Skip to main content
Log in

Analysis of a Contact Problem Problem Involving an Elastic Body with Dual-Phase-Lag

  • Published:
Applied Mathematics & Optimization Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  1. Bazarra, N., Copetti, M.I.M., Fernández, J.R., Quintanilla, R.: Numerical analysis of some dual-phase-lag models. Comput. Math. Appl. 77, 407–426 (2019)

    Article  MathSciNet  Google Scholar 

  2. Borgmeyer, K., Quintanilla, R., Racke, R.: Phase-lag heat condition: decay rates for limit problems and well-posedness. J. Evol. Equ. 14, 863–884 (2014)

    Article  MathSciNet  Google Scholar 

  3. Campo, M., Fernández, J.R., Kuttler, K.L., Shillor, M., Viaño, J.M.: Numerical analysis and simulations of a dynamic frictionless contact problem with damage. Comput. Methods Appl. Mech. Eng. 196(1–3), 476–488 (2006)

    Article  MathSciNet  Google Scholar 

  4. Chandrasekharaiah, D.: Hyperbolic thermoelasticity: a review of recent literature. ASME Appl. Mech. Rev. 51(12), 705–729 (1998)

    Article  Google Scholar 

  5. Chirita, S.: On the time differential dual-phase-lag thermoelastic model. Meccanica 52, 349–361 (2017)

    Article  MathSciNet  Google Scholar 

  6. Chirita, S., Ciarletta, M., Tibullo, V.: On the thermomechanical consistency of the time differential dual-phase-lag models of heat conduction. Int. J. Heat Mass Transf. 114, 277–285 (2017)

    Article  Google Scholar 

  7. Ciarlet, P.G.: The finite element method for elliptic problems. In: Ciarlet, P.G., Lions, J.L. (eds.) Handbook of Numerical Analysis, vol. 2, pp. 17–352. North-Holland, Amsterdam (1991)

    Google Scholar 

  8. Fabrizio, M., Lazzari, B.: Stability and second law of thermodynamics in dual-phase-lag heat conduction. Int. J. Heat Mass Transf. 74, 484–489 (2014)

    Article  Google Scholar 

  9. Glowinski, R.: Numerical Methods for Nonlinear Variational Problems. Springer, New York (1984)

    Book  Google Scholar 

  10. Martins, J.A.C., Oden, J.T.: A numerical analysis of a class of problems in elastodynamics with friction. Comput. Methods Appl. Mech. Eng. 40(3), 327–360 (1983)

    Article  MathSciNet  Google Scholar 

  11. Muñoz Rivera, J.E., de Lacerda Oliveira, M.: Exponential stability for a contact problem in thermoelasticity. IMA J. Appl. Math. 58(1), 71–82 (1997)

    Article  MathSciNet  Google Scholar 

  12. Quintanilla, R.: Exponential stability in the dual-phase-lag heat conduction theory. J. Non-Equilb. Thermodyn. 27, 217–227 (2002)

    MATH  Google Scholar 

  13. Quintanilla, R., Racke, R.: Qualitative aspects in dual-phase-lag thermoelasticity. SIAM J. Appl. Math. 66(3), 977–1001 (2006)

    Article  MathSciNet  Google Scholar 

  14. Quintanilla, R., Racke, R.: Qualitative aspects in dual-phase-lag heat conduction. Proc. R. Soc. Lond. A 463, 659–674 (2007)

    MathSciNet  MATH  Google Scholar 

  15. Simon, J.: Compact sets in the space \({L}^p(0,{T};{B})\). Ann. Mat. Pura Appl. (4) 146(1), 65–96 (1987)

    Article  MathSciNet  Google Scholar 

  16. Tzou, D.: The generalized lagging response in small-scale and high-rate heating. Int. J. Heat Mass Transf. 38, 3231–3240 (1995)

    Article  Google Scholar 

  17. Tzou, D.: A unified field approach for heat conduction from macro- to micro-scales. ASME. J. Heat Transf. 117(1), 8–16 (1995)

