}_0\\ {}-\left(\frac{R^4\cdot \left(1+v\right)}{r^2}+{r}^2\cdot \left(5+v\right)\right)\cdot \cos \left(\theta \right)\cdot {d_1}^{\prime}\\ {}+\left(\frac{R^4\cdot \left(1+v\right)}{r^2}+{r}^2\cdot \left(5+v\right)\right)\cdot \sin \left(\theta \right)\cdot {d}_1\\ {}+\left(3\cdot r\cdot {R}^2\cdot \left(1+v\right)-{R}^6\cdot {r}^{-3}\cdot \left(1+v\right)-2\cdot {r}^3\cdot \left(3+v\right)\right)\cdot \cos \left(2\cdot \theta \right)\cdot {b_2}^{\prime}\\ {}+\left(\left({R}^{-2}\cdot r+{R}^2\cdot {r}^{-3}\right)\cdot \left(1+v\right)+2\cdot {r}^{-1}\cdot \left(1-v\right)\right)\cdot \cos \left(2\cdot \theta \right)\cdot {d_2}^{\prime}\\ {}+\left(\frac{12\cdot {r}^2\cdot \left(1+v\right)}{R^6}-\frac{r^4}{R^8}\left(21+9\cdot v\right)-\frac{3}{r^4}\cdot \left(1+v\right)\right)\cdot \cos \left(3\cdot \theta \right)\cdot {c_3}^{\prime}\\ {}+\left(\frac{9\cdot {r}^2\cdot \left(1+v\right)}{R^4}-\frac{2\cdot {r}^4}{3\cdot {R}^6}\left(21+9\cdot v\right)+\frac{1}{r^2}\left(1-3\cdot v\right)\right)\cdot \cos \left(3\cdot \theta \right)\cdot {d_3}^{\prime}\\ {}+{\displaystyle \sum_{n=4,5,6}^N\left[\left(\begin{array}{l}\left(n+1\right)\cdot {r}^{\left(n-1\right)}\cdot {R}^2\cdot \left(1+v\right)-{r}^{n+1}\cdot \left(n+4+n\cdot v\right)\\ {}-\cdot {r}^{-\left(n+1\right)}\cdot {R}^{\left(2\cdot n+2\right)}\cdot \left(1+v\right)\end{array}\right)\cdot \cos \left(n\cdot \theta \right)\cdot {b_n}^{\prime}\right]}\\ {}+{\displaystyle \sum_{n=4,5,6}^N\left[\left(\begin{array}{l}{r}^{\left(n-1\right)}\cdot {R}^{-\left(2\cdot n-2\right)}\cdot \left(1+v\right)-\left(1-n\right)\cdot {r}^{-\left(n+1\right)}\cdot {R}^2\cdot \left(1+v\right)\\ {}-{r}^{\left(-n+1\right)}\left(n-4+n\cdot v\right)\end{array}\right)\cdot \cos \left(n\cdot \theta \right)\cdot {d_n}^{\prime}\right]}\\ {}+{\displaystyle \sum_{n=2,3,4}^N\left[\left(\begin{array}{l}-\left(n+1\right)\cdot {r}^{\left(n-1\right)}\cdot {R}^2\cdot \left(1+v\right)+{r}^{n+1}\cdot \left(n+4+n\cdot v\right)\\ {}+{r}^{-\left(n+1\right)}\cdot {R}^{\left(2\cdot n+2\right)}\cdot \left(1+v\right)\end{array}\right)\cdot \sin \left(n\cdot \theta \right)\cdot {b}_n\right]}\\ {}+{\displaystyle \sum_{n=2,3,4}^N\left[\left(\begin{array}{l}-{r}^{\left(n-1\right)}\cdot {R}^{-\left(2\cdot n-2\right)}\cdot \left(1+v\right)+\left(1-n\right)\cdot {r}^{-\left(n+1\right)}\cdot {R}^2\cdot \left(1+v\right)\\ {}+{r}^{\left(-n+1\right)}\left(n-4+n\cdot v\right)\end{array}\right)\cdot \sin \left(n\cdot \theta \right)\cdot {d}_n\right]}\\ {}-{S}_1\cdot \sin \left(\theta \right)-{S}_2\cdot \cos \left(\theta \right)+{R}^{*}\cdot r\end{array}\right] $$
(A6)

where g(θ) = S1cos θ – S2sin θ. S 1 and S 2 represent rigid body translations and R* represents a rigid body rotation [7]. For a physical member loaded in a testing machine, S 1, S 2 and R* can be equated to zero.

