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Research Papers

A Characteristic-Based Constitutive Law for Dispersed Fibers

[+] Author and Article Information
Liang Ge

Department of Surgery,
University of California San Francisco,
San Francisco Veterans Affairs Medical Center,
San Francisco, CA 94121
e-mail: liang.ge@va.gov

Manuscript received December 3, 2015; final manuscript received April 21, 2016; published online June 7, 2016. Assoc. Editor: Hai-Chao Han.

J Biomech Eng 138(7), 071006 (Jun 07, 2016) (8 pages) Paper No: BIO-15-1624; doi: 10.1115/1.4033517 History: Received December 03, 2015; Revised April 21, 2016

Biological tissues are typically constituted of dispersed fibers. Modeling the constitutive laws of such tissues remains a challenge. Direct integration over all fibers is considered to be accurate but requires very expensive numerical integration. A general structure tensor (GST) model was previously developed to bypass this costly numerical integration step, but there are concerns about the model's accuracy. Here we estimate the approximation error of the GST model. We further reveal that the GST model ignores strain energy induced by shearing motions. Subsequently, we propose a new characteristic-based constitutive law to better approximate the direct integration model. The new model is very cost-effective and closely approximates the “true” strain energy as calculated by the direct integration when stress–strain nonlinearity or fiber dispersion angle is small.

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Figures

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Fig. 1

Fiber direction vector ef defined by two Eulerian angles Θ and Φ

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Fig. 2

Relationship between fiber dispersion angle and b

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Fig. 3

Relationship between κi,(i=1,5), and b

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Fig. 4

Strain energy as calculated by direct integration (DI-solid lines), general structure tensor (GST-dash-dotted lines), and characteristic (CH-dashed lines) models for k2=1: (a) b = 10, (b) b = 5, and (c) b = 1

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Fig. 6

Strain energy as calculated by direct integration (DI-solid lines), general structure tensor (GST-dash-dotted lines), and characteristic (CH-dashed lines) models for k2=50: (a) b = 10, (b) b = 5, and (c) b = 1

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Fig. 5

Strain energy as calculated by direct integration (DI-solid lines), general structure tensor (GST-dash-dotted lines) and characteristic (CH-dashed lines) models for k2=10: (a) b = 10, (b) b = 5, and (c) b = 1

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