Data-driven micromorphic mechanics for materials with strain localization

TL;DR

This work addresses the poor length-scale representation of classical Cauchy continua in strain-localizing materials by developing a data-driven micromorphic framework that couples a macro displacement with a micro-deformation field. Generalized stresses and strains are stored in a data set and matched to mechanically admissible states via a set-valued, fixed-point procedure that enforces balance laws without predefined constitutive models. Through 1D gradient-damage tests and 2D damage and plasticity problems, the method demonstrates accurate global responses and correct spatial distributions, effectively extracting length-scale information from data. The approach enables data-driven homogenization and failure-mode analysis in microstructured solids, with future prospects for data acquisition, dimensionality reduction, and adaptive sampling to handle high-dimensional generalized state spaces.

Abstract

This paper explores the role of generalized continuum mechanics, and the feasibility of model-free data-driven computing approaches thereof, in solids undergoing failure by strain localization. Specifically, we set forth a methodology for capturing material instabilities using data-driven mechanics without prior information regarding the failure mode. We show numerically that, in problems involving strain localization, the standard data-driven framework for Cauchy/Boltzmann continua fails to capture the length scale of the material, as expected. We address this shortcoming by formulating a generalized data-driven framework for micromorphic continua that effectively captures both stiffness and length-scale information, as encoded in the material data, in a model-free manner. These properties are exhibited systematically in a one-dimensional softening bar problem and further verified through selected plane-strain problems.

Figures (18)

• Figure 1: Schematic representation of a BVP in a micromorphic continuum (left) with a generic microstructure (right). The dashed blue arrows represent second-order tensors. The first-order theory adopted in this work assumes that the micro-deformation ${{\raisebox{\depth}{$\chi$}}}_{ij}$ is homogeneous in $\Omega^\mathrm{m}$ but varies in $\Omega$.
• Figure 2: BVP for the 1D bar with constant cross-section.
• Figure 3: Data gathered from the solution of the 1D bar with constant cross-section, using the damage model in \ref{['sec:app1']} simplified to the 1D case, to be used in standard data-driven computations. The data is taken from all the material points in the bar, yielding 15 300 points (100 elements and 153 load steps). Note that $\sigma+\tau$ plays the role of Cauchy's stress.
• Figure 4: Data-driven results using the standard framework and comparison with the reference simulation (solid lines) for the 1D bar with constant cross-section, showing (a) the force-displacement response and the spatial distribution of (b) damage, (c) displacements, and (d) strains. The strain profile shows both the mechanical and material states, which visually overlap.
• Figure 5: Verification of equilibrium conditions in the data-driven simulations of the 1D bar with constant cross-section, showing the results of the standard framework (dots) and the micromorphic framework (x marks). The color evolution corresponds to the load step $\bar{u}$. The spatial distribution of (a) $\sigma+\tau$ satisfies the equilibrium equation \ref{['eq:eqsig_disc']} with uniform profiles. The spatial distribution of (b) $\mathrm{d}\mu/\mathrm{d}\mathrm{x}+\tau$ satisfies the double-stress equilibrium equation \ref{['eq:eqmu_disc']} with vanishing profiles.

Theorems & Definitions (4)

• Remark 2.1
• Remark 4.1
• Remark 4.2
• Remark 4.3