A neural network constitutive model for hyperelasticity based on molecular dynamics simulations

Abstract

Numerical analysis of the hyperelastic behavior of polymer materials has drawn significant interest from within the field of mechanical engineering. Currently, hyperelastic models based on the energy density function, such as the Neo-Hookean, Mooney-Rivlin, and Ogden models, are used to investigate the hyperelastic responses of materials. Conventionally, constants relating to materials were determined from experimental data by using global least-squares fitting. However, formulating a constitutive equation to capture the complex behavior of hyperelastic materials was difficult owing to the limitations of the analytical model and experimental data. This study addresses these limitations by using a system of neural networks (NNs) to design a data-driven surrogate model without a specific function formula, and employs molecular dynamics (MD) simulations to calculate the massive amount of combined loading data of hyperelastic materials. Thus, MD simulations were used to propose an NN constitutive model for hyperelasticity to derive the constitutive equation to model the complex hyperelastic response. In addition, the probability distributions of the numerical solutions of hyperelasticity are used to characterize the uncertainty of the MD models. These statistical finite element results not only present numerical results with reliability ranges but also scattered distributions of the solution obtained from the MD-based probability distributions.

Automated constitutive modeling of isotropic hyperelasticity based on artificial neural networks

“…Defining the criteria max i | i mean | ≤ 0.015 % and max i i max ≤ 0.015 % lead to the selection of an ANN with N = 11. Regarding other ANN-based approaches, this is a rather small network, see [6,13]. Moreover, the transformation step D → D red enables to use a set containing only n ≈ 300 tuples D red i for the training which is, again, comparatively small, cf.…”

“…In this context, ANN-based models are predestined for the approximation of the homogenized data basis. For instance, this has been done for the scale bridging of composites with cubic microstructures [37], the simulation of the elasticplastic deformation behavior of open-cell foam structures [52] and to link stresses and strains or traction-separation curves obtained from molecular dynamics simulations to the continuum scale [6,14], respectively. A combination of classical FE 2 simulations with on-the-fly adaptive switching to ANN-based surrogate models is shown by Fritzen et al [16].…”

“…the free energy function can be given by = (I 1 , I 2 , I 3 ) including I 1 := tr C, I 2 := 1 − tr C 2 and I 3 := det C,(6) …”

Material modeling for parametric, anisotropic finite strain hyperelasticity based on machine learning with application in optimization of metamaterials

“…In summary, it is concluded (i) that the proposed hyperelastic parametric model can generalize well over several order of magnitude for the highly nonlinear buckling behavior of the parametric cells and (ii) that the newly proposed weights (31) and (32) transforming the standard weighted MSE to the relative error 𝜖 incorporating function (W) and gradient (P) information enhances the prediction quality over the entire range of weak and stiff structures.…”

“…In Reference 29 an approach for the correction of finite hyperelastic models was presented and then showcased in interesting biomechanical applications in Reference 30, but again limited to isotropic material behavior. Molecular dynamics simulations of polymers have been used in Reference 31 in order to generate data and train a feed‐forward neural network (FFNN) as a constitutive model in the context of isotropic finite hyperelasticity. Starting from a small strain highly nonlinear model, the radial numerically explicit potentials were suggested in Reference 32 as an extension of the NEXP proposed first in Reference 33 which employs a tensor decomposition into amplitude and direction followed by kernel interpolation.…”