Quasi-trivial solutions for composite laminates: recent developments




=1. Introduction=

Continuous fiber reinforced composite laminates are excellent materials for lightweight structures. Their mechanical behavior can be tailored through the selection of constituents and the stacking sequence, giving designers a very wide design space of possibilities. This flexibility, however, is rarely fully exploited. Often, simple design constraints (e.g. enforcing mid-plane symmetry of the stacking sequence to achieve bending–extension uncoupling) or simplistic “one-size-fits-all” design approaches (e.g. using quasi-isotropic laminates as “black aluminum”) are applied. Such practices drastically reduce the available design space and consequently limit the attainable performance.

To fully exploit the potential of composite laminates, novel design strategies offering more flexibility than traditional methods are required. One promising approach is the use of ''quasi-trivial'' (QT) solutions [1-3]. QT solutions can be understood as a special class of stacking sequences that inherently exhibit certain desirable elastic behaviors. Specifically, laminates made exploiting QT solutions can be designed to exhibit '''bending–extension uncoupling''' or '''bending–extension homogeneity''' (the latter means the laminate’s in-plane stiffness matrix is proportional to its out-of-plane bending stiffness matrix, as predicted by classical lamination theory). Remarkably, a QT sequence can also achieve both bending-extension uncoupling and bending-extension homogeneity simultaneously, a condition known as '''quasi-homogeneity'''. Notably, the family of QT sequences that yield bending-extension uncoupling is much broader than the set of conventional symmetric laminates (in fact, symmetric sequences are a subset of QT uncoupled solutions). Furthermore, by exploiting bending-extension homogeneity or quasi-homogeneity, designers can achieve specialized laminate properties that are not attainable with simpler design approaches.

Despite their potential, QT solutions have seen limited adoption in practice. This is largely due to the limited understanding of their characteristics and the scarce availability of known QT stacking sequences. In other words, few QT solutions are documented or readily accessible, and their properties have not been widely studied or disseminated.

In this work, we present and discuss three key recent developments aimed at overcoming these challenges and advancing the use of QT solutions in composite laminate design:

:* '''A novel analytical framework''' that formally describes QT solutions and related laminate design concepts, providing a rigorous mathematical foundation for their analysis and manipulation;

:* '''Newly identified properties of QT solutions''' derived using this framework, including analytical conditions on allowable number orientations and ply counts per orientation in QT laminates;

:* '''A new algorithmic strategy''' to efficiently generate QT stacking sequences, which exploits the above properties (and uses a recursive search approach) to greatly improve computational efficiency.

Together, these developments expand the knowledge and understanding of QT solutions and provide practical tools for leveraging them in design. We hope that this work will help unlock the full potential of QT stacking sequences, encouraging their adoption and enabling composite designers to better exploit the extraordinary tailorability of composite materials.

=2. Recent developments on quasi-trivial solutions for composite laminate design=

'''A novel formal framework for QT solutions'''. QT solutions were first introduced in 2001 by Vannucci and Verchery [1]. In their seminal paper, the authors demonstrated the existence of these special stacking sequences, provided a definition for them, and gave an initial description of their properties and how they can be obtained. In view of their peculiar features and the novel design possibilities they offer, QT sequences have since attracted significant research interest. For example, QT stacking sequences have been used in combination with optimization strategies for designing variable angle-tow laminates [4,5]. More recently, they enabled the design of fully-uncoupled multidirectional (FUMD) laminate specimens for interlaminar fracture testing of non-unidirectional interfaces [6-8]. These applications highlight the practical value of QT sequences in advanced composite design.

However, since the original work of Vannucci and Verchery, relatively little effort has been devoted to a deeper, more systematic analysis of QT solutions. Until now, descriptions of QT sequences remained largely heuristic, and no formal theoretical framework existed. This lack of a rigorous framework has likely hindered broader exploration and adoption of QT solutions in the composites community.

To address this problem, we developed a '''new analytical framework''' for describing QT solutions, drawing on concepts from set theory to ensure mathematical rigor. Within this framework, we formally defined several important concepts relevant to laminates and QT sequences, including:

:* '''Common laminate design concepts''' that were previously defined only informally (for example, the notion of a stacking sequence itself, and typical coefficients used in classical lamination theory);

:* '''Key concepts from the original QT literature''' that had only heuristic definitions (for instance, the concept of orientation groups/sets, introduced by Vannucci and Verchery [1], which we have now formalized mathematically);

:* '''Novel concepts''', introduced for the first time in this work.

