Research Papers

Poisson's Contraction and Fiber Kinematics in Tissue: Insight From Collagen Network Simulations

[+] Author and Article Information
R. C. Picu

Department of Mechanical, Aerospace
and Nuclear Engineering,
Rensselaer Polytechnic Institute,
Troy, NY 12180
e-mail: picuc@rpi.edu

S. Deogekar

Department of Mechanical, Aerospace and
Nuclear Engineering,
Rensselaer Polytechnic Institute,
Troy, NY 12180
e-mail: deoges@rpi.edu

M. R. Islam

Department of Mechanical, Aerospace and
Nuclear Engineering,
Rensselaer Polytechnic Institute,
Troy, NY 12180
e-mail: islamm3@rpi.edu

1Corresponding author.

Manuscript received June 10, 2017; final manuscript received November 1, 2017; published online January 12, 2018. Editor: Victor H. Barocas.

J Biomech Eng 140(2), 021002 (Jan 12, 2018) (12 pages) Paper No: BIO-17-1252; doi: 10.1115/1.4038428 History: Received June 10, 2017; Revised November 01, 2017

Connective tissue mechanics is highly nonlinear, exhibits a strong Poisson's effect, and is associated with significant collagen fiber re-arrangement. Although the general features of the stress–strain behavior have been discussed extensively, the Poisson's effect received less attention. In general, the relationship between the microscopic fiber network mechanics and the macroscopic experimental observations remains poorly defined. The objective of the present work is to provide additional insight into this relationship. To this end, results from models of random collagen networks are compared with experimental data on reconstructed collagen gels, mouse skin dermis, and the human amnion. Attention is devoted to the mechanism leading to the large Poisson's effect observed in experiments. The results indicate that the incremental Poisson's contraction is directly related to preferential collagen orientation. The experimentally observed downturn of the incremental Poisson's ratio at larger strains is associated with the confining effect of fibers transverse to the loading direction and contributing little to load bearing. The rate of collagen orientation increases at small strains, reaches a maximum, and decreases at larger strains. The peak in this curve is associated with the transition of the network deformation from bending dominated, at small strains, to axially dominated, at larger strains. The effect of fiber tortuosity on network mechanics is also discussed, and a comparison of biaxial and uniaxial loading responses is performed.

