Abstract
An approximate analytical solution for the displacement and microrotation vector fields is derived for pure torsion of a prismatic bar with square cross section comprised of homogeneous, isotropic linear Cosserat elastic material. This is accomplished by analytical simplification coupled with use of the principle of minimum potential energy together with polynomial representations for the desired field components. Explicit approximate expressions are derived for cross section warp and for applied torque versus angle of twist of the bar. These show that torsional rigidity exceeds the classical elasticity value, the difference being larger for slender bars, and that cross section warp is less than the classical amount. Experimental measurements on two sets of 3D printed square cross section polymeric bars, each set having a different microstructure and four different cross section sizes, revealed size effects not captured by classical elasticity but consistent with the present analysis for physically sensible values of the Cosserat moduli. The warp can allow inference of Cosserat elastic constants independently of any sensitivity the material may have to dilatation gradients; warp also facilitates inference of Cosserat constants that are difficult to obtain via size effects.
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References
Gauthier, R.D., Jahsman, W.E.: A quest for micropolar elastic constants. J. Appl. Mech. 42, 369–374 (1975)
Krishna Reddy, G.V., Venkatasubramanian, N.K.: On the flexural rigidity of a micropolar elastic circular cylinder. J. Appl. Mech. 45, 429–431 (1978)
Yang, J.F.C., Lakes, R.S.: Transient study of couple stress in compact bone: torsion. J. Biomech. Eng. 103, 275–279 (1981)
Lakes, R.S.: Experimental microelasticity of two porous solids. Int. J. Solids Struct. 22, 55–63 (1986)
Cosserat, E., Cosserat, F.: Theorie des Corps Deformables. Hermann et Fils, Paris (1909)
Eringen, A.C.: Theory of micropolar elasticity. In: Liebowitz, H. (ed.) Fracture, pp. 621–729. Academic Press, New York (1968)
Mindlin, R.D.: Stress functions for a Cosserat continuum. Int. J. Solids Struct. 1, 265–271 (1965)
Timoshenko, S., Goodier, J.N.: Theory of Elasticity. McGraw-Hill, New York (1970)
Rueger, Z., Li, D., Lakes, R.S.: Observation of Cosserat elastic effects in a tetragonal negative Poisson’s ratio lattice. Phys. Status Solidi B 254, 1600840 (2017). https://doi.org/10.1002/pssb.201600840
Li, D., Ma, J., Dong, L., Lakes, R.S.: Three-dimensional stiff cellular structures with negative Poisson’s ratio. Phys. Status Solidi B 254, 1600785 (2017). https://doi.org/10.1002/pssb.201600785
Rueger, Z., Lakes, R.S.: Cosserat elasticity in lattices (in preparation)
Prall, D., Lakes, R.S.: Properties of a chiral honeycomb with a Poisson’s ratio -1. Int. J. Mech. Sci. 39, 305–314 (1997)
Spadoni, A., Ruzzene, M.: Elasto-static micropolar behavior of a chiral auxetic lattice. J. Mech. Phys. Solids 60, 156–171 (2012)
Rueger, Z., Lakes, R.S.: Experimental Cosserat elasticity in open cell polymer foam. Philos. Mag. 96, 93–111 (2016)
Askar, A., Cakmak, A.S.: A structural model of a micropolar continuum. Int. J. Eng. Sci. 6, 583–589 (1968)
Iesan, D.: Torsion of micropolar elastic beams. Int. J. Eng. Sci. 9, 1047–1060 (1971)
Shmoylova, E., Potapenko, S., Dorfmann, A.: Weak solutions to anti-plane boundary value problems in a linear theory of elasticity with microstructure. Arch. Mech. 59(6), 519–539 (2007)
Park, H.C., Lakes, R.S.: Torsion of a micropolar elastic prism of square cross section. Int. J. Solids Struct. 23, 485–503 (1987)
Eringen, A.C.: Theory of thermo-microstretch elastic solids. Int. J. Eng. Sci. 28(12), 1291–1301 (1990)
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Drugan, W.J., Lakes, R.S. Torsion of a Cosserat elastic bar with square cross section: theory and experiment. Z. Angew. Math. Phys. 69, 24 (2018). https://doi.org/10.1007/s00033-018-0913-1
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DOI: https://doi.org/10.1007/s00033-018-0913-1