Through this theoretic framework, QT solutions are for the first time described as precise mathematical objects. This formalization makes it possible to study and manipulate them in a systematic way. By laying a rigorous foundation, the framework paves the way for developing robust design tools for QT laminates and is expected to catalyze further research and interest in this area.

'''Properties of QT solutions'''. The introduction of the formal framework above also enabled the discovery of several previously unknown properties that characterize QT solutions. In particular, we derived analytical constraints on the allowable number of distinct fiber orientations in a QT solution and on the number of plies of each orientation. These constraints are expressed as functions of two factors: the total number of plies in the laminate, and the type of QT solution (whether it is bending–extension uncoupled, bending–extension homogeneous, or quasi-homogeneous). In essence, we obtained clear rules for what combinations of ply counts and orientations can (or cannot) form a valid QT solution of a given type and total ply number. These new properties deepen the understanding of the internal structure of QT sequences and set explicit boundaries on their composition.

'''Novel search algorithm to obtain QT solutions'''. One of the main challenges in working with QT solutions is finding them in the first place. Identifying QT solutions requires specialized search algorithms, and the computational cost of brute-force searches grows dramatically as the number of plies increases. In fact, for laminates with many plies, an exhaustive search for QT solutions becomes infeasible with naive methods.

In this work, we developed a new algorithm to efficiently obtain QT solutions. The algorithm leverages the analytical properties derived from our formal framework to prune the search space and avoid unnecessary computations. By incorporating the constraints on ply counts and orientations mentioned above, the algorithm can immediately eliminate many candidate sequences that cannot yield QT behavior, thereby focusing only on promising regions of the design space. Moreover, the algorithm employs a novel recursive strategy to construct QT solutions. This recursive approach is unprecedented in the QT literature and significantly improves the search efficiency allowing to obtain longer QT sequences than in the past.

Using this novel algorithmic approach, we were able to generate an extensive database of QT solutions. Notably, the new algorithm’s efficiency allowed us to find QT stacking sequences with a higher total number of plies than any previously reported solutions. This growing database includes QT sequences for all three types (bending–extension uncoupled, bending–extension homogeneous, and quasi-homogeneous), providing a rich resource of design candidates for various applications.

=3. Discussion =

'''Formal framework for QT solutions'''. QT solutions represent a high-potential tool for composite laminate design, offering new opportunities to fine-tune mechanical behavior beyond what is possible with conventional designs. Realizing this potential in practice requires easy-to-use yet rigorous design frameworks and tools centered on QT sequences. The analytical framework developed in this work addresses this need by providing a solid mathematical description of QT solutions. It lays the groundwork for future developments: for example, we are already extending this framework in ongoing research on interlaminar fracture of multidirectional laminates and the design of FUMD test specimens [9]. By formalizing the concept of QT sequences, the framework is an essential first step toward integrating QT solutions into composite design methodologies.

'''On the properties of QT solutions'''. The newly derived properties of QT stacking sequences prove useful in multiple ways. First, we directly utilized these properties to improve the efficiency of our QT search algorithm, which was critical for enabling the discovery of longer and more complex QT sequences than were previously attainable. More generally, the constraints defined by these properties can be built into laminate design and optimization frameworks that incorporate QT solutions. By applying such rules, designers can dramatically narrow down the search space and avoid exploring infeasible designs, simplifying the design space exploration and reducing computational effort. Finally, because these properties are straightforward and easy to evaluate, they can guide early-stage design decisions. For instance, during preliminary laminate design, an engineer can use these rules to know if the number of orientation and/or of plies per orientation offered by QT solutions with a given total ply number may satisfy his design needs, before running any complex optimization.

'''QT solutions search algorithm'''. The novel search algorithm developed in this work has, thanks to its increased efficiency, allowed us to create a broad database of QT solutions. This database encompasses bending–extension uncoupled, bending–extension homogeneous, and quasi-homogeneous sequences across a range of total ply counts. Given the unique design opportunities that QT solutions offer, we expect this database to become a valuable resource for the community. Designers and researchers can draw on it for reference, using known QT sequences as building blocks or inspiration for tackling new laminate design problems. In essence, the algorithm and resulting catalog of solutions lower the barrier to adopting QT sequences in practical applications by providing readily available examples and saving others from the need to reobtain QT solution through costly computations.

