Interpretable data‐driven modeling of hyperelastic membranes

Abstract

We proposed an approach for interpretable data‐driven modeling of an isotropic incompressible hyperelastic membrane deformation. The approach is based on the response functions in terms of the Laplace stretch and the finite element method, where response functions are partial derivatives of a hyperelastic potential with respect to the chosen strain measure. The Laplace stretch as the strain measure allows us to recover directly the response functions from experimental data and construct automatically data‐driven constitutive relations. All needed formulas were obtained explicitly. We tested our approach for membrane inflation with data‐driven constitutive relations based on the perforated membrane extension tests.

Data-Driven Anisotropic Biomembrane Simulation Based on the Laplace Stretch

“…For the approximate solution of Equation ( 2) we use the hyperelastic nodal force version of the P 1 finite element method [26][27][28]. For the reader's convenience, we briefly review its membrane variant in terms of the Laplace stretch [29]. We consider the deformation of a consistent triangulation of the initial planar configuration Ω 0 .…”

“…where ∂ψ/∂ξ s are so-called response functions [8,9]. The other factors ∂ξ s /∂Q i are defined completely by concise formulas [29]. Let the initially flat membrane in flat configuration Ω 0 be deformed so that its current configuration Ω t be defined in the three-dimensional space with basis vectors e 1 , e 2 , e 3 of global (fixed) Cartesian coordinates.…”

“…To perform data-driven finite element forward simulations, we use the hyperelastic nodal force version of the P 1 finite element method [26][27][28] with a simple interpolation method for response functions in the 3D space of the Laplace stretch variables that is requested in the iterative solution of the equilibrium problem. For the case of an isotropic material such approach was proposed in [29]. However, the advantage of using the Laplace stretch as the strain measure and the response function based approach is its applicability to both isotropic and anisotropic materials.…”

“…However, the advantage of using the Laplace stretch as the strain measure and the response function based approach is its applicability to both isotropic and anisotropic materials. In this study we test our approach for datadriven simulation of anisotropic membranes using the similar steps proposed in [29] for the isotropic material, except for virtual experimental protocols applicable to real-world scenarios and material anisotropy. To this end, we use synthetic data based on virtual experiments for Holzapfel-Gasser-Ogden model [30] with parameters corresponding to porcine skin [31].…”

“…It is important to highlight that the utilization of the actual protocols established for cruciform specimens [32] serves multiple purposes. First and foremost, the authors [32] have proposed a "series of mechanical tests designed to maximize inhomogeneous strain fields and in-plane shear forces", which means that it is not necessary to introduce holes in order to achieve a heterogeneous deformation field as it was done previously for isotropic case by [13,29]. Furthermore, through the implementation of high resolution imaging and digital image correlation techniques, it becomes possible to capture accurately the complete displacement field.…”