The problem of the wrinkling of plane isotropic membranes characterized by a Fung type constitutive model in biaxial tension has been formulated and solved within the framework of finite strain hyperelasticity. The formulation follows the approach of Pipkin [ Pipkin, A.C., 1986, IMA J. Appl. Math., 36, pp. 85–99 ; 1994, ibid., 52, pp. 297–308 ], and the out of plane geometric nonlinearities are treated as constitutive nonlinearities through a modification of the elastic potential. The wrinkling criteria are based on the natural contraction of a membrane in simple tension. Both the natural contraction and the modified elastic potential are defined in closed form. The model has been implemented in a finite element code and the numerical solution validated using study cases with analytical solution. Applications are presented that simulate the response of stretched membranes, where distinct regions of behavior (taut, wrinkled, and slack or inactive) develop during loading, and a simple procedure of reconstructive surgery, characterized by the excision of a circular portion of the skin and the suture of the wound edges, where the wrinkling of the skin causes the extrusion of the edges (dog-ear formation).

The mechanical behavior of the membrane of the red blood cell is governed by two primary microstructural features: the lipid bilayer and the underlying spectrin network. The lipid bilayer is analogous to a two-dimensional fluid in that it resists changes to its surface area, yet poses little resistance to shear. A skeletal network of spectrin molecules is cross-linked to the lipid bilayer and provides the shear stiffness of the membrane. Here, a general continuum level constitutive model of the large stretch behavior of the red blood cell membrane that directly incorporates the microstructure of the spectrin network is developed. The triangulated structure of the spectrin network is used to identify a representative volume element (RVE) for the model. A strain energy density function is constructed using the RVE together with various representations of the underlying molecular chain force-extension behaviors where the chain extensions are kinematically determined by the macroscopic deformation gradient. Expressions for the nonlinear finite deformation stress-strain behavior of the membrane are obtained by proper differentiation of the strain energy function. The stress-strain behaviors of the membrane when subjected to tensile and simple shear loading in different directions are obtained, demonstrating the capabilities of the proposed microstructurally detailed constitutive modeling approach in capturing the small to large strain nonlinear, anisotropic mechanical behavior. The sources of nonlinearity and evolving anisotropy are delineated by simultaneous monitoring of the evolution in microstructure including chain extensions, forces and orientations as a function of macroscopic stretch. The model captures the effect of pretension on the mechanical response where pretension is found to increase the initial modulus and decrease the limiting extensibility of the networked membrane.

Based on fiber reinforced continuum mechanics theory, an anisotropic hyperelastic constitutive model for the human annulus fibrosus is developed. A strain energy function representing the anisotropic elastic material behavior of the annulus fibrosus is additively decomposed into three parts nominally representing the energy contributions from the matrix, fiber and fiber-matrix shear interaction, respectively. Taking advantage of the laminated structure of the annulus fibrosus with one family of aligned fibers in each lamella, interlamellar fiber-fiber interaction is eliminated, which greatly simplifies the constitutive model. A simple geometric description for the shearing between the fiber and the matrix is developed and this quantity is used in the representation of the fiber-matrix shear interaction energy. Intralamellar fiber-fiber interaction is also encompassed by this interaction term. Experimental data from the literature are used to obtain the material parameters in the constitutive model and to provide model validation. Determination of the material parameters is greatly facilitated by the partition of the strain energy function into matrix, fiber and fiber-matrix shear interaction terms. A straightforward procedure for computation of the material parameters from simple experimental tests is proposed.