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Strain-stiffening behavior of materials such as rubberlike materials and biological soft tissues is an important phenomenon. In this paper, we proposed a surface Green’s function tensor to describe the strain-stiffening behavior of the stretchable elastomer based on the Gent constitutive model. The surface Green’s function tensor of the Gent constitutive model can be recovered to that of neo-Hookean model, and applied to the indentation problem with a flat-ended cylindrical indenter. The relation between the indentation force and strain-stiffening parameter is analytically derived for equi-biaxial prestretched elastomers. The study shows that the strain-stiffening of the elastomer has a great impact on indentation behaviors, especially for the cases of large prestretches. For a given indentation depth, the indentation force decreases with the increase of the strain-stiffening parameter. For a given stiffening parameter, the indentation force increases with the increase of the prestretches. The proposed surface Green’s function tensor has also potential applications in other fields, such as wetting, cell migration, self-assembly on strain-stiffening materials, etc.

Mechanical responsiveness is prevalent in biological systems and plays an essential role in many biomechanical processes. The past two decades have witnessed enormous effort devoted to the developm...

Collagen networks are usually modeled as an inelastic material that accentuates the role of bond dynamics in their viscoelastic response. However, permanent cross-links can be used to fix these fib...

This work discusses the first set of rheometric measurements carried out on commercially accessible juvenile and aged skin models under large amplitude oscillatory shear deformations. The results were compared with those of native male whole human and dermis-only foreskin specimens, catering to a few ages from 0.5 to 68 years, including specimens from a 23-year-old male abdomen. At large strains, strain thinning was more pronounced for the dermis of the young skins and for their whole skin counterparts. An inverse qualitative tendency was observed for the adult skins and the skin models. This can be explained by the high dermal collagen compactness associated with an incomplete epidermal proliferation. The qualitative Lissajous plots as well as the quantitative dimensionless indices analyzed using the MITlaos software indicated predominant nonlinear intracycle elastic strain stiffening and viscous shear thinning for all the native specimens at the maximum deformation. For the full thickness models, we have evidence of structure collapse and yielding under similar conditions. The whole skin specimen from the 68-year-old male showed smaller age-dependent nonlinear elastic contributions than the dermis, which we relate to the epidermal degeneration taking place during aging. Regardless of the age group, the models manifested more pronounced intercycle and intracycle elastic nonlinearities, and their magnitudes were significantly larger. The nonlinear elastic trends will serve as advanced standards for understanding and delineating the mechanical limits of destructive and non-destructive deformations of such unique biomaterials.

The paper aims at evaluation of mechanical tests of soft tissues and creation of their representative stress–strain responses and respective constitutive models. Interpretation of sets of experimental results depends highly on the approach to the data analysis. Their common representation through mean and standard deviation may be misleading and give nonrealistic results. In the paper, raw data of seven studies consisting of 11 experimental data sets (concerning carotid wall and atheroma tissues) are re-analyzed to show the importance of their rigorous analysis. The sets of individual uniaxial stress–stretch curves are evaluated using three different protocols: stress-based, stretch-based, and constant-based, and the population-representative response is created by their mean or median values. Except for nearly linear responses, there are substantial differences between the resulting curves, being mostly the highest for constant-based evaluation. But also the stretch-based evaluation may change the character of the response significantly. Finally, medians of the stress-based responses are recommended as the most rigorous approach for arterial and other soft tissues with significant strain stiffening.

The structural organization of peripheral nerves enables them to adapt to different body postures and movements by varying their stiffness. Indeed, they could become either compliant or stiff in response to the amount of external solicitation. In this work, the global response of nerves to axial stretch was deterministically derived from the interplay between the main structural constituents of the nerve connective tissue. In particular, a theoretical framework was provided to explicitly decouple the action of the ground matrix and the contribution of the collagen fibrils on the macroscopic stiffening of stretched nerves. To test the overall suitability of this approach, as a matter of principle, the change of the shape of relevant curves was investigated for changes of numerical parameters, while a further sensitivity study was performed to better understand the dependence on them. In addition, dimensionless stress and curvature were used to quantitatively account for both the matrix and the fibril actions. Finally, the proposed framework was used to investigate the stiffening phenomenon in different nerve specimens. More specifically, the proposed approach was able to explicitly and deterministically model the nerve stiffening of porcine peroneal and canine vagus nerves, closely reproducing (R2&gt, 0.997) the experimental data.

