Conference Agenda

We conduct a multiscale study of a composite material comprising two solid phases undergoing a remodelling process of its microstructure. This remodelling is characterised by the evolution of inelastic distortions, described within the context of the Bilby-Kröner-Lee decomposition of the deformation gradient tensor [1] in a multiscale framework [2-4]. The study begins with the derivation of governing equations that capture the behaviour of the composite's constituents, rooted in purely mechanical setting. To bridge the gap between the evolution of the microstructure, dictated by the progression of the inelastic distortions, and macroscopic properties, we employ the two-scale Asymptotic Homogenisation (AH) technique [2-4]. This mathematical framework enables the upscaling of fine-scale inelastic distortions—responsible for variations in the elasticity moduli of the composite—to coarse-scale descriptions, as well as the geometrical characteristics of the composite's internal structure. Our ultimate objective is to address computational challenges associated with predicting the effective mechanical properties of remodelling composites [2]. These challenges arise due to the complex interactions between cell-level problems and homogenised macro-scale behaviours. By leveraging our approach, we derive analytical expressions for the effective coefficients, parameterised spatially and temporally by the evolving tensor of anelastic distortions. Particular attention is given to cases involving multilayered systems [3] or uniaxially fiber-reinforced composites [4].

References:
[1] Micunovic M., Thermomechanics of Viscoplasticity - Fundamentals and Applications. Springer, Heidelberg, Germany, 2009.
[2] Ramírez-Torres A., Di Stefano S., Grillo A., Rodríguez-Ramos R., Merodio J., Penta R., An asymptotic homogenization approach to the microstructural evolution of heterogeneous media. International Journal of Non-Linear Mechanics, 106:245–257, 2018.
[3] Giammarini A., Ramírez-Torres A., Grillo A., Effective elasto-(visco)plastic coefficients of a biphasic composite material with scale-dependent size effects. Mathematical Methods in the Applied Sciences, 48:926–979, 2025.
[4] Ramírez-Torres A., Roque-Piedra A., Giammarini A., Grillo A., Rodríguez-Ramos R., Analytical expressions for the effective coefficients of fibre-reinforced composite materials under the influence of inelastic distortions. Journal of Applied Mathematics and Mechanics (ZAMM), Accepted, 2025.

Acknowledgements:

A. Ramírez-Torres is supported by the Engineering and Physical Sciences Research Council [grant number EP/Y001583/1]. This grant is funded by the International Science Partnerships Fund (ISPF) and the UK Research and Innovation (UKRI). A. Grillo is partially supported by MIUR (Italian Ministry of Education, University and Research) through the PRIN project n. 2017KL4EF3 on “Mathematics of active materials: From mechanobiology to smart devices”, the PRIN project n. 2020F3NCPX on “Mathematics for industry 4.0 (Math4I4)”. A. Grillo's research group is funded by the European Union ‐ Next Generation EU. A. Grillo and A. Giammarini have been supported by the Research Project Prin2022 PNRR of National Relevance P2022KHFNB on “Innovative multi‐scale approaches, possibly based on Fractional Calculus, for the effective constitutive modeling of cell mechanics, engineered tissues, and metamaterials in Biomedicine and related fields” granted by the Italian MUR. This study was carried out within the MICS (Made in Italy – Circular and Sustainable) Extended Partnership and A. Giammarini received funding from the European Union Next-GenerationEU (PIANO NAZIONALE DI RIPRESA E RESILIENZA (PNRR) – MISSIONE 4 COMPONENTE 2, INVESTIMENTO 1.3 – D.D. 1551.11-10-2022, PE00000004). This manuscript reflects only the authors’ views and opinions, neither the European Union nor the European Commission can be considered responsible for them. R. Rodríguez‐Ramos thanks to Chamada CNPq 09/2023 PQ‐2 Productividade em Pesquisa, processo . 307188/2023‐0 e ao Edital UFF PROPPI . 05/2022.

In the context of cell mechanics, we depict a multi-scale continuum approach to study the exchange of mechnanical actions between focal adhesions (FAs), the extra-cellular matrix (ECM) and living cells [1]. In particular, we investigate the role of force- and stress-type stimuli in determining the remodelling, or structural time evolution, of both FAs and ECM, and we investigate how possible heterogeneities attainable at the micro-scale, i.e., the scale characterising the internal structure of each component of focal adhesions, may influence their behaviour [2, 3, 5]. With reference to some structural models available in the literature, we describe a focal adhesion as a sandwich structure, accounting for three main components: the adhesion plaque, the integrins receptors and the ECM [1, 3]. Following previous works [1, 2, 3], we employ a mono-dimensional shear lag model [2, 3], so that, both the adhesion plaque and the ECM are modelled as linear elastic straight fibres subjected to axial deformation only, while the family of integrins receptors is represented by a system of elastic and non-elastic forces. Furthermore, we consider the description of focal adhesions’ dynamics by accounting for their remodelling and micro-scale inhomogeneities. To achieve this, we follow and adapt to our scopes some tools of non-linear elastoplasticity and we adhere to techniques of Asymptotic Homogenization to elucidate how the micro-structure of focal adhesions influences their overall behaviour [3, 4, 5]. In this regard, we obtain closed form of the effective coefficients characterising the FA-ECM complex, with the latter containing both constitutive and geometric information available at the micro-structure.