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Hulmes, D. , 2008, Collagen Diversity, Synthesis and Assembly, Springer, Boston, MA, pp. 15–47.
Hansen, K. A. , Weiss, J. A. , and Barton, J. K. , 2002, “ Recruitment of Tendon Crimp With Applied Tensile Strain,” ASME J. Biomech. Eng., 124(1), pp. 72–77. [CrossRef]
Thorpe, C. T. , Klemt, C. , Riley, G. P. , Birch, H. L. , Clegg, P. D. , and Screen, H. R. , 2013, “ Helical Sub-Structures in Energy-Storing Tendons Provide a Possible Mechanism for Efficient Energy Storage and Return,” Acta Biomater., 9(8), pp. 7948–7956. [CrossRef] [PubMed]
Szczesny, S. E. , Peloquin, J. M. , Cortes, D. H. , Kadlowec, J. A. , Soslowsky, L. J. , and Elliott, D. M. , 2012, “ Biaxial Tensile Testing and Constitutive Modeling of Human Supraspinatus Tendon,” ASME J. Biomech. Eng., 134(2), p. 021004. [CrossRef]
Gusachenko, I. , Tran, V. , Houssen, Y. G. , Allain, J.-M. , and Schanne-Klein, M.-C. , 2012, “ Polarization-Resolved Second-Harmonic Generation in Tendon upon Mechanical Stretching,” Biophys. J., 102(9), pp. 2220–2229. [CrossRef] [PubMed]
Mansfield, J. C. , Winlove, C. P. , Moger, J. , and Matcher, S. J. , 2008, “ Collagen Fiber Arrangement in Normal and Diseased Cartilage Studied by Polarization Sensitive Nonlinear Microscopy,” J. Biomed. Opt., 13(4), p. 044020. [CrossRef] [PubMed]
Meng, Q. , An, S. , Damion, R. A. , Jin, Z. , Wilcox, R. , Fisher, J. , and Jones, A. , 2017, “ The Effect of Collagen Fibril Orientation on the Biphasic Mechanics of Articular Cartilage,” J. Mech. Behav. Biomed. Mater., 65, pp. 439–453. [CrossRef] [PubMed]
Zhang, S. , Bassett, D. S. , and Winkelstein, B. A. , 2016, “ Stretch-Induced Network Reconfiguration of Collagen Fibres in the Human Facet Capsular Ligament,” J. R. Soc. Interface, 13(114), p. 20150883. [CrossRef] [PubMed]
Mauri, A. , Ehret, A. E. , Perrini, M. , Maake, C. , Ochsenbein-Kölble, N. , Ehrbar, M. , Oyen, M. L. , and Mazza, E. , 2015, “ Deformation Mechanisms of Human Amnion: Quantitative Studies Based on Second Harmonic Generation Microscopy,” J. Biomech., 48(9), pp. 1606–1613. [CrossRef] [PubMed]
Perrini, M. , Mauri, A. , Ehret, A. E. , Ochsenbein-Kölble, N. , Zimmermann, R. , Ehrbar, M. , and Mazza, E. , 2015, “ Mechanical and Microstructural Investigation of the Cyclic Behavior of Human Amnion,” ASME J. Biomech. Eng., 137(6), p. 061010. [CrossRef]
Bancelin, S. , Lynch, B. , Bonod-Bidaud, C. , Ducourthial, G. , Psilodimitrakopoulos, S. , Dokládal, P. , Allain, J.-M. , Schanne-Klein, M.-C. , and Ruggiero, F. , 2015, “ Ex Vivo Multiscale Quantitation of Skin Biomechanics in Wild-Type and Genetically-Modified Mice Using Multiphoton Microscopy,” Sci. Rep., 5, p. 17635.
Jayyosi, C. , Affagard, J.-S. , Ducourthial, G. , Bonod-Bidaud, C. , Lynch, B. , Bancelin, S. , Ruggiero, F. , Schanne-Klein, M.-C. , Allain, J.-M. , and Bruyère-Garnier, K. , 2017, “ Affine Kinematics in Planar Fibrous Connective Tissues: An Experimental Investigation,” Biomech. Model. Mechanobiol., 16(4), pp. 1459–1473. [CrossRef] [PubMed]
