Inflation of a circular elastomeric membrane into a horizontally semi-infinite liquid reservoir of finite vertical depth: Estimation of material parameters from volume–pressure data

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Abstract

A method is introduced for estimating values of the hyperelastic material parameters and the residual membrane tension (prestretch) that best describe a set of volume-pressure inflation response data acquired during a cell-elastomer composite diaphragm inflation (CDI) experiment. Based on a model for the quasi-static inflation of a prestretched clamped circular isotropic incompressible hyperelastic membrane into a horizontally semi-infinite incompressible liquid reservoir of finite vertical depth, the fit methodology presented here is developed explicitly for a Mooney-Rivlin (MR) strain energy function. The parameter fit is formulated as a nonlinear least-squares optimization problem (NLSOP), and two empirical constraints are postulated to restrict the domain of model fit parameters to physically relevant values. A numerical least-squares regression analysis utilizing the Nelder-Mead simplex algorithm is proposed for solving the constrained NLSOP. To demonstrate the capabilities of the method, the fit procedure is utilized to estimate the residual membrane tensions and the MR constitutive parameters C1 and C2 that characterize the pseudoelastic loading and unloading responses of a polydimethylsiloxane elastomer membrane as observed in the context of a CDI experiment. The material parameter estimates are verified as being both accurate and repeatable, and the MR non-uniform inflation model is shown to produce better fits (in a least-squares sense) than direct polynomial fitting of the original data set. Extension of the fit methodology to volume-pressure data measured for cell-elastomer composite diaphragm specimens, though possible, requires the development of new constitutive models that can better account for the complex rheological behavior of a living epithelial sheet.

Introduction

Material parameter estimation is one of the cornerstones of constitutive modeling in finite elasticity. A great deal of literature exists on the topic since parameter fit methodologies have been used to model the behavior of a diverse array of materials, including both synthetic rubber elastomers [1], [2], [3], [4], [5], [6] and biological tissues [7], [8], [9], [10]. Despite some subtle differences, most all fit routines adhere to the same basic precepts [7]. First, data is collected from an experiment that corresponds to a tractable (complex) boundary value problem, e.g., planar biaxial stretch tests [11], superposed axial extension and torsion on a cylindrical rod [12], [13], or axisymmetric membrane inflation [1], [6], [14], [15], [16], [17]. For an assumed form of strain energy function, W, solutions of these boundary value problems are written in terms of the unknown material parameters. Next, an optimization problem is formulated for determining the best-fit values of the unknown material parameters that globally minimize some measure of the error, usually in a least-squares sense, between theoretically predicted and experimentally measured quantities of displacement, stretch, and/or stress resultants [18]. The final step of a good regression methodology involves an estimation of the variability of the best-fit parameter values and an assessment as to whether or not the assumed form of W was over-parameterized.

In recent work, we have demonstrated that a variation of the classic circular membrane inflation experiment, referred to as cell-elastomer composite diaphragm inflation (CDI), can be used to investigate the mechanobiological properties of cytoskeletal protein networks and cell-cell anchoring junctions within a living epithelial sheet [19], [20], [21], [22]. As our initial CDI experiments have shown, a living epithelial sheet exhibits complex rheological behaviors that are mediated by mechanical interactions between the cell layer and the polydimethylsiloxane (PDMS) elastomer membrane as well as cytoskeletal reorganization within the cell sheet itself [19]. From a standpoint of mechanics, such tissue remodeling is necessarily accompanied by adaptations in both the constitutive material properties and the residual tensions present within the freestanding cell sheet and the PDMS elastomer that compose a composite diaphragm specimen. Therefore, as motivation for this work, we wondered whether or not it was possible to estimate values for each of these material parameters given only a set of quasi-static (V, p(r = a)) volume-pressure measurements recorded during an inflation experiment [22]. In comparison to previous circular membrane inflation studies that utilized load, deformation, curvature, and/or single-point displacement measurements from discrete inflation states, material parameter estimation [1], [6], [14], [16], [17] from volume–pressure data represents a largely unexplored problem.

