Far Eastern Mathematical Journal

To content of the issue

Thermodynamically consistent equations of the couple stress elasticity

Sadovskii V.M.

2016, issue 2, P. 209-222

Abstract

To describe motion in a micropolar medium, with the concurrent translational degrees of freedom and independent particle rotations, it is chosen to use natural measure of curvature that is a strain state characteristic independent of deformation method. It is shown that the common Lagrangian curvature measure with the rate of change equal to a tensor of angular velocity gradients is only applicable under geometrically linear approximation. The nonlinear constitutive equations of the couple stress theory are constructed using the method of internal thermodynamic parameters of state. The linearization of these equations in isotropic case yields the Cosserat continuum equations, where material resistance to the change in curvature is characterized by a single coefficient as against the three independent coefficients of the classical theory. The complete system of equations for the dynamics of a medium with couple stresses under finite strains and particle rotations reduces to a thermodynamically consistent system of laws of conservation. This system allows to obtain the integral estimates that guarantee the uniqueness and continuous dependence on the initial data of solutions of the Cauchy problems and the boundary-value problems with dissipative boundary conditions.

Keywords: elasticity, Cosserat continuum, couple stresses, curvature tensor, thermodynamically consistent system

Download the article (PDF-file)

References

[1] E. Cosserat, F. Cosserat, Theorie des Corps Deformables, Librairie Scientifique Hermann et Fils, Paris, 1909. [2] V.A. Pal'mov, “Osnovnye uravneniia teorii nesimmetrichnoi uprugosti”, Prikl. matem. i mekhan., 28:3 (1964), 401–408. [3] V.T. Koiter, “Momentnye napriazheniia v teorii uprugosti”, Mekhanika: Sb. perevodov. T. 3, 1965, 89–112. [4] E.L. Aero, A.N. Bulygin, E.V. Kuvshinskii, “Asimmetricheskaia gidromekhanika” , Prikl. matem. i mekhan., 29:2 (1965), 297–308. [5] V.I. Kondaurov, “O nelineinykh uravneniiakh dinamiki uprugoi mikropoliarnoi sredy” , Prikl. matem. i mekhan., 48:3 (1984), 404–413. [6] E. Nikitin, L.M. Zubov, “Conservation laws and conjugate solutions in the elasticity of simple materials and materials with couple stress”, J. Elast., 51 (1998), 1–22. [7] W. Pietraszkiewicz, V.A. Eremeev, “On natural strain measures of the non-linear micropolar continuum”, Int. J. Solids Struct., 46 (2009), 774–787. [8] A.E. Green, P.M. Naghdi, W.L. Wainwright, “A general theory of a Cosserat surface”, Arch. Rat. Mech. Anal., 20:4 (1965), 287–308. [9] L.I. Shkutin, Mekhanika deformatsii gibkikh tel, Nauka, Sib. otd-nie, Novosibirsk, 1988. [10] Kh. Al'tenbakh, P.A. Zhilin, “ Obshchaia teoriia uprugikh prostykh obolochek”, Uspekhi mekhaniki, 11:4 (1988), 107–148. [11] V.M. Sadovskii, “Termodinamicheski samosoglasovannaia sistema zakonov sokhraneniia nesimmetrichnoi teorii uprugosti”, Dal'nevost. matem. zhurn., 11:2 (2011), 201–212. [12] V.I. Kondaurov, V.E. Fortov, Osnovy termomekhaniki kondensirovannoi sredy, Izd-vo MFTI, M., 2002. [13] S.K. Godunov, E.I. Romenskii, Elementy mekhaniki sploshnykh sred i zakony sokhraneniia, Nauchnaia kniga, Novosibirsk, 1998. [14] O. Sadovskaya, V. Sadovskii, Mathematical Modeling in Mechanics of Granular Materials, Ser.: Advanced Structured Materials. V. 21, Springer, Heidelberg – New York – Dordrecht – London, 2012. [15] A.G. Kulikovskii, N.V. Pogorelov, A.Iu. Semenov, Matematicheskie voprosy chislennogo resheniia giperbolicheskikh sistem uravnenii, Fizmatlit, M., 2001. [16] S.K. Godunov, Uravneniia matematicheskoi fiziki, Nauka, M., 1979.

To content of the issue