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Research Papers

Elastic Characterization of Transversely Isotropic Soft Materials by Dynamic Shear and Asymmetric Indentation

[+] Author and Article Information
R. Namani, Y. Feng, R. J. Okamoto, G. M. Genin

Department of Mechanical Engineering and Materials Science,  Washington University in St. Louis, St. Louis, MO 63130

N. Jesuraj, S. E. Sakiyama-Elbert

Department of Biomedical Engineering,  Washington University in St. Louis, St. Louis, MO 63130

P. V. Bayly1

Department of Mechanical Engineering and Materials Science, Washington University in St. Louis, St. Louis, MO 63130; Department of Biomedical Engineering,  Washington University in St. Louis, St. Louis, MO 63130baylyp@seas.wustl.edu

1

Corresponding author.

J Biomech Eng 134(6), 061004 (Jun 08, 2012) (11 pages) doi:10.1115/1.4006848 History: Received February 14, 2012; Revised May 03, 2012; Posted May 18, 2012; Published June 08, 2012; Online June 08, 2012

The mechanical characterization of soft anisotropic materials is a fundamental challenge because of difficulties in applying mechanical loads to soft matter and the need to combine information from multiple tests. A method to characterize the linear elastic properties of transversely isotropic soft materials is proposed, based on the combination of dynamic shear testing (DST) and asymmetric indentation. The procedure was demonstrated by characterizing a nearly incompressible transversely isotropic soft material. A soft gel with controlled anisotropy was obtained by polymerizing a mixture of fibrinogen and thrombin solutions in a high field magnet (B = 11.7 T); fibrils in the resulting gel were predominantly aligned parallel to the magnetic field. Aligned fibrin gels were subject to dynamic (20–40 Hz) shear deformation in two orthogonal directions. The shear storage modulus was 1.08 ± 0. 42 kPa (mean ± std. dev.) for shear in a plane parallel to the dominant fiber direction, and 0.58 ± 0.21 kPa for shear in the plane of isotropy. Gels were indented by a rectangular tip of a large aspect ratio, aligned either parallel or perpendicular to the normal to the plane of transverse isotropy. Aligned fibrin gels appeared stiffer when indented with the long axis of a rectangular tip perpendicular to the dominant fiber direction. Three-dimensional numerical simulations of asymmetric indentation were used to determine the relationship between direction-dependent differences in indentation stiffness and material parameters. This approach enables the estimation of a complete set of parameters for an incompressible, transversely isotropic, linear elastic material.

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Copyright © 2012 by American Society of Mechanical Engineers
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Figures

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Figure 3

Experiment setup for asymmetric indentation of aligned fibrin gels. (a) Schematic diagram of disk-shaped gel sample (diameter: 18 mm; thickness: 3.0 mm) and an indenter with a rounded rectangular tip 19.1 mm in length and 1.0 mm to 1.6 mm in width. The gel is submerged in a PBS solution and rests on the bottom of a glass dish. (b) Top view of indentation with fibers aligned perpendicular or parallel to the long axis of the indenter. Lines indicate the direction of the magnetic alignment. (c) The indentation protocol consisting of a series of imposed displacements during which force and displacement are measured. A preload and hold (force-relaxation) step is followed by the actual indentation step which was used for data analysis. A third displacement step is performed to observe the relaxation behavior of the fibrin gel.

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Figure 2

(a) Schematic diagram of the dynamic shear testing (DST). The sample is deformed in simple shear by the harmonic displacement of the base, while the force on the stationary upper surface is measured. (b) Fibrin gel orientation for the DST. The vertical and horizontal lines indicate the dominant fiber directions of the aligned gel. When the imposed displacement is parallel to the dominant fiber axis, shear is imposed in a plane normal to the plane of isotropy. When displacement is perpendicular to the dominant fiber axis, the plane of isotropy undergoes shear deformation.

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Figure 1

(a) An element of a transversely isotropic material. The plane of isotropy is the 2-3 plane, and the vector normal to the plane of isotropy a is aligned with the e1 unit vector. (b) A 3-D finite element (FE) model of indentation of an elastic material with a rectangular tip. Only one quarter of the circular gel is modeled. (c) The material coordinate system is aligned with the global model (eX , eY , eZ ) coordinate system so that the predominant fiber direction is perpendicular to the long axis of the indenter. (d) The material coordinate system is rotated 90 deg about eZ so that predominant fiber direction is parallel to the long axis of the indenter.

