Continuum Mechanics and Thermodynamics

, Volume 31, Issue 1, pp 331–340

# A polynomial way to control the decay of solutions for dipolar bodies

• Marin Marin
• Andreas Öchsner
Original Article

## Abstract

In our paper, we consider a combination of two sub-cylinders coupled by an interface in a semi-infinite cylinder. Both sub-cylinders are made of dipolar elastic materials. For one of the two sub-cylinders, we will consider the elastostatic problem, and for the other the elastodynamic problem. Thus, the spatial behaviors of the sub-cylinders are of different kind and the question arises whether the evolution of this combination can be controlled. By using a polynomial way, we prove that the decay of solutions for the two problems can be controlled.

## Keywords

Dipolar bodies Elastostatics Elastodynamics Spatial estimates Upper bound Polynomial decay

## References

1. 1.
Flavin, J.N., Knops, R.J.: Some spatial decay estimates in continuum dynamics. J. Elast. 17, 249–264 (1987)
2. 2.
Flavin, J.N., Knops, R.J., Payne, L.E.: Energy bounds in dynamical problems for a semi-infinite elastic beam. In: Eason, G., Ogden, R.W. (eds.) Elasticity: Mathematical Methods and Applications, pp. 101–111. Ellis-Horwood, Chichester (1990)Google Scholar
3. 3.
Flavin, J.N., Rionero, S.: Qualitative Estimates for Partial Differential Equations. An Introduction. CRC Press, Boca Raton (1996)
4. 4.
Horgan, C.O., Quintanilla, R.: Spatial decay of transient end effects in functionally graded heat conducting materials. Q. Appl. Math. 59, 529542 (2001)
5. 5.
Quintanilla, R.: Spatial estimates for an equation with delay. Z. Angew. Math. Phys. 61(2), 381–388 (2010)
6. 6.
Quintanilla, R.: Spatial stability for the quasi-static problem of thermo-elasticity. J. Elast. 76, 93105 (2004)
7. 7.
Quintanilla, R., Saccomandi, G.: Quasistatic anti-plane motion in the simplest theory of nonlinear viscoelasticity. Nonlinear Anal. Ser. B Real World Appl. 9, 14991517 (2008)
8. 8.
Chirita, S.: On the spatial decay estimates in certain time-dependent problems of continuum mechanics. Arch. Mech. 47, 755–771 (1995)
9. 9.
Horgan, C.O., Payne, L.E., Wheeler, L.T.: Spatial decay estimates in transient heat conduction. Q. Appl. Math. 42, 119–127 (1984)
10. 10.
Nunziato, J.W.: On the spatial decay of solutions in the nonlinear theory of heat conduction. J. Math. Anal. Appl. 48, 687–698 (1974)
11. 11.
Eringen, A.C.: Theory of thermo-microstretch elastic solids. Int. J. Eng. Sci. 28, 1291–1301 (1990)
12. 12.
Eringen, A.C.: Microcontinuum Field Theories. Springer, New York (1999)
13. 13.
Mindlin, R.D.: Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 16, 51–78 (1964)
14. 14.
Green, A.E., Rivlin, R.S.: Multipolar continuum mechanics. Arch. Ration. Mech. Anal. 17, 113–147 (1964)
15. 15.
Fried, E., Gurtin, M.E.: Thermomechanics of the interface between a body and its environment. Contin. Mech. Thermodyn. 19(5), 253–271 (2007)
16. 16.
Vallée, C., Rădulescu, V., Atchonouglo, K.: New variational principles for solving extended Dirichlet–Neumann problems. J. Elast. 123, 1–18 (2016)
17. 17.
Marin, M.: Weak solutions in elasticity of dipolar porous materials. Math. Probl. Eng. 1–8, 158908 (2008)
18. 18.
Marin, M., Öchsner, A.: The effect of a dipolar structure on the Holder stability in Green–Naghdi thermoelasticity. Contin. Mech. Thermodyn. 29(6), 1365–1374 (2017)
19. 19.
Marin, M., Agarwal, R.P., Mahmoud, S.R.: Modeling a microstretch thermo-elastic body with two temperatures. Abstr. Appl. Anal. 2013, 1–7 (2013)
20. 20.
Marin, M.: An approach of a heat-flux dependent theory for micropolar porous media. Meccanica 51(5), 1127–1133 (2016)
21. 21.
Abbas, I.A.: A GN model based upon two-temperature generalized thermoelastic theory in an unbounded medium with a spherical cavity. Appl. Math. Comput. 245, 108–115 (2014)
22. 22.
Abbas, I.A.: Eigenvalue approach for an unbounded medium with a spherical cavity based upon two-temperature generalized thermoelastic theory. J. Mech. Sci. Technol. 28(10), 4193–4198 (2014)
23. 23.
Othman, M.I.A.: State space approach to generalized thermoelasticity plane waves with two relaxation times under the dependence of the modulus of elasticity on reference temperature. Can. J. Phys. 81(12), 1403–1418 (2003)
24. 24.
Sharma, J.N., Othman, M.I.A.: Effect of rotation on generalized thermo-viscoelastic Rayleigh–Lamb waves. Int. J. Solids Struct. 44(13), 4243–4255 (2007)
25. 25.
Craciun, E.M., Carabineanu, A., Peride, N.: Antiplane interface crack in a pre-stressed fiber-reinforced elastic composite. Comput. Mater. Sci. 43(1), 184–189 (2008)
26. 26.
Sadowski, T., Craciun, E.M., Rabaea, A., Marsavina, L.: Mathematical modeling of three equal collinear cracks in an orthotropic solid. Meccanica 51(2), 329–339 (2016)

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

## Authors and Affiliations

• Marin Marin
• 1
• Andreas Öchsner
• 2