Lynch, B. , Bancelin, S. , Bonod-Bidaud, C. , Gueusquin, J.-B. , Ruggiero, F. , Schanne-Klein, M.-C. , and Allain, J.-M. , 2017, “ A Novel Microstructural Interpretation for the Biomechanics of Mouse Skin Derived From Multiscale Characterization,” Acta Biomater., 50, pp. 302–311. [CrossRef] [PubMed]
Holzapfel, G. A. , and Ogden, R. W. , 2009, Biomechanical Modelling at the Molecular, Cellular and Tissue Levels, Springer Science & Business Media, Vienna, Austria. [CrossRef]
Ault, H. , and Hoffman, A. , 1992, “ A Composite Micromechanical Model for Connective Tissues—Part I: Theory,” ASME J. Biomech. Eng, 114(1), pp. 137–141. [CrossRef]
Cortes, D. H. , and Elliott, D. M. , 2012, “ Extra-Fibrillar Matrix Mechanics of Annulus Fibrosus in Tension and Compression,” Biomech. Model. Mechanobiol., 11(6), pp. 781–790. [CrossRef] [PubMed]
Guerin, H. L. , and Elliott, D. M. , 2007, “ Quantifying the Contributions of Structure to Annulus Fibrosus Mechanical Function Using a Nonlinear, Anisotropic, Hyperelastic Model,” J. Orthopaedic Res., 25(4), pp. 508–516. [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. Elast. Phys. Sci. Solids, 61(1–3), pp. 1–48.
Lanir, Y. , 1979, “ A Structural Theory for the Homogeneous Biaxial Stress-Strain Relationships in Flat Collagenous Tissues,” J. Biomech., 12(6), pp. 423–436. [CrossRef] [PubMed]
Martufi, G. , and Gasser, T. C. , 2011, “ A Constitutive Model for Vascular Tissue That Integrates Fibril, Fiber and Continuum Levels With Application to the Isotropic and Passive Properties of the Infrarenal Aorta,” J. Biomech., 44(14), pp. 2544–2550. [CrossRef] [PubMed]
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]
Head, D. A. , Levine, A. J. , and MacKintosh, F. , 2003, “ Deformation of Cross-Linked Semiflexible Polymer Networks,” Phys. Rev. Lett., 91(10), p. 108102. [CrossRef] [PubMed]
Wilhelm, J. , and Frey, E. , 2003, “ Elasticity of Stiff Polymer Networks,” Phys. Rev. Lett., 91(10), p. 108103. [CrossRef] [PubMed]
DiDonna, B. , and Lubensky, T. , 2005, “ Nonaffine Correlations in Random Elastic Media,” Phys. Rev. E, 72(6), p. 066619. [CrossRef]
Hatami-Marbini, H. , and Picu, R. , 2008, “ Scaling of Nonaffine Deformation in Random Semiflexible Fiber Networks,” Phys. Rev. E, 77(6), p. 062103. [CrossRef]
Picu, R. , 2011, “ Mechanics of Random Fiber Networks—A Review,” Soft Matter, 7(15), pp. 6768–6785. [CrossRef]
Broedersz, C. P. , and MacKintosh, F. C. , 2014, “ Modeling Semiflexible Polymer Networks,” Rev. Mod. Phys., 86(3), p. 995. [CrossRef]
Billiar, K. , and Sacks, M. , 1997, “ A Method to Quantify the Fiber Kinematics of Planar Tissues Under Biaxial Stretch,” J. Biomech., 30(7), pp. 753–756. [CrossRef] [PubMed]
