Thermal-stress analysis of a damaged solid sphere using hyperbolic two-temperature generalized thermoelasticity theory

Hamdy M Youssef,Alaa A El-Bary,Eman A N Al-Lehaibi

doi:10.1038/s41598-021-82127-1

Abstract

This work aims to study the influence of the rotation on a thermoelastic solid sphere in the context of the hyperbolic two-temperature generalized thermoelasticity theory based on the mechanical damage consideration. Therefore, a mathematical model of thermoelastic, homogenous, and isotropic solid sphere with a rotation based on the mechanical damage definition has been constructed. The governing equations have been written in the context of hyperbolic two-temperature generalized thermoelasticity theory. The bounding surface of the sphere is thermally shocked and without volumetric deformation. The singularities of the studied functions at the center of the sphere have been deleted using L’Hopital’s rule. The numerical results have been represented graphically with various mechanical damage values, two-temperature parameters, and rotation parameter values. The two-temperature parameter has significant effects on all the studied functions. Damage and rotation have a major impact on deformation, displacement, stress, and stress–strain energy, while their effects on conductive and dynamical temperature rise are minimal. The thermal and mechanical waves propagate with finite speeds on the thermoelastic body in the hyperbolic two-temperature theory and the one-temperature theory (Lord-Shulman model).

Highlights

This work aims to study the influence of the rotation on a thermoelastic solid sphere in the context of the hyperbolic two-temperature generalized thermoelasticity theory based on the mechanical damage consideration
Authors and researchers have provided many mathematical models in which they studied the transmission of thermomechanical waves in solid materials
It requires a large space not limited to one research to talk about thermomechanical transition mathematical models by using elastic materials

Summary

The governing equations

Consider a perfect thermal thermoelastic, isotropic, and spherical body that fills the region = {(r, ψ, φ) : 0 ≤ r ≤ a, 0 ≤ ψ ≤ 2π, 0 ≤ φ < 2π }. Where D = 0 is devoted to the undamaged case while D = 1 describes the fully damaged case formally with a total loss of stress carrying capacity. In the case of isotropic damage, the effective stresses are given by[20]: σij = (1 − D)σij,. Where σij are the stresses components in the undamaged material. The displacement components have the form ur , uψ , uφ = (u(r, t), 0, 0 ). The constitutive equations with mechanical damage parameter[28]: σrr = (1 − D)(2μerr + e) − γ (1 − D)(TD − T0),.

The strain components are err

The diagonalization method

Numerical results and discussion

Conclusions

Author contributions

Additional information