On the estimates for the elastic moduli of random Voronoi triclinic polycrystals

Duc-Chinh Pham


Our major new results and the previous ones on the bounds for elastic random polycrystals, and most advanced 3D finite element results for random 3D Voronoi polycrystals are resumed and analysed (together for the first time). Recently obtained numerical Dirichlet and Neumann simulation results for the effective elastic moduli of a large 10000-grain-size random Voronoi polycrystal representative volume element (RVE) for a number of triclinic and monoclinic base crystals (Mursheda and Ranganathan, 2017) are compared critically with the bounds on the moduli. Though major parts within the simulation results fall within the bounds of Pham (2011), some Dirichlet upper estimates still lie outside the bounds. Many more RVEs are needed to represent the Voronoi polycrystal on the same RVE-size-level, and larger RVEs are needed for checking the convergence and comparisons with the bounds.


effective elastic moduli; random Voronoi polycrystal; triclinic crystal; scatter measures of the estimates

Full Text:



W. Voigt. Lehrbuch der kristallphysik:(mit ausschluss der kristalloptik). Teubner, Leipzig, (1928).

A. Reuß. Berechnung der fließgrenze von mischkristallen auf grund der plastizitätsbedingung für einkristalle. ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik, 9, (1), (1929), pp. 49–58. https://doi.org/10.1002/zamm.19290090104.

R. Hill. The elastic behaviour of a crystalline aggregate. Proceedings of the Physical Society. Section A, 65, (5), (1952), p. 349. https://doi.org/10.1088/0370-1298/65/5/307.

Z. Hashin and S. Shtrikman. A variational approach to the theory of the elastic behavior of polycrystals. Journal of the Mechanics and Physics of Solids, 10, (4), (1962), pp. 343–352. https://doi.org/10.1016/0022-5096(62)90005-4.

R. Zeller and P. H. Dederichs. Elastic constants of polycrystals. Physica Status Solidi, 55, (2), (1973), pp. 831–842. https://doi.org/10.1002/pssb.2220550241.

M. W. M. Willemse and W. J. Caspers. Electrical conductivity of polycrystalline materials. Journal of Mathematical Physics, 20, (8), (1979), pp. 1824–1831. https://doi.org/10.1063/1.524284.

E. Kroner. Graded and perfect disorder in random media elasticity. Journal of the Engineering Mechanics Division, 106, (5), (1980), pp. 889–914.

J. J. McCoy. Macroscopic response of continua with random microstructures. In Mechanics Today, pp. 1–40. Elsevier, (1981). https://doi.org/10.1016/b978-0-08-024749-6.50012-0.

P. D. Chinh. Bounds on the effective shear modulus of multiphase materials. International Journal of Engineering Science, 31, (1), (1993), pp. 11–17. https://doi.org/10.1016/0020-7225(93)90060-8.

P. D. Chinh. On the scatter ranges for the elastic moduli of random aggregates of general anisotropic crystals. Philosophical Magazine, 91, (4), (2011), pp. 609–627. https://doi.org/10.1080/14786435.2010.528459.

P. D. Chinh. Bounds on the elastic moduli of statistically isotropic multicomponent materials and random cell polycrystals. International Journal of Solids and Structures, 49, (18), (2012), pp. 2646–2659. https://doi.org/10.1016/j.ijsolstr.2012.05.021.

J. M. Brown. Determination of Hashin–Shtrikman bounds on the isotropic effective elastic moduli of polycrystals of any symmetry. Computers & Geosciences, 80, (2015), pp. 95–99. https://doi.org/10.1016/j.cageo.2015.03.009.

C. M. Kube and A. P. Arguelles. Bounds and self-consistent estimates of the elastic constants of polycrystals. Computers & Geosciences, 95, (2016), pp. 118–122. https://doi.org/10.1016/j.cageo.2016.07.008.

D. C. Pham, C. H. Le, and T. M. H. Vuong. Estimates for the elastic moduli of d-dimensional random cell polycrystals. Acta Mechanica, 227, (10), (2016), pp. 2881–2897. https://doi.org/10.1007/s00707-016-1653-y.

M. Nygårds. Number of grains necessary to homogenize elastic materials with cubic symmetry. Mechanics of Materials, 35, (11), (2003), pp. 1049–1057. https://doi.org/10.1016/s0167-6636(02)00325-3.

S. I. Ranganathan and M. Ostoja-Starzewski. Scaling function, anisotropy and the size of RVE in elastic random polycrystals. Journal of the Mechanics and Physics of Solids, 56, (9), (2008), pp. 2773–2791. https://doi.org/10.1016/j.jmps.2008.05.001.

J. Besson, G. Cailletaud, J.-L. Chaboche, and S. Forest. Non-linear mechanics of materials, Vol. 167. Springer Science & Business Media, (2009).

M. R. Murshed and S. I. Ranganathan. Hill–Mandel condition and bounds on lower symmetry elastic crystals. Mechanics Research Communications, 81, (2017), pp. 7–10. https://doi.org/10.1016/j.mechrescom.2017.01.005. [

A. G. Every and A. K. McCurdy. Second and higher-order elastic constants. In Landolt-Börnstein Numerical Data and Functional Relationships in Science and Technology New Series Group III: Crystal and Solid State Physics, Springer-Verlag, Berlin, Vol. 29, (1992), pp. 1–743.

S. I. Ranganathan and M. Ostoja-Starzewski. Universal elastic anisotropy index. Physical Review Letters, 101, (5), (2008). https://doi.org/10.1103/PhysRevLett.101.055504.

DOI: https://doi.org/10.15625/0866-7136/14795 Display counter: Abstract : 97 views. PDF : 62 views.


  • There are currently no refbacks.

Copyright (c) 2020 Vietnam Academy of Science and Technology


Editorial Office of Vietnam Journal of Mechanics

3rd Floor, A16 Building, 18B Hoang Quoc Viet Street, Cau Giay District, Hanoi, Vietnam
Tel: (+84) 24 3791 7103
Email: vjmech@vjs.ac.vn