Gilbert, T. W. , Sacks, M. S. , Grashow, J. S. , Woo, S. L.-Y. , Badylak, S. F. , and Chancellor, M. B. , 2006, “ Fiber Kinematics of Small Intestinal Submucosa Under Biaxial and Uniaxial Stretch,” ASME J. Biomech. Eng., 128(6), pp. 890–898. [CrossRef]
Guerin, H. A. L. , and Elliott, D. M. , 2006, “ Degeneration Affects the Fiber Reorientation of Human Annulus Fibrosus Under Tensile Load,” J. Biomech., 39(8), pp. 1410–1418. [CrossRef] [PubMed]
Lake, S. P. , Cortes, D. H. , Kadlowec, J. A. , Soslowsky, L. J. , and Elliott, D. M. , 2012, “ Evaluation of Affine Fiber Kinematics in Human Supraspinatus Tendon Using Quantitative Projection Plot Analysis,” Biomech. Model. Mechanobiol., 11(1–2), pp. 197–205. [CrossRef] [PubMed]
Sander, E. A. , Stylianopoulos, T. , Tranquillo, R. T. , and Barocas, V. H. , 2009, “ Image-Based Multiscale Modeling Predicts Tissue-Level and Network-Level Fiber Reorganization in Stretched Cell-Compacted Collagen Gels,” Proc. Natl. Acad. Sci., 106(42), pp. 17675–17680. [CrossRef]
Storm, C. , Pastore, J. J. , MacKintosh, F. C. , Lubensky, T. C. , and Janmey, P. A. , 2004, “Nonlinear Elasticity in Biological Gels,” preprint arXiv: cond-mat/0406016. https://arxiv.org/ftp/cond-mat/papers/0406/0406016.pdf
Stylianopoulos, T. , and Barocas, V. H. , 2007, “ Multiscale, Structure-Based Modeling for the Elastic Mechanical Behavior of Arterial Walls,” ASME J. Biomech. Eng., 129(4), pp. 611–618. [CrossRef]
Zhang, L. , Lake, S. P. , Lai, V. K. , Picu, C. R. , Barocas, V. H. , and Shephard, M. S. , 2012, “ A Coupled Fiber-Matrix Model Demonstrates Highly Inhomogeneous Microstructural Interactions in Soft Tissues Under Tensile Load,” ASME J. Biomech. Eng., 135(1), p. 011008. [CrossRef]
Zhang, L. , Lake, S. , Barocas, V. , Shephard, M. , and Picu, R. , 2013, “ Cross-Linked Fiber Network Embedded in an Elastic Matrix,” Soft Matter, 9(28), pp. 6398–6405. [CrossRef] [PubMed]
Ban, E. , Barocas, V. H. , Shephard, M. S. , and Picu, R. C. , 2016, “ Softening in Random Networks of Non-Identical Beams,” J. Mech. Phys. Solids, 87, pp. 38–50. [CrossRef] [PubMed]
Borodulina, S. , Motamedian, H. R. , and Kulachenko, A. , 2016, “ Effect of Fiber and Bond Strength Variations on the Tensile Stiffness and Strength of Fiber Networks,” Int. J. Solids Struct., epub.
Kallmes, O. , and Corte, H. , 1960, “ The Structure of Paper—I: The Statistical Geometry of an Ideal Two Dimensional Fiber Network,” Tappi J., 43(9), pp. 737–752.
Shahsavari, A. , and Picu, R. , 2013, “ Size Effect on Mechanical Behavior of Random Fiber Networks,” Int. J. Solids Struct., 50(20), pp. 3332–3338. [CrossRef]
Heussinger, C. , and Frey, E. , 2007, “ Role of Architecture in the Elastic Response of Semiflexible Polymer and Fiber Networks,” Phys. Rev. E, 75(1), p. 011917. [CrossRef]
Lake, S. P. , Hadi, M. F. , Lai, V. K. , and Barocas, V. H. , 2012, “ Mechanics of a Fiber Network Within a Non-Fibrillar Matrix: Model and Comparison With Collagen-Agarose Co-Gels,” Ann. Biomed. Eng., 40(10), pp. 2111–2121. [CrossRef] [PubMed]
