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A novel hybrid technique to decompose in-plane thermoelastic displacement fields into thermal and structural displacement fields

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Abstract

Structural health monitoring techniques assess structural responses by retrieving total displacement fields encompassing thermal and structural displacement fields. However, techniques to decompose a total displacement field into individual displacement fields—thermal- and structural-load induced fields—have not been explored. To address this research gap, the present work proposes and demonstrates a novel hybrid technique—coupling a low-fidelity FEM and an analytical technique formulated using complex variables. The technique incorporates partial coarse-mesh FEM boundary data with field variables expressions—presented as Laurent series—to compute unknown constants in the series. The technique is illustrated for thermoelastic problems including circular and elliptical rings and a plate with a hole. On the other hand, non-thermoelastic problems of practical utility—a special case of thermoelastic problems—are presented to demonstrate the versatility of the technique. The individual decomposed displacement fields are plotted as contour plots over the domains and are corroborated with high-fidelity FEM. \( L^{2} \) norms indicate a very good correspondence for thermoelastic problems, indicating the efficacy of the technique. The non-thermoelastic cases show higher deviation but within reasonable limits. Subsequently, the extension of the technique to experiments and evaluation of the stress fields are briefly discussed.

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Abbreviations

\( \mathrm {iFEM} \) :

Inverse finite element method

\( \mathrm {MAM} \) :

Modal analysis method

\(\mathrm {FEM}\) :

Finite element method

\(\mathrm {BEM}\) :

Boundary element method

\( \mathrm {RIM} \) :

Radial integration method

\(\mathrm {DIC}\) :

Digital image correlation

\(\mathrm {IRT}\) :

Infrared thermography

\(\mathrm {HM}\) :

Hybrid method

\(\mathrm {BCs}\) :

Boundary conditions

\(\mathrm {BVP}\) :

Boundary value problem

T :

Temperature field

\( \omega \left( z \right) \), \(g'\left( {{z_t}} \right) \) :

Temperature functions

\( {q_x} \), \( {q_y} \) :

Heat flux components in cartesian coordinate

z, \( z_{t} \) :

Complex variables

k :

Thermal conductivity

Q :

Total or resultant heat flow

\(\phi \) :

Airy stress function (ASF)

\(r, \theta \) :

Polar coordinate

\( \gamma (z) \), \( \psi (z) \) :

Kolosov-Muskhelishvili (K-M) potentials.

\({u_{x}}, {u_{y} }\) :

Displacement components in cartesian coordinate

\( \kappa \), \( \alpha \) :

Material parameters

E :

Young’s modulus

G :

Rigidity modulus

\(\nu \) :

Poisson’s ratio

\( \alpha _{0} \) :

Thermal expansion coefficient

\(S^n\), \(S^n_{x}\), \(S^n_{y}\) :

Traction vector and its components in cartesian coordinate

F, F x, \(F_{y}\) :

Force vector and its components in cartesian coordinate

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

Displacement components in polar coordinate

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

Stress components in polar coordinate

\({\sigma _{x}}, {\sigma _{y}, \tau _{xy}}\) :

Stress components in cartesian coordinate

\( \Gamma \) :

Boundary

\( \Omega \) :

Domain

\({{{X}}_{{\mathrm{FEM}}}}\), \( {{{X}}_{{{HM}}}}\) :

Field variables from FEM and HM

\( n_{t} \), \( n_{s} \) :

Best choice of n from thermal- and structural- parts

P :

Uniformly applied pressure

\(\beta \) :

Half-angle of a patch-load

\( u_x^{0} \), \( u_y^{0} \) :

Rigid body translational terms

\( {w_z} \) :

Rigid body rotational terms

\( {\sigma _{ij}}\), \( {\sigma _{kl}} \) :

Stresses in indicial notation

\({e_{ij}}\), \({e_{kl}}\) :

Strains in indicial notation

\( {C_{ijkl}} \) :

Elasticity tensor

\( {S_{ijkl}} \) :

Elastic compliance tensor

\( \mu \), \( \mu _{t} \) :

Parameters in characteristic equation

\( {F_i} \) :

Arbitrary functions where, i = 1 to 4

\({\Phi _i(z_i)}\) :

Complex potentials where, i =1 to 2

\(q_{i}\) :

Heat flux vector where, i =1 to 2.

\(k_{ij}\) :

Thermal conductivity tensor where, i, j =1 to 2.

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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, Dharwad, Karnataka, 580011, India

    Yogesh S. Thube & Tejas P. Gotkhindi

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  1. Yogesh S. Thube
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Correspondence to Tejas P. Gotkhindi.

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Thube, Y.S., Gotkhindi, T.P. A novel hybrid technique to decompose in-plane thermoelastic displacement fields into thermal and structural displacement fields. Acta Mech 233, 3747–3776 (2022). https://doi.org/10.1007/s00707-022-03298-0

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