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Special Section Papers

Effect of Volume Element Geometry on Convergence to a Representative Volume

[+] Author and Article Information
Katherine Acton

Department of Mechanical Engineering,
University of St. Thomas,
St. Paul, MN 55105
e-mail: kacton@stthomas.edu

Connor Sherod

Department of Mechanical Engineering,
University of St. Thomas,
St. Paul, MN 55105
e-mail: sher7302@stthomas.edu

Bahador Bahmani

Department of Mechanical, Aerospace, and
Biomedical Engineering,
University of Tennessee Knoxville (UTK)/
Space Institute (UTSI),
Tullahoma, TN 37388
e-mail: bbahmani@vols.utk.edu

Reza Abedi

Department of Mechanical, Aerospace, and
Biomedical Engineering,
University of Tennessee Knoxville (UTK)/
Space Institute (UTSI),
Tullahoma, TN 37388
e-mail: rabedi@utk.edu

1Corresponding author.

Manuscript received February 18, 2019; final manuscript received May 3, 2019; published online June 5, 2019. Assoc. Editor: George Stefanou.

ASME J. Risk Uncertainty Part B 5(3), 030907 (Jun 06, 2019) (8 pages) Paper No: RISK-19-1014; doi: 10.1115/1.4043753 History: Received February 18, 2019; Revised May 03, 2019

To accurately simulate fracture, it is necessary to account for small-scale randomness in the properties of a material. Apparent properties of statistical volume element (SVE) can be characterized below the scale of a representative volume element (RVE). Apparent properties cannot be defined uniquely for an SVE, in the manner that unique effective properties can be defined for an RVE. Both constitutive behavior and material strength properties in SVE must be statistically characterized. The geometrical partitioning method can be critically important in affecting the probability distributions of mesoscale material property parameters. Here, a Voronoi tessellation-based partitioning scheme is applied to generate SVE. Resulting material property distributions are compared with those from SVE generated by square partitioning. The proportional limit stress of the SVE is used to approximate SVE strength. Superposition of elastic results is used to obtain failure strength distributions from boundary conditions at variable angles of loading.

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Figures

Grahic Jump Location
Fig. 1

Partitioning of RVE microstructure (top) into square and Voronoi SVE (middle and bottom figures, respectively) each with side length S

Grahic Jump Location
Fig. 2

Calculation of failure strength s̃n. Failure occurs when the maximum stress on any inclusion boundary, in the direction normal to the inclusion, reaches a threshold value σTH [19].

Grahic Jump Location
Fig. 3

Normalized tensile strength as a function of load angle θ for square SVE size δ = 1/8

Grahic Jump Location
Fig. 4

Normalized tensile strength as a function of load angle θ for Voronoi SVE size δ = 1/8

Grahic Jump Location
Fig. 5

Normalized shear strength as a function of load angle θ for square SVE size δ = 1/8

Grahic Jump Location
Fig. 6

Normalized shear strength as a function of load angle θ for Voronoi SVE size δ = 1/8

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Fig. 7

Mean tensile strength as a function of load angle θ for square SVE. Strength is averaged over all SVE for a given SVE size δ = 1/4, δ = 1/8, and δ = 1/16.

Grahic Jump Location
Fig. 8

Mean tensile strength as a function of load angle θ for Voronoi SVE. Strength is averaged over all SVE for a given SVE size δ = 1/4, δ = 1/8, and δ = 1/16.

Grahic Jump Location
Fig. 9

Mean shear strength as a function of load angle θ for square SVE. Strength is averaged over all SVE for a given SVE size δ = 1/4, δ = 1/8, and δ = 1/16.

Grahic Jump Location
Fig. 10

Mean shear strength as a function of load angle θ for Voronoi SVE. Strength is averaged over all SVE for a given SVE size δ = 1/4, δ = 1/8, and δ = 1/16.

Grahic Jump Location
Fig. 11

Mean maximum wave speed as a function of load angle θ for square SVE. Strength is averaged over all SVE for a given SVE size δ = 1/4, δ = 1/8, and δ = 1/16.

Grahic Jump Location
Fig. 12

Mean maximum wave speed as a function of load angle θ for Voronoi SVE. Strength is averaged over all SVE for a given SVE size δ = 1/4, δ = 1/8, and δ = 1/16.

Grahic Jump Location
Fig. 13

Tensile strength as a function of SVE size δ for square and Voronoi SVE. Triple lines indicate minimum, mean, and maximum property values recovered from population of SVE with given size.

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Fig. 14

Shear strength as a function of SVE size δ for square and Voronoi SVE. Triple lines indicate minimum, mean, and maximum property values recovered from population of SVE with given size.

Grahic Jump Location
Fig. 15

Maximum wave speed as a function of SVE size δ for square and Voronoi SVE. Triple lines indicate minimum, mean, and maximum property values recovered from population of SVE with given size.

Grahic Jump Location
Fig. 16

Coefficient of variation of tensile strength as a function of SVE size for square and Voronoi SVE. The three lines represent the minimum, mean, and maximum of this anisotropy index across all SVEs of a given size.

Grahic Jump Location
Fig. 17

Coefficient of variation of shear strength as a function of SVE size for square and Voronoi SVE. The three lines represent the minimum, mean, and maximum of this anisotropy index across all SVEs of a given size.

Grahic Jump Location
Fig. 18

Coefficient of variation of maximum wave speed as a function of SVE size for square and Voronoi SVE. The three lines represent the minimum, mean, and maximum of this anisotropy index across all SVEs of a given size.

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