Skip to main content
SpringerLink
Log in

Scaling laws in elastic polycrystals with individual grains belonging to any crystal class

  • Original Paper
  • Published:
Acta Mechanica Aims and scope Submit manuscript

Abstract

In this paper, we develop unifying scaling laws describing the response of elastic polycrystals at finite mesoscales. These polycrystals are made up of individual grains belonging to any crystal class (from cubic to triclinic) and are generated by Voronoi tessellations with varying grain sizes. Rigorous scale-dependent bounds are then obtained by setting up and solving Dirichlet and Neumann boundary value problems consistent with the Hill–Mandel homogenization condition. The results generated are benchmarked with existing numerical results in special cases and the effect of grain shape on the scaling behavior is investigated. The convergence to the effective elastic properties with increasing number of grains is established by analyzing 5180 boundary value problems. This leads to the notion of an elastic scaling function which takes a power law form in terms of the universal anisotropy index and the mesoscale. Based on the scaling function, a material scaling diagram is constructed using which the convergence to the effective properties can be analyzed for any elastic microstructure.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Murshed, M.R., Ranganathan, S.I., Abed, F.H.: Design maps for fracture resistant functionally graded materials. Eur. J. Mech. A Solids 58, 31–41 (2016)

    Article  MathSciNet  Google Scholar 

  2. Schumacher, C., Bickel, B., Rys, J., Marschner, S., Daraio, C., Gross, M.: Microstructures to control elasticity in 3D printing. ACM Trans. Graph. 34(4), 136 (2015)

    Article  Google Scholar 

  3. Oxman, N.: Variable property rapid prototyping: inspired by nature, where form is characterized by heterogeneous compositions, the paper presents a novel approach to layered manufacturing entitled variable property rapid prototyping. Virtual Phys. Prototyp. 6(1), 3–31 (2011)

    Article  Google Scholar 

  4. Sevostianov, I., Kachanov, M.: On some controversial issues in effective field approaches to the problem of the overall elastic properties. Mech. Mater. 69(1), 93–105 (2014)

    Article  Google Scholar 

  5. Sevostianov, I.: On the shape of effective inclusion in the Maxwell homogenization scheme for anisotropic elastic composites. Mech. Mater. 75, 45–59 (2014)

    Article  Google Scholar 

  6. Hill, R.: Elastic properties of reinforced solids: some theoretical principles. J. Mech. Phys. Solids 11(5), 357–372 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  7. Mandel, J.: Contribution théorique à l’étude de l’écrouissage et des lois de l’écoulement plastique. In: Görtler, H. (ed.) Applied Mechanics, pp. 502–509. Springer, Berlin (1966)

  8. Raghavan, B.V., Ranganathan, S.I.: Bounds and scaling laws at finite scales in planar elasticity. Acta Mech. 225(11), 3007–3022 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ranganathan, S.I., Ostoja-Starzewski, M.: Towards scaling laws in random polycrystals. Int. J. Eng. Sci. 47(11), 1322–1330 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ranganathan, S.I., Ostoja-Starzewski, M.: Scaling function, anisotropy and the size of RVE in elastic random polycrystals. J. Mech. Phys. Solids 56(9), 2773–2791 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Khisaeva, Z., Ostoja-Starzewski, M.: On the size of RVE in finite elasticity of random composites. J. Elast. 85(2), 153–173 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. Jiang, M., Alzebdeh, K., Jasiuk, I., Ostoja-Starzewski, M.: Scale and boundary conditions effects in elastic properties of random composites. Acta Mech. 148(1–4), 63–78 (2001)

    Article  MATH  Google Scholar 

  13. Dalaq, A.S., Ranganathan, S.I.: Invariants of mesoscale thermal conductivity and resistivity tensors in random checkerboards. Eng. Comput. 32(6), 1601–1618 (2015)

    Article  Google Scholar 

  14. Kale, S., Saharan, A., Koric, S., Ostoja-Starzewski, M.: Scaling and bounds in thermal conductivity of planar gaussian correlated microstructures. J. Appl. Phys. 117(10), 104301 (2015)

