Abstract
In this work we study a contact problem between a thermoelastic body with dual-phase-lag and a deformable obstacle. The contact is modelled using a modification of the well-known normal compliance contact condition. An existence and uniqueness result is proved applying the Faedo–Galerkin method and Gronwall’s inequality. The exponential stability is also shown. Then, we introduce a fully discrete approximation by using the implicit Euler scheme and the finite element method. A discrete stability property and a priori error estimates are obtained, from which the linear convergence of the algorithm is derived under suitable regularity conditions. Finally, some numerical examples are presented to demonstrate the accuracy of the approximation and the behaviour of the solution.
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The work of N. Bazarra and J.R. Fernández has been supported by the Ministerio de Economía y Competitividad under the research project MTM2015-66640-P with FEDER Funds.
Appendix
Appendix
The classical theory of heat conduction is based on the Fourier’s law. As it is well known this model predicts the problem of infinite propagation speed effect. This was a paradox because, in fact, heat transmission at low temperature has been observed to propagate by means of waves. Many authors tried to overcome this drawback proposing different alternative constitutive equations.
According to Fourier’s law, the heat flux \(\mathbf q \) is the instantaneous result of a temperature gradient, that is the heat flux occurs simultaneously with the establishment of a temperature gradient. There are several decades ago, Chandrasekharaiah [4] and Tzou (see [16, 17]) proposed a modification of the theory of the heat equation replacing the Fourier law by an approximation of the equation
where \(\kappa >0\) is the thermal conductivity, and in which the gradient of temperature, at a point in the material at time \(t+\tau _{\theta }\), corresponds to the heat flux vector at the same point at time \(t+\tau _{q}\). Hence, \(\tau _{q}>0\) is the phase-lag of the heat flux and \(\tau _{\theta }>0\) of the gradient of temperature. The delay time \(\tau _{\theta }\) is caused by microstructural interaction and the delay time \(\tau _{q}\) is interpreted as the relaxation time due to fast transient effect of thermal inertia. We note that the relation (76) allows either the temperature gradient or the heat flux to become the effect and remaining one the cause. For materials with \(\tau _q>\tau _\theta \), the heat flux vector is the result of a temperature gradient. It is the other way round for materials with \(\tau _\theta >\tau _q\). If \(\tau _q=\tau _\theta \) (not necessarily equal to zero), the response between the temperature gradient and the heat flux is instantaneous; in this case, the relation (76) becomes identical with the classical Fourier law [4]. Tzou [16] refers to the relation (76) as the dual-phase-lag model of the constitutive equation connecting the heat flux vector and the temperature gradient. In addition, he has shown that this model is admissible within the framework of the second law of the extended irreversible thermodynamics.
Following [4, 13], the thermoelastic model can be written by adding to (76) the following linear momentum balance equation and the equation of conservation of energy:
where \(m\ne 0\), \(\rho \) (the mass density), c (the specific heat for unit mass) and \(\theta ^0\) are positive constants. Moreover \(\mu >0\) and \(\lambda \) are the Lamé moduli satisfying
Expanding both sides of (76) by Taylor’s series and retaining terms up to the second order in \(\tau _q\), but only the term of the first order in \(\tau _{\theta }\), we obtain the following generalization of the heat conduction law:
Eliminating \(\mathbf q \) from this law and the energy equation (77)\(_2\) we obtain in one dimension the equation
Now, using the notation \(\tilde{f}=f+\tau _q f_t+ \frac{\tau _q^2}{2}f_{tt}\) and applying this differential operator \(\,\tilde{}\,\) to the differential equation (77)\(_1\), we obtain the hyperbolic coupled one-dimensional homogeneous isotropic model under the assumption \(\rho =c=1\):
We remark that finding a solution \((\tilde{u}, \theta )\) allows to determine the desired solutions \((u, \theta )\) to the original system. Throughout the paper, it will be useful to write the model in the following equivalent form:
where we used the notation \(\hat{f}= f+\tau _\theta f_t\).
In addition, we suppose that the rod, which has natural length \(\ell \), is fixed at \(x = 0\) and may come into contact with a deformable obstacle on \(x=\ell \) as shown in Fig. 9 where we recall that the stress field \(\sigma \) is given by
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Bazarra, N., Bochicchio, I., Fernández, J.R. et al. Analysis of a Contact Problem Problem Involving an Elastic Body with Dual-Phase-Lag. Appl Math Optim 83, 939–977 (2021). https://doi.org/10.1007/s00245-019-09574-1
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DOI: https://doi.org/10.1007/s00245-019-09574-1