Spider silk is an extraordinary protein material that exhibits counterintuitive mechanical behaviors such as a reduction in stiffness of several orders of magnitude, supercontraction (i.e. a shortening of up to ~60% in length), and twist upon exposure to high humidity. These non-trivial responses originate from a unique polymeric structure made of crystalline domains that are embedded in a highly aligned amorphous matrix. Broadly, high humidity leads to water uptake by the silk, which motivates the dissociation of intermolecular hydrogen cross-linking bonds. In this talk, I will present energetically motivated models that explain the origin of supercontraction and twist in spider silk. Using tools from statistical mechanics, I will show that the dissociation of intermolecular bonds gives rise to a transition from a glassy to a rubbery phase, an increase in entropy, and a decrease in free energy. These factors shed light on the underlying mechanisms that govern supercontraction and agree with experimental findings. In addition, I will employ a continuum-based framework to show that the twist behavior originates from helical features that exist in a glassy spider silk fiber. The merit of these works is two-fold: (1) they account for the microstructural evolution of spider silk in response to water uptake and (2) they provide a method to characterize the microstructural evolution of hydrogen-bond dominated networks. The insights from the presented models pave the way to the design of novel biomimetic fibers with non-trivial properties.

1) N. Cohen, “The underlying mechanisms behind the hydration-induced and mechanical response of spider silk”, Journal of the Mechanics and Physics of Solids, 172:105141, 2023.

2) N. Cohen and C.D. Eisenbach, “Humidity-Driven Supercontraction and Twist in Spider Silk”, Physical Review Letters, 128:098101, 2022.

3) N. Cohen, M. Levin, and C.D. Eisenbach, “On the origin of supercontraction in spider silk”, Biomacromolecules, 22:993-1000, 2021.

The study of the mechanical properties of viscoelastic composites has been of great interest due to their unique characteristics. Following the methodology proposed in [1,2], we study the effective properties of nonlinear viscoelastic heterogeneous materials. With this goal, we employ the asymptotic homogenisation technique to decouple the equilibrium equation into a cell and a homogenised problem. The theory developed in this work is specialised to the case of a strain energy density of Saint-Venant type, with the second Piola-Kirchhoff stress tensor also featuring a viscous contribution. Within this setting, we frame the general theory in the case of infinitesimal displacements to make use of the correspondence principle which results from the employment of the Laplace transform. This choice is also advantageous to avoid the numerical complications arising in a finite theory. Furthermore, it permits obtaining the classical cell and homogenised problems in linear viscoelasticity as a special case. We frame our analysis by considering the case of uniaxially fibre-reinforced composites and, taking inspiration from [3], we write short formulae for the effective coefficients associated with the antiplane problem. Finally, after selecting different constitutive models for the terms associated with the memory of the constituents, our results evidence that the approximations of the semi-analytical method converge rapidly and comparisons with data available in the literature show a good agreement. Further developments of this work aim to generalise the model using the covariant formulation of continuum mechanics and to include a broader analysis of different microstructural geometric arrangements of nonlinear viscoelastic composites. A further scope is to frame the general theory in biological scenarios of interest. These include but are not limited to, biological fibrous tissues, such as muscles and connective tissue. It is expected that further research in this area will lead to new research questions in materials science and biomathematics.

[1] Pruchnicki, E. Hyperelastic homogenized law for reinforced elastomer at finite strain with edge effects. Acta Mechanica 1998, 129, 139–162. https://doi.org/10.1007/bf01176742.

[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 2018, 106, 245–257. https://doi.org/10.1016/j.ijnonlinmec.2018.06.012

[3] Rodríguez-Ramos, R.; Otero, J.; Cruz-González, O.; Guinovart-Díaz, R.; Bravo-Castillero, J.; Sabina, F.; Padilla, P.; Lebon, F.; Sevostianov, I. Computation of the relaxation effective moduli for fibrous viscoelastic composites using the asymptotic homogenization method. International Journal of Solids and Structures 2020, 190, 281–290. https://doi.org/10.1016/j.ijsolstr.2019.11.014.

