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2-D mathematical model of hyperbolic two-temperature generalized thermoelastic solid cylinder under mechanical damage effect

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Abstract

This paper presents the construction of a two-dimensional mathematical model of a thermoelastic, homogeneous, and isotropic solid cylinder with its bounding surface subjected to a thermal shock. It has been discovered that the governing differential equations may be constructed in the framework of the hyperbolic two-temperature generalized thermoelasticity theory based on the study of the damage mechanics variable. It has been demonstrated visually that different values of the two-temperature parameter, damage mechanics variable and cylindrical length may produce numerically significant increases in dynamical and conductive temperatures, strain, and the average of the major stress components. The two-temperature parameter, as well as the diameter of the cylindrical axis, has substantial effects on all the functions under consideration. The damage mechanics variable has very little impact on the conductive and dynamical temperatures, but it has large implications on the strain and stress distributions, as seen in the graphs. The thermal wave can propagate at a limited pace using the hyperbolic two-temperature theory. The hyperbolic two-temperature theory offers a finite speed of propagation to the thermal wave.

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Abbreviations

\(C_{{\text{E}}}\) :

Specific heat at constant strain

\(c_{o} \,\,\) :

\(= \sqrt {\frac{\lambda + 2\,\mu }{\rho }}\) Longitudinal wave speed

D :

The mechanical damage variable

\(e_{ij}\) :

The strain components

\(K\) :

Thermal conductivity

\(T_{{\text{D}}} ,\,T_{{\text{C}}}\) :

Dynamical and conductive temperature, respectively

\(T_{o}\) :

Reference temperature

\(t\,,\tau_{0}\) :

Time and thermal relaxation time, respectively

\(u\,\) :

\(u\, = \left( {u_{r} ,u_{\psi } ,u_{z} } \right)\) The displacement functions

\(\alpha_{{\text{T}}}\) :

Coefficient of linear thermal expansion

\( \beta \) :

\( = \left( {\frac{{\lambda + 2\mu }}{\mu }} \right)^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}}} \)

\( \gamma \) :

\(= \left( {3\lambda + 2\mu } \right)\alpha _{T} \)

\(\varepsilon\) :

\(= \frac{\gamma }{{\rho \,C_{{\text{E}}} }}\) The mechanical coupling constant (dimensionless)

\(\varepsilon_{1}\) :

\(= \frac{{\gamma T_{o} }}{\mu }\) The thermoelastic coupling constant (dimensionless)

\(\eta\) :

\(= \frac{{\rho \,C_{{\text{E}}} }}{K}\) The thermal viscosity

\(\lambda \;,\;\mu \,\) :

Lamé’s constants

\(\rho \,\) :

Density

\(\sigma_{ij} \,\) :

Components of the stress tensor

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Authors and Affiliations

  1. Mathematics Department, Faculty of Education, Alexandria University, Alexandria, Egypt

    Hamdy M. Youssef

  2. Mechanical Engineering Department, College of Engineering and Islamic Architecture, Umm Al-Qura University, Makkah, 21955, Kingdom of Saudi Arabia

    Hamdy M. Youssef

  3. Mathematics Department, Al-Lith University College, Umm Al-Qura University, Al-Lith, Kingdom of Saudi Arabia

    Eman A. N. Al-Lehaibi

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  1. Hamdy M. Youssef
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Correspondence to Hamdy M. Youssef.

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Youssef, H.M., Al-Lehaibi, E.A.N. 2-D mathematical model of hyperbolic two-temperature generalized thermoelastic solid cylinder under mechanical damage effect. Arch Appl Mech 92, 945–960 (2022). https://doi.org/10.1007/s00419-021-02083-0

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