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Research Papers

Elastic Microplane Formulation for Transversely Isotropic Materials

[+] Author and Article Information
Congrui Jin

Department of Mechanical Engineering,
State University of New York at Binghamton,
Binghamton, NY 13902
e-mail: cjin@binghamton.edu

Marco Salviato

William E. Boeing Department of Aeronautics
and Astronautics,
University of Washington,
Seattle, WA 98195
e-mail: salviato@aa.washington.edu

Weixin Li

Department of Civil
and Environmental Engineering,
Northwestern University,
Evanston, IL 60208
e-mail: w.li@u.northwestern.edu

Gianluca Cusatis

Department of Civil and
Environmental Engineering,
Northwestern University,
Evanston, IL 60208
e-mail: g-cusatis@northwestern.edu

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received August 11, 2016; final manuscript received September 1, 2016; published online October 5, 2016. Editor: Yonggang Huang.

J. Appl. Mech 84(1), 011001 (Oct 05, 2016) (14 pages) Paper No: JAM-16-1400; doi: 10.1115/1.4034658 History: Received August 11, 2016; Revised September 01, 2016

This contribution investigates the extension of the microplane formulation to the description of transversely isotropic materials such as shale rock, foams, unidirectional composites, and ceramics. Two possible approaches are considered: (1) the spectral decomposition of the stiffness tensor to define the microplane constitutive laws in terms of energetically orthogonal eigenstrains and eigenstresses and (2) the definition of orientation-dependent microplane elastic moduli. The first approach, as demonstrated previously, provides a rigorous way to tackle anisotropy within the microplane framework, which is reviewed and presented herein in a clearer manner; whereas the second approach represents an approximation which, however, makes the formulation of nonlinear constitutive equations much simpler. The efficacy of the second approach in modeling the macroscopic elastic behavior is compared to the thermodynamic restrictions of the anisotropic parameters showing that a significant range of elastic properties can be modeled with excellent accuracy. Further, it is shown that it provides a very good approximation of the microplane stresses provided by the first approach, with the advantage of a simpler formulation. It is concluded that the spectral stiffness decomposition represents the best approach in such cases as for modeling composites, in which accurately capturing the elastic behavior is important. The introduction of orientation-dependent microplane elastic moduli provides a simpler framework for the modeling of transversely isotropic materials with remarked inelastic behavior, as in the case, for example, of shale rock.

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Figures

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Fig. 1

(a) Coordinate system for transversely isotropic materials and (b) spherical coordinate system

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Fig. 2

The contour plot of t for each case. The results are compared with the thermodynamic restrictions on elastic constants of transversely isotropic materials. The figure appears in color in the electronic version of this article.

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Fig. 3

The values of t(α,β,γ), ν(α,β,γ), and ν′(α,β,γ) for any α > 0, β > 0, and γ > 0 for each case with different values of t indicated by different colors. The figure appears in color in the electronic version of this article.

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Fig. 4

The ranges of ν and ν′ obtained from microplane model based on case C and case D when 0<E′/E≤1, respectively. The ranges of ν and ν′ for various types of shale studied by the existing literature [4045] are also plotted. The figure appears in color in the electronic version of this article.

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Fig. 5

The results for the values and the ratios of Ei (i = N, M, L) as a function of θ obtained from case C and case D for example A, respectively

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Fig. 6

The variation of apparent Young's modulus with anisotropy angle in comparison with experimental data provided by Cho et al. [42]

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Fig. 7

The results for the values and the ratios of Ei (i = N, M, L) as a function of θ obtained from case C and case D for example B, respectively

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Fig. 8

The distribution of the normal strain component, εN, on a generic microplane sphere caused by different types of macroscopic strains for the Boryeong shale with E = 37.3 GPa, E′=18.4 GPa, ν=0.15, ν′=0.16, and G = 12.0 GPa

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Fig. 9

The distribution of the normalized normal stress component, σN, on a generic microplane sphere caused by different types of macroscopic strains for the Boryeong shale with E = 37.3 GPa, E′=18.4 GPa, ν=0.15, ν′=0.16, and G = 12.0 GPa

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Fig. 10

The distributions of the normalized normal stress component σN based on the assumption that σN=ENεN, where EN=1/(a1 sin2θ+a2 cos2θ) as given in case C, caused by different types of macroscopic strains for the Boryeong shale with E = 37.3 GPa, E′=18.4 GPa, ν=0.15, ν′=0.16, and G = 12.0 GPa

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Fig. 11

(a) plots of the actual EN under ε11=ε22=ε33=1 and ε13=ε23=ε12=0, (b) plots of the actual EN under ε11=ε22=ε33/4=1 and ε13=ε23=ε12=0, (c) plots of the actual EN under ε11=ε22=ε33=ε12=1 and ε13=ε23=0, and (d) plots of EN=1/(a1 sin2θ+a2 cos2θ) as assumed in case C

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