Computational multiscale methods can predict instability-induced failure of fiber-reinforced composites — such as fiber microbuckling and kink-band formation under compression — at a fraction of the cost of fully resolved simulations, but the most efficient published hybrid schemes require the analyst to decide in advance which regions of the structure need explicit microstructural resolution. Greco, Leonetti, Lonetti, Luciano and Pranno (Composite Structures 251, 2020) showed that such an a-priori hybrid hierarchical/concurrent scheme reproduces critical loads and instability modes of finite-strain periodic composites to within about 0.2–0.5% of direct numerical simulation, while explicitly acknowledging that its efficiency hinges on a manual partitioning whose automatic counterpart is missing. The open problem, stated by the authors as the natural extension of their work, is to equip this hybrid multiscale framework with adaptive capabilities: rigorous zooming-in criteria that automatically detect incipient microscopic instability during the macroscopic solve and insert fine-scale critical regions without a-priori partitioning, preserving critical-load and mode accuracy while maximizing computational efficiency. That the microscopically derived constitutive database used in the homogenized subdomains has been demonstrated only in the planar, radial-path sampling setting is a related derived limitation of the published scheme, not part of the authors' stated future work. No adaptive framework for finite-strain microscopic instability has been reported to date.

Background

Fiber-reinforced composite structures often fail through microscopic instability: under compressive or shear loading, individual fibers or layers buckle locally (fiber microbuckling, often developing into kink bands) well before the surrounding matrix fails, and this local event can trigger structural collapse. Simulating it directly is expensive: a 'direct numerical simulation' (DNS) resolves every fiber explicitly at the structural scale, so the mesh is enormous. Multiscale mechanics avoids this by splitting the problem across scales. A representative volume element (RVE) is a small periodic cell of the microstructure whose homogenized (averaged) response stands in for the material at the macro scale; 'computational homogenization' solves a microscopic boundary-value problem on the RVE whenever the macroscopic model asks for stresses or tangent stiffness. In hierarchical (offline) schemes the RVE response is precomputed into a database before the structural analysis; in concurrent (online, often called FE2) schemes the RVE is solved on the fly at every macroscopic integration point and time step — accurate but costly, and unable to represent boundary-layer effects near constraints where scale separation breaks down. Greco, Leonetti, Lonetti, Luciano and Pranno (Composite Structures 251, 2020, 112529; arXiv:2108.03983) combined the two in a hybrid hierarchical/concurrent scheme: the structural domain is partitioned into 'critical' subdomains, where the microstructure is modeled explicitly with all its nonlinear, large-deformation, instability-prone physics, and 'noncritical' subdomains, which use a macroscopic stress/strain law extracted at the preprocessing stage from RVE calculations and stored as a 'reduced database' interpolated during the solve. To keep this database affordable in their planar (two-dimensional, plane-strain) examples, they exploited frame-indifference of the homogenized energy (independence of the stored energy under superposed rigid-body rotations) to parametrize the data on the macroscopic right stretch tensor U‾\overline{\boldsymbol{U}} and sample it along radial loading paths in the resulting three-dimensional strain space, rather than over a four-dimensional deformation-gradient space. Validated against DNS on fiber-reinforced beams and plates under large deformations, the hybrid scheme predicted critical loads within 0.2–0.5% and reproduced the competing local/global instability modes — but only because the critical regions were marked in advance. Two facts make the follow-on problem hard and non-routine. First, the authors show that deciding where microscopic instability will initiate, in more complex geometries or under more general loading, is itself nontrivial — exactly the information the partitioning needs, and available only after an expensive analysis unless cheap, reliable precursors ('zooming-in criteria') exist. Adaptive model-refinement ideas were available in the literature only for small-deformation damage problems (e.g. the same group's adaptive multiscale strategy for masonry, Composite Structures 153, 2016), not for finite-strain microscopic instability, where the trigger must anticipate a microscopic bifurcation. Second, a derived limitation concerns the reduced database on which the homogenized subdomains rely: its construction and interpolation have been demonstrated only in the planar, radial-path sampling setting, and outside that setting the database lives in a higher-dimensional strain space (a fully three-dimensional parametrization on the symmetric right stretch tensor has six independent components, versus three in the planar setting), so sampling density and interpolation accuracy — on which the accuracy of every homogenized subdomain depends — become the binding constraint for extensions beyond the demonstrated setting. Later work has addressed neighboring pieces but not the combination: adaptive concurrent multiscale with an explicit activation criterion was demonstrated in 2025 only for masonry damage with a phase-field cohesive zone model (a fracture description in which cracks are represented by a smooth damage field rather than sharp surfaces; a small-strain setting); deep-neural-network surrogates have replaced interpolated databases in finite-strain FE2 loops and in a 2025 damage-surface model for a two-dimensional perforated microstructure, but for damage evolution rather than instability onset; and microscopic stability analyses of periodic continua (e.g. via high-fidelity generalized method of cells micromechanics, a semi-analytical scheme for computing the response of periodic composites) remain RVE-level rather than structural, adaptive frameworks.

Problem Statement

Develop an adaptive hybrid hierarchical/concurrent multiscale framework for microstructural-instability-induced failure of locally periodic fiber-reinforced composites under large deformations and general loading, replacing the a-priori manual partitioning of the published hybrid scheme (Greco et al., Composite Structures 251, 2020) with an automatic mechanism. Concretely, the framework must: (a) define rigorous zooming-in criteria — computationally cheap indicators evaluated during the macroscopic solve that reliably detect where microscopic instability is incipient, in complex geometries and under general loading, and trigger the insertion of explicit fine-scale (fully resolved) critical subdomains there, neither missing local instabilities nor refining indiscriminately (adaptive removal of fine-scale regions may be included as a design element where accuracy permits); and (b) preserve the validated accuracy of the non-adaptive scheme — predicted critical loads and microscopic instability modes within the error bands established against direct numerical simulation (fractions of a percent in the source's benchmarks), including the competition between local and global modes and boundary-layer effects near constraints — while reducing the computational cost relative to both direct numerical simulation and semi-concurrent FE2-type analysis. Components (a) and (b) are jointly in service of the single objective — efficient, automatic, accuracy-preserving multiscale prediction of instability-induced composite failure — and a solution must demonstrate them working together on the source's benchmark class (finite-strain periodic fiber-reinforced structures) with comparisons to direct numerical simulation. The homogenized (noncritical) subdomains may use the microscopically derived constitutive database constructed as in the published scheme; extending the construction and interpolation of that database beyond the demonstrated planar, radial-path setting — in particular to three-dimensional microstructures and the higher-dimensional macroscopic strain spaces that arise under fully three-dimensional loading paths — is a recognized derived limitation of the published scheme rather than a completion requirement of this problem.

References

  1. Fabrizio Greco, Lorenzo Leonetti, Paolo Lonetti, Raimondo Luciano, Andrea Pranno, 'A multiscale analysis of instability-induced failure mechanisms in fiber-reinforced composite structures via alternative modeling approaches', Composite Structures 251 (2020) 112529. DOI: 10.1016/j.compstruct.2020.112529; arXiv:2108.03983. https://arxiv.org/abs/2108.03983
  2. Fabrizio Greco, Lorenzo Leonetti, Raimondo Luciano, Paolo Nevone Blasi, 'An adaptive multiscale strategy for the damage analysis of masonry modeled as a composite material', Composite Structures 153 (2016) 972–988. DOI: 10.1016/j.compstruct.2016.06.066. https://doi.org/10.1016/j.compstruct.2016.06.066
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