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Figure 8

(a) The stiffness of fibrin gel samples is the slope of the indentation force-displacement loading curve (Figs. 7 and 7). The perpendicular stiffness k⊥exp and the parallel stiffness k‖exp were significantly different for the aligned gels (n = 8, paired Student’s t-test, p = 0.013). The indentation stiffness of control gels was slightly but significantly higher for the first test kAexp than the second test kBexp (n = 6, paired Student’s t-test, p = 0.04). (b) Normalized stiffness during the loading ramp and at equilibrium (after relaxation) in the aligned and control gels. The normalized stiffness during loading was significantly different from the normalized stiffness at equilibrium for the aligned gels (n = 8, paired Student’s t-test, p = 0.04), but not for the control gels.

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Figure 4

Force-displacement curves illustrating the effects of transversely isotropic material parameter ratios μ1 /μ2 and β/μ2 : (a) and (b) β/μ2  = 0, 10, 20, or 40; μ1  = μ2  = 0.5 kPa; (c) and (d) μ1 /μ2  = 1, 1.5, 2, or 2.5, μ2  = 0.5 kPa and β = 0. The left column ((a) and (c)) shows results obtained with the dominant fiber direction parallel to the long axis of the indenter tip (see Fig. 1) and the right column ((b) and (d)) shows the result obtained with the dominant fiber axis perpendicular to the long axis of the indenter tip (see Fig. 1). The force-displacement curves for each set of parameters are approximately linear and the slopes of the force-displacement curves represent indentation stiffness with the dominant fiber axis parallel (k‖) and perpendicular (k⊥ ) to the long axis of the indenter.

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Figure 5

Normal stress components in the global coordinate system (eX , eY , eZ ) for asymmetric indentation to a depth of 0.2 mm. (a), (d), (g) Stresses for isotropic material (μ, E, ν) = (0.5, 1.498, 0.4975). (b), (e), (h) Stresses for a transversely isotropic elastic material with parameters (μ1 , μ2 , β) = (0.5, 1.0, 5.0) or, equivalently (E1 , E2 , ν12 , ν2 ) = (8.4975, 1.881, 0.4975, 0.881), indented with the fibers parallel to the long axis of the indenter (eY ). (c), (f), (i) Stresses for the same transversely isotropic material indented with the fibers perpendicular to the long axis of the indenter. All moduli are stated in kPa; Poisson’s ratios are dimensionless.

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Figure 6

Storage (elastic) and loss (viscous) components of the complex shear modulus μ*=μ′+iμ′′ measured using the DST for (a) a representative control gel tested in one orientation (μA ) and then rotated about the vertical axis by 90 deg (μB ), and (b) a representative aligned gel tested with shear loading applied in a plane parallel to the dominant fiber axis (μ1 ), or in a plane normal to the dominant fiber axis (μ2 ). Data are shown over the frequency range of 20–40 Hz. Samples were tested at 0%, and 5% precompression; data is shown only for 5% precompression. Comparison of the components of the complex shear modulus of the (c) control gels (n = 5), and (d) aligned gels (n = 13) samples, estimated by the DST over the range of 20–40 Hz. Differences between the storage moduli (μ1′ and μ2′) and between the loss moduli (μ1′′ and μ2′′) for the aligned gels were statistically significant (p values as shown; Student’s t-test), indicating material anisotropy. Error bars show one standard deviation.

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Figure 7

(a) and (b) Force-displacement measurements during indentation of (a) control (nonaligned) fibrin gels (open circles, first test; closed squares, second test) and (b) aligned fibrin gels (open circles, indenter perpendicular to dominant fiber direction; closed squares, indenter aligned with dominant fiber direction). The indentation loading ramp duration was 0.33 s. (c) and (d) Force relaxation for 240 s after indentation of control fibrin gels and aligned fibrin gels. Relaxation time is plotted on a logarithmic scale. Both control and aligned fibrin gels lose more than 90% of their peak indentation force after 240 s. Inset in panel (d) shows force relaxation for aligned gels on a linear time scale.

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