Micropolar elastoplasticity using a fast Fourier transform-based solver

TL;DR

This work extends fast Fourier transform (FFT)–based spectral methods to elastoplastic micropolar (Cosserat) continua, enabling full-field and homogenized analyses of size-dependent composites. A closed-form radial-return mapping is derived from thermodynamics-based constitutive equations, providing an explicit update for macro- and micro-plastic strains within an EB time-stepping scheme. The authors implement a nonlinear FFT-based solver that solves the micropolar Lippmann–Schwinger equations using precomputed Green’s tensors and a fixed-point iteration with a convergence criterion, and they verify accuracy via numerically manufactured solutions. Results demonstrate the model’s ability to capture micro-plasticity, micro–macro coupling, and length-scale effects, including microplastic ratcheting under cyclic loading and size-dependent responses. The approach offers efficient generation of large datasets for complex microstructures, with potential extensions to high-contrast materials via augmented-Lagrangian schemes.

Abstract

This work presents a micromechanical spectral formulation for obtaining the full-field and homogenized response of elastoplastic micropolar composites. A closed-form radial-return mapping is derived from thermodynamics-based micropolar elastoplastic constitutive equations to determine the increment of plastic strain necessary to return the generalized stress state to the yield surface, and the algorithm implementation is verified using the method of numerically manufactured solutions. Then, size-dependent material response and micro-plasticity are shown as features that may be efficiently simulated in this micropolar elastoplastic framework. The computational efficiency of the formulation enables the generation of large datasets in reasonable computing times.

Figures (9)

• Figure 1: Plastic strain and averaged stress-strain curves for four geometries. Each geometry is simulated with micropolar and classical Cauchy parameters. Solid lines show the composite response, while dashed lines show the response of the matrix (softer black material phase).
• Figure 1: Absolute errors per voxel over time between each component of the FFT output and the manufactured solutions. Different colors correspond to different tensor components (see legend on right). Different lines of the same color correspond to the $4\times 4\times 4 = 64$ voxels that were simulated over time. All errors are bounded by the FFT error threshold of $\varepsilon = 1\times10^{-9}$.
• Figure 2: Convergence study for the error threshold $\varepsilon$. Metric used is the time-averaged distance of $T_{12}$ from the reference solution $T_{12}$ which is taken to be the reference solution found for an error threshold $\varepsilon = 1\times10^{-9}$. The metric is defined in Eq. \ref{['eqn:trial_stresses2']}. The solution converges with respect to this metric after $\varepsilon = 1\times10^{-5}$, and the time the simulation takes to run increases.
• Figure 2: Convergence study for two a geometries: (left) laminate with 50% volume fraction, (right) centered spherical inclusion with radius $1/4$ the cubic side length. Color represents the absolute error of the spatially averaged $t_{12}$ component of the stress tensor at the last time step, where reference is taken as the most resolved solution in space and time, shown as the white "reference" square in the top right. The laminate geometry is exactly represented in voxels for all resolutions considered; the spherical inclusion is never perfectly represented by voxels, due to the curvature of it's surface.
• Figure 3: Top row: $T_{12}(t) := \langle t_{12}(\mathbf{x},t)\rangle$ as a function of $E_{12}(t)$ showing hysteresis that emerges on the macro-scale due entirely to micro-plastic dissipation. The line corresponding to the non-dissipative elastic response is shown for comparison. Bottom row: first is $T_{\text{eq}}(t) := \langle t_{\text{eq}}(\mathbf{x},t)\rangle$ as a function of time, showing that the macro-scale does not plastify; second is $M_{\text{eq}}(t) := \langle m_{\text{eq}}(\mathbf{x},t)\rangle$ as a function of time, demonstrating that the micro-scale variables are the only source of plastic dissipation in this study.