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Thread: Compressible versus Nearly Incompressible Materials: Beam Bending

  1. #1
    Join Date
    Dec 2007
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    906

    Default Compressible versus Nearly Incompressible Materials: Beam Bending

    An incompressible solid does not change in volume when loaded. Incompressibility is an idealization of the real behavior of materials. Enforcing incompressibility exactly within a finite element formulation is not commonly done. Instead, in FEBio, a nearly-incompressible material behavior is implemented using a strain energy formulation that uncouples the volumetric and distortional components of the deformation.

    For linear isotropic elasticity, Hooke's law provides two material constants, e.g., Young's modulus E and Poisson's ratio v, and it is well known that v=0.5 represents an incompressible material. In FEBio, there are many isotropic materials that reduce to Hooke's law in the limit of infinitesimal strains, including "isotropic elastic" (St-Venant-Kirchhoff), "neo-Hookean", "Mooney-Rivlin", and "Holmes-Mow" materials. To model nearly-incompressible behavior, it may be tempting to set Poisson's ratio to an elevated value, e.g., v=0.499 in the "isotropic elastic", "neo-Hookean" or "Holmes-Mow" materials. However, as v approaches 0.5, the response produced with these models will become inaccurate. For describing a material with v=0.499, it is necessary to use one of the nearly-incompressible materials in the FEBio library, such as the "Mooney-Rivlin" solid.

    To illustrate this issue, consider the problem of pure beam bending of a linear isotropic elastic solid, which accepts an exact solution in the theory of elasticity. A beam of length L=10, with a rectangular cross-section of width b=0.4 and height h=1, is subjected to bending moments M=-2 at -L/2 and M=2 at L/2. Young's modulus is set to E=1e6, while Poisson's ratio is either set to v=0.3 or v=0.499. With v=0.3, it is appropriate to use one of the compressible material models, e.g., the "neo-Hookean" material. In the finite element analysis, only a quarter model is employed by accounting for symmetry. The beam length is along X, the height is along Y and the depth (cross-section width) is along Z. The moment is prescribed about Z.

    The theoretical solution for the axial normal strain is EXX = M*Y/(E*I), and that for the transverse normal strains is EYY = EZZ = -v*M*Y/(E*I), where I=b*h^3/12 is the second area moment of the cross-section about the Z-axis. The figure below confirms the agreement between the FEBio solution and the theoretical solution.
    NHp3.png
    However, if this analysis is repeated with a "neo-Hookean" solid using v=0.499, the FEBio solution is no longer accurate, as shown in the next figure.
    NHp499.png
    To remediate this problem, it is necessary to use a "Mooney-Rivlin" solid, whose material constants are c1, c2 and k. Here, it is important to figure out the equivalence between these three material constants and the more familiar parameters E and v. First, recall that alternative material constants for linear isotropic elastic solids are the Lamé constants lambda and mu (where mu is the shear modulus). These parameters are related according to E = 2*mu*(1+v) and lambda = 2*mu*v/(1-2*v).

    In the Mooney-Rivlin material, mu = 2*(c1+c2) and lambda = k - 4*(c1+c2)/3. After some manipulation, it can be shown that E=1e6 and v=0.499 is equivalent to c1+c2=166778 and k=1.66667e8. The analysis may be performed by setting c2=0, so that c1=166778. Results shown below demonstrate that agreement is now achieved between FEBio and the theoretical solution.
    MRp499.png
    In summary, near-incompressibility cannot be enforced simply by adjusting Poisson's ratio. It is necessary to employ one of the uncoupled formulations available in FEBio, such as the Mooney-Rivlin solid, and suitably adjust the bulk modulus k. If agreement is sought with theoretical solutions that are formulated in terms of Poisson's ratio v, it is necessary to find the equivalencies between the uncoupled material parameters and v, as shown above.

    Gerard

  2. #2
    Join Date
    Jul 2019
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    8

    Default

    Hi,

    Thanks for the nice description of how we can obtain the required material parameters for the Mooney-Rivlin model. I have a question about unit consistency while deriving these parameters. I am using the E with MPa unit, should I use it the same to derive the lambda, mu, c1, and c2 values? Also, in most of the literature, I see C01, C10, and C11 as the constant of the Mooney-Rivlin material. Is there any relation between these parameters and c1 and c2 used in FEBio formulations?

    Best

  3. #3
    Join Date
    Dec 2007
    Posts
    906

    Default

    Hi,

    should I use it the same to derive the lambda, mu, c1, and c2 values?
    Yes, all moduli have the same units.
    Is there any relation between these parameters and c1 and c2 used in FEBio formulations?
    Using the standard notation found on Wikipedia, the correspondence is C10 = c1, C01 = c2, C11 = 0.

    Best,

    Gerard

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