Session K26: Mechanics of Soft Disordered Networks: From Remodeling to Fracture

Mechanics and fracture properties of soft disordered networks in cartilage and cartilage inspired materials

Articular cartilage (AC) is a soft tissue that provides a smooth cushion and distributes the mechanical load in joints. AC’s mechanical properties mainly arise from a composite matrix made up of extracellular collagen and aggrecan networks.  As a material, AC is remarkable. It is only a few millimeters thick, can bear up to ten times our body weight over 100-200 million loading cycles despite minimal regenerative capacity, and still avoids fracturing. The simultaneous strength, fracture resistance, and longevity of native AC remain unmatched in synthetic materials. Such properties are desperately needed for tissue engineering, tissue repair, and even soft robotics applications. I will discuss the structural origins of AC’s exceptional mechanical properties using the framework of rigidity percolation theory applied to composite materials and compare our predictions with experiments. Our results provide an understanding of the tissue depth-dependent mechanical properties and how tissue mechanics changes in response to changes in tissue composition during diseases such as osteoarthritis. This framework also offers insights into how structure, composition, and constitutive mechanical properties can be tuned to resist and blunt cracks in AC and cartilage-inspired soft materials. The flexibility in resulting material properties and ease of implementation can be harnessed to fabricate artificial tissue constructs with tunable mechanics. I will discuss results that are an important step towards achieving this future.   

Rigidity percolation in a random tensegrity via analytic graph theory

Tensegrities are mechanical structures that include cable-like elements that are strong and lightweight relative to rigid rods yet support only extensile stress. From suspension bridges to the musculoskeletal system to individual biological cells, humanity makes excellent use of tensegrities, yet the sharply nonlinear response of cables presents serious challenges to analytical theory. Here we consider large tensegrity structures with randomly placed cables (and struts) overlaid on a regular rigid backbone whose corresponding system of inequalities is reduced via analytic theory to an exact graph theory. We identify a novel coordination number that controls two rigidity percolation transitions: one in which global interactions between cables first support external loads and one in which the structure becomes fully rigid.  We show that even the addition of a few cables strongly modifies conventional rigidity percolation, both by modifying the sharpness of the transition and by introducing avalanche effects in which a single constraint can eliminate multiple floppy modes.    

Self-organized buckling patterns underlie transition from macroscopic extension to contraction in active nonlinear elastic networks.

Many fundamental cellular processes require exquisitely orchestrated large-scale reorganization of structural filaments. One mechanism of reorganization is via internal forces generated by motor proteins. The transmission of these forces is mediated by a highly non-linear network of fiber-like filaments. To understand the role of buckling and failure in such networks, we examine a model nonlinear elastic network subjected to an internal force dipole. Such networks exhibit non-monotonic elastic deformation in response to the applied force. We observe a transition from linear and non-linear extensility to global contractile behavior. We demonstrate this emergence of contractile behavior is associated with a large-scale transformation of the underlying lattice structure. These results recapitulate observations of active microtubule/actin gels which transition from extensile flows to global contraction [J. Berezney et. al., arXiv 2110.00166]. This work underscores the importance of cytoskeletal networks and metamaterials whose failure modes and nonlinear mechanics can be engineered to generate complex and adaptive large-scale phenomena.   

Deformation and Break-up of 3D-Printed Soft Elastic Solids

Bioprinting and tissue manufacturing requires the precise placement of cells and extracellular matrix into macroscopic sized structures with microscopic resolution. However, cell-generated stresses within these 3D-printed structures can result in deformations and instabilities that will alter the shape as it matures. Similarly, capillary forces acting at the interface of soft elastic solids can drive deformations over the length-scale of the elasto-capillary length. These interfacial instabilities provide an opportunity to explore the shape-evolution of 3D-printed structural elements, including beams, sheets, and tubes and can be leveraged to develop new design principles for 3D-bioprinting and tissue manufacturing that capture the deformations resulting from cell-generated stresses. Here, we explore the effects of capillary driven deformation on soft elastic beams, using packed granular microgels as our printed ink. By leveraging the highly tunable material properties of these packed microgel systems, we systematically explore the stability of 3D-printed beams across a range of yield stresses and beam diameters. Furthermore, we compare these interfacially-driven instabilities to the deformation of 3D-printed cellular microbeams under cell-generated stresses.   

