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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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Figures

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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. 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. 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. 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.

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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. 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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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. 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).

Grahic Jump Location
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.

Grahic Jump Location
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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