Nachtrab, S. , Kapfer, S. C. , Arns, C. H. , Madadi, M. , Mecke, K. , Schröder , and Turk, G. E. , 2011, “ Morphology and Linear‐Elastic Moduli of Random Network Solids,” Adv. Mater., 23(22–23), pp. 2633–2637. [CrossRef] [PubMed]
Pritchard, R. H. , Huang, Y. Y. S. , and Terentjev, E. M. , 2014, “ Mechanics of Biological Networks: From the Cell Cytoskeleton to Connective Tissue,” Soft Matter, 10(12), pp. 1864–1884. [CrossRef] [PubMed]
Provenzano, P. P. , and Vanderby, R. , 2006, “ Collagen Fibril Morphology and Organization: Implications for Force Transmission in Ligament and Tendon,” Matrix Biol., 25(2), pp. 71–84. [CrossRef] [PubMed]
Motte, S. , and Kaufman, L. J. , 2013, “ Strain Stiffening in Collagen—I: Networks,” Biopolymers, 99(1), pp. 35–46. [CrossRef] [PubMed]
Yang, Y-L. , Leone, L. M. , and Kaufman, L. J. , 2009, “ Elastic Moduli of Collagen Gels Can Be Predicted From Two-Dimensional Confocal Microscopy,” Biophys. J., 97(7), pp. 2051–2060. [CrossRef] [PubMed]
Lai, V. K. , Frey, C. R. , Kerandi, A. M. , Lake, S. P. , Tranquillo, R. T. , and Barocas, V. H. , 2012, “ Microstructural and Mechanical Differences Between Digested Collagen–Fibrin Co-Gels and Pure Collagen and Fibrin Gels,” Acta Biomater., 8(11), pp. 4031–4042. [CrossRef] [PubMed]
Lindström, S. B. , Vader, D. A. , Kulachenko, A. , and Weitz, D. A. , 2010, “ Biopolymer Network Geometries: Characterization, Regeneration, and Elastic Properties,” Phys. Rev. E, 82(5), p. 051905. [CrossRef]
Lindström, S. B. , Kulachenko, A. , Jawerth, L. M. , and Vader, D. A. , 2013, “ Finite-Strain, Finite-Size Mechanics of Rigidly Cross-Linked Biopolymer Networks,” Soft Matter, 9(30), pp. 7302–7313. [CrossRef]
D'Amore, A. , Stella, J. A. , Wagner, W. R. , and Sacks, M. S. , 2010, “ Characterization of the Complete Fiber Network Topology of Planar Fibrous Tissues and Scaffolds,” Biomaterials, 31(20), pp. 5345–5354. [CrossRef] [PubMed]
Koh, C. , and Oyen, M. , 2012, “ Branching Toughens Fibrous Networks,” J. Mech. Behav. Biomed. Mater., 12, pp. 74–82. [CrossRef] [PubMed]
Yang, L. , van der Werf, K. O. , Fitié, C. F. , Bennink, M. L. , Dijkstra, P. J. , and Feijen, J. , 2008, “ Mechanical Properties of Native and Cross-Linked Type I Collagen Fibrils,” Biophys. J., 94(6), pp. 2204–2211. [CrossRef] [PubMed]
Arevalo, R. C. , Urbach, J. S. , and Blair, D. L. , 2010, “ Size-Dependent Rheology of Type-I Collagen Networks,” Biophys. J., 99(8), pp. L65–L67. [CrossRef] [PubMed]
Shahsavari, A. , and Picu, R. , 2012, “ Model Selection for Athermal Cross-Linked Fiber Networks,” Phys. Rev. E, 86(1), p. 011923. [CrossRef]
Dutov, P. , Antipova, O. , Varma, S. , Orgel, J. P. , and Schieber, J. D. , 2016, “ Measurement of Elastic Modulus of Collagen Type I Single Fiber,” PloS One, 11(1), p. e0145711. [CrossRef] [PubMed]
Stauffer, D. , and Aharony, A. , 1994, Introduction to Percolation Theory, CRC Press, Boca Raton, FL.
