0
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.

FIGURES IN THIS ARTICLE
<>
Copyright © 2016 by ASME
Your Session has timed out. Please sign back in to continue.

References

Holzapfel, G. A. , and Ogden, R. W. , 2009, “ Constitutive Modelling of Passive Myocardium: A Structurally Based Framework for Material Characterization,” Philos. Trans. R. Soc. London A, 367(1902), pp. 3445–3475. [CrossRef]
Wong, V. M. , Wenk, J. F. , Zhang, Z. , Cheng, G. , Acevedo-Bolton, G. , Burger, M. , Saloner, D. A. , Wallace, A. W. , Guccione, J. M. , Ratcliffe, M. B. , and Ge, L. , 2012, “ The Effect of Mitral Annuloplasty Shape in Ischemic Mitral Regurgitation: A Finite Element Simulation,” Ann. Thorac. Surg., 93(3), pp. 776–782. [CrossRef] [PubMed]
Genet, M. , Lee, L. C. , Nguyen, R. , Haraldsson, H. , Acevedo-Bolton, G. , Zhang, Z. , Ge, L. , Ordovas, K. , Kozerke, S. , and Guccione, J. M. , 2014, “ Distribution of Normal Human Left Ventricular Myofiber Stress at End Diastole and End Systole: A Target for In Silico Design of Heart Failure Treatments,” J. Appl. Physiol., 117(2), pp. 142–152. [CrossRef] [PubMed]
Ferruzzi, J. , Vorp, D. A. , and Humphrey, J. D. , 2011, “ On Constitutive Descriptors of the Biaxial Mechanical Behaviour of Human Abdominal Aorta and Aneurysms,” J. R. Soc. Interface, 8(56), pp. 435–450. [CrossRef] [PubMed]
Riveros, F. , Martufi, G. , Gasser, T. C. , and Rodriguez-Matas, J. F. , 2015, “ On the Impact of Intraluminal Thrombus Mechanical Behavior in AAA Passive Mechanics,” Ann. Biomed. Eng., 43(9), pp. 1–12. [CrossRef] [PubMed]
Raut, S. S. , Jana, A. , De Oliveira, V. , Muluk, S. C. , and Finol, E. A. , 2014, “ The Effect of Uncertainty in Vascular Wall Material Properties on Abdominal Aortic Aneurysm Wall Mechanics,” Computational Biomechanics for Medicine, Springer, Berlin, pp. 69–86.
Gasser, T. C. , Auer, M. , Labruto, F. , Swedenborg, J. , and Roy, J. , 2010, “ Biomechanical Rupture Risk Assessment of Abdominal Aortic Aneurysms: Model Complexity Versus Predictability of Finite Element Simulations,” Eur. J. Vasc. Endovasc. Surg., 40(2), pp. 176–185. [CrossRef] [PubMed]
Yucesoy, C. A. , Koopman, B. H. , Huijing, P. A. , and Grootenboer, H. J. , 2002, “ Three-Dimensional Finite Element Modeling of Skeletal Muscle Using a Two-Domain Approach: Linked Fiber-Matrix Mesh Model,” J. Biomech., 35(9), pp. 1253–1262. [CrossRef] [PubMed]
Miller, K. , and Chinzei, K. , 2002, “ Mechanical Properties of Brain Tissue in Tension,” J. Biomech., 35(4), pp. 483–490. [CrossRef] [PubMed]
Holzapfel, G. A. , Sommer, G. , Gasser, C. T. , and Regitnig, P. , 2005, “ Determination of Layer-Specific Mechanical Properties of Human Coronary Arteries With Nonatherosclerotic Intimal Thickening and Related Constitutive Modeling,” Am. J. Physiol. Heart Circ. Physiol., 289(5), pp. H2048–H2058. [CrossRef] [PubMed]
Kroon, M. , and Holzapfel, G. A. , 2008, “ A New Constitutive Model for Multi-Layered Collagenous Tissues,” J. Biomech., 41(12), pp. 2766–2771. [CrossRef] [PubMed]
Sacks, M. S. , 2003, “ Incorporation of Experimentally-Derived Fiber Orientation Into a Structural Constitutive Model for Planar Collagenous Tissues,” ASME J. Biomech. Eng., 125(2), pp. 280–287. [CrossRef]
Holzapfel, G. A. , Gasser, T. C. , and Ogden, R. W. , 2000, “ A New Constitutive Framework for Arterial Wall Mechanics and a Comparative Study of Material Models,” J. Elasticity Phys. Sci. Solids, 61(1–3), pp. 1–48. [CrossRef]
Lanir, Y. , 1983, “ Constitutive Equations for Fibrous Connective Tissues,” J. Biomech., 16(1), pp. 1–12. [CrossRef] [PubMed]
Cortes, D. H. , Lake, S. P. , Kadlowec, J. A. , Soslowsky, L. J. , and Elliott, D. M. , 2010, “ Characterizing the Mechanical Contribution of Fiber Angular Distribution in Connective Tissue: Comparison of Two Modeling Approaches,” Biomech. Model. Mechanobiol., 9(5), pp. 651–658. [CrossRef] [PubMed]
