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Arbitrarily-shaped layered polar orthotropic domains: elastostatics using analytical and coupled analytical-FE approaches

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Abstract

Polar orthotropic materials as wood and filament-wound composites are ubiquitous in nature and technological applications, respectively. With annular domains rendering themselves naturally for polar orthotropy, applications featuring annular domains have received considerable attention; however, arbitrarily-shaped domains are less explored. In an attempt to address this gap, the present work explores elastostatics of arbitrarily-shaped layered 2D domains. After identifying the refinement required in an earlier work to yield a physically and mathematically consistent solution, the work considers analytically homogenous and layered polar orthotropic annular domains and corroborates the results with FEA. Subsequently, the arbitrarily-shaped layered domains—including octagonal-shaped, cruciform-shaped and annular domains—are analyzed by employing a coupled FE-analytical technique combining coarse-mesh boundary displacement-data from FEA and the refined solution. Results for the illustrative cases under symmetric and anti-symmetric loadings are depicted as contours over the domains for the resultant displacements and von Mises stresses. A comparison with converged FE results and \(L^{2}\)-error norms indicate a good correspondence between the two—demonstrating the efficacy of the coupled technique. The potential extension of the technique to hybrid-experimental strategies, asymmetric cases, composite cylindrical FGMs and a complex-variable based formulation for the coupled technique is briefly discussed.

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Abbreviations

\(\psi\) :

Airy stress function

\(r, \theta\) :

Polar coordinate

\({\sigma _{rr}}, {\sigma _{\theta \theta }}, {\tau _{r\theta }}\) :

Stress components in polar coordinate

\({u_{rr}}, {u_{\theta \theta } }\) :

Displacement components in polar coordinate

\({{{U}}_{{\textrm{FEM}}}},{{{U}}_{{\textrm{HM}}}}\) :

Resultant displacements from FEM and HM, respectively

\(E_r\),\(E_\theta\) :

Young’s moduli

\(\nu _{r\theta }\),\(\nu _{\theta r}\) :

Poisson’s ratios

\(G_{r\theta }\) :

Rigidity modulus

\({\sigma _0}\left( \theta \right)\) :

Uniformly distributed radial pressure over an angle \(\theta\)

P :

Uniformly applied pressure

\(\beta\) :

Half-angle of a patch-load

\(\textrm{AM}\) :

Analytical method

\(\textrm{ASF}\) :

Airy stress function

\(\textrm{BCs}\) :

Boundary conditions

\(\textrm{BVP}\) :

Boundary value problem

\(\textrm{DIC}\) :

Digital image correlation

\(\textrm{FEM}\) :

Finite element method

\(\textrm{FGM}\) :

Functionally graded material

\(\textrm{HM}\) :

Hybrid method

\(\textrm{TSA}\) :

Thermoelastic stress analysis

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Author notes

  1. Yogesh S. Thube and Tejas P. Gotkhindi have contributed equally to this work.

Authors and Affiliations

  1. Department of Mechanical, Materials and Aerospace Engineering, Indian Institute of Technology Dharwad, WALMI Campus, PB Road, Near High Court Bench, Dharwad, Karnataka, 580011, India

    Yogesh S. Thube & Tejas P. Gotkhindi

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  1. Yogesh S. Thube
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  2. Tejas P. Gotkhindi
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YST: Conceptualization; Writing—original draft; Formal analysis. TPG: Conceptualization; Formal analysis; Writing—review and editing; Supervision.

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Correspondence to Tejas P. Gotkhindi.

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Thube, Y.S., Gotkhindi, T.P. Arbitrarily-shaped layered polar orthotropic domains: elastostatics using analytical and coupled analytical-FE approaches. Int J Adv Eng Sci Appl Math 16, 1–24 (2024). https://doi.org/10.1007/s12572-023-00330-x

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