Encyclopedia of Continuum Mechanics

Living Edition
| Editors: Holm Altenbach, Andreas Öchsner

Scaling Function in Mechanics of Random Materials

  • Shivakumar I. Ranganathan
  • Muhammad Ridwan Murshed
Living reference work entry
DOI: https://doi.org/10.1007/978-3-662-53605-6_72-1

Synonyms

Definitions

The concept of a dimensionless scaling function is introduced and its role is discussed in the context of multiscale mechanics of random composites. The proposed scaling function stems from the scalar contraction of the ensemble averaged tensors obtained using Dirichlet and Neumann type boundary conditions. In its most generic form, the scaling function depends upon the phase contrast, volume fraction, material anisotropy, and mesoscale. The scaling function essentially quantifies the departure of a random medium from a homogeneous continuum.

Introduction

Recent advances in computational mechanics have dramatically changed the landscape of engineering and science. The primary driving force is due to a rapid decrease in the computational cost which is estimated as a billion-fold reduction during the last 40 years (Belytschko et al., 2007). In particular, computational mechanics has led to...

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References

  1. Belytschko T, Hughes T, Patankar N, Herakovich C, Bakis C (2007) Research directions in computational and composite mechanics. A report of the United States National Committee on theoretical and applied mechanicsGoogle Scholar
  2. Dalaq AS, Ranganathan SI (2015) Invariants of mesoscale thermal conductivity and resistivity tensors in random checkerboards. Eng Comput 32(6):1601–1618CrossRefGoogle Scholar
  3. Dalaq AS, Ranganathan SI, Ostoja-Starzewski M (2013) Scaling function in conductivity of planar random checkerboards. Comput Mater Sci 79:252–261CrossRefGoogle Scholar
  4. Donea J, Huerta A (2003) Finite element methods for flow problems. Wiley, ChichesterCrossRefGoogle Scholar
  5. Du X, Ostoja-Starzewski M (2006) 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, The Royal Society, vol 462, pp 2949–2963Google Scholar
  6. Gorb L, Kuz’min V, Muratov E (2014) Application of computational techniques in pharmacy and medicine, vol 17. Springer, DordrechtGoogle Scholar
  7. Hill R (1963) Elastic properties of reinforced solids: some theoretical principles. J Mech Phys Solids 11(5): 357–372CrossRefzbMATHGoogle Scholar
  8. Huang YG, Shiota Y, Wu MY, Su SQ, Yao ZS, Kang S, Kanegawa S, Li GL, Wu SQ, Kamachi T et al (2016) Superior thermoelasticity and shape-memory nanopores in a porous supramolecular organic framework. Nat Commun 7:11564CrossRefGoogle Scholar
  9. Huebner KH, Dewhirst DL, Smith DE, Byrom TG (2008) The finite element method for engineers. Wiley, HobokenGoogle Scholar
  10. Kale S, Saharan A, Koric S, Ostoja-Starzewski M (2015) Scaling and bounds in thermal conductivity of planar gaussian correlated microstructures. J Appl Phys 117(10):104301CrossRefGoogle Scholar
  11. Khisaeva Z, Ostoja-Starzewski M (2006) Mesoscale bounds in finite elasticity and thermoelasticity of random composites. In: Proceedings of the royal society of London A: mathematical, physical and engineering sciences, The Royal Society, vol 462, pp 1167–1180Google Scholar
  12. Kuneš J (2012) Dimensionless physical quantities in science and engineering. Elsevier, LondonGoogle Scholar
  13. Mandel J (1966) Contribution théorique à létude de lécrouissage et des lois de lécoulement plastique. In: Görtler H (ed) Applied mechanics. Springer, Berlin/Heidelberg, pp 502–509CrossRefGoogle Scholar
  14. Mukherjee T, Manvatkar V, De A, DebRoy T (2017) Dimensionless numbers in additive manufacturing. J Appl Phys 121(6):064904CrossRefGoogle Scholar
  15. Murshed MR, Ranganathan SI (2017a) Hill–Mandel condition and bounds on lower symmetry elastic crystals. Mech Res Commun 81:7–10CrossRefGoogle Scholar
  16. Murshed MR, Ranganathan SI (2017b) Scaling laws in elastic polycrystals with individual grains belonging to any crystal class. Acta Mechanica 228(4):1525–1539MathSciNetCrossRefGoogle Scholar
  17. Murshed MR, Ranganathan SI, Abed FH (2016) Design maps for fracture resistant functionally graded materials. Eur J Mech A Solids 58:31–41MathSciNetCrossRefGoogle Scholar
  18. Oden JT, Belytschko T, Babuska I, Hughes T (2003) Research directions in computational mechanics. Comput Methods Appl Mech Eng 192(7):913–922MathSciNetCrossRefzbMATHGoogle Scholar