Trainability in biopolymer-inspired disordered spring networks

Disordered networks are ubiquitous in nature, particularly in biological systems where they serve as the mechanical foundation of structures ranging from proteins to the cytoskeleton. Previously it was shown that disordered spring networks can be tuned to exhibit target mechanical properties by selectively removing individual bonds, but breaking events in biological networks usually involve more complicated changes of the network topology. Inspired by this, we introduce a disordered spring network model that shows the basic features observed in biopolymer networks, including force/strain-induced changes in topology. Nodes can be removed/broken, analogous to crosslink unbinding, with the edges attached to it being redistributed to neighboring nodes while preserving topology and conserving rest length (akin to preserving mass). We simulate these networks under cyclic strain protocols and determine the mechanical properties of the resulting networks after breaking events. The resulting networks are stressed even in the unstrained state due to rearrangement of the edges. The networks studied are trainable to develop different desired mechanical properties when the initial state has been prepared to have suitable disorder and internal stresses.    

Relaxation in network materials: the contribution of the network

Stress relaxation in network materials is controlled by the relaxation of the network structure, the migration of solvent and the viscoelasticity of the embedding matrix. In this work we focus on the contribution of the network to this process. To this end, we consider athermal random networks in which fibers behavior is described by a Maxwell model and monitor the relaxation on network scale. We observe a behavior similar to that reported in many glass formers close to Tg: two relaxation regimes, of which the early time relaxation is exponential, while the late times are described by a stretched exponential. We study the dependence of the stretch exponent on network parameters and observe the maximum slowdown in networks at the transition between the affine and non-affine regimes. The parameter controlling the physical nature of relaxation in this athermal system is a non-dimensional group which is also used to distinguish between affine and non-affine networks. This parameter replaces temperature which plays a similar role in thermal systems. Direct microscopic observations of the dynamics support this physical picture.   

Strain-induced critical slowing of stress relaxation in disordered networks

Common biological materials contain embedded fibrous networks that stiffen significantly in response to applied strain, a phenomenon that provides living tissues with enhanced mechanical resilience and enables long-range force transmission by cells. Prior work has shown that strain-induced stiffening in mechanically under-constrained elastic networks corresponds to the traversal of a “critical point” in strain, at which the energy-minimizing network configuration becomes marginally stable. Near this transition, various quantities measured via quasistatic deformation exhibit power law scaling with respect to the reduced strain. However, the dynamics of systems near this transition remain poorly understood. Here, we model the small-amplitude oscillatory shear and stress relaxation of under-constrained, fluid-immersed elastic networks subjected to applied extensional pre-strain (e.g. swelling and simple shear). We show that the rheology of these networks is controlled by a pre-strain-controlled correlation length and corresponding characteristic relaxation time that both diverge at a connectivity-controlled critical point, giving rise to weak power law scaling of the complex modulus with frequency and diverging strain fluctuations.   