Excellent elasticity and thermal stability are both essential for the applications of ultra-high temperature sealing materials. Since strain-stiffening behavior guarantees the materials can sustain large strains without initiation of irreversible defects, seeking for strain-stiffening materials with good thermal stability is significant for the development of thermal sealing at ultra-high temperatures. Strain-stiffening behaviors of crystalline Y 2 B 3 C 2 , a new member of ultra-high temperature ceramics, are predicted under (010)[001] and (010)[100] shear deformation. The moduli against these two shears are extremely low at first, but gradually increase to handsome values of 62 and 101 GPa. The ultra-high ideal shear strain and considerable ideal stress indicate the promising applications of Y 2 B 3 C 2 in thermal sealing at ultra-high temperatures. It is noticed that the strain-stiffening effect is originated from the deformable planar B-C network. The special bonding features in the CBC-chain are found to be responsible for the easy distortion of the covalent network.

Biological tissues can accurately differentiate external mechanical stresses and actively select suitable strategies (e.g., reversible strain-stiffening, self-healing) to sustain or restore their integrity and related functionalities as required. Synthetic materials that can imitate the characteristics of biological tissues have a wide range of engineering and bioengineering applications. However, no success has been demonstrated to realize such strain-stiffening behavior in synthetic networks, particularly using flexible polymers, which has remained a great challenge. Here, we present one such synthetic hydrogel material prepared from two flexible polymers (polyethylene glycol and branched polyethylenimine) that exhibits both strain-stiffening and self-healing capabilities. The developed synthetic hydrogel network not only mimics the main features of biological mechanically responsive systems but also autonomously self-heals after becoming damaged, thereby recovering its full capacity to perform its normal physiological functions.

Polymer nanocomposites-materials in which a polymer matrix is blended with nanoparticles (or fillers)-strengthen under sufficiently large strains. Such strain hardening is critical to their function, especially for materials that bear large cyclic loads such as car tires or bearing sealants. Although the reinforcement (i.e., the increase in the linear elasticity) by the addition of filler particles is phenomenologically understood, considerably less is known about strain hardening (the nonlinear elasticity). Here, we elucidate the molecular origin of strain hardening using uniaxial tensile loading, microspectroscopy of polymer chain alignment, and theory. The strain-hardening behavior and chain alignment are found to depend on the volume fraction, but not on the size of nanofillers. This contrasts with reinforcement, which depends on both volume fraction and size of nanofillers, potentially allowing linear and nonlinear elasticity of nanocomposites to be tuned independently.

openaire: EC/H2020/742829/EU//DRIVEN Strain-stiffening is one of the characteristic properties of biological hydrogels and extracellular matrices, where the stiffness increases upon increased deformation. Whereas strain-stiffening is ubiquitous in protein-based materials, it has been less observed for polysaccharide and synthetic polymer gels. Here we show that agarose, that is, a common linear polysaccharide, forms helical fibrillar bundles upon cooling from aqueous solution. The hydrogels with these semiflexible fibrils show pronounced strain-stiffening. However, to reveal strain-stiffening, suppressing wall slippage turned as untrivial. Upon exploring different sample preparation techniques and rheological architectures, the cross-hatched parallel plate geometries and in situ gelation in the rheometer successfully prevented the slippage and resolved the strain-stiffening behavior. Combining with microscopy, we conclude that strain-stiffening is due to the semiflexible nature of the agarose fibrils and their geometrical connectivity, which is below the central-force isostatic critical connectivity. The biocompatibility and the observed strain-stiffening suggest the potential of agarose hydrogels in biomedical applications.

