Comparison between spring network models and continuum constitutive laws: Application to the large deformation of a capsule in shear flow

Abstract

A capsule is a liquid drop enclosed by a solid, deformable membrane. To analyze the deformation of a capsule accurately, both the fluid mechanics of the internal and external fluids and the solid mechanics of the membrane must be solved precisely. Recently, many researchers have used discrete spring network models to express the membrane mechanics of capsules and biological cells. However, it is unclear whether such modeling is sufficiently accurate to solve for capsule deformation. This study examines the correlations between the mechanical properties of the discrete spring network model and continuum constitutive laws. We first compare uniaxial and isotropic deformations of a two-dimensional (2D) sheet, both analytically and numerically. The 2D sheet is discretized with four kinds of mesh to analyze the effect of the spring network configuration. We derive the relationships between the spring constant and continuum properties, such as the Young modulus, Poisson ratio, area dilation modulus, and shear modulus. It is found that the mechanical properties of spring networks are strongly dependent on the mesh configuration. We then calculate the deformation of a capsule under inflation and in a simple shear flow in the Stokes flow regime, using various membrane models. To achieve high accuracy in the flow calculation, a boundary-element method is used. Comparing the results between the different membrane models, we find that it is hard to express the area incompressibility observed in biological membranes using a simple spring network model.

Investigation of membrane mechanics using spring networks: Application to red-blood-cell modelling

“…7(b) shows that all spring networks are vulnerable to surface dilation. The results agree with the prediction by Omori et al [18] that the surface area of a spring network dilates twice as fast as the dilation force in an isotropic loading test, i.e. dS=dT≈2.…”

“…ΔT = (Δt 2 )/(2 m). This modified equation cannot accurately predict the transient motion of a node, but is well suited for the calculation of an equilibrium position [18] subject to a convergence tolerance, e.g. |∑F pi(t) × ΔT | b 10 −8 .…”

“…In this work the springnetwork, i.e. discrete membrane model, mechanics are investigated through the network elasticity [18], which is measured by the slope of tension-strain and tension-dilation curves of the membrane under uniaxial and isotropic loadings respectively. Uniaxial deformation is a combination of shearing and dilation while isotropic deformation is solely due to dilation, see Fig.…”

“…A number of discrete RBC models yield a good representation of the cell mechanics [12] and capture the characteristics of the cell's response in flow [13][14][15][16]. However, discrete RBC models have been criticised for their anisotropic membrane properties [17,18], i.e. spring-network properties.…”

“…This new spring type showed the great potential of spring networks for accurate 2D membrane and 3D solid simulations [23]. Omori et al [18] performed a comparison between discrete and continuum membrane models both for 2D and 3D applications. The paper made the important conclusion that a spring network is mechanically anisotropic and strain-softening, and thus is not suitable for modelling strain-hardening material, e.g.…”