Model-free data-driven inelasticity in Haigh-Westergaard space -- a study how to obtain data points from measurements

TL;DR

This work extends model-free data-driven computational mechanics to inelastic materials with isotropic hardening by incorporating tangent-space information through Haigh–Westergaard coordinates. The method uses a fixed-point data projection framework with enhanced projections $P_\mathcal{C}$ and $P_\mathcal{D}$ and classifies states into elastic or inelastic via a yield rule, while sampling the yield surface in the octahedral plane using a combined tension–torsion test and deriving the tangent from a tensile test. The core contribution is the data-driven tangent formulation in Haigh–Westergaard space, including a data-enforced normal on the yield surface and an adapted projection that reduces data requirements and computational effort. The approach demonstrates accuracy and efficiency gains on a 3D elastoplastic benchmark with nonlinear isotropic hardening, and discusses practical considerations for noisy data and extensions to physics-informed neural networks for further improvements.

Abstract

Model-free data-driven computational mechanics, first proposed by Kirchdoerfer and Ortiz, replaces phenomenological models with numerical simulations based on sample data sets in strain-stress space. Recent literature extended the approach to inelastic problems using structured data sets, tangent space information, and transition rules. From an application perspective, the coverage of qualified data states and calculating the corresponding tangent space is crucial. In this respect, material symmetry significantly helps to reduce the amount of necessary data. This study applies the data-driven paradigm to elasto-plasticity with isotropic hardening. We formulate our approach employing Haigh-Westergaard coordinates, providing information on the underlying material yield surface. Based on this, we use a combined tension-torsion test to cover the knowledge of the yield surface and a single tensile test to calculate the corresponding tangent space. The resulting data-driven method minimizes the distance over the Haigh-Westergaard space augmented with directions in the tangent space subject to compatibility and equilibrium constraints.

Figures (9)

• Figure 1: Visualisation of a stress tensor $\hbox{\boldmath $\sigma$}$ and its deviatoric part in the Haigh-Westergaard stress space.
• Figure 2: Schematic illustration of a normal vector $\hat{\hbox{\boldmath $N$}}$ at a random point $\hbox{\boldmath $s$}$ of a parametrized yield surface $\Phi$ in Haigh-Westergaard coordinates
• Figure 3: Schematic illustration of data points $(\hat{\rho}_i, \hat{\theta}_i)$ which are gained by combined tensile-torsion test lying on the initial yield surface $\Phi(\theta)$.
• Figure 4: Boundary conditions and dimensions of a quadratic plate under increasing extension $\bar{u}$ on the grey area and uniformly distributed pressure $p$ over the thickness direction.
• Figure 5: Visualization of yield surface function $\Phi(\theta)$ for (a) $k = 1$ and (b) $k = 0.75$.