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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

## Abstract

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.

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## Figures

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

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.

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.

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.

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.

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