A projection method to extract biological membrane models from 3D material models.

Area of Science:

• Computational mechanics
• Solid mechanics
• Material modeling

Background:

• Deriving lower-dimensional models from higher-dimensional ones is crucial for computational efficiency.
• Membrane theories require careful consideration of kinematic and constitutive assumptions.
• Anisotropic material models, such as the Holzapfel-Gasser-Ogden model, present unique challenges in reduced-order modeling.

Purpose of the Study:

• To present a novel projection method for deriving two-dimensional (2D) membrane models from three-dimensional (3D) material models.
• To simplify the formulation by eliminating the incompressibility constraint through the assumption of zero transverse stress.
• To validate the proposed method using a specific example of an anisotropic Holzapfel-Gasser-Ogden model.

Main Methods:

• A projection procedure based on general membrane theory for a 2D manifold.
• Elimination of the Lagrange multiplier for incompressibility by assuming zero transverse stress.
• Discretization and linearization of the nonlinear membrane model using the finite element method (FEM).

Main Results:

• The proposed projection method successfully derives membrane models from 3D material models.
• Numerical examples using quadratic Lagrange and NURBS finite elements demonstrate good agreement.
• The results align well with analytical solutions and full 3D FEM computations.

Conclusions:

• The developed projection method provides an efficient and accurate way to model membranes, particularly those made of anisotropic materials.
• The method simplifies complex 3D problems into manageable 2D membrane formulations.
• This approach is validated for the anisotropic Holzapfel-Gasser-Ogden model and shows promise for broader applications in computational mechanics.