Beyond Representative Volume Elements for Random Heterogeneous Materials

Project Information

BeyondRVE

Grant agreement ID: 101040238

DOI

10.3030/101040238

EC signature date 4 February 2022

Start date 1 July 2022

End date 30 June 2027

Funded under

• European Research Council (ERC)

Total cost

€ 1 499 651,00

EU contribution

€ 1 499 651,00

Coordinated by

UNIVERSITAET DUISBURG-ESSEN
Germany

Periodic Reporting for period 1 - BeyondRVE (Beyond Representative Volume Elements for Random Heterogeneous Materials)

Reporting period: 2022-07-01 to 2024-12-31

Summary of the context and overall objectives of the project

To create materials with superior properties, different materials with specific advantages are combined to form composites. Traditional experiments are oftentimes costly and time-consuming. Therefore, computational micromechanics comes into play which is based on an explicit description of the microstructure as well as dedicated models for the constituent materials and permits to replace some of the experiments with simulations.
As a consequence of modern production processes, the microstructures of composite materials are typically characterized by a high degree of complexity. Digital volume images of microstructures are essential for conducting an accurate micromechanical analysis. However, it was realized that working with such digital images comes with specific boundary-layer effects which pollute both the computed local solution fields and the extracted effective mechanical properties. A key objective of the project BeyondRVE is to provide an engineering methodology to screen these boundary effects. Such a piece of technology would permit to work with smaller sample sizes, accelerating the material analysis and design process significantly.
The second objective concerns the nature of the computed quantities associated to the composites under consideration. Traditionally, the effective properties of a composite materials are deterministic quantities which emerge on volume elements of sufficient size. Within BeyondRVE, dedicated stochastic models are planned to be identified from computational analyses based on material microstructures.
The third objective is concerned with the simulation technology itself. Due to complexity of industrial microstructures, the simulation approach prevalent on component scale, boundary-conforming finite elements, is outperformed by more specialized methods based on voxels - VOlume piXELS. These exploit a matrix-free formulation to analyze materials with digital microstructures rapidly, e.g. by making use of the fast Fourier transform (FFT). These extremely efficient computational techniques, however, fail to provide accurate stresses at material interfaces. Within BeyondRVE, a novel computational approach is to be designed which unites the best of both worlds: the accuracy of boundary-conforming finite elements and the computational prowess of voxel-based approaches.
Last but not least, modern microstructure modeling tools are to be developed within the projects, e.g. for long-fiber reinforced composites.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

Serious progress was made within BeyondRVE on fast micromechanics solvers. We started with investigating the theoretical basics of FFT-based methods: we uncovered the superconvergence phenomenon of the effective stresses responsible for the success of regular grid based methods, we provided a sharp analysis of the convergence rate of the original Moulinec-Suquet discretization and connected the composite-voxel method to assumed/enhanced strain methods.
Aside from these findings, we extended FFT-based methods to exploit adaptivity via an octree-based coarsening, together with an elastic stabilization technique.
Traditionally, methods based on the fast Fourier transform (FFT) impose periodic boundary conditions naturally for the sought fluctuation fields under consideration. Although Dirichlet boundary conditions are readily imposed for conducting composites, the extension to small-strain mechanics seemed impossible due to problems with constructing Green's operator. We introduced a trick to side-step this inherent limitation with immediate applications, e.g. for nano-indentation.
The work on microstructure models focused on fiber composites. Fiber-orientation tensors (FOTs) serve as a critical mathematical tool for modeling injection molding. Unfortunately, filling simulations only provide the second-order FOT, whereas the fourth-order FOT is required for a mechanics. To close this apparent gap, closure approximations are used to guesstimate the fourth-order FOT based on the second-order one. Within BeyondRVE, we contributed to the understanding of FOTs and the design of closures in a number of ways. For a start, we gave a complete mathematical description of the set of realizable fourth-order FOTs in dimension three filling a decade-long gap (and requiring a result of D. Hilbert from 1890!). With this description at hand, we provided two novel closure approximations. The symmetrized implicit quadratic closure is a computationally cheap version of the maximum-entropy closure. Also, within BeyondRVE, a groundbreaking closure was introduced which couples length to orientation and captures experimental results more more accurately than length-agnostic closures. Last but not least, the project BeyondRVE made a breakthrough discovery by providing a robust LFT (long fiber thermoplastics) generator, regarding the phase space of curved fibers as a smooth manifold. With the technology at hand, LFT microstructures with industrial volume fractions may be generated.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

The central example of a material where the project BeyondRVE can show its strengths is LFT (longe-fiber reinforced thermoplastics). Due to the excessive length of the cylindrical fibers, microstructures of LFT materials are extremely challenging to handle, let alone generate. The only known successful attempt (for LFT at industrial volume fraction) was based on an explicit finite-element simulation of a compression process, taking weeks to finish on a high-performance computing cluster.
Within BeyondRVE, we were able to develop a geometry-based LFT generator which reliably generated periodic unit cells of LFT at industrial volume fractions on the order of hours on a conventional laptop. The central idea is to consider fibers as chains of spherocylinders whose configuration space is described by a smooth manifold. The movement of the fibers is governed by the geodesic spray on Riemannian manifolds, i.e. Lagrangian mechanics without external potential. Then, both the non-overlapping condition of the fibers and the prescribed orientation tensor may be encoded by an optimization framework.
There are further results which are upcoming but not yet to be disclosed (after two years).

Long-fiber microstructure with two layers and associated stress field