An introduction to the Ogden model in biomechanics: benefits, implementation tools and limitations

Abstract

Constitutive models are important to biomechanics for two key reasons. First, constitutive modelling is an essential component of characterizing tissues' mechanical properties for informing theoretical and computational models of biomechanical systems. Second, constitutive models can be used as a theoretical framework for extracting and comparing key quantities of interest from material characterization experiments. Over the past five decades, the Ogden model has emerged as a popular constitutive model in soft tissue biomechanics with relevance to both informing theoretical and computational models and to comparing material characterization experiments. The goal of this short review is threefold. First, we will discuss the broad relevance of the Ogden model to soft tissue biomechanics and the general characteristics of soft tissues that are suitable for approximating with the Ogden model. Second, we will highlight exemplary uses of the Ogden model in brain tissue, blood clot and other tissues. Finally, we offer a tutorial on fitting the one-term Ogden model to pure shear experimental data via both an analytical approximation of homogeneous deformation and a finite-element model of the tissue domain. Overall, we anticipate that this short review will serve as a practical introduction to the use of the Ogden model in biomechanics. This article is part of the theme issue 'The Ogden model of rubber mechanics: Fifty years of impact on nonlinear elasticity'.

Keywords: blood clot; brain; constitutive modelling; hyperelasticity; open data; thrombus.

Figure 1.

Pure shear as an idealized (homogeneous) and inhomogeneous problem, i.e. the deformation field is a function of the spatial coordinates. Note, pure shear specimens are typically designed with a large width ($w$) to height ($h$) ratio such that the deformed width may be considered unchanged, while the thickness may change to preserve volume. (Online version in colour.)

Figure 2.

(One-term) Ogden model response to pure shear. (a) Sensitivity to the ‘nonlinearity’ parameter $\alpha$ with ${\mu }_0=1$. (b) Sensitivity to the conventional shear modulus ${\mu }_0$ with $\alpha =5$. (Online version in colour.)

Figure 3.

Example blood clot dataset. Force–displacement curve as measured under pure shear and the corresponding digital image correlation-based visualization of the Euler–Almansi strain. (Online version in colour.)

Figure 4.

Schematic of the pipeline for fitting the Ogden model parameters to an experimental dataset. In §3b, we provide details on (i) the analytical solution and in §3c we provide details on the forward finite-element (FE) simulation (ii). (Online version in colour.)

Figure 5.

Ogden material parameter identification. (a) Curve fit between the analytical solution of the homogeneous pure shear problem and our example dataset. (b) Curve fit between the numerical solution of the inhomogeneous pure shear problem and our example dataset. RMSE, root mean square error. (Online version in colour.)

Figure 6.

Numerical model and simulation results for the inhomogeneous pure shear problem. (a) We took advantage of pure shear’s symmetry and modelled only one-eighth of the full sample geometry. (b) Euler–Almansi strain at 5 mm of displacement in the two-direction. (c) Euler–Almansi strain at 5 mm of displacement in the one-direction. (Online version in colour.)