Strain-stiffening is a well-documented behavior in soft biological materials such as liver and brain tissue. Measuring and characterizing this nonlinear response, which is commonly considered as a mechanism for damage prevention, is of great interest to engineers for design of better biomimetic materials, and to physicians for diagnostic purposes. However, probing the elastic response of soft or biological materials at large deformation in their natural habitat, is an arduous task. Here, we present the Volume Controlled Cavity Expansion (VCCE) technique as a measurement method that offers the ability of characterizing the local stiffening response of materials in addition to identifying their shear modulus. By employing minimal constitutive representations involving only two constants (Mooney–Rivlin, Gent, and Ogden) we show that for the conventional PDMS samples, this technique and an accompanying data analysis method capture the shear modulus, as well as providing reliable measures of the stiffening behavior of the samples.

Heart valves are characterized to be highly flexible yet tough, and exhibit complex deformation characteristics such as nonlinearity, anisotropy, and viscoelasticity, which are, at best, only partially recapitulated in scaffolds for heart valve tissue engineering (HVTE). These biomechanical features are dictated by the structural properties and microarchitecture of the major tissue constituents, in particular collagen fibers. In this study, the unique capabilities of melt electrowriting (MEW) are exploited to create functional scaffolds with highly controlled fibrous microarchitectures mimicking the wavy nature of the collagen fibers and their load-dependent recruitment. Scaffolds with precisely-defined serpentine architectures reproduce the J-shaped strain stiffening, anisotropic and viscoelastic behavior of native heart valve leaflets, as demonstrated by quasistatic and dynamic mechanical characterization. They also support the growth of human vascular smooth muscle cells seeded both directly or encapsulated in fibrin, and promote the deposition of valvular extracellular matrix components. Finally, proof-of-principle MEW trileaflet valves display excellent acute hydrodynamic performance under aortic physiological conditions in a custom-made flow loop. The convergence of MEW and a biomimetic design approach enables a new paradigm for the manufacturing of scaffolds with highly controlled microarchitectures, biocompatibility, and stringent nonlinear and anisotropic mechanical properties required for HVTE.

Gels represent an increasingly important class of soft materials with applications ranging from regenerative medicine to commodity materials. However, gels typically exhibit relative mechanical weakness, which worsens under repeated strain. Here we report a new class of responsive gels with latent crosslinking moieties that exhibit strain-stiffening behavior. This property results from the lability of disulfides, initially isolated in a protected state, then activated to crosslink on-demand. The thiol groups are induced to form inter-chain crosslinks when subjected to mechanical compression, resulting in a gel that strengthens under strain. Molecular shielding design elements regulate the strain-sensitivity and spontaneous crosslinking tendencies of the polymer network. These strain-responsive gels represent a rational design of new advanced materials with on-demand stiffening properties and potential applications in elastomers, adhesives, foams, films, and fibers.

In this paper, nonlinear resonance characteristics of a dielectric elastomer actuator are investigated with special consideration on the thermal effects. A finite thermo-elasticity model based on the Gent model is constructed to analyze the vibrational response of the system. The equation of motion is derived via the Euler–Lagrange method. The multiple scales method and the Taylor series expansion are used to solve the governing equation. Nonlinear resonant responses of the system such softening/hardening and jump are explored. Furthermore, the influences of different system parameters including temperature, limiting stretch, damping, mechanical load, relative permittivity and voltage on the frequency response curves are explored. The results are compared with those obtained in the isothermal state, and those solved by numerical methods. It is found that both softening and hardening-type nonlinearities occur in the system in both non-thermal and thermal conditions.

Filamentous biomaterials such as fibrin or collagen networks exhibit an enormous stiffening of their elastic moduli upon large deformations. This pronounced nonlinear behavior stems from a significant separation between the stiffnesses scales associated with bending versus stretching the material's constituent elements. Here we study a simple model of such materials, floppy networks of hinged rigid bars embedded in an elastic matrix, in which the effective ratio of bending to stretching stiffnesses vanishes identically. We introduce a theoretical framework and build upon it to construct a numerical method with which the model's micro- and macromechanics can be carefully studied. Our model, numerical method and theoretical framework allow us to robustly observe and fully understand the critical properties of the athermal strain-stiffening transition that underlies the nonlinear mechanical response of a broad class of biomaterials.

