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TECHNICAL PAPERS

# Constitutive Modeling of the Finite Deformation Behavior of Membranes Possessing a Triangulated Network Microstructure

[+] Author and Article Information
M. Arslan, M. C. Boyce

Department of Mechanical Engineering, Institute for Soldier Nanotechnologies, Massachusetts Institute of Technology, Cambridge, MA

Pretension is also present in analogous three-dimensional formulations of classical statistical mechanics of rubber elasticity (Treloar (14), James and Guth (17)) where the initial network chain tension is balanced by internal pressure carried, for example, by intermolecular van der Waal interactions (17). The spectrin network pretension will be balanced by a few possible sources including, for example, lipid bilayer stress, cytosol interactions, interactions with other protein molecules.

An alternative, equally general formulation has been provided in the supplemental materials of the recent parallel work of Li et al. (13), where the Worm-like chain model is used for the constituents and the network behavior is obtained using the virial stress theorem which is commonly used in atomistic and molecular level simulations to obtain stress (see, for example, Bergstrom and Boyce (19)).

These uniaxial tension stress-stretch results were also reported in Arslan and Boyce (20).

We note that Arruda and Boyce (15-16) and Boyce (24) have shown that higher order $I1$-based models are phenomenological equivalents to non-Gaussian statistical models, in particular, to the eight-chain model of Arruda and Boyce (15).

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Corresponding author.

J. Appl. Mech 73(4), 536-543 (Sep 30, 2005) (8 pages) doi:10.1115/1.2130360 History: Received May 17, 2005; Revised September 30, 2005

## Abstract

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.

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## Figures

Figure 1

Electron micrograph of a spread human erythrocyte cytoskeleton (6)

Figure 2

Schematic of the triangulated network in (a) the undeformed state, also depicting Voronoi tessellation (the superposed hexagon) to identify the area of the RVE; (b) when stretched in the 2 direction (surface area is preserved); (c) the representative volume element.

Figure 3

Schematic of the RVE in undeformed configuration (solid lines) and when subjected to an arbitrary deformation gradient (dashed lines)

Figure 4

(a) Uniaxial tensile stress vs stretch behavior in the 1 direction and the 2 direction for constituent linear and constituent non-Gaussian chain behaviors. (b) Percent change in the 10% strain secant modulus vs the angle of the uniaxial tension applied for various angles with respect to axis 1 for linear chain and non Gaussian chain behavior.

Figure 5

(a) Evolution of chain orientation and chain stretch with respect to axis 1 for chains A, B and C for uniaxial tension in the 1 direction. (b) Evolution of force in chains A, B and C with axial stretch in the 1 direction.

Figure 6

(a) Evolution of chain orientation and chain stretch with respect to axis 1 for chains A, B and C for uniaxial stress in the 2 direction. (b) Evolution of force in chains A, B and C with axial stretch in the 2 direction.

Figure 7

(a) Simple shear stress vs shear strain behavior in the 12 direction and in the 21 direction for constituent linear and constituent non-Gaussian chain behaviors. (b) Percent change in the 10% strain secant modulus vs the angle of the simple shear applied for various angles with respect to axis 1 for linear chain and non-Gaussian chain behavior.

Figure 8

(a) Evolution of chain orientation and chain stretch with respect to axis 1 for chains A, B and C for simple shear in the 12 direction. (b) Evolution of force in chains A, B and C with simple shear in the 12 direction.

Figure 9

(a) Evolution of chain orientation and chain stretch with respect to axis 1 for chains A, B and C for simple shear in the 21 direction. (b) Evolution of force in chains A, B and C with simple shear in the 21 direction.

Figure 10

Effect of pretension on (a) the uniaxial stress-stretch behavior and on (b) the initial axial modulus

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