    Article  Google Scholar 

  15. Ranganathan, S.I., Ostoja-Starzewski, M.: Mesoscale conductivity and scaling function in aggregates of cubic, trigonal, hexagonal, and tetragonal crystals. Phys. Rev. B 77(21), 214308 (2008)

    Article  Google Scholar 

  16. Ostoja-Starzewski, M., Schulte, J.: Bounding of effective thermal conductivities of multiscale materials by essential and natural boundary conditions. Phys. Rev. B 54(1), 278 (1996)

    Article  Google Scholar 

  17. Raghavan, B.V., Ranganathan, S.I., Ostoja-Starzewski, M.: Electrical properties of random checkerboards at finite scales. AIP Adv. 5(1), 017131 (2015)

    Article  Google Scholar 

  18. Du, X., Ostoja-Starzewski, M.: On the scaling from statistical to representative volume element in thermoelasticity of random materials. Netw. Heterog. Media 1(2), 259 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  19. Khisaeva, Z., Ostoja-Starzewski, M.: Mesoscale bounds in finite elasticity and thermoelasticity of random composites, In: Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 462, pp. 1167–1180. The Royal Society (2006)

  20. Ostoja-Starzewski, M.: Material spatial randomness: From statistical to representative volume element. Probab. Eng. Mech. 21(2), 112–132 (2006)

    Article  Google Scholar 

  21. Karim, M., Krabbenhoft, K.: Extraction of effective cement paste diffusivities from X-ray microtomography scans. Transp. Porous Media 84(2), 371–388 (2010)

    Article  Google Scholar 

  22. Ostoja-Starzewski, M., Du, X., Khisaeva, Z., Li, W.: Comparisons of the size of the representative volume element in elastic, plastic, thermoelastic, and permeable random microstructures. Int. J. Multiscale Comput. Eng. 5(2), 73–82 (2007)

  23. Du, X., Ostoja-Starzewski, M.: On the size of representative volume element for darcy law in random media. In: Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 462, pp. 2949–2963. The Royal Society (2006)

  24. Ostoja-Starzewski, M.: Microstructural Randomness and Scaling in Mechanics of Materials. CRC Press, Boca Raton (2007)

    Book  MATH  Google Scholar 

  25. Ostoja-Starzewski, M., Ranganathan, S.I.: Scaling and homogenization in spatially random composites. Mathematical Methods and Models in Composites. Imperial College Press, London (2013)

    MATH  Google Scholar 

  26. Ranganathan, S.I., Ostoja-Starzewski, M.: Scale-dependent homogenization of inelastic random polycrystals. J. Appl. Mech. 75(5), 051008 (2008)

    Article  Google Scholar 

  27. Ostoja-Starzewski, M.: Scale effects in plasticity of random media: status and challenges. Int. J. Plast 21(6), 1119–1160 (2005)

    Article  MATH  Google Scholar 

  28. Jiang, M., Ostoja-Starzewski, M., Jasiuk, I.: Scale-dependent bounds on effective elastoplastic response of random composites. J. Mech. Phys. Solids 49(3), 655–673 (2001)

    Article  MATH  Google Scholar 

  29. Kanit, T., Forest, S., Galliet, I., Mounoury, V., Jeulin, D.: Determination of the size of the representative volume element for random composites: statistical and numerical approach. Int. J. Solids Struct. 40(13), 3647–3679 (2003)

    Article  MATH  Google Scholar 

  30. El Houdaigui, F., Forest, S., Gourgues, A.-F., Jeulin, D.: On the size of the representative volume element for isotropic elastic polycrystalline copper. In: IUTAM Symposium on Mechanical Behavior and Micro-Mechanics of Nanostructured Materials, pp. 171–180. Springer, Berlin (2007)

  31. Dalaq, A.S., Ranganathan, S.I., Ostoja-Starzewski, M.: Scaling function in conductivity of planar random checkerboards. Comput. Mater. Sci. 79, 252–261 (2013)

