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Soft biological tissues exhibit a remarkable resilience to large mechanical loads, a property which is associated with the strain stiffening capability of the biopolymer networks that structurally support the tissues. Yet, recent studies have shown that composite systems such as tissues and blood clots exhibit mechanical properties that contradict those of the polymer matrix - demonstrating stiffening in compression, but softening in shear and tension. The microscopic basis of this apparent paradox remains poorly understood. We show that composite hydrogels and tissues do indeed exhibit non-linear elastic stiffening in shear - which is governed by the stretching of the polymer chains in the matrix - and that it is driven by the same mechanism that drives compression stiffening. However, we show that the non-linear elastic stiffening in composite hydrogels and tissues is masked by mechanical dissipation arising from filler-polymer interactions known as the Mullins effect, and we introduce a method to characterize the non-linear elasticity of the composites in isolation from this overall strain softening response through large-amplitude oscillatory shear experiments. We present a comprehensive characterization of the non-linear elastic strain stiffening of composite hydrogels and soft tissues, and show that the strain stiffening in shear and compression are both governed by universal strain amplification factors that depend on essential properties of the composite system, such as the filler concentration and the filler-polymer interaction strength. These results elucidate the microscopic mechanisms governing the non-linear mechanics of tissues, which provides design principles for engineering tissue-mimetic soft materials, and have broad implications for cell-matrix mechanotransduction in living tissues.

Viscoelastic stress relaxation is a basic characteristic of soft matter systems such as colloids, gels, and biological networks. Although the Maxwell model of linear viscoelasticity provides a classical description of stress relaxation, the Maxwell model is often not sufficient for capturing the complex relaxation dynamics of soft matter. In this Tutorial, we introduce and dis-cuss the physics of non-Maxwellian linear stress relaxation as observed in soft materials, the ascribed origins of this effect in different systems, and appropriate models that can be used to capture this relaxation behavior. We provide a basic toolkit that can assist the understanding and modeling of the mechanical relaxation of soft materials for diverse applications.

Gels made of telechelic polymers connected by reversible crosslinkers are a versatile design platform for biocompatible viscoelastic materials. Their linear response to a step strain displays a fast, near-exponential relaxation when using low valence crosslinkers, while larger supramolecular crosslinkers bring about much slower dynamics involving a wide distribution of time scales whose physical origin is still debated. Here, we propose a model where the relaxation of polymer gels in the dilute regime originate from elementary events in which the bonds connecting two neighboring crosslinkers all disconnect. Larger crosslinkers allow for a greater average number of bonds connecting them, but also generate more heterogeneity. We characterize the resulting distribution of relaxation time scales analytically, and accurately reproduce stress relaxation measurements on metal-coordinated hydrogels with a variety of crosslinker sizes including ions, metal-organic cages, and nanoparticles. Our approach is simple enough to be extended to any crosslinker size and could thus be harnessed for the rational design of complex viscoelastic materials., Comment: 6 pages 5 figures 1 table for the main text and 9 pages 7 figures for the supplement