Sharma, A. , Licup, A. , Rens, R. , Vahabi, M. , Jansen, K. , Koenderink, G. , and MacKintosh, F. , 2016, “ Strain-Driven Criticality Underlies Nonlinear Mechanics of Fibrous Networks,” Phys. Rev. E, 94(4), p. 042407. [CrossRef] [PubMed]
Wyart, M. , Liang, H. , Kabla, A. , and Mahadevan, L. , 2008, “ Elasticity of Floppy and Stiff Random Networks,” Phys. Rev. Lett., 101(21), p. 215501. [CrossRef] [PubMed]
Licup, A. J. , Münster, S. , Sharma, A. , Sheinman, M. , Jawerth, L. M. , Fabry, B. , Weitz, D. A. , and MacKintosh, F. C. , 2015, “ Stress Controls the Mechanics of Collagen Networks,” Proc. Natl. Acad. Sci., 112(31), pp. 9573–9578. [CrossRef]
Rezakhaniha, R. , Agianniotis, A. , Schrauwen, J. T. , Griffa, A. , Sage, D. , Bouten, C. V. , van de Vosse, F. , 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), pp. 461–473. [CrossRef] [PubMed]
Sherman, V. R. , Yang, W. , and Meyers, M. A. , 2015, “ The Materials Science of Collagen,” J. Mech. Behav. Biomed. Mater., 52, pp. 22–50. [CrossRef] [PubMed]
Depalle, B. , Qin, Z. , Shefelbine, S. J. , and Buehler, M. J. , 2015, “ Influence of Cross-Link Structure, Density and Mechanical Properties in the Mesoscale Deformation Mechanisms of Collagen Fibrils,” J. Mech. Behav. Biomed. Mater., 52, pp. 1–13. [CrossRef] [PubMed]
Eppell, S. , Smith, B. , Kahn, H. , and Ballarini, R. , 2006, “ Nano Measurements With Micro-Devices: Mechanical Properties of Hydrated Collagen Fibrils,” J. R. Soc. Interface, 3(6), pp. 117–121. [CrossRef] [PubMed]
Svensson, R. B. , Hassenkam, T. , Hansen, P. , and Magnusson, S. P. , 2010, “ Viscoelastic Behavior of Discrete Human Collagen Fibrils,” J. Mech. Behav. Biomed. Mater., 3(1), pp. 112–115. [CrossRef] [PubMed]
Miyazaki, H. , and Hayashi, K. , 1999, “ Tensile Tests of Collagen Fibers Obtained From the Rabbit Patellar Tendon,” Biomed. Microdev., 2(2), pp. 151–157. [CrossRef]
Gentleman, E. , Lay, A. N. , Dickerson, D. A. , Nauman, E. A. , Livesay, G. A. , and Dee, K. C. , 2003, “ Mechanical Characterization of Collagen Fibers and Scaffolds for Tissue Engineering,” Biomaterials, 24(21), pp. 3805–3813. [CrossRef] [PubMed]
Lake, S. P. , and Barocas, V. H. , 2011, “ Mechanical and Structural Contribution of Non-Fibrillar Matrix in Uniaxial Tension: A Collagen-Agarose Co-Gel Model,” Ann. Biomed. Eng., 39(7), pp. 1891–1903. [CrossRef] [PubMed]
Žagar, G. , Onck, P. R. , and van der Giessen, E. , 2015, “ Two Fundamental Mechanisms Govern the Stiffening of Cross-Linked Networks,” Biophys. J., 108(6), pp. 1470–1479. [CrossRef] [PubMed]
Reese, S. P. , Maas, S. A. , and Weiss, J. A. , 2010, “ Micromechanical Models of Helical Superstructures in Ligament and Tendon Fibers Predict Large Poisson's Ratios,” J. Biomech., 43(7), pp. 1394–1400. [CrossRef] [PubMed]
Rigby, B. J. , Hirai, N. , Spikes, J. D. , and Eyring, H. , 1959, “ The Mechanical Properties of Rat Tail Tendon,” J. General Physiol., 43(2), pp. 265–283. [CrossRef]
