A computational approach to identify the material parameters of the relaxed micromorphic model

TL;DR

This work introduces a computational framework to identify the micro-scale elasticity $\mathbb{C}_{\mathrm{micro}}$ and curvature length $L_{\mathrm{c}}$ of the relaxed micromorphic model by matching the total energy of the homogeneous relaxed micromorphic continuum to that of fully resolved heterogeneous unit-cell samples via a Hill-Mandel energy equivalence and least-squares fitting. By leveraging the Curl-based curvature and a two-scale interpretation bounded by $\mathbb{C}_{\mathrm{micro}}$ and $\mathbb{C}_{\mathrm{macro}}$, the method identifies a small set of parameters (e.g., in 2D cubic symmetry, $\mu_{\mathrm{micro}}, \mu^{*}_{\mathrm{micro}}, \lambda_{\mathrm{micro}}, \mu L_{\mathrm{c}}^{2}$) across multiple specimen sizes and deformation modes. The results show that the relaxed micromorphic model can closely reproduce the heterogeneous response with fewer parameters than full micromorphic or Cosserat models, and that boundary-condition choices and physically meaningful upper bounds on micro stiffness improve robustness. The study demonstrates the practical viability of a two-scale, energy-based calibration approach for generalized continua and highlights the potential for explicit micro–macro homogenization relations to be developed for the relaxed micromorphic framework in future work.

Abstract

We determine the material parameters in the relaxed micromorphic generalized continuum model for a given periodic microstructure in this work. This is achieved through a least squares fitting of the total energy of the relaxed micromorphic homogeneous continuum to the total energy of the fully-resolved heterogeneous microstructure, governed by classical linear elasticity. The relaxed micromorphic model is a generalized continuum that utilizes the $\Curl$ of a micro-distortion field instead of its full gradient as in the classical micromorphic theory, leading to several advantages and differences. The most crucial advantage is that it operates between two well-defined scales. These scales are determined by linear elasticity with microscopic and macroscopic elasticity tensors, which respectively bound the stiffness of the relaxed micromorphic continuum from above and below. While the macroscopic elasticity tensor is established a priori through standard periodic first-order homogenization, the microscopic elasticity tensor remains to be determined. Additionally, the characteristic length parameter, associated with curvature measurement, controls the transition between the micro- and macro-scales. Both the microscopic elasticity tensor and the characteristic length parameter are here determined using a computational approach based on the least squares fitting of energies. This process involves the consideration of an adequate number of quadratic deformation modes and different specimen sizes. We conduct a comparative analysis between the least square fitting results of the relaxed micromorphic model, the fitting of a skew-symmetric micro-distortion field (Cosserat-micropolar model), and the fitting of the classical micromorphic model with two different formulations for the curvature...

Figures (16)

• Figure 1: The stiffness of the relaxed micromorphic model (RMM) is bounded from above and below. Other generalized continua exhibit unbounded stiffness for small sizes. For large values of the characteristic length, linear elasticity with a micro elasticity tensor is recovered (one UC) while linear elasticity with a macro elasticity tensor is obtained for small values of the characteristic length (many UCs).
• Figure 2: A depiction of the scaling $L_{\mathrm{c}}/n$. This scaling delivers the intended "smaller is stiffer" effect for computations on a domain of fixed size and constant $L_{\mathrm{c}}$.
• Figure 3: Unit-cell with the material and geometrical parameters.
• Figure 4: Illustration of the homogenization procedure to identify the parameters of an equivalent linear elastic medium under affine kinematic boundary condition.
• Figure 5: Results for the parameter identification algorithm for linear elasticity under affine boundary conditions.