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Thermal stress distribution in a hyperbolic disk made of rubber/brass material

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Abstract

This article deals with the study of the thermal stress distribution in a hyperbolic disk made of natural rubber/brass material under the effect of density, thickness and temperature. Seth’s transition theory and generalised strain measure are used for finding the governing equation. Analytical solutions are presented for the hyperbolic disk (k < 0, convergent; k = 0, uniform; k > 0, divergent) made of natural rubber/brass material. The effects of different pertinent parameters (i.e., temperature, density and thickness) are considered for the hyperbolic disk made of rubber/brass material. The behaviour of stress distribution, angular speed and temperature distribution are investigated. From the obtained results, it is noticed that convergent disk (i.e., k < 0) made of natural rubber material requires higher angular speed at the center of the disk in comparison to the uniform/divergent disk (i.e., k = 0/k > 0). By applying the thermal condition, the value of circumferential stress is also increasing at the inner surface of the hyperbolic disk made of natural rubber/brass material. The radial stress of hyperbolic disk (i.e., convergent/uniform/divergent) made of natural rubber/brass material increases throughout under the thermal effect and density profile. The convergent disk (i.e., k < 0) made of natural rubber/brass material is more convenient than that of the uniform/divergent disk (i.e., k = 0/k > 0).

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Abbreviations

\(\tau_{ij}\) :

Stress tensor (N/m2)

\(\varepsilon_{ij}\) :

Strain tensor (dimensionless)

\(r_{i}\) :

Inner radius (m)

\(r_{o}\) :

Outer radius (m)

\(m\) :

Density parameter (dimensionless)

\(k\) :

Thickness parameter (dimensionless)

\(u,v,w\) :

Displacement components (m)

d, A, B :

Constants (dimensionless)

E :

Young modulus (N/m2)

c :

Compressibility (m2/N)

\(n\) :

Strain measure coefficient (dimensionless)

\(\zeta\) :

Transition function

\(r\) :

Function of x and y

\(r,\theta ,z\) :

Polar co-ordinates

\(\eta\) :

Function of r only

\(\alpha\) :

Thermal moduli (N/m2 oF)

\(\beta\) :

Thermal moduli (N/m2 oF)

\(\lambda ,\mu\) :

Lame’s constants

\(\Omega^{2}\) :

Speed factor (dimensionless)

\(\Theta\) :

Temperature (oF)

\(\rho\) :

Density (kg/m3)

\(\omega\) :

Angular velocity (1/s)

\(R_{o} = r_{i} /r_{o} ,\quad R = r/r_{o}\) :

Radii ratio

\(\sigma_{r} = \tau_{rr} /Y\) :

Radial stress component

\(\sigma_{\theta } = \tau_{\theta \theta } /Y\) :

Circumferential stress component

\(\beta = c\xi \Theta_{o} /Y\) :

Temperature

\(\Omega^{2} = \rho_{o} \omega^{2} r_{o}^{2} /Y\) :

Angular speed

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Funding

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. This research was conducted using internal grants.

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Authors and Affiliations

  1. Department of Mathematics, ICFAI University, Baddi, Himachal Pradesh, India

    Pankaj Thakur & Naresh Kumar

  2. Department of Physics, ICFAI University, Baddi, Himachal Pradesh, India

    Kanav Gupta

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  1. Pankaj Thakur
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  2. Naresh Kumar
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Correspondence to Pankaj Thakur.

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Thakur, P., Kumar, N. & Gupta, K. Thermal stress distribution in a hyperbolic disk made of rubber/brass material. J Rubber Res 25, 27–37 (2022). https://doi.org/10.1007/s42464-022-00147-6

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