Einstein, D. R. , 2002, “ Nonlinear Acoustic Analysis of the Mitral Valve,” Ph.D. thesis, University of Washington, Seattle, WA.
Freed, A. D. , Einstein, D. R. , and Vesely, I. , 2005, “ Invariant Formulation for Dispersed Transverse Isotropy in Aortic Heart Valves,” Biomech. Model. Mechanobiol., 4(2–3), pp. 100–117. [CrossRef] [PubMed]
Gasser, T. C. , Ogden, R. W. , and Holzapfel, G. A. , 2006, “ Hyperelastic Modelling of Arterial Layers With Distributed Collagen Fibre Orientations,” J. R. Soc. Interface, 3(6), pp. 15–35. [CrossRef] [PubMed]
Holder, O. , 1889, “ Uber Einen Mittelwertssatz,” Nachr. Akad. Wiss. Gottingen Math.-Phys. Kl, 2, pp. 38–47.
Stein, W. A. , Abbott, T. , and Abshoff, M. , 2012, “ SageMath,” http://www.sagemath.org/
Raghupathy, R. , and Barocas, V. H. , 2009, “ A Closed-Form Structural Model of Planar Fibrous Tissue Mechanics,” J. Biomech., 42(10), pp. 1424–1428. [CrossRef] [PubMed]
Zeinali-Davarani, S. , Choi, J. , and Baek, S. , 2009, “ On Parameter Estimation for Biaxial Mechanical Behavior of Arteries,” J. Biomech., 42(4), pp. 524–530. [CrossRef] [PubMed]
Rodríguez, J. F. , Martufi, G. , Doblaré, M. , and Finol, E. A. , 2009, “ The Effect of Material Model Formulation in the Stress Analysis of Abdominal Aortic Aneurysms,” Ann. Biomed. Eng., 37(11), pp. 2218–2221. [CrossRef] [PubMed]
Sommer, G. , and Holzapfel, G. A. , 2012, “ 3D Constitutive Modeling of the Biaxial Mechanical Response of Intact and Layer-Dissected Human Carotid Arteries,” J. Mech. Behav. Biomed. Mater., 5(1), pp. 116–128. [CrossRef] [PubMed]
Karlon, W. J. , Covell, J. W. , Mcculloch, A. D. , Hunter, J. J. , and Omens, J. H. , 1998, “ Automated Measurement of Myofiber Disarray in Transgenic Mice With Ventricular Expression of RAS,” Anat. Rec., 252(4), pp. 612–625. [CrossRef] [PubMed]
Canham, P. B. , Finlay, H. M. , Dixon, J. G. , Boughner, D. R. , and Chen, A. , 1989, “ Measurements From Light and Polarised Light Microscopy of Human Coronary Arteries Fixed at Distending Pressure,” Cardiovasc. Res., 23(11), pp. 973–982. [CrossRef] [PubMed]
Rezakhaniha, R. , Agianniotis, A. , Schrauwen, J. T. C. , Griffa, A. , Sage, D. , Bouten, C. V. C. , Van de Vosse, F. N. , Unser, M. , and Stergiopulos, N. , 2012, “ Experimental Investigation of Collagen Waviness and Orientation in the Arterial Adventitia Using Confocal Laser Scanning Microscopy,” Biomech. Model. Mechanobiol., 11(3–4), pp. 461–473. [CrossRef] [PubMed]
Gasser, T. C. , Gallinetti, S. , Xing, X. , Forsell, C. , Swedenborg, J. , and Roy, J. , 2012, “ Spatial Orientation of Collagen Fibers in the Abdominal Aortic Aneurysm's Wall and Its Relation to Wall Mechanics,” Acta Biomater., 8(8), pp. 3091–3103. [CrossRef] [PubMed]
Billiar, K. L. , and Sacks, M. S. , 1997, “ A Method to Quantify the Fiber Kinematics of Planar Tissues Under Biaxial Stretch,” J. Biomech., 30(7), pp. 753–756. [CrossRef] [PubMed]
Chandran, P. L. , and Barocas, V. H. , 2006, “ Affine Versus Non-Affine Fibril Kinematics in Collagen Networks: Theoretical Studies of Network Behavior,” ASME J. Biomech. Eng., 128(2), pp. 259–270. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

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

Grahic Jump Location
Fig. 2

Relationship between fiber dispersion angle and b

Grahic Jump Location
Fig. 3

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

Grahic Jump Location
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

Grahic Jump Location
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

Grahic Jump Location
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

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