Stress shielding minimization is one of the major issues in implant design. Currently, aseptic loosening brought on by stress shielding is one of the primary causes of revision surgery. As bone regeneration is triggered by a stress stimulus, poor load transmission to the bone can result in a low stimulus, which can cause bone decay and other issues.

The development and evolution of additive manufacturing techniques have made it possible to significantly reduce stiffness by introducing porosity into implant materials as virtually any shape can be manufactured through those processes, regardless of shape complexity. In order to encourage cell adhesion and proliferation, porous geometries are frequently used in the construction of scaffolds. Additionally, porous geometries allow for the fine tuning of mechanical properties through changes to its topological design.

Feed-forward neural networks are able to do complex non-linear mapping between the input and output data. It has been shown that a neural network with one hidden layer and a number n of neurons in capable of representing any function. When a neural network presents several hidden layers, it is usually considered a deep learning approach.

The objective of this study is to create an optimal design by training a neural network to produce the ideal unit cell topology for a given constitutive elastic matrix. As a result, the network has the ability to reverse the homogenization process my mapping the relationship between the constitutive elastic properties and the unit cell geometry. A feed-forward neural network was created and trained in MATLAB with data generated from a set of several different geometries.

To each of these geometries, homogenization with periodic boundary conditions was performed. The lattice was modelled as a biphasic material where the solid phase was modeled with the properties of the material and the remainder area of the representative volume element (RVE) was considered to be a void phase with 1e-06 of the Young’s modulus of the solid material in order to reduce the influence of these elements to the homogenized constitutive matrix. A uniform mesh of square 2D elements allows to directly impose the periodic boundary conditions. The original geometry is therefore simplified to fit the uniform mesh where each element, a structured square, is either attributed solid or void properties.

The constitutive matrix used as the input of the network was obtained by applying a deformation gradient leading to a strain state where all components, but one is null, and thus, the macro-stress tensor of the RVE is equal to one line in the constitutive matrix. The linear-elastic analysis is run using ABAQUS as the solver.

Composite or microstructured materials have been long since considered as important means to engineer and optimize mechanical properties for specific applications. With the advent of additive manufacturing (also known as 3D-printing), production of artificial constructs conceived to possess specific optimal properties is now becoming possible, with applications ranging from construction to biomimetic materials. The architecture of such composites is typically based on designing features at a small scale, that lead to the desired large-scale behavior of structural elements. In light of this, theoretical studies of composites often involve asymptotic homogenization or alternative upscaling techniques. Most often, a periodic assembly of basic units is a practical way to build metamaterials.

In the design and analysis of composite materials based on periodic arrangements of sub-units it is of paramount importance to control the emergent material symmetry in relation to the elastic response. In numerous applications it would be useful to obtain effectively isotropic materials. While these typically emerge from a random microstructure, it is not obvious how to achieve isotropy with a periodic order. I has been long since recognized that inclusions distributed on a two-dimensional hexagonal lattice orthogonally extruded in the third dimension can give rise to transversely isotropic materials. Nevertheless, in spite of some computational evidence emerged in recent years, a generalization of this result to full three-dimensional isotropy has so far remained elusive.

We present a rigorous and yet simple proof of the fact that a periodic arrangement on a face-centered cubic lattice of spherical inclusions of an isotropic solid within an isotropic matrix gives rise to a large-scale isotropic response [Mech. Res. Commun. 126 (2022) 104001]. In doing so, we also show that any rhomboidal computational cell that generates such a lattice can be used to successfully design homogenized solids in which the material symmetry is not affected by the periodicity of the construction, since the latter would preserve even the largest possible symmetry group. It is significant to observe that the geometric symmetry group of such rhomboidal cells is strictly smaller than the symmetry group of the lattice they generate, but the lattice and not the cell is the geometrically relevant structure when analyzing large-scale properties.

We frame our discussion in the context of linear elasticity by introducing a normalized Voigt representation of the elasticity tensor which is very convenient for the identification of material parameters and symmetries for inclusion lattices. Such a representation hinges on the definition of a basis for the space of symmetric tensors that is linked to the generators of the periodic lattice. In this way, the coefficients of the 6 by 6 elasticity matrix that describes the linear stress-strain relation acquire a material meaning which is independent of the coordinate basis chosen to represent the tensors.