The obtained multi-scale model, and the related field equations, are solved and the forces exchanged by focal adhesions with the ECM and cells are studied. Detailed findings from this investigation are summarized in [5].

References

[1] Cao X., et al., “A chemomechanical model of matrix and nuclear rigidity regulation of focal

adhesion size,” Biophys. J., 109.9, 1807-1817 (2015).

[2] Di Stefano S., et al., “On the role of elasticity in focal adhesion stability within the passive

regime,” Int J Non Linear Mech, 146, 104157 (2022).

[3] Di Stefano S., et al., “On the role of friction and remodelling in cell–matrix interactions: a

continuum mechanical model,” Int J Non Linear Mech, 142, 103966 (2022).

[4] Ramírez-Torres-Torres A., et al., “An asymptotic homogenization approach to the microstructural evolution of heterogeneous media,” Int J Non Linear Mech, 106, 245-257 (2018).

[5] Di Stefano S., et al., ''Homogenised structural behaviour of remodelling cell-matrix systems: the case of focal adhesions''. In preparation.

Within the framework of "Hybrid Mixture Theory" [1], multicellular aggregates can be formalized as biphasic continuum media featuring a fluid phase, typically identified with an interstitial fluid carrying nutrients, and a solid phase representing, e.g., cells, protein filaments, extracellular matrix and other biological components.

Experimental evidence [2] shows that, under external loading, the solid phase of multicellular aggregates can undergo irreversible deformations, so that, after relaxation, these biological structures tend not to recover their shape prior to the application of the load [3]. Some authors [4,5] have proposed to attribute this behavior to internal structural reorganizations, which, in some degree, share similarities with the inelastic processes occurring in non-biological media. While these structural transformations belong to the class of phenomena named “remodeling” in the biomechanics community, their presumed resemblance with inelasticity has suggested, among other possibilities, to describe them by multiplicatively decomposing the deformation gradient tensor of a given multicellular aggregate into an elastic and an inelastic (remodeling) part (see, e.g., [3] and the references therein). On the other hand, also the fluid phase contributes to the overall dissipative behavior of the systems under study, for example through its exchange interactions with the solid phase. In the literature, the inelastic aspects of remodeling and the dissipation introduced by the fluid are usually addressed by resorting to models that are of grade zero in the distortions and that rely on flow laws of Darcian type for the fluid. In spite of their utility, however, these models are unable to resolve explicitly the interactions between the multicellular aggregates and the surfaces of the apparatuses with which they are in contact during experiments. An example of these interactions could be the change in ductility localized near the contact regions of a specimen with the testing device.

To cope with this lack of information, an interesting, yet relatively unexplored, research direction could be the introduction of higher-order descriptors both for the inelastic distortions and for the fluid kinematics. In particular, the flow of the interstitial fluid may be characterized by a non-negligible viscosity, thereby undermining the hypothesis of Darcian-like regimes. In this case, one possibility is to account for the so-called Brinkman correction in the expression of the overall fluid stress tensor [6].

In this presentation, we take a step in this direction by extending the model of Gurtin&Anand [7] to the theory of biphasic mixtures, and coupling it with a Darcy–Brinkman model for the fluid flow. We follow the paradigm of the Principle of Virtual Power [8] to obtain the dynamic equations of the system, and we discuss some related aspects of configurational and analytical mechanics. Finally, we compare numerically some preliminary results [9] with some well established models taken from the literature.

References:

[1] Bennethum, L.S., et al.: Transport in Porous Media 39(2): 187–225 (2000).

[2] Marmottant, P. et al.: Proc Natl Acad Sci USA. 106(41): 17271-5 (2009).

[3] Di Stefano, S., Giammarini, A., Giverso, C., et al.: Z. Angew. Math. Phys. 73: 79 (2022).

[4] Giverso, C., Preziosi, L.: Math Med Bio 29: 181-204 (2012).

[5] Ambrosi, D., Preziosi, L.: Biomech Model Mechanobiol 8(5): 397-413 (2009).

[6] Brinkman, H. C.: Applied Scientific Research. 1(1): 27–34 (1949).

[7] Gurtin, M. E., Anand, L.: Int. J. Plast., 21(12): 2297-2318 (2005).

[8] Germain, P.: SIAM J. Appl. Math. 25(3): 556–575 (1973).

[9] Giammarini, A., Pastore, A., Ramìrez-Torres, A., Grillo, A.: To be submitted.