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Vibrational analysis of transversely isotropic hollow cylinder based on fractional generalized thermoelastic diffusion models with nonlocal effects

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Abstract

Due to numerous applications of micro-/nano-sized structures in medical, electrical, mechanical, and aeronautical engineering, etc., mathematical modelling of thermoelastic diffusive responses of these structures has become a hot topic of investigation. The inclusion of fractional-order derivatives in these models brings out more productive results. Thus considering fractional generalized thermoelastic diffusion model with nonlocal elastic effects, transient responses of a transversely isotropic hollow cylinder are analysed. The medium is held at undisturbed state initially and periodically varying thermal and continuous concentration loadings applied at the outer boundary of the cylinder. Assuming the plane strain and axisymmetry in the cylinder, the problem is reduced to one dimension which is solved using Laplace transformation along with inversion technique. The objective of this work is to theoretically investigate the impact of nonlocal and fractional-order parameters on thermophysical quantities of a transversely isotropic hollow cylinder and consideration of diffusion phenomenon along with the said effects augments the novelty of this work. Thus, the graphical representation of the results reveals that displacement is less perturbed with the introduction of the nonlocal elastic parameter. The fractional-order parameter has an increasing impact on thermal and diffusion profiles. Time has varying degrees of influence on all the physical quantities.

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Abbreviations

b :

Measure of diffusive effect

c :

Measure of thermodiffusive effect

C :

Mass concentration

P :

Chemical potential per unit mass

S :

Entropy per unit mass

T :

Temperature (above the reference temperature \(T_0\))

\(\alpha \) :

Fractional parameter

\(\rho \) :

Mass density

\(\xi \) :

Nonlocal elastic parameter

\(a_{ij}\) :

Thermal modulus tensor

\(b_{ij}\) :

Diffusion modulus tensor

\(c_E\) :

Specific heat at constant strain

\(c_{ijkl}\) :

Elastic parameters

\(d_{ij}\) :

Diffusivity coefficient tensor

\(e_{ij}\) :

Strain tensor

\(K_{ij}\) :

Thermal conductivity tensor

\(u_i\) :

Components of displacement vector

\(q_i\) :

Components of heat flux vector

\(\delta _{ij}\) :

Kronecker’s delta

\(\eta _i\) :

Components of mass flux vector

\(\sigma _{ij}\) :

Stress tensor

\(\tau _0,\tau _1\) :

Thermal relaxation times

\(\tau ^0,\tau ^1\) :

Diffusion relaxation times

\(\nabla \) :

Gradient operator

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The authors are highly thankful to the reviewers for their valuable suggestions and comments that helped to improve the quality of this work.

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    Geetanjali Geetanjali & P. K. Sharma

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Geetanjali, G., Sharma, P.K. Vibrational analysis of transversely isotropic hollow cylinder based on fractional generalized thermoelastic diffusion models with nonlocal effects. Acta Mech 235, 147–166 (2024). https://doi.org/10.1007/s00707-023-03738-5

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