arXiv subjects

Herbert Baaser

Publications and source records attributed to Herbert Baaser.

5 recordsLinked to original sources

arXiv2026

Concurrent enforcement of polyconvexity and true-stress-true-strain monotonicity in incompressible isotropic hyperelasticity: application to neural network constitutive models

The design of physics-augmented neural networks (PANNs) for the purposes of constitutive modeling has received considerable attention as of late for a variety of material behaviors. Here, we revisit the classical framework of isotropic incompressible hyperelasticity in light of recent advances in the study of constitutive inequalities. We show that polyconvexity implies true-stress-true-strain monotonicity for a large class of incompressible strain-energy functions. The resulting elastic law obeys the physically reasonable Legendre-Hadamard (or ellipticity) condition as well as the notion of increasing stress with increasing strain. These results then inform the architecture of four distinct PANNs which are subsequently calibrated to three different sets of experimental data each. We show that different PANN parametrizations - satisfying the same constitutive constraints a priori - have varying approximation power for the description of material behavior. Moreover, even when distinct parametrizations perform comparatively well within the calibration regime, they show pronounced differences in extrapolation. This observation motivates a critical discussion about the predictive power of PANNs which also has implications for the modeling of more complex material behavior by virtue of neural networks.

math-ph↗

arXiv2025

Hyperelastic stability landscape: A check for HILL stability of isotropic, incompressible hyperelasticity depending on material parameters

In this paper, we describe a uniform and standardized approach for analytically verifying the stability of isotropic, incompressible hyperelastic material models. Here, we address {\sl stability} as fulfillment of the {\sc Hill} condition -- i.e.\ the positive definiteness of the material modulus in the {\sc Kirchhoff} stress -- log--strain relation. For incompressible material behavior, all mathematically and mechanically possible deformations lie within a range bounded, on the one hand, by uniaxial states and, on the other hand, by biaxial states; shear {deformation} states lie in between. This becomes particularly clear when the possible states are represented in the invariant plane. This very representation is now also used to visualize the regions of unstable material behavior depending on the selected strain energy function and the respective data set of material parameters. This demonstrates how, for some constellations of energy functions, with appropriate selection or calibration of parameters, stable and unstable regions can be observed. If such cases occur, it is no longer legitimate to use them to initiate, for example, finite element simulations. This is particularly striking when, for example, a fit appears stable in uniaxial tension, but the same parameter set for shear states results in unstable behavior without this being specifically investigated. The presented approach can reveal simple indicators for this.

cond-mat.mtrl-sci↗

arXiv2019

Inconsistency of uhyper and umat in Abaqus for compressible hyperelastic materials

In this article, we revisited Bažant's comments on the implementation of hyperelastic material models in commercial finite element software. We would like to clarify that our assertions only apply if the material models are implemented as hypoelastic, i.e. by incremental stress updates, in common interfaces (including, in particular, umat in Abaqus). This assumption was not made sufficiently clear in the article. If, on the other hand, the stress calculations are implemented using the umat interface with absolute (or "total") stress updates, as is also assumed in the uhyper interface, there is no difference in the internal processes or the results between the umat and the uhyper implementation. This applies to highly compressible formulations as well, where the Kirchhoff and Cauchy stress tensors are clearly distinguished.

math.NA↗

arXiv2018

Again anti-plane shear

We reconsider anti-plane shear deformations of the form $φ(x)=(x_1,\,x_2,\,x_3+u(x_1,x_2))$ based on prior work of Knowles and relate the existence of anti-plane shear deformations to fundamental constitutive concepts of elasticity theory like polyconvexity, rank-one convexity and tension-compression symmetry. In addition, we provide finite-element simulations to visualize our theoretical findings.

math.AP↗

arXiv2017

A finite element implementation of the isotropic exponentiated Hencky-logarithmic model and simulation of the eversion of elastic tubes

We investigate a finite element formulation of the exponentiated Hencky-logarithmic model whose strain energy function is given by \[ W_\mathrm{eH}(\boldsymbol{F}) = \dfrac{\mu}{k}\, e^{\displaystyle k \left\lVert\mbox{dev}_n \log\boldsymbol{U}\right\rVert^2} + \dfrac{\kappa}{2 \hat{k}}\, e^{\displaystyle \hat{k} [\mbox{tr} (\log\boldsymbol{U})]^2 }\,, \] where $\mu>0$ is the (infinitesimal) shear modulus, $\kappa>0$ 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.

math.NA↗