We consider the problem corresponding to the torsion of a cylindrical beam which incorporates the effects of material microstructure. The beam is modelled using micropolar or Cosserat elasticity and the boundary integral equation method is used to solve the corresponding boundary value problem. The exact solution is found in the form of an integral potential which is then approximated numerically using a Fourier series based on the corresponding singular solutions of the governing equations. We give an example, which demonstrates the effectiveness of the numerical method.
Eringen, A. C.Linear theory of micropolar elasticity . Journal of Mathematical Mechanics, 15, 909-923 (1966).
2.
Horgan, C. O.Anti-plane shear deformations in linear and nonlinear solid mechanics . SIAM Review, 37, 53-81 (1995).
3.
Schiavone, P.Integral equation methods in plane asymmetric elasticity , Journal of Elasticity, 43, 31-43 (1996).
4.
Eringen, A. C.Theory of micropolar plates . Journal of Applied Mathematics and Physics, 18, 12-30 (1967).
5.
Potapenko, S. , Schiavone, P. and Mioduchowski, A.Antiplane shear deformations in a linear theory of elasticity with microstructure . Zeitschrift für Angewandte Mathematik und Physik, (to appear).
6.
Potapenko, S. , Schiavone, P. and Mioduchowski, A.On the solution of mixed problems in antiplane micropolar elasticity . Mathematics and Mechanics of Solids, 8, 151-160 , (2003).
7.
Potapenko, S. Application of the boundary integral equation method the problem of torsion of micropolar bars, Ph.D. Thesis, University of Alberta, Edmonton, Canada, 2002.
8.
Smith, A. C.Torsion and vibrations of cylinders of a micro-polar elastic solid, in Recent Advances in Engineering Science, Vol. 5, pp.129-137, ed. A. C. Eringen , 1970.
9.
Iesan, D.Torsion of micropolar elastic beams . International Journal of Engineering Science, 9, 1047-1060 (1971).
10.
Schiavone, P.On existence theorems in the theory of extensional motions of thin micropolar plates . International Journal of Engineering Science, 27, 1129-1133 (1989).
11.
Constanda, C.Some comments on the integration of certain systems of partial differential equations in continuum mechanics . Journal of Applied Mathematics and Physics, 29, 835-839 (1978).
12.
Iesan, D.Existence theorems in the theory of micropolar elasticity . International Journal of Engineering Science, 8, 777-791 (1970).
13.
Kupradze V.D.et al. , in Three-Dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity, North-Holland, Amsterdam , 1979.
14.
Muskhelishvili, N.Singular Integral Equations, Noordhoff, Groningen , 1953.
15.
Schiavone, P. and Constanda, C.Existence theorems in the theory of thin micropolar plates . International Journal of Engineering Science, 27, 463-468 (1989).
16.
Schiavone, P. and Constanda, C.Uniqueness in the elastostatic problem of bending of thin micropolar plates . Archives of Mechanics, 41, 785-791 (1989).
17.
Schiavone, P.A generalized Fourier approximation in micropolar elasticity . Zeitschrift für Angewandte Mathematik und Physik, 40, 839-845 (1989).
18.
Schiavone, P.Fundamental sequences of functions in the approximation of solutions to exterior problems in the bending of thin micropolar plates . Applicable Analysis, 35, 263-274 (1990).
19.
Schiavone, P.Generalized Fourier series for exterior problems in extensional motions of thin micropolar plates . International Journal of Engineering Science, 28, 1067-1072 (1990).
20.
Schiavone, P.Uniqueness in dynamic problems of thin micropolar plates , Applied Mathematics Letters, 4, 81-83 (1991).
21.
Gauthier, R. D. and Jahsman, W. E.A quest for micropolar elastic constants . Journal of Applied Mechanics (ASME), June, 369-374 (1975).
Constanda, C.A Mathematical Analysis of Bending of Plates with Transverse Shear Deformation, Longmans, Harlow , 1990.
24.
Miranda, C.Partial Differential Equations of Elliptic Type, Springer, Berlin , 1970.
25.
Lakes, R.Experimental methods for study of Cosserat elastic solids and other generalized elastic continua, in Continuum Models for Materials with Microstructure, ed. H. B. Muhlhaus , Wiley, New York , 1995.