In this paper, we present a method for estimating values of the Mooney–Rivlin (MR) material parameters and residual membrane tension (prestretch) that best describe the pseudoelastic volume-pressure (V, p(r = a)) response of a prestretched clamped circular isotropic incompressible PDMS elastomer membrane as it is quasi-statically inflated into a horizontally semi-infinite incompressible liquid reservoir of finite vertical depth. We make no attempt at material parameter estimation for the complex time-dependent rheological behavior of an actual cell-elastomer composite diaphragm specimen [18]. Such work remains open to future investigation [22]. The paper is organized as follows. In Section 2, the nonlinear boundary value problem associated with the non-uniform axisymmetric MR membrane inflation model is reviewed, and a method for calculating volume–pressure response curves is introduced [23]. In Section 3, the parameter fit corresponding to the model is formulated as a nonlinear least-squares optimization problem (NLSOP), and a numerical least-squares regression analysis utilizing the Nelder–Mead simplex algorithm is proposed for solving the NLSOP. Physically relevant domains of the model fit parameters are deliberated in Section 4, and the NLSOP fit methodology is utilized in Section 5 to estimate the MR constitutive parameters and residual membrane tension characterizing the baseline pseudoelastic inflation response of a PDMS elastomer membrane specimen as measured in a CDI experiment (“compliant recipe B” [21]). Section 6 summarizes the major contributions of this study while highlighting areas of future research.

Section snippets

Inflation of a Mooney–Rivlin membrane in a CDI experiment

A general model for the quasi-static inflation of a prestretched clamped circular isotropic incompressible hyperelastic membrane into a horizontally semi-infinite incompressible liquid reservoir of finite vertical depth is described in detail in a separate publication [23]. Here we present only the introductory problem statement along with a summary of the nonlinear boundary value problem that governs the deformation of a Mooney-Rivlin (MR) membrane in this class of non-uniform axisymmetric

Parameter estimation from (V, p(r = a)) inflation response data

In previous work, the non-uniform axisymmetric inflation model (cf. Section 2) was used to simulate the (V, p(r = a)) response of a PDMS elastomer membrane in a CDI experiment, assuming a MR strain energy function [23]. Unfortunately, with respect to the inverse problem of material parameter estimation from volume–pressure data, the inflation model represents a complex boundary value problem where there is no available analytical solution on which to base a standard nonlinear regression analysis.

Physically relevant domain of the Mooney–Rivlin model fit parameters

In the context of the present non-uniform inflation model, the variables ΓMR,ΔMR,andεlMR in the MR parameter set ΛMR=[ΓMR,ΔMR,εlMR]T can vary over the domains 0ΓMR1, ΔMR1, and εlMR0. However, when attempting to fit ΓMR,ΔMR,andεlMR to (V, p(r = a)) inflation response data using the optimization procedure described in Section 3, it is necessary to further restrict the domains of ΓMR,ΔMR,andεlMR such that fit parameter estimates derived from the volume-pressure data will yield only physically

Results and discussion

Consider the 6-cycle (V, p(r = a)) data set depicted in Fig. 7 illustrating the baseline inflation response of a “compliant recipe B” PDMS membrane specimen with radius a = 2.45 mm and clamped membrane thickness hcMR = 10.6 ± 1.4 μm (data originally presented in reference [21]). The data were collected as part of a standard 6-cycle volume-controlled composite diaphragm inflation experiment in which the membrane specimen was submerged within a near horizontally semi-infinite liquid reservoir containing an

Conclusions

The results presented in this study demonstrate that volume-pressure (V, p(r = a)) data can be used to estimate the classical constants C1 and C2 and the residual tension (or prestretch) that best characterize the mechanical behavior of a clamped circular Mooney–Rivlin membrane quasi-statically inflated into a horizontally semi-infinite incompressible liquid reservoir of finite vertical depth. Although the nonlinear least-squares optimization problem, fit procedure, and parameter estimates shown

Acknowledgements

J.C. Selby thanks the Whitaker Foundation for their support of his graduate studies through the predoctoral fellowship program. This work was funded in part by the Critical Research Initiatives Program at the University of Illinois.

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    • Inflation of a circular elastomeric membrane into a horizontally semi-infinite liquid reservoir of finite vertical depth: Quasi-static deformation model

      2009, International Journal of Engineering Science
      Citation Excerpt :

      Although previous models have considered the effects of prestretch when predicting deformation-load behavior [15], in a companion paper [39], we will demonstrate how to use such a model to solve the inverse problem: finding optimized values of the constitutive material parameters and residual membrane tension (prestretch) that best characterize a set of volume–pressure (V, p(r = a)) response data acquired during an inflation experiment.

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