Diamant, J. , Keller, A. , Baer, E. , Litt, M. , and Arridge, R. , 1972, “ Collagen; Ultrastructure and Its Relation to Mechanical Properties as a Function of Ageing,” Proc. R. Soc. London B: Biol. Sci., 180(1060), pp. 293–315. [CrossRef]
Ban, E. , Barocas, V. H. , Shephard, M. S. , and Picu, C. R. , 2016, “ Effect of Fiber Crimp on the Elasticity of Random Fiber Networks With and Without Embedding Matrices,” ASME J. Appl. Mech., 83(4), p. 041008. [CrossRef]
Clark, J. M. , 1991, “ Variation of Collagen Fiber Alignment in a Joint Surface: A Scanning Electron Microscope Study of the Tibial plateau in Dog, Rabbit, and Man,” J. Orthop. Res., 9(2), pp. 246–257. [CrossRef] [PubMed]
Kääb, M. , Ap Gwynn, I. , and Nötzli, H. , 1998, “ Collagen Fibre Arrangement in the Tibial Plateau Articular Cartilage of Man and Other Mammalian Species,” J. Anat., 193(1), pp. 23–34. [CrossRef] [PubMed]
Wu, J. P. , Kirk, T. B. , and Zheng, M. H. , 2008, “ Study of the Collagen Structure in the Superficial Zone and Physiological State of Articular Cartilage Using a 3D Confocal Imaging Technique,” J. Orthop. Surg. Res., 3(1), p. 29. [CrossRef] [PubMed]
Elliott, D. M. , Narmoneva, D. A. , and Setton, L. A. , 2002, “ Direct Measurement of the Poisson's Ratio of Human Patella Cartilage in Tension,” ASME. J. Biomech. Eng., 124(2), pp. 223–228. [CrossRef]
Chang, D. , Lottman, L. , Chen, A. , Schinagl, R. , Albrecht, D. , Pedowitz, R. , Brossman, J. , Frank, L. , and Sah, R. , 1999, “ The Depth-Dependent, Multi-Axial Properties of Aged Human Patellar Cartilage in Tension,” Trans. Annu. Meet. Orthop. Res. Soc., 24, p. 644. https://www.ors.org/Transactions/45/0644.PDF
Huang, C.-Y. , Stankiewicz, A. , Ateshian, G. A. , and Mow, V. C. , 2005, “ Anisotropy, Inhomogeneity, and Tension–Compression Nonlinearity of Human Glenohumeral Cartilage in Finite Deformation,” J. Biomech., 38(4), pp. 799–809. [CrossRef] [PubMed]
Puxkandl, R. , Zizak, I. , Paris, O. , Keckes, J. , Tesch, W. , Bernstorff, S. , Purslow, P. , and Fratzl, P. , 2002, “ Viscoelastic Properties of Collagen: Synchrotron Radiation Investigations and Structural Model,” Philos. Trans. R. Soc. London B, 357(1418), pp. 191–197. [CrossRef]
Kukreti, U. , and Belkoff, S. M. , 2000, “ Collagen Fibril D-Period May Change as a Function of Strain and Location in Ligament,” J. Biomech., 33(12), pp. 1569–1574. [CrossRef] [PubMed]
Chen, H. , Zhao, X. , Berwick, Z. C. , Krieger, J. F. , Chambers, S. , and Kassab, G. S. , 2016, “ Microstructure and Mechanical Property of Glutaraldehyde-Treated Porcine Pulmonary Ligament,” ASME J. Biomech. Eng., 138(6), p. 061003. [CrossRef]
Mauri, A. , Perrini, M. , Ehret, A. E. , De Focatiis, D. S. , and Mazza, E. , 2015, “ Time-Dependent Mechanical Behavior of Human Amnion: Macroscopic and Microscopic Characterization,” Acta Biomater., 11, pp. 314–323. [CrossRef] [PubMed]


