Local response and emerging nonlinear elastic length scale in biopolymer matrices.

Nonlinear stiffening is a ubiquitous property of major types of biopolymers that make up the extracellular matrices (ECM) including collagen, fibrin, and basement membrane. Within the ECM, many types of cells such as fibroblasts and cancer cells have a spindle-like shape that acts like two equal and opposite force monopoles, which anisotropically stretch their surroundings and locally stiffen the matrix. Here, we first use optical tweezers to study the nonlinear force-displacement response to localized monopole forces. We then propose an effective-probe scaling argument that a local point force application can induce a stiffened region in the matrix, which can be characterized by a nonlinear length scale R* that increases with the increasing force magnitude; the local nonlinear force-displacement response is a result of the nonlinear growth of this effective probe that linearly deforms an increasing portion of the surrounding matrix. Furthermore, we show that this emerging nonlinear length scale R* can be observed around living cells and can be perturbed by varying matrix concentration or inhibiting cell contractility.

Flexoelectric Enhancement of Strain Gradient Elasticity Across a Ferroelectric-to-Paraelectric Phase Transition.

We study the temperature dependent elastic properties of Ba0.8Sr0.2TiO3 freestanding membranes across the ferroelectric-to-paraelectric phase transition using an atomic force microscope. The bending rigidity of thin membranes can be stiffer compared to stretching due to strain gradient elasticity (SGE). We measure the Young's modulus of freestanding Ba0.8Sr0.2TiO3 drumheads in bending and stretching dominated deformation regimes on a variable temperature platform, finding a peak in the difference between the two Young's moduli obtained at the phase transition. This demonstrates a dependence of SGE on the dielectric properties of a material and alludes to a flexoelectric origin of an effective SGE.

Uniformity of anti-plane stresses inside a nonlinear elastic elliptical or parabolic inhomogeneity.

We study the anti-plane strain problem associated with a p-Laplacian nonlinear elastic elliptical inhomogeneity embedded in an infinite linear elastic matrix subjected to uniform remote anti-plane stresses. A full-field exact solution is derived using complex variable techniques. It is proved that the stress field inside the elliptical inhomogeneity is nevertheless uniform. The uniformity of stresses is also observed inside a p-Laplacian nonlinear elastic parabolic inhomogeneity.

Accretion-ablation mechanics.

In this paper, we formulate a geometric nonlinear theory of the mechanics of accreting-ablating bodies. This is a generalization of the theory of accretion mechanics of Sozio & Yavari (Sozio & Yavari 2019 J. Nonlinear Sci. 29, 1813-1863 (doi:10.1007/s00332-019-09531-w)). More specifically, we are interested in large deformation analysis of bodies that undergo a continuous and simultaneous accretion and ablation on their boundaries while under external loads. In this formulation, the natural configuration of an accreting-ablating body is a time-dependent Riemannian [Formula: see text]-manifold with a metric that is an unknown a priori and is determined after solving the accretion-ablation initial-boundary-value problem. In addition to the time of attachment map, we introduce a time of detachment map that along with the time of attachment map, and the accretion and ablation velocities, describes the time-dependent reference configuration of the body. The kinematics, material manifold, material metric, constitutive equations and the balance laws are discussed in detail. As a concrete example and application of the geometric theory, we analyse a thick hollow circular cylinder made of an arbitrary incompressible isotropic material that is under a finite time-dependent extension while undergoing continuous ablation on its inner cylinder boundary and accretion on its outer cylinder boundary. The state of