Active camouflage from a polymer Human skin is soft and compliant, but it can quickly become stiff when deformed to prevent injury. Chameleon skin can change color when the animal goes from a relaxed to an excited state. Although these properties can be captured individually in synthetic materials, the combination of different dynamic responses can be hard to control. Vatankhah-Varnosfaderani et al. created triblock copolymers of the ABA variety, where the A blocks have a linear structure and the B blocks are like bottlebrushes. When strained, these polymers stiffened like human skin and changed color, thus giving the materials a range of adaptive properties. Science , this issue p. 1509

We present a novel approach to understand geometric-incompatibility-induced rigidity in under-constrained materials, including sub-isostatic 2D spring networks and 2D and 3D vertex models for dense biological tissues. We show that in all these models a geometric criterion, represented by a minimal length $\bar\ell_\mathrm{min}$, determines the onset of prestresses and rigidity. This allows us to predict not only the correct scalings for the elastic material properties, but also the precise {\em magnitudes} for bulk modulus and shear modulus discontinuities at the rigidity transition as well as the magnitude of the Poynting effect. We also predict from first principles that the ratio of the excess shear modulus to the shear stress should be inversely proportional to the critical strain with a prefactor of three, and propose that this factor of three is a general hallmark of geometrically induced rigidity in under-constrained materials and could be used to distinguish this effect from nonlinear mechanics of single components in experiments. Lastly, our results may lay important foundations for ways to estimate $\bar\ell_\mathrm{min}$ from measurements of local geometric structure, and thus help develop methods to characterize large-scale mechanical properties from imaging data., Comment: 10 pages, 5 figures

When a confined elastic layer is under tension, undulations can occur at its exposed surfaces, giving the fingering or fringe instability. These instabilities are of great concern in the design of robust adhesives, since they not only initiate severe local deformations in adhesive layers but also cause non-monotonic overall stress vs. stretch relations of the layers. Here, we show that the strain stiffening of soft elastic materials can significantly delay and even suppress the fringe and fingering instabilities, and give monotonic stress vs. stretch relations. Instability development requires local large deformation, which can be inhibited by material-stiffening. We provide a quantitative phase diagram to summarize the stiffening's effects on the instabilities and stress vs. stretch relations in confined elastic layers. We further use numerical simulations and experiments to validate our findings.

Transition-metal light-element compounds are a class of designer materials tailored to be a new generation of superhard solids, but indentation strain softening has hitherto limited their intrinsic load-invariant hardness to well below the 40 GPa threshold commonly set for superhard materials. Here we report findings from first-principles calculations that two tungsten nitrides, hP4-WN and hP6-WN_{2}, exhibit extraordinary strain stiffening that produces remarkably enhanced indentation strengths exceeding 40 GPa, raising exciting prospects of realizing the long-sought nontraditional superhard solids. Calculations show that hP4-WN is metallic both at equilibrium and under indentation, marking it as the first known intrinsic superhard metal. An x-ray diffraction pattern analysis indicates the presence of hP4-WN in a recently synthesized specimen. We elucidate the intricate bonding and stress response mechanisms for the identified structural strengthening, and the insights may help advance rational design and discovery of additional novel superhard materials.

Structure-property relationships of silk is an intriguing topic for silk-based biomaterials research since these features are related to biomimicking the processing in natural silk fiber formation which results in excellent mechanical properties. Strain-stiffening is common for spider silks and nonmulberry silkworm silks. However, the structural origin of strain-stiffening remains unclear. In this paper, the strain-dependent structural change of Antheraea pernyi silkworm silk is studied by X-ray fiber diffraction and Fourier transform infrared spectroscopy under stretching. Based on a combination of mechanical and structural analysis, the molecular origins of strain-stiffening in A. pernyi silk were determined. The relatively high content of the β-sheets within the amorphous domains in A. pernyi silk is responsible for strain-stiffening, where "molecular spindles" enhance the extensibility and toughness of the fiber.

