Robert J. Martin

Dominik K. Klein, Mauricio Fernández, Robert J. Martin et al.

In the present work, two machine learning based constitutive models for finite deformations are proposed. Using input convex neural networks, the models are hyperelastic, anisotropic and fulfill the polyconvexity condition, which implies ellipticity and thus ensures material stability. The first constitutive model is based on a set of polyconvex, anisotropic and objective invariants. The second approach is formulated in terms of the deformation gradient, its cofactor and determinant, uses group symmetrization to fulfill the material symmetry condition, and data augmentation to fulfill objectivity approximately. The extension of the dataset for the data augmentation approach is based on mechanical considerations and does not require additional experimental or simulation data. The models are calibrated with highly challenging simulation data of cubic lattice metamaterials, including finite deformations and lattice instabilities. A moderate amount of calibration data is used, based on deformations which are commonly applied in experimental investigations. While the invariant-based model shows drawbacks for several deformation modes, the model based on the deformation gradient alone is able to reproduce and predict the effective material behavior very well and exhibits excellent generalization capabilities. In addition, the models are calibrated with transversely isotropic data, generated with an analytical polyconvex potential. For this case, both models show excellent results, demonstrating the straightforward applicability of the polyconvex neural network constitutive models to other symmetry groups.

Boumediene Nedjar, Herbert Baaser, Robert J. Martin et al.

We investigate a finite element formulation of the exponentiated Hencky-logarithmic model whose strain energy function is given by \[ W_\mathrm{eH}(\boldsymbol{F}) = \dfracμ{k}\, e^{\displaystyle k \left\lVert\mbox{dev}_n \log\boldsymbol{U}\right\rVert^2} + \dfracκ{2 \hat{k}}\, e^{\displaystyle \hat{k} [\mbox{tr} (\log\boldsymbol{U})]^2 }\,, \] where $μ>0$ is the (infinitesimal) shear modulus, $κ>0$ is the (infinitesimal) bulk modulus, $k$ and $\hat{k}$ are additional dimensionless material parameters, $\boldsymbol{U}=\sqrt{\boldsymbol{F}^T\boldsymbol{F}}$ and $\boldsymbol{V}=\sqrt{\boldsymbol{F}\boldsymbol{F}^T}$ are the right and left stretch tensor corresponding to the deformation gradient $\boldsymbol{F}$, $\log$ denotes the principal matrix logarithm on the set of positive definite symmetric matrices, $\mbox{dev}_n \boldsymbol{X} = \boldsymbol{X}-\frac{\mbox{tr} \boldsymbol{X}}{n}\boldsymbol{1}$ and $\lVert \boldsymbol{X} \rVert = \sqrt{\mbox{tr}\boldsymbol{X}^T\boldsymbol{X}}$ are the deviatoric part and the Frobenius matrix norm of an $n\times n$-matrix $\boldsymbol{X}$, respectively, and $\mbox{tr}$ denotes the trace operator. To do so, the equivalent different forms of the constitutive equation are recast in terms of the principal logarithmic stretches by use of the spectral decomposition together with the undergoing properties. We show the capability of our approach with a number of relevant examples, including the challenging "eversion of elastic tubes" problem.