Abstract
This study is the first to use the diagonalization method for the new modelling of a homogeneous, thermoelastic, and isotropic solid sphere that has been subjected to mechanical damage. The fundamental equations were derived using the hyperbolic two-temperature generalized thermoelasticity theory with mechanical damage taken into account. The outer surface of the sphere has been assumed to have been shocked thermally without cubical dilatation. The numerical results for the dynamical and conductive temperatures increment, strain, displacement, and average of the principal stresses components have been represented graphically with different values of the hyperbolic two-temperature parameter and mechanical damage parameters. The two-temperature model parameter and the mechanical damage parameter have significant effects. The propagations of the thermomechanical waves take place at finite speeds in the context of the hyperbolic two-temperature theory as well as in the usual context of the Lord–Shulman theory with one-temperature.
Highlights
The can mechanical andvalue thermal waves propagate with finite time
Wesee less time intervals can seethe that the value ofhas time has significant effects limited speeds under the hyperbolic two-temperature thermoelasticity model, and inthe studied functions
The mechanical and thermal waves propagate with finite or limited on all the studied functions
Summary
Because of the spherical symmetry, the components of displacement take the form:. The heat conduction equation under the hyperbolic two-temperature theory takes the forms. Σψψ = (1 − D ) 2μeψψ + λe − γθ σφφ = (1 − D ) 2μeφφ + λe − γθ Equation (14) can be modified to be in the form h i (1 − D ) (λ + 2μ)∇2 e − γ∇2 θ = ρe (15). We will use the following dimensionless variables [5,22]: σ (19). By using the forms in Equation (26) in Equations (20)–(22), we obtain the following equations:. When the following zero initial conditions have been used: e(r, 0) = φ(r, 0) = θ (r, 0) =.
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