=4. Conclusions=

In this work, we summarized a number of recent advances concerning QT solutions for the design of composite laminates. Specifically, we discussed:

:* '''A novel analytical framework''' that provides a formal, rigorous description of laminate stacking sequence concepts (which were traditionally defined heuristically) and leads to a precise mathematical definition of QT solutions;

:* '''Newly discovered properties of QT solutions''', focusing on explicit conditions that the orientation set of a QT solution and the distribution of plies must satisfy, given the total number of plies and the QT solution type (uncoupled, homogeneous, or quasi-homogeneous);

:* '''A new algorithmic strategy''' for generating QT solutions, which exploits the above properties and introduces a recursive approach to dramatically increase computational efficiency. This advancement has allowed us to obtain QT solutions with a larger number of plies than was previously possible.

We also outlined the importance and implications of these developments, highlighting that:

:* The formal analytical framework establishes a foundation for developing robust design methodologies and tools that leverage QT solutions in practice;

:* The derived properties of QT sequences can be useful in preliminary design stages (guiding initial design choices) and can impose helpful constraints to simplify design space exploration by eliminating impractical configurations early;

:* The efficient search algorithm enabled the creation of an extensive QT solution database, which may become an essential reference for engineers and researchers in composite laminate design.

By strengthening the theoretical underpinnings of QT solutions and providing practical means to obtain them, this work aims to facilitate more widespread use of QT stacking sequences. We believe that these contributions will help unlock new, superior design options for composite structures and inspire further research into advanced laminate tailoring strategies

=5. Acknowledgements=

Torquato Garulli acknowledges the funding from the European Union’s Horizon Europe research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 101061912. Marco Picchi Scardaoni acknowledges the support from the project PRA2022-69 "Advanced modelling of ultra-lightweight materials and structures" and from the project PRIN 2022-20229BM9EL “NutShell”. Marco Picchi Scardaoni acknowledges also the Italian National Group of Mathematical Physics INdAM-GNFM.

=6.    Bibliografía=

[1] P. Vannucci and G. Verchery, «A special class of uncoupled and quasi-homogeneous laminates,» Composites Science and Technology, vol. 61, nº 10, 2001, DOI: 10.1016/S0266-3538(01)00039-2.

[2] T. Garulli, A. Catapano, M. Montemurro, J. Jumel, D. Fanteria, «Quasi-trivial stacking sequences for the design of thick laminates,» Composite Structures, Volume 200, 2018, DOI: 10.1016/j.compstruct.2018.05.120.

[3] T. Garulli, A. Catapano, M. Montemurro, J. Jumel, D. Fanteria, «Quasi-trivial solutions for uncoupled, homogeneous and quasi-homogeneous laminates with high number of plies,» ECCM VI, International Center for Numerical Methods in Engineering (CIMNE), 2018.

[4] M. Montemurro and A. Catapano, «On the effective integration of manufacturability constraints within the multi-scale methodology for designing variable angle-tow laminates,» Composite Structures, vol. 161, ,2017, DOI: 10.1016/j.compstruct.2016.11.018.

[5] M. Montemurro and A. Catapano, «A general b-spline surfaces theoretical framework for optimization of variable angle-tow laminates,» Composite Structures, vol. 209, 2019, DOI: 10.1016/j.compstruct.2018.10.094.

[6] T. Garulli, A. Catapano, D. Fanteria, J. Jumel, E. Martin, «Design and finite element assessment of fully uncoupled multi-directional layups for delamination tests. Journal of Composite Materials, vol. 54, nº 6, 2020, DOI: 10.1177/0021998319868293.

[7] T. Garulli, A. Catapano, D. Fanteria, W. Huang, J. Jumel, E. Martin, «Experimental assessment of Fully-Uncoupled Multi-Directional specimens for mode I delamination tests,» Composites Science and Technology, vol 200, 2020, DOI: 10.1016/j.compscitech.2020.108421.

[8] T. Garulli, «Design and validation of Fully-Uncoupled Multi-Directional lay-ups to evaluate interlaminar fracture toughness,» Doctoral dissertation, Université de Bordeaux; Università di Pisa, 2020.

[9] T. Garulli, A. Arteiro, A., N. Blanco Villaverde, J. Renart Canalias, «Interlaminar fracture testing of multidirectional laminates: on finite width effect and residual stresses,» European Society of Composite Materials (ESCM), Proceedings of the 21st European Conference on Composite Materials: Vol. 8: Special sessions, 2024.