    Article  Google Scholar 

  18. Vermeersch, B., Mey, G.: Non-fourier thermal conduction in nano-scaled electronic structures. Analog. Int. Circ. Sig. Process. 55, 197–204 (2008)

    Article  Google Scholar 

  19. Wang, L., Xu, M.: Well-posedness of dual-phase-lagging heat equation: higher dimensions. Int. J. Heat Mass Transf. 45, 1055–1061 (2002)

    Article  Google Scholar 

  20. Zhou, J., Zhang, Y., Chen, J.: An axisymmetric dual-phase-lag bioheat model for laser heating of living tissues. Int. J. Therm. Sci. 49, 1477–1485 (2009)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to José R. Fernández.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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

$$\begin{aligned} \mathbf{q}(\mathbf x , t+\tau _q) = - \kappa \nabla \theta (\mathbf x , t+\tau _\theta ) \end{aligned}$$
(76)

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:

$$\begin{aligned}&\mu u_{i,jj}+(\lambda + \mu ) u_{j,ji}-m \theta _{,i} = ({u}_{i})_{tt}, \nonumber \\&\quad -q_{i,i}-m \theta ^0 ({u}_{i,i})_t= c \theta _t \end{aligned}$$
(77)

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

$$\begin{aligned} \alpha = 2 \mu + \lambda >0 . \end{aligned}$$

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:

$$\begin{aligned} \mathbf{q} + \tau _q \frac{\partial \mathbf q }{\partial t} + \frac{1}{2} \tau _q^2 \frac{\partial ^2 \mathbf q }{\partial t^2}= - \kappa \left( \nabla \theta + \tau _{\theta } \frac{\partial (\nabla \theta )}{\partial t} \right) . \end{aligned}$$
(78)

Eliminating \(\mathbf q \) from this law and the energy equation (77)\(_2\) we obtain in one dimension the equation

$$\begin{aligned} \theta _{ttt}+ \dfrac{2}{\tau _q}\theta _{tt}+ \dfrac{2}{\tau _q^2}\theta _{t} -\dfrac{2\,\tau _\theta \,\kappa }{c \tau _q^2}\theta _{txx}-\dfrac{2\kappa }{ c \tau _q^2}\theta _{xx} +\dfrac{2\,m\,\theta ^0}{ c \tau _q^2}\left( u_{xt} + \tau _q u_{xtt}+ \frac{\tau _q^2}{2} u_{xttt}\right) =0. \end{aligned}$$

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 \(\rho =c=1\):

$$\begin{aligned}&\tilde{u}_{tt}-\alpha \,\tilde{u}_{xx}+ \dfrac{\tau _q^2\, m}{2}\theta _{ttx}+\tau _q\,m\,\theta _{tx}+m\,\theta _x=0,\\&\theta _{ttt}+ \dfrac{2}{\tau _q}\theta _{tt}+ \dfrac{2}{\tau _q^2}\theta _{t}+\dfrac{2\,m\,\theta ^0}{\tau _q^2}\tilde{u}_{tx} -\dfrac{2\,\tau _\theta \,\kappa }{\tau _q^2}\theta _{txx}-\dfrac{2\kappa }{\tau _q^2}\theta _{xx}=0 . \end{aligned}$$

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:

$$\begin{aligned} \tilde{u}_{tt}(x,t)-\alpha \tilde{u}_{xx}(x,t)+m \tilde{\theta }_{x}(x,t)= & {} 0, \\ \tilde{\theta }_{t}(x,t)-\kappa \hat{\theta }_{xx}(x,t)+m\theta ^0 \tilde{u}_{xt}(x,t)= & {} 0, \end{aligned}$$

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

$$\begin{aligned} \begin{array}{rl} \sigma (x,t) &{}= \alpha \tilde{u}_{x}(x,t)-m[\theta (x,t)+\tau _q \theta _t(x,t)+\frac{\tau _q^2}{2}\theta _{tt}(x,t)] \\ &{}\qquad \quad \text {for a.e. } x\in (0,\ell ) \text { and } t\in (0,T). \end{array} \end{aligned}$$
Fig. 9
figure 9

The physical setting

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00245-019-09574-1

Keywords

Navigation