0
Research Papers

Random Bulk Properties of Heterogeneous Rectangular Blocks With Lognormal Young's Modulus: Effective Moduli

[+] Author and Article Information
Leon S. Dimas, Tristan Giesa

Laboratory for Atomistic and Molecular
Mechanics (LAMM),
Department of Civil and
Environmental Engineering,
Massachusetts Institute of Technology,
77 Massachusetts Avenue,
Cambridge, MA 02139
Department of Civil and
Environmental Engineering,
Massachusetts Institute of Technology,
77 Massachusetts Avenue,
Cambridge, MA 02139

Daniele Veneziano

Department of Civil and
Environmental Engineering,
Massachusetts Institute of Technology,
77 Massachusetts Avenue,
Cambridge, MA 01239

Markus J. Buehler

Laboratory for Atomistic and Molecular
Mechanics (LAMM),
Department of Civil and
Environmental Engineering,
Massachusetts Institute of Technology,
77 Massachusetts Avenue,
Cambridge, MA 02139
Department of Civil and
Environmental Engineering,
Massachusetts Institute of Technology,
77 Massachusetts Avenue,
Cambridge, MA 02139
e-mail: mbuehler@MIT.EDU

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received September 21, 2014; final manuscript received October 9, 2014; accepted manuscript posted October 13, 2014; published online November 14, 2014. Editor: Yonggang Huang.

J. Appl. Mech 82(1), 011003 (Jan 01, 2015) (9 pages) Paper No: JAM-14-1434; doi: 10.1115/1.4028783 History: Received September 21, 2014; Revised October 09, 2014; Accepted October 13, 2014; Online November 14, 2014

We investigate the effective elastic properties of disordered heterogeneous materials whose Young's modulus varies spatially as a lognormal random field. For one-, two-, and three-dimensional (1D, 2D, and 3D) rectangular blocks, we decompose the spatial fluctuations of the Young's log-modulus F=lnE into first- and higher-order terms and find the joint distribution of the effective elastic tensor by multiplicatively combining the term-specific effects. The analytical results are in good agreement with Monte Carlo simulations. Through parametric analysis of the analytical solutions, we gain insight into the effective elastic properties of this class of heterogeneous materials. The results have applications to structural/mechanical reliability assessment and design.

FIGURES IN THIS ARTICLE
<>
Copyright © 2015 by ASME
Your Session has timed out. Please sign back in to continue.