$$ u=\frac{1}{E}\left[\begin{array}{l}\left(\begin{array}{l}\left(-\frac{1}{r}\cdot \cos \left(\theta \right)-\frac{3}{4}\frac{r^2}{R^3}\cdot \left( \cos \left(\theta \right)+3\cdot \theta \sin \left(\theta \right)\right)\right)\cdot \left(1+v\right)\\ {}+\left(\frac{r^4}{8\cdot {R}^5}\right)\cdot \left(\left(1+5\cdot v\right)\cdot \cos \left(\theta \right)+\left(21+9\cdot v\right)\cdot \theta \cdot \sin \left(\theta \right)\right)\end{array}\right)\cdot {b}_0+\\ {}\left(\begin{array}{l}2\cdot r\cdot \left(1-v\right)\cdot \cos \left(\theta \right)-\frac{3\cdot {r}^2}{2\cdot R}\cdot \left(1+v\right)\cdot \left( \cos \left(\theta \right)+3\cdot \theta \sin \left(\theta \right)\right)\\ {}+\frac{r^4}{4\cdot {R}^3}\cdot \left( \cos \left(\theta \right)\cdot \left(1+5\cdot v\right)+\theta \cdot \sin \left(\theta \right)\cdot \left(21+9\cdot v\right)\right)\end{array}\right)\cdot {c}_0\\ {}+\left(\begin{array}{l}\frac{r^2\cdot \left(1+v\right)}{4\cdot {R}^3}\left( \cos \left(\theta \right)\cdot \tan \left(3\cdot \theta \right)- \ln \left( \cos \left(3\cdot \theta \right)\right)\cdot \sin \left(\theta \right)\right)\\ {}-\frac{r^4}{8\cdot {R}^5}\left( \cos \left(\theta \right)\cdot \tan \left(3\cdot \theta \right)\cdot \left(1+5\cdot v\right)-\frac{\left(21+9\cdot v\right)}{3} \ln \left( \cos \left(3\cdot \theta \right)\right)\cdot \sin \left(\theta \right)\right)\end{array}\right)\cdot {A}_0\\ {}+\left(\frac{R^4\cdot \left(1+v\right)}{r^2}+{r}^2\cdot \left(3-v\right)\right)\cdot \sin \left(2\cdot \theta \right)\cdot {d_1}^{\prime}\\ {}+\left(\frac{R^4\cdot \left(1+v\right)}{r^2}\cdot \cos \left(2\cdot \theta \right)+{r}^2\cdot \left(\left(5+v\right)\cdot \cos \left(2\cdot \theta \right)-\left(4+4v\right)\cdot { \cos}^2\left(\theta \right)\right)\right)\cdot {d}_1\\ {}+\left(\begin{array}{l}3\cdot r\cdot {R}^2\cdot \left(1+v\right)\cdot \sin \left(\theta \right)-{R}^6\cdot {r}^{-3}\cdot \left(1+v\right)\cdot \sin \left(3\cdot \theta \right)\\ {}-2\cdot {r}^3\left(\left(3\cdot \cos \left(2\cdot \theta \right)\cdot \sin \left(\theta \right)\right)+\left(\left( \sin \left(2\cdot \theta \right)\cdot \cos \left(\theta \right)+\left( \sin \left(3\cdot \theta \right)\right)\right)\cdot v\right)\right)\end{array}\right){b_2}^{\prime}\\ {}+\left(\begin{array}{l}\left({R}^{-2}\cdot r\cdot \sin \left(\theta \right)-{R}^2\cdot {r}^{-3}\cdot \sin \left(3\cdot \theta \right)\right)\cdot \left(1+v\right)\\ {}+2\cdot {r}^{-1}\cdot \left( \sin \left(2\cdot \theta \right)\cdot \cos \left(\theta \right)+v\cdot \cos \left(2\cdot \theta \right)\cdot \sin \left(\theta \right)+ \sin \left(\theta \right)\right)\end{array}\right)\cdot {d_2}^{\prime}\\ {}+\left(\frac{12\cdot {r}^2\cdot \left(1+v\right)}{R^6}\cdot \sin \left(2\cdot \theta \right)-\frac{r^4}{R^8}\left(\left(3+15\cdot v\right)\cdot \sin \left(2\cdot \theta \right)-\left(18-6\cdot v\right)\cdot \cos \left(3\cdot \theta \right)\cdot \sin \left(\theta \right)\right)+\frac{3}{r^4}\left( \sin \left(4\cdot \theta \right)\right)\right)\cdot {c_3}^{\prime}\\ {}+\left(\begin{array}{l}\frac{9\cdot {r}^2\cdot \left(1+v\right)}{R^4}\cdot \sin \left(2\cdot \theta \right)-\frac{r^4}{R^6}\left(\left(14+6\cdot v\right)\cdot \sin \left(2\cdot \theta \right)-\left(12-4\cdot v\right)\cdot \sin \left(3\cdot \theta \right)\cdot \cos \left(\theta \right)\right)\\ {}+\frac{1}{r^2}\left(\left(5+v\right)\cdot \sin \left(2\cdot \theta \right)-\left(4+4\cdot v\right)\cdot \cos \left(3\cdot \theta \right)\cdot \sin \left(\theta \right)\right)\end{array}\right)\cdot {d_3}^{\prime}\\ {}+{\displaystyle \sum_{n=4,5,6}^N\left[\left(\begin{array}{l}\left(n+1\right)\cdot {r}^{\left(n-1\right)}\cdot {R}^2\cdot \left(1+v\right)\cdot \sin \left(\left(n-1\right)\cdot \theta \right)\\ {}-{r}^{n+1}\cdot \left(\left(n+n\cdot v+4\right)\cdot \sin \left(\left(n+1\right)\cdot \theta \right)-2\cdot \left(3-v\right)\cdot \sin \left(n\cdot \theta \right)\cdot \cos \left(\theta \right)\right)\\ {}+{r}^{-\left(n+1\right)}\cdot {R}^{\left(2\cdot n+2\right)}\cdot \left(1+v\right)\cdot \sin \left(\left(n+1\right)\cdot \theta \right)\end{array}\right){b_n}^{\prime}\right]}\\ {}+{\displaystyle \sum_{n=4,5,6}^N\left[\left(\begin{array}{l}{r}^{\left(n-1\right)}\cdot {R}^{-\left(2\cdot n-2\right)}\cdot \left(1+v\right)\cdot \sin \left(\left(n-1\right)\cdot \theta \right)+\left(1-n\right)\cdot {r}^{-\left(n+1\right)}\cdot {R}^2\cdot \left(1+v\right)\cdot \sin \left(\left(n+1\right)\cdot \theta \right)\\ {}+{r}^{\left(-n+1\right)}\left(\left(n+n\cdot v-4\right)\cdot \sin \left(\left(n+1\right)\cdot \theta \right)+2\cdot \left(3-v\right)\cdot \sin \left(n\cdot \theta \right)\cdot \cos \left(\theta \right)\right)\end{array}\right)\cdot {d_n}^{\prime}\right]}\\ {}+{\displaystyle \sum_{n=2,3,4}^N\left[\left(\begin{array}{l}\left(n+1\right)\cdot {r}^{\left(n-1\right)}\cdot {R}^2\cdot \left(1+v\right)\cdot \cos \left(\left(n-1\right)\cdot \theta \right)\\ {}-{r}^{n+1}\cdot \left(\left(n+n\cdot v+4\right)\cdot \cos \left(\left(n-1\right)\cdot \theta \right)-2\cdot \left(3-v\right)\cdot \cos \left(n\cdot \theta \right)\cdot \cos \left(\theta \right)\right)\\ {}+{r}^{-\left(n+1\right)}\cdot {R}^{\left(2\cdot n+2\right)}\cdot \left(1+v\right)\cdot \cos \left(\left(n+1\right)\cdot \theta \right)\end{array}\right)\cdot {b}_n\right]}\\ {}+{\displaystyle \sum_{n=2,3,4}^N\left[\left(\begin{array}{l}{r}^{\left(n-1\right)}\cdot {R}^{-\left(2\cdot n-2\right)}\cdot \left(1+v\right)\cdot \cos \left(\left(n-1\right)\cdot \theta \right)+\left(1-n\right)\cdot {r}^{-\left(n+1\right)}\cdot {R}^2\cdot \left(1+v\right)\cdot \cos \left(\left(n+1\right)\cdot \theta \right)+\\ {}{r}^{\left(-n+1\right)}\left(\left(n+n\cdot v-4\right)\cdot \cos \left(\left(n+1\right)\cdot \theta \right)+2\cdot \left(3-v\right)\cdot \cos \left(n\cdot \theta \right)\cdot \cos \left(\theta \right)\right)\end{array}\right)\cdot {d}_n\right]}\end{array}\right] $$
(A7)

and

