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Hi Leo,
For a continuous fiber distribution these relations are not generally valid. In a continuous fiber distribution the confined (H-A) and unconfined (E) compression moduli depend on the properties of the ground matrix and the fiber modulus. H-A represents only the properties of the ground matrix whereas E combines ground matrix and fiber. To better understand this concept please see this paper, particularly section 2.5.
For a conewise-linear-elastic (CLE) solid matrix (Curnier et al., Journal of Elasticity, v 37, n 1, p 1-38, 1994-1995), the exact relations between H-A, lambda2 and compressive values of E and v are given in the Soltz paper, see eq. 24. A CLE model has four elastic constants (H-A, H+A, lambda2 and mu), whereas a solid mixture consisting of an isotropic neo-Hookean ground matrix (with properties E and v) and three identical and mutually orthogonal fiber bundles (with a tensile modulus related to ksi) has only three elastic properties (E, v and ksi). Therefore, in a strict sense, it is not possible to establish an exact equivalence between the Soltz model and the neo-Hookean + 3 fibers solid mixture. The relations proposed above represent a best approximation.
Basically, the reason that an exact equivalence cannot be found is because the neo-Hookean material has isotropic symmetry whereas the Soltz embodiment of the CLE theory uses cubic symmetry (a special case of the more general orthotropic symmetry CLE model proposed by Curnier et al.). If one would like to reproduce the Soltz model more accurately in FEBio, I suppose that it would be better to use an orthotropic elastic solid rather than the neo-Hookean solid I had originally suggested in this thread.
Please keep in mind that I myself no longer advocate using only three mutually orthogonal fiber bundles for modeling cartilage. In other words, even though Michael Soltz and I originally proposed to use the CLE model to describe cartilage, since then we have come to realize that a continuous fiber distribution provides even better agreement with observed experimental responses, as explained in this paper. Therefore I recommend using a spherical or ellipsoidal fiber distribution for the fibers, with an isotropic ground matrix, to model cartilage. These models are available in FEBio.
A solid mixture containing a neo-Hookean solid and a fiber-exp-pow fiber simply superposes the strain energy densities of these two types of materials. The tensile fiber modulus in the limit of approaching zero strain from the positive side is equal to 4*ksi when beta=2 and alpha=0. This can be deduced from the description of this material in the User's Manual.
Best,
Gerard
considering a continuous fiber distribution and a "Holmes-Mow" solid, are these relations always valid?:
The properties of a "neo-Hookean" solid are "E" (Young's modulus) and "v" (Poisson's ratio). They are related to H-A and lambda2 as follows:
H-A = (1-v)E/(1-2v)(1+v)
lambda2 = v E/(1-2v)(1+v)
The properties of a "neo-Hookean" solid are "E" (Young's modulus) and "v" (Poisson's ratio). They are related to H-A and lambda2 as follows:
H-A = (1-v)E/(1-2v)(1+v)
lambda2 = v E/(1-2v)(1+v)
For a conewise-linear-elastic (CLE) solid matrix (Curnier et al., Journal of Elasticity, v 37, n 1, p 1-38, 1994-1995), the exact relations between H-A, lambda2 and compressive values of E and v are given in the Soltz paper, see eq. 24. A CLE model has four elastic constants (H-A, H+A, lambda2 and mu), whereas a solid mixture consisting of an isotropic neo-Hookean ground matrix (with properties E and v) and three identical and mutually orthogonal fiber bundles (with a tensile modulus related to ksi) has only three elastic properties (E, v and ksi). Therefore, in a strict sense, it is not possible to establish an exact equivalence between the Soltz model and the neo-Hookean + 3 fibers solid mixture. The relations proposed above represent a best approximation.
Basically, the reason that an exact equivalence cannot be found is because the neo-Hookean material has isotropic symmetry whereas the Soltz embodiment of the CLE theory uses cubic symmetry (a special case of the more general orthotropic symmetry CLE model proposed by Curnier et al.). If one would like to reproduce the Soltz model more accurately in FEBio, I suppose that it would be better to use an orthotropic elastic solid rather than the neo-Hookean solid I had originally suggested in this thread.
Please keep in mind that I myself no longer advocate using only three mutually orthogonal fiber bundles for modeling cartilage. In other words, even though Michael Soltz and I originally proposed to use the CLE model to describe cartilage, since then we have come to realize that a continuous fiber distribution provides even better agreement with observed experimental responses, as explained in this paper. Therefore I recommend using a spherical or ellipsoidal fiber distribution for the fibers, with an isotropic ground matrix, to model cartilage. These models are available in FEBio.
The properties of all three "fiber-exp-pow" solids are "ksi", "alpha" and "beta". To reproduce the Krishnan paper, you need to select
alpha = 0
beta = 2
ksi = (H+A - H-A)/4
alpha = 0
beta = 2
ksi = (H+A - H-A)/4
Best,
Gerard
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