  19. Ostoja-Starzewski M (2006) Material spatial randomness: from statistical to representative volume element. Probab Eng Mech 21(2):112–132MathSciNetCrossRefGoogle Scholar
  20. Ostoja-Starzewski M (2007) Microstructural randomness and scaling in mechanics of materials. CRC Press, Boca RatonCrossRefzbMATHGoogle Scholar
  21. Ostoja-Starzewski M, Ranganathan SI (2013) Scaling and homogenization in spatially random composites. In: Mantic̆ V (ed) Mathematical methods and models in composites. Imperial College Press, London, pp 61–101CrossRefGoogle Scholar
  22. Ostoja-Starzewski M, Du X, Khisaeva Z, Li W (2007) Comparisons of the size of the representative volume element in elastic, plastic, thermoelastic, and permeable random microstructures. Int J Multiscale Comput Eng 5(2):73–82CrossRefGoogle Scholar
  23. Ostoja-Starzewski M, Costa L, Ranganathan SI (2015) Scale-dependent homogenization of random hyperbolic thermoelastic solids. J Elast 118(2):243–250MathSciNetCrossRefzbMATHGoogle Scholar
  24. Ostoja-Starzewski M, Kale S, Karimi P, Malyarenko A, Raghavan B, Ranganathan SI, Zhang J (2016) Chapter two-scaling to RVE in random media. Adv Appl Mech 49:111–211CrossRefGoogle Scholar
  25. Panigrahi PK (2016) Transport phenomena in microfluidic systems. Wiley, SingaporeCrossRefGoogle Scholar
  26. Quey R, Dawson P, Barbe F (2011) Large-scale 3D random polycrystals for the finite element method: generation, meshing and remeshing. Comput Methods Appl Mech Eng 200(17):1729–1745CrossRefzbMATHGoogle Scholar
  27. Raghavan BV, Ranganathan SI (2014) Bounds and scaling laws at finite scales in planar elasticity. Acta Mechanica 225(11):3007–3022MathSciNetCrossRefzbMATHGoogle Scholar
  28. Raghavan BV, Ranganathan SI, Ostoja-Starzewski M (2015) Electrical properties of random checkerboards at finite scales. AIP Adv 5(1):017131CrossRefGoogle Scholar
  29. Ranganathan SI, Ostoja-Starzewski M (2008a) Mesoscale conductivity and scaling function in aggregates of cubic, trigonal, hexagonal, and tetragonal crystals. Phys Rev B 77(21):214308CrossRefGoogle Scholar
  30. Ranganathan SI, Ostoja-Starzewski M (2008b) Scale-dependent homogenization of inelastic random polycrystals. J Appl Mech 75(5):051008CrossRefGoogle Scholar
  31. Ranganathan SI, Ostoja-Starzewski M (2008c) Scaling function, anisotropy and the size of RVE in elastic random polycrystals. J Mech Phys Solids 56(9): 2773–2791MathSciNetCrossRefzbMATHGoogle Scholar
  32. Ranganathan SI, Ostoja-Starzewski M (2008d) Universal elastic anisotropy index. Phys Rev Lett 101(5):055504CrossRefGoogle Scholar
  33. Ranganathan SI, Ostoja-Starzewski M (2009) Towards scaling laws in random polycrystals. Int J Eng Sci 47(11):1322–1330MathSciNetCrossRefzbMATHGoogle Scholar
  34. Ranganathan SI, Decuzzi P, Wheeler LT, Ferrari M (2010a) Geometrical anisotropy in biphase particle reinforced composites. J Appl Mech 77(4):041017CrossRefGoogle Scholar
  35. Ranganathan SI, Yoon DM, Henslee AM, Nair MB, Smid C, Kasper FK, Tasciotti E, Mikos AG, Decuzzi P, Ferrari M (2010b) Shaping the micromechanical behavior of multi-phase composites for bone tissue engineering. Acta Biomaterialia 6(9):3448–3456CrossRefGoogle Scholar
  36. Reuss A (1929) 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):49–58CrossRefzbMATHGoogle Scholar
  37. Shore SN (2012) An introduction to astrophysical hydrodynamics. Academic, San DiegoGoogle Scholar
  38. Splinter R (2010) Handbook of physics in medicine and biology. CRC Press, Boca RatonCrossRefGoogle Scholar
  39. Voigt W (1928) Lehrbuch der Kristallphysik (mit Ausschluss der Kristalloptik). Teubner, LeipzigzbMATHGoogle Scholar
  40. White FM (2003) Fluid mechanics. McGraw-Hill, New YorkGoogle Scholar
  41. Zhang J, Ostoja-Starzewski M (2016) 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, The Royal Society, vol 472Google Scholar

Copyright information

© Springer-Verlag GmbH Germany 2018

Authors and Affiliations

  • Shivakumar I. Ranganathan
    • 1
  • Muhammad Ridwan Murshed
    • 2
  1. 1.Department of Mechanical Engineering, Virginia Polytechnic Institute and State UniversityNorthern Virginia CenterFalls ChurchUSA
  2. 2.Department of Mechanical EngineeringRowan UniversityGlassboroUSA

Section editors and affiliations

  • Martin Ostoja-Starzewski
    • 1
  1. 1.Department of Mechanical Science & Engineering, Institute for Condensed Matter Theory and Beckman InstituteUniversity of Illinois at Urbana–ChampaignUrbanaUSA