A Microstructurally Based Orthotropic Hyperelastic Constitutive Law

[+] Author and Article Information
J. E. Bischoff

Department of Engineering Science, University of Auckland, Auckland, New Zealande-mail: j.bischoff@Auckland.ac.nz

E. A. Arruda, K. Grosh

Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109-2125

J. Appl. Mech 69(5), 570-579 (Aug 16, 2002) (10 pages) doi:10.1115/1.1485754 History: Received January 15, 2001; Revised September 11, 2001; Online August 16, 2002
Copyright © 2002 by ASME
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The freely-jointed chain approximation of a macromolecule as a series of rigid links, with one end pinned at the origin O and the other end located by the chain vector R in its reference configuration and by the chain vector r in its deformed configuration
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The change in strain energy accompanying deformation of a single macromolecular chain from its rms length. Inset: a close-up of the curves near their respective rms lengths, showing that a decrease in chain length below the reference rms length results in a decrease in strain energy.
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Nominal stress (in the X-direction) versus stretch plots extracted from finite element simulations of constrained uniaxial extension with constant initial orientation of a and sinusoidally varying initial orientation of a as shown in Figs. 10(a) and 11(a).
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Simulation of constrained uniaxial simulation in which the initial orientation of a varies sinusoidally with X (Fig. 11(a)). Parameters used in the simulation are the same as for the simulation in Fig. 10. In Figs. 11(bd) the contour definitions and corresponding locations on the global stress-stretch curve (Fig. 12) are the same as for the simulation in Fig. 10.
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Simulation of constrained uniaxial simulation in which a is initially fixed at a 30-deg orientation throughout the domain (Fig. 10(a)). Parameters used in the simulation are n=2⋅1024/m3,N=1.1,a=1.4,b=1.0,c=1.2, and B=1 MPa. Figures 10(bd), corresponding to the global stress-stretch states marked (b)–(d), respectively, in Fig. 12, show deformed meshes with contours of the stress σ11 (kPa). Contour lines are shown in increments of 3 kPa, 10 kPa, and 15 kPa for Figs. 10(b), 10(c), and 10(d), respectively.
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Finite element simulation (using a single linear brick element) and numerical simulation of load-controlled biaxial tension. Parameters used for the simulations are n=2⋅1024/m3,N=2.5,a=1.8,b=2,c=1.66, and B=1 MPa. The ratio of the applied loads was fixed at Fr=F2/F1=5. Figure 9(a) shows the load-stretch responses in the directions of the applied loads and Fig. 9(b) shows the variation of the transverse stretch λ3 as a function of the stretch λ1. The inset in Fig. 9 shows the constitutive responses closer to zero deformation.
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Finite element simulations (using a single linear brick element) and numerical simulations of uniaxial tension. Parameters used for the simulations are n=2⋅1024/m3,N=3,a=2,b=2.5,c=1.32, and B=1 MPa. Figure 8(a) shows the stress-stretch response in the direction of the applied load and Fig. 8(b) shows the variations of the transverse stretches λ2 and λ3 as functions of the applied stretch λ1.
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Equibiaxial tension data from Billiar and Sacks 28 and corresponding fits using the orthotropic model. Data are plotted as symbols and represent the constitutive response for fresh and glutaraldehyde-fixed aortic valve cusp samples in the two material directions in which loads were applied. Fits are plotted as lines and were generated using the following parameters: for fresh tissue data, n=6⋅1017/m3,N=1.96,B=100 kPa,a=2.05,b=1.7, and c=0.865; for fixed tissue data, n=7⋅1017/m3,N=1.48,B=500 kPa,a=1.85,b=1.35, and c=0.822.
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Simulations of load-controlled biaxial tension for various values of the load ratio Fr. Other parameters in the simulations were fixed: n=8⋅1024/m3,N=2,B=40 MPa,a=1.2,b=1.5, and c=2.076. Figure 6(a) shows the T11 versus λ1 response and Fig. 6(b) shows the T22 versus λ2 response for a given value of Fr.
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Data from uniaxial tests on rabbit skin from Lanir and Fung 1 and the corresponding fits using the orthotropic model. Data are represented by symbols and the fits by solid lines. The parameters used to fit the data are n=3.75⋅1022/m3,N=1.25,B=50 kPa,a=1.37,b=1.015, and c=1.447.
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Response of the orthotropic eight-chain model to uniaxial deformation in each of three directions (X1,X2,X3) for two sets of aspect ratios: (1) a=2,b=3,c=4, and (2) a=1.8,b=3.124,c=4. The material axes are mutually orthogonal and aligned with the coordinate axes. All simulations were performed using parameters n=8⋅1024/m3,N=7.25, and B=1 MPa.
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Eight-chain, three-dimensional orthotropic unit cell. The eight curved lines in the unit cell represent the constituent macromolecules and the straight lines represent the boundaries of the unit cell. The cell has dimensions a×b×c along the material axes a, b, c, respectively, oriented with respect to the reference coordinate system Xi.




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