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Fig. 1

Realizations of models with (a) randomly oriented straight fibers, (b) randomly oriented tortuous fibers, and (c) preferentially oriented straight fibers

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Fig. 2

(a) Nominal stress–stretch curves for networks of density equivalent to collagen concentration of c = 1 mg/ml and c = 4 mg/ml. The three regimes discussed in text are indicated by vertical bars and red filled circles. (b) Data in (a) replotted as tangent stiffness versus stress. The red filled circles indicate the transition between the regimes indicated in (a). The figure includes experimental data for reconstructed collagen gels (orange circles) [68], human amnion (green squares) [9], and mouse skin dermis (blue triangles) [11].

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Fig. 4

Variation of the incremental Poisson's ratio with the stretch ratio for models with c = 1 and 4 mg/ml. The red filled squares mark the transition between the regimes indicated in Fig. 2(a). The figure includes experimental data for reconstructed collagen gels (orange circles) [68], human amnion (green squares) [9], and mouse skin (blue triangles) [11].

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Fig. 8

Variation of the incremental orientation index, dP2/dλ1, for models with collagen concentration c = 1 mg/ml (crosses) and c = 4 mg/ml (open circles), along with the energy partition for the same models and loading history. The energy stored in the bending and axial modes is shown. The contribution of the shear and torsional modes is smaller than 10% in all cases and is not shown. The transition from the bending-dominated state at small stretches to the axial dominance observed at large stretches (crossing of curves in the lower figure) corresponds to the peak in the incremental P2 (shown by the red arrow).

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Fig. 9

Variation of the incremental orientation index, dP2/dλ1, for models with collagen concentration c = 4 mg/ml and various levels of pre-alignment, along with the energy partition for the same models and loading history. Only the energy stored in thebending and axial modes is shown. The transitions from the bending-dominated state at small stretches to the axial dominance observed at large stretches (crossings in the lower figure) correspond to the peaks in the incremental P2, as indicated by the red arrows.

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Fig. 10

Comparison of the affine model prediction (continuous red line), calculated (c = 1 mg/ml (filled diamonds) and c = 4 mg/ml (open circles)), and experimental (blue triangles, [11]) variation of the orientation index with the ratio λ2/λ1

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Fig. 3

Tangent stiffness versus stress for models with tortuosity parameter τ = 0 (reproduced from Fig. 2(b)), τ = 1.15 and τ = 1.3. The vertical axis is normalized by the small strain modulus of regime 1, E0.

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Fig. 5

Variation of the 3D orientation index P2 with the stretch ratio for networks with c = 1 and 4 mg/ml. The red filled squares indicate the transition between regimes indicated in Fig. 2(a). The figure includes the 2D orientation index for mouse skin (blue triangles) [11].

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Fig. 6

Relationship between the incremental Poisson's ratio and the orientation index for models with c = 4 mg/ml and various levels of pre-alignment in the initial, unloaded configuration (P20). The red dots on the P20=0 curve indicate the bounds of regime 2. The dotted blue line shows the variation of the small strain Poisson's ratio with the degree of pre-alignment. The blue triangles represent the measured in-plane incremental Poisson's ratio for a pre-aligned sample of mouse skin [11]. The curves corresponding to the four values of P20 are shown with both markers and lines to emphasize the trends.

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Fig. 7

Relationship between the incremental Poisson's ratio and the orientation index for models with c = 1 mg/ml and various levels of tortuosity. The orientation index is evaluated based on the end-to-end vectors of the crimped fibers.

Grahic Jump Location
Fig. 11

(a) Nominal stress–stretch curves for networks subjected to uniaxial (from Fig. 2(a)) and biaxial loading. (b) Data in (a) replotted as tangent stiffness versus stress.

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Fig. 12

Data in Fig. 11(a) replotted as σ11t3, where t3 is the model thickness in the direction perpendicular to the plane of biaxial tension, x3. The collapse of the two curves indicates that the difference seen in Fig. 11(a) is due to the more pronounced Poisson's contraction in the x3 direction in the biaxial loading case.

Grahic Jump Location
Fig. 13

Relationship between the incremental Poisson's ratio and the orientation index for models with c = 4 mg/ml subjected to uniaxial (reproduced from Fig. 6) and biaxial loading. In the uniaxial case, P2 is computed in 3D. In the biaxial case, both νi and P2 are computed as averages over the x1 − x3 and x2 − x3 faces of the model. Biaxial stretch is applied in the x1 and x2 directions.

Grahic Jump Location
Fig. 14

Dependence of the 2D and 3D measures of fiber orientation, P22D and P23D, on parameter σ representing the broadness of the distribution function of fiber orientation relative to the stretch direction. The inset shows the ratio P22D/P23D function of P23D, i.e., the error made by using the 2D version of the orientation index. It is assumed that deformation is uniaxial and no anisotropy develops in the plane perpendicular to the loading direction during deformation, i.e., the x1 − x2 and x1 − x3 planes are equivalent.



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