Elasticity Models for the Spherical Indentation of Gels and Soft Biological Tissues

AFM microor nanoindentation is a powerful technique for mapping the elasticity of materials at high resolution. When applied to soft matter, however, its accuracy is equivocal. The sources of the uncertainty can be methodological or analytical in nature. In this paper, we address the lack of practicable nonlinear elastic contact models, which frequently compels the use of Hertzian models in analyzing force curves. We derive and compare approximate force­ indentation relations based on a number of hyperelastic general strain energy functions. These models were applied to existing data from the spherical indentation of native mouse cartilage tissue as well as chemically crosslinked poly( vinyl alcohol) gels. For the biological tissue, the Fung and single-term Ogden models were found to provide the best fit of the data while the Mooney-Rivlin and van der Waals models were most suitable for the synthetic gels. The other models (neo-Hookean, two-term reduced polynomial, Fung, van der Waals, and Hertz) were effective to varying degrees. The Hertz model proved to be acceptable for the synthetic gels at small strains ( <20% for the samples tested). Although this finding supports the generally accepted view that many soft elastic materials can be assumed to be linear elastic at small strains, we propose the use of the nonlinear models when evaluating the large-strain indentation response of gels and tissues. INTRODUCTION In numerical simulations or uniaxial and biaxial mechanical tests, polymer gels and biological tissues are often modeled successfully using linear elasticity theory at small strains and rubber elasticity theory at large strains. For measurement of elasticity at micron and submicron length scales, the prevalence of atomic force microscopy in materials research has established nanoindentation as one of the leading techniques. However, despite advancements in instrumentation and analysis methods, its application to soft matter is still complicated by tip­ sample interactions and the lack of practical nonlinear contact mechanics models. Many investigators rely on models based on the Hertz theory to analyze force curves. Consequently, errors are frequently incurred by applying these linear elastic representations beyond their validity range or at the small-strain range where the indentation process is most prone to noise. We have developed approximate relations for the non-interactive, spherical indentation of non-Hookean materials. From the Hertz equation and various strain energy functions, force­ indentation relationships were formulated in the following forms: neo-Hookean [1], Mooney­ Rivlin [1], two-term reduced polynomial [2], single-term Ogden [3], Fung [4,5], and van der Waals [6]. In this paper, we first introduce these contact mechanics equations. Results of testing each model by fitting it to data obtained from the large-strain indentation of swollen poly( vinyl alcohol) gels and of native cartilage samples are then presented. Limiting the number of fitting parameters in each equation to two, we identify those models that were found to be most suitable for rubber-like gels and biological extracellular matrices and cells. THEORY Strain energy potentials, uniaxial stress-strain relationships, and the derived force­ indentation equations are listed in Table I. The derivation approach has been described elsewhere [7] and will only be briefly summarized here. We define the indentation stress (a* = F I mi, where F is the force applied to the indenter and a the contact radius) and strain(£!'= a I R, where R is the radius of the indenter) such that they are linearly proportional for Hertzian contact. The uniaxial stress-strain relations are then transformed into force-indentation equations by assuming that the contact radius varies with indentation depth( 8) in the Hertzian manner. Tip-sample interactions are assumed to be negligible. Table I. Strain energy potentials and the stress-strain and contact equations derived from them. Name Strain energy potential (V) Uniaxial stress (σ) – stretch (λ) equation Force (E) – indentation (δ) equation 'EO9 ehsshmf o`q`ldsdqr: D hr hmhsh`k. hmehmhsdrhl`k Xntmf'r lnctktr ( Mooney-Ri vlin Neo-Hookean [1] VLQ = B0 H0 − 2 ( )+ B1 H1 − 2 ( ): σ = 1B0 λ − λ ( )* 1B1 0− λ ( ) E = πQ 1A0 δ 4 . 1 − 2Q δ 1 + 2Qδ 2 . 1 δ − 1Q δ 1 + Q ⎛