$$ v=\frac{1}{E}\left[\begin{array}{l}\left(\left(-\frac{1}{r}\cdot \sin \left(\theta \right)-\frac{3}{4}\frac{r^2}{R^3}\cdot \left( \sin \left(\theta \right)-3\cdot \theta \cos \left(\theta \right)\right)\right)\cdot \left(1+v\right)+\left(\frac{r^4}{8\cdot {R}^5}\right)\cdot \left(\left(1+5\cdot v\right)\cdot \sin \left(\theta \right)-\left(21+9\cdot v\right)\cdot \theta \cdot \cos \left(\theta \right)\right)\right)\cdot {b}_0\\ {}+\left(2\cdot r\cdot \left(1-v\right)\cdot \sin \left(\theta \right)-\frac{3\cdot {r}^2}{2\cdot R}\cdot \left(1+v\right)\cdot \left( \sin \left(\theta \right)-3\cdot \theta \cos \left(\theta \right)\right)+\frac{r^4}{4\cdot {R}^3}\cdot \left( \sin \left(\theta \right)\cdot \left(1+5\cdot v\right)-\theta \cdot \cos \left(\theta \right)\cdot \left(21+9\cdot v\right)\right)\right)\cdot {c}_0\\ {}+\left(\begin{array}{l}\frac{r^2\cdot \left(1+v\right)}{4\cdot {R}^3}\left( \sin \left(\theta \right)\cdot \tan \left(3\cdot \theta \right)+ \ln \left( \cos \left(3\cdot \theta \right)\right)\cdot \cos \left(\theta \right)\right)\\ {}-\frac{r^4}{8\cdot {R}^5}\left( \sin \left(\theta \right)\cdot \tan \left(3\cdot \theta \right)\cdot \left(1+5\cdot v\right)+\frac{\left(21+9\cdot v\right)}{3} \ln \left( \cos \left(3\cdot \theta \right)\right)\cdot \cos \left(\theta \right)\right)\end{array}\right)\cdot {A}_0\\ {}-\left(\frac{R^4\cdot \left(1+v\right)}{r^2}\cdot \cos \left(2\cdot \theta \right)+{r}^2\cdot \left(2\cdot \left(1+v\right)+\left(3-v\right)\cdot \cos \left(2\cdot \theta \right)\right)\right)\cdot {d_1}^{\prime }+\left(\frac{R^4\cdot \left(1+v\right)}{r^2}\cdot \sin \left(2\cdot \theta \right)+{r}^2\cdot \left(\left(3-v\right)\cdot \sin \left(2\cdot \theta \right)\right)\right)\cdot {d}_1\\ {}+\left(r\cdot {R}^2\cdot \left(1+v\right)\cdot \left(3\cdot \cos \left(\theta \right)-{r}^{-4}\cdot {R}^4\cdot \cos \left(3\theta \right)\right)-\left(2\cdot \cos \left(\theta \right)\cdot {r}^3\cdot \left( \cos \left(2\theta \right)\cdot \left(3-v\right)+2v\right)\right)\right){b_2}^{\prime}\\ {}+\left(\left({R}^{-2}\cdot r\cdot \cos \left(\theta \right)+{R}^2\cdot {r}^{-3}\cdot \cos \left(3\cdot \theta \right)\right)\cdot \left(1+v\right)+2\cdot \cos \left(\theta \right)\cdot {r}^{-1}\cdot \left(- \cos \left(2\cdot \theta \right)\cdot \left(1+v\right)+2\right)\right)\cdot {d_2}^{\prime}\\ {}+\left(\frac{12\cdot {r}^2\cdot \left(1+v\right)}{R^6}\cdot \cos \left(2\cdot \theta \right)+\frac{24\cdot {r}^4}{R^8}\left(-\left(3-v\right)\cdot { \cos}^4\left(\theta \right)+2\cdot \left(1-v\right)\cdot { \cos}^2\left(\theta \right)+\left(\frac{1+5\cdot v}{8}\right)\right)-\frac{3}{r^4}\left( \cos \left(4\cdot \theta \right)\cdot \left(1+v\right)\right)\right)\cdot {c_3}^{\prime}\\ {}+\left(\begin{array}{l}\frac{9\cdot {r}^2\cdot \left(1+v\right)}{R^4}\cdot \cos \left(2\cdot \theta \right)-\frac{2\cdot {r}^4}{R^6}\left(8\cdot \left(3-v\right)\cdot { \cos}^4\left(\theta \right)-16\cdot \left(1-v\right)\cdot { \cos}^2\left(\theta \right)-\left(1+5\cdot v\right)\right)\\ {}+\frac{1}{r^2}\left(-16\cdot \left(1+v\right)\cdot { \cos}^4\left(\theta \right)+2\cdot { \cos}^2\left(\theta \right)\cdot \left(11+7\cdot v\right)-\left(5+v\right)\right)\end{array}\right)\cdot {d_3}^{\prime}\\ {}+{\displaystyle \sum_{n=4,5,6}^N\left[\left(\begin{array}{l}\left(n+1\right)\cdot {r}^{\left(n-1\right)}\cdot {R}^2\cdot \left(1+v\right)\cdot \cos \left(\left(n-1\right)\cdot \theta \right)\\ {}-{r}^{n+1}\cdot \left(\left(n\cdot \left(1+v\right)\right)\cdot \cos \left(\left(n-1\right)\cdot \theta \right)-2\cdot \sin \left(n\cdot \theta \right)\cdot \sin \left(\theta \right)\cdot \left(1-v\right)+4\cdot \cos \left(n\cdot \theta \right)\cdot \cos \left(\theta \right)\right)\\ {}-{r}^{-\left(n+1\right)}\cdot {R}^{\left(2\cdot n+2\right)}\cdot \left(1+v\right)\cdot \cos \left(\left(n+1\right)\cdot \theta \right)\end{array}\right){b_n}^{\prime}\right]}\\ {}+{\displaystyle \sum_{n=4,5,6}^N\left[\left(\begin{array}{l}{r}^{\left(n-1\right)}\cdot {R}^{-\left(2\cdot n-2\right)}\cdot \left(1+v\right)\cdot \cos \left(\left(n-1\right)\cdot \theta \right)\\ {}-\left(1-n\right)\cdot {r}^{-\left(n+1\right)}\cdot {R}^2\cdot \left(1+v\right)\cdot \cos \left(\left(n+1\right)\cdot \theta \right)\\ {}-{r}^{\left(-n+1\right)}\left(\left(n+n\cdot v\right)\cdot \cos \left(\left(n+1\right)\cdot \theta \right)-2\cdot \left(1-v\right)\cdot \sin \left(n\cdot \theta \right)\cdot \sin \left(\theta \right)-4\cdot \cos \left(n\cdot \theta \right)\cdot \cos \left(\theta \right)\right)\end{array}\right)\cdot {d_n}^{\prime}\right]}\\ {}+{\displaystyle \sum_{n=2,3,4}^N\left[\left(\begin{array}{l}-\left(n+1\right)\cdot {r}^{\left(n-1\right)}\cdot {R}^2\cdot \left(1+v\right)\cdot \sin \left(\left(n-1\right)\cdot \theta \right)\\ {}+{r}^{n+1}\cdot \left(\left(n\cdot \left(1+v\right)\right)\cdot \sin \left(\left(n-1\right)\cdot \theta \right)+2\cdot \cos \left(n\cdot \theta \right)\cdot \sin \left(\theta \right)\cdot \left(1-v\right)+4\cdot \sin \left(n\cdot \theta \right)\cdot \cos \left(\theta \right)\right)\\ {}+{r}^{-\left(n+1\right)}\cdot {R}^{\left(2\cdot n+2\right)}\cdot \left(1+v\right)\cdot \sin \left(\left(n+1\right)\cdot \theta \right)\end{array}\right){b}_n\right]}\\ {}+{\displaystyle \sum_{n=2,3,4}^N\left[\left(\begin{array}{l}-{r}^{\left(n-1\right)}\cdot {R}^{-\left(2\cdot n-2\right)}\cdot \left(1+v\right)\cdot \sin \left(\left(n-1\right)\cdot \theta \right)\\ {}+\left(1-n\right)\cdot {r}^{-\left(n+1\right)}\cdot {R}^2\cdot \left(1+v\right)\cdot \sin \left(\left(n+1\right)\cdot \theta \right)\\ {}+{r}^{\left(-n+1\right)}\left(\left(n+n\cdot v\right)\cdot \sin \left(\left(n+1\right)\cdot \theta \right)+2\cdot \left(1-v\right)\cdot \cos \left(n\cdot \theta \right)\cdot \sin \left(\theta \right)-4\cdot \sin \left(n\cdot \theta \right)\cdot \cos \left(\theta \right)\right)\end{array}\right)\cdot {d}_n\right]}\end{array}\right] $$
(A8)

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Paneerselvam, S., Samad, W.A., Venkatesh, R. et al. Displacement-Based Experimental Stress Analysis of a Circularly-Perforated Asymmetrical Isotropic Structure. Exp Mech 57, 129–142 (2017). https://doi.org/10.1007/s11340-016-0218-2

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