References

Beran, M. J., 1968, “Monographs in Statistical Physics and Thermodynamics,” Statistical Continuum Theories, Vol. xv, Interscience Publishers, New York, p. 424.
Kachanov, M., and Sevostianov, I., 2013, Effective Properties of Heterogeneous Materials, Vol. 193, Springer, Dordrecht, The Netherlands.
Nemat-Nasser, S., and Hori, M., 1993, Micromechanics: Overall Properties of Heterogeneous Materials, Vol. xx (North-Holland Series in Applied Mathematics and Mechanics), North-Holland, Amsterdam, The Netherlands, p. 687.
Sanchez-Palencia, E., Zaoui, A., and International Centre for Mechanical Sciences, 1987, “Homogenization Techniques for Composite Media: Lectures Delivered at the CISM International Center for Mechanical Sciences,” (Lecture Notes in Physics, Udine, Italy, July 1–5, 1985, Vol. ix), Springer-Verlag, Berlin, Germany, p. 397.
Torquato, S., 2002, Random Heterogeneous Materials: Microstructure and Macroscopic Properties, Vol. 16, Springer, New York.
Gupta, H. S., Stachewicz, U., Wagermaier, W., Roschger, P., Wagner, H. D., and Fratzl, P., 2006, “Mechanical Modulation at the Lamellar Level in Osteonal Bone,” J. Mater. Res., 21(8), pp. 1913–1921. [CrossRef]
Tai, K., Dao, M., Suresh, S., Palazoglu, A., and Ortiz, C., 2007, “Nanoscale Heterogeneity Promotes Energy Dissipation in Bone,” Nature Mater., 6(6), pp. 454–462. [CrossRef]
Younis, S., Kauffmann, Y., Blouch, L., and Zolotoyabko, E., 2012, “Inhomogeneity of Nacre Lamellae on the Nanometer Length Scale,” Cryst. Growth Des., 12(9), pp. 4574–4579. [CrossRef]
Zhang, T., Li, X., Kadkhodaei, S., and Gao, H., 2012, “Flaw Insensitive Fracture in Nanocrystalline Graphene,” Nano Lett., 12(9), pp. 4605–4610. [CrossRef] [PubMed]
Dimas, L. S., Giesa, T., and Buehler, M. J., 2014, “Coupled Continuum and Discrete Analysis of Random Heterogeneous Materials: Elasticity and Fracture,” J. Mech. Phys. Solids, 63, pp. 481–490. [CrossRef]
Veneziano, D., 2003, Computational Fluid and Solid Mechanics, Elsevier, Cambridge, MA.
Ostoja-Starzewski, M., 1998, “Random Field Models of Heterogeneous Materials,” Int. J. Solids Struct., 35(19), pp. 2429–2455. [CrossRef]
Ghanem, R., and Spanos, P. D., 1990, “Polynomial Chaos in Stochastic Finite-Elements,” ASME J. Appl. Mech., 57(1), pp. 197–202. [CrossRef]
Stefanou, G., 2009, “The Stochastic Finite Element Method: Past, Present and Future,” Comput. Methods Appl. Mech. Eng., 198(9–12), pp. 1031–1051. [CrossRef]
Vanmarcke, E., and Grigoriu, M., 1983, “Stochastic Finite-Element Analysis of Simple Beams,” ASCE J. Eng. Mech., 109(5), pp. 1203–1214. [CrossRef]
Ngah, M. F., and Young, A., 2007, “Application of the Spectral Stochastic Finite Element Method for Performance Prediction of Composite Structures,” Compos. Struct., 78(3), pp. 447–456. [CrossRef]
Pellissetti, M. F., and Ghanem, R. G., 2000, “Iterative Solution of Systems of Linear Equations Arising in the Context of Stochastic Finite Elements,” Adv. Eng. Software, 31(8–9), pp. 607–616. [CrossRef]
Yamazaki, F., Shinozuka, M., and Dasgupta, G., 1988, “Neumann Expansion for Stochastic Finite-Element Analysis,” ASCE J. Eng. Mech., 114(8), pp. 1335–1354. [CrossRef]
Huyse, L., and Maes, M. A., 2001, “Random Field Modeling of Elastic Properties Using Homogenization,” ASCE J. Eng. Mech., 127(1), pp. 27–36. [CrossRef]
Ostojastarzewski, M., and Wang, C., 1989, “Linear Elasticity of Planar Delaunay Networks—Random Field Characterization of Effective Moduli,” Acta Mech., 80(1–2), pp. 61–80. [CrossRef]
Arwade, S. R., and Deodatis, G., 2011, “Variability Response Functions for Effective Material Properties,” Probab. Eng. Mech., 26(2), pp. 174–181. [CrossRef]
Graham-Brady, L., 2010, “Statistical Characterization of Meso-Scale Uniaxial Compressive Strength in Brittle Materials With Randomly Occurring Flaws,” Int. J. Solids Struct., 47(18–19), pp. 2398–2413. [CrossRef]
Ma, J. A., Temizer, I., and Wriggers, P., 2011, “Random Homogenization Analysis in Linear Elasticity Based on Analytical Bounds and Estimates,” Int. J. Solids Struct., 48(2), pp. 280–291. [CrossRef]
Shinozuka, M., and Deodatis, G., 1988, “Response Variability of Stochastic Finite-Element Systems,” ASCE J. Eng. Mech., 114(3), pp. 499–519. [CrossRef]
Shinozuka, M., 1987, “Structural Response Variability,” ASCE J. Eng. Mech., 113(6), pp. 825–842. [CrossRef]
Teferra, K., Arwade, S. R., and Deodatis, G., 2012, “Stochastic Variability of Effective Properties Via the Generalized Variability Response Function,” Comput. Struct., 110, pp. 107–115. [CrossRef]
Teferra, K., Arwade, S. R., and Deodatis, G., 2014, “Generalized Variability Response Functions for Two-Dimensional Elasticity Problems,” Comput. Methods Appl. Mech. Eng., 272, pp. 121–137. [CrossRef]
Veneziano, D., and Tabaei, A., 2001, “Analysis of Variance Method for the Equivalent Conductivity of Rectangular Blocks,” Water Resour. Res., 37(12), pp. 2919–2927. [CrossRef]
See [CrossRef] at for detailed derivations of the equations and supplementary results.
Karhunen, K., 1946, Zur Spektraltheorie Stochastischer Prozesse, Ann. Acad. Sci. Fenn., Ser. A, 37.

Figures

Grahic Jump Location
Fig. 1

Realizations of 2D normal log-stiffness fields with a simple exponential correlation kernel for normalized correlation lengths of (a) 0.125 and (b) 0.5

Grahic Jump Location
Fig. 2

(a) 1D rod. Comparison of theoretical (lineplots and ellipses) and numerically predicted distributions (histograms and scatter plots) of F1 for r0/L = 0.125, σF = 0.3 and correlation function e-r/r0 or e-(r/r0)2^. (b) Comparison of theoretical and simulated distributions of the 2D elastic tensor for a rectangular block with parameters L2/L1 = 100, r0/L1 = 2, σF = 0.5 and correlation function e-r/r0. (c) Similar comparison for a cubic specimen with r0/L = 0.25, σF = 0.3 and correlation function e-r/r0.

Grahic Jump Location
Fig. 3

Normalized mean and standard deviation of the effective Young's modulus of a 1D rod as a function of the dimensionless specimen length L/r0, for correlation functions e-(r/r0) and e-(r/r0)2. As L/r0→∞ the mean value tends to the deterministic ergodic limit, while the standard deviation tends to 0.

Grahic Jump Location
Fig. 4

Parameters of the effective elastic tensor of a 2D rectangular block as a function of the aspect ratio L2/L1 and the normalized correlation distance r0/L1. The correlation function is e-r/r0. The correlation coefficients are shown for σF = 0.5. The longitudinal, square-like, and transversal regimes are indicated in the low left panel.

Grahic Jump Location
Fig. 5

Parameters of the effective elastic tensor for a 3D rectangular block with side lengths L1 = L2 = L and L3, as a function of the aspect ratio L3/L and the normalized correlation distance r0/L1. The correlation function is e-r/r0. The correlation coefficients are for σF = 0.5. The plate-like, cube-like, and elongated regimes are indicated in the center low panel.

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In