    Article  Google Scholar 

  32. Zhang, J., Ostoja-Starzewski, M.: Frequency-dependent scaling from mesoscale to macroscale in viscoelastic random composites. In: Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 472. The Royal Society (2016)

  33. Suquet, P.: Elements of homogenization for inelastic solid mechanics. Homog. Tech. Compos. Media 272, 193–278 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  34. Hill, R.: The Mathematical Theory of Plasticity. Oxford University Press, Oxford (1950)

    MATH  Google Scholar 

  35. Quey, R., Dawson, P., Barbe, F.: Large-scale 3D random polycrystals for the finite element method: generation, meshing and remeshing. Comput. Methods Appl. Mech. Eng. 200(17), 1729–1745 (2011)

    Article  MATH  Google Scholar 

  36. Ranganathan, S.I., Ostoja-Starzewski, M.: Universal elastic anisotropy index. Phys. Rev. Lett. 101(5), 055504 (2008)

    Article  Google Scholar 

  37. Sab, K.: On the homogenization and the simulation of random materials. Eur. J. Mech. A Solids 11(5), 585–607 (1992)

    MathSciNet  MATH  Google Scholar 

  38. Huet, C.: Application of variational concepts to size effects in elastic heterogeneous bodies. J. Mech. Phys. Solids 38(6), 813–841 (1990)

    Article  MathSciNet  Google Scholar 

  39. Ostoja-Starzewski, M., Kale, S., Karimi, P., Malyarenko, A., Raghavan, B., Ranganathan, S. I., Zhang, J.: Scaling to RVE in Random Media. Adv. Appl. Mech. 49, 111–211 (2016). doi:10.1016/bs.aams.2016.07.001

  40. Hill, R.: The elastic behaviour of a crystalline aggregate. Proc. Phys. Soc. Lond. Sect. A 65(5), 349–354 (1952)

    Article  Google Scholar 

  41. Bhattacharya, K., Suquet, P.: A model problem concerning recoverable strains of shape-memory polycrystals. In: Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 461, pp. 2797–2816. The Royal Society (2005)

  42. Berryman, J.G.: Bounds and self-consistent estimates for elastic constants of random polycrystals with hexagonal, trigonal, and tetragonal symmetries. J. Mech. Phys. Solids 53(10), 2141–2173 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  43. Watt, J.P.: Hashin-Shtrikman bounds on the effective elastic moduli of polycrystals with monoclinic symmetry. J. Appl. Phys. 51(3), 1520–1524 (1980)

    Article  Google Scholar 

  44. Lide, D.R.: Handbook of Chemistry and Physics. CRC Press, Boca Raton (2004)

    Google Scholar 

  45. Brown, J.M., Angel, R.J., Ross, N.: Elasticity of plagioclase feldspars. J. Geophys. Res. Solid Earth 121, 663–675 (2016)

  46. Watt, J.P., Peselnick, L.: Clarification of the Hashin-Shtrikman bounds on the effective elastic moduli of polycrystals with hexagonal, trigonal, and tetragonal symmetries. J. Appl. Phys. 51(3), 1525–1531 (1980)

    Article  Google Scholar 

  47. Ledbetter, H., Kim, S.: Monocrystal elastic constants and derived properties of the cubic and the hexagonal elements. Handb. Elast. Prop. Solids Liquids Gases 2, 97–106 (2001)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, Rowan University, 201 Mullica Hill Road, Glassboro, NJ, 08028, USA

    Muhammad Ridwan Murshed & Shivakumar I. Ranganathan

  2. Department of Biomedical Engineering, Rowan University, 201 Mullica Hill Road, Glassboro, NJ, 08028, USA

    Shivakumar I. Ranganathan

Authors
  1. Muhammad Ridwan Murshed
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shivakumar I. Ranganathan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shivakumar I. Ranganathan.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Murshed, M.R., Ranganathan, S.I. Scaling laws in elastic polycrystals with individual grains belonging to any crystal class. Acta Mech 228, 1525–1539 (2017). https://doi.org/10.1007/s00707-016-1774-3

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00707-016-1774-3

Search

Navigation