Despite the accepted notion that erythrocytes are uniquely deformable cells, the apparent Young’s modulus values reported in the literature do not differ so much from those of other cells. We devised to measure the local deformability of living immobilized human erythrocytes at a low force, in contact-free mode, using an application of Scanning Ion Conductance Microscopy (SICM) previously developed in our laboratory. Reversible indentations were induced by forces of up to few hundreds pN. The indentation did not grow linearly with the force. The apparent Young’s modulus varied from 0.2 to 1.5 kPa applying forces from 20 to 500 pN on a cell surface area of about 0.2 μm2, exhibiting a progressive stiffening at increasing force. Control measurements showed that A549 cells exhibit a constant value of the apparent Young’s modulus (about 2 kPa) for forces up to about 800 pN. These findings show that SICM is a suitable tool to investigate cell mechanical properties, when forces in the range of tens of pN are required, in the absence of mechanical contact between probe and sample. The nonlinear deformation of the erythrocyte has to be taken into account in modeling the complex regulation mechanism of the microvascular beds.

Intermediate filaments are common structural elements found in abundance in all metazoan cells, where they form networks that contribute to the elasticity. Here, we report measurements of the linear and nonlinear viscoelasticity of networks of two distinct intermediate filaments, vimentin and neurofilaments. Both exhibit predominantly elastic behavior with strong nonlinear strain stiffening. We demonstrate that divalent ions behave as effective cross-linkers for both networks, and that the elasticity of these networks is consistent with the theory for that of semiflexible polymers. © 2010 The American Physical Society.

We develop a model for gels and entangled solutions of semiflexible biopolymers such as F-actin. Such networks play a crucial structural role in the cytoskeleton of cells. We show that the rheologic properties of these networks can result from nonclassical rubber elasticity. This model can explain a number of elastic properties of such networks {\em in vitro}, including the concentration dependence of the storage modulus and yield strain., Comment: Uses RevTeX, full postscript with figures available at http://www.umich.edu/~fcm/preprints/agel/agel.html

The purpose of this paper is to investigate the effects of strain-stiffening for the classical problems of axial and azimuthal shearing of a hollow circular cylinder composed of an incompressible isotropic non-linearly elastic material. For some specific strain-energy densities that give rise to strain-stiffening in the stress–stretch response, the stresses and resultant axial forces are obtained in explicit closed form. While such results are well known for classical constitutive models such as the Mooney–Rivlin and neo-Hookean models, our main focus is on materials that undergo severe strain-stiffening in the stress–stretch response. In particular, we consider in detail two phenomenological constitutive models that reflect limiting chain extensibility at the molecular level and involve constraints on the deformation. The amount of shearing that tubes composed of such materials can sustain is limited by the constraint. Numerical results are also obtained for an exponential strain-energy that exhibits a less abrupt strain-stiffening effect. Potential applications of the results to the biomechanics of soft tissues are indicated.

Materials such as elastomeric foams, lattices, and cellular solids are capable of undergoing large elastic volume changes. Although many hyperelastic constitutive formulations have been proposed for deviatoric (shape changing) behaviour, few variations exist for large deformation volumetric behaviour. The first section of this paper presents a critical analysis of current volumetric hyperelastic models and highlights their limitations for large volumetric strains. In the second section of the paper we propose three novel volumetric strain energy density functions, which: (1) are valid for large volumetric deformations, (2) offer separate control of the volumetric strain stiffening behaviour during shrinkage (volume reduction) and expansion (volume increase), and (3) provide precise control of non-monotonic volumetric strain stiffening. To illustrate the ability of the novel formulations to capture complex volumetric material behaviour they are fitted and compared to a range of published experimental data.