Mechanics of sub-isostatic fiber networks at a finite temperature

Fibrous networks are a ubiquitous component of physiological systems, e.g., the interconnected collagen protein in the extracellular matrix (ECM). These networks are responsible for the mechanical stability of cells and tissues. It has been shown that a simple coarse-grained model of spring networks with bending interactions can capture the rheology of real biopolymers. Maxwell showed that the athermal spring networks with an average connectivity below a threshold (isostatic point) are unstable under small deformations. The experimental studies confirm that the average connectivity of real biopolymers is far below the isostatic connectivity. Under a finite applied shear strain, these sub-isostatic networks undergo a transition from a floppy to a rigid state at a critical strain that depends on the connectivity and geometry of the network structure. In the linear regime, on the other hand, sub-isostatic networks at finite temperature exhibit a non-zero shear modulus that, in contrast to entropic elasticity, has an anomalous dependence on temperature. Using a Monte Carlo method, we study this temperature-dependence elasticity and the corresponding critical exponents in sub-isostatic networks near their critical strain in both 2D and 3D models. Interestingly, our results will shed light on the analogy between this floppy-to-rigid phase transition and the zero-temperature criticality in quantum systems.   

Hole closure in sheets of active deformable particles

Holes created in sheets of epithelial cells provide an in vitro experimental system to gain insight into wound healing in biological tissues. Experimental studies have found that holes close as a result of collective cell motion that is influenced by cell shape, motility, and adhesion. We propose the deformable particle (DP) model to study the mechanics of hole closure in epithelial sheets, where the cells are modeled as individual deformable polygons with N_v vertices whose positions are determined by a shape-energy function with terms that control the cell area, perimeter, and curvature. To model the dynamics of hole closure, we include adhesive protrusions and collective contractile tensions for cells along the perimeter of the hole.  An important advantage of the DP model for describing hole closure is the ability for individual cells to take on a wide range of complex shapes.  We vary the cell adhesion, shape parameter, rate of protrusion, and contractile tension to determine how they affect collective cell motion and the rate of hole closure.   

Phase separation and viscoelastic ripening in crosslinked networks

The process of phase separation in elastic solids and viscous fluids is of fundamental importance to the stability and function of soft materials. We explore the dynamics of phase separation and domain growth in a viscoelastic material such as a polymer gel. Using analytical theory and Monte Carlo simulations we find a new domain-growth regime where the domain size increases as a power law, with a ripening exponent α that depends on the viscoelastic properties of the material. For example, α = 1 in a Maxwell material, which is markedly different from the well-known Ostwald ripening process with α = 1/3. We generalize our theory to systems with arbitrary power-law relaxation behavior and discuss our findings in the context of the long-term stability of materials and recent experimental results on phase separation in cross-linked networks and cell cytoskeleton.   

Disordered Topological Metamaterials

Current designs of metamaterials achieve unique mechanical properties such as phononic bandgaps and topological phonon modes by precise control of unit-cell structures, which poses challenges for miniaturization. Recently novel material design techniques have been developed to utilize disordered materials and the slow evolution through their rugged energy landscapes, referred to as directed aging. This kind of design has been used to produce metamaterials with specialized bulk properties. In this talk, we will present our investigation on mechanisms for encoding topological material properties into packing-derived disordered networks. Topological edge floppy modes arise in ordered and disordered materials close to the verge of mechanical instability, and we will show how these modes may be encoded in disordered materials via directed aging without the need to precisely control the microstructure.   

Fast detection of early-stage damage in soft elastomers

In order to investigate the fracture mechanism of soft polymer networks, it is necessary to cover a wide range of length scales: from the molecular scale with bond breaking in stretched chains to the macroscopic scale with the crack propagation and rupture of the material. In this work, we developed time- and space-resolved multi-speckle diffusing wave spectroscopy (MSDWS) coupled with molecular scale measurement of bond breaking, to provide a new approach to studying fracture mechanics in multiple length scales.

Based on a single-edge notched PDMS elastomer with the additional mechanophore and probe particles, strain rate distribution over the whole sample and bond breaking around the crack can be simultaneously mapped until macroscopic fracture. During induction period of macroscopic fracture, we discovered a large scale strain rate localization (~1 cm²), which is related to local damage in the scales of tens of microns. Applying the results, we developed a detection method of microscopic damage long time before macroscopically observable. Our insight of the coupling between ongoing molecule damage and dynamic mechanical response will also provide us complete vision to the research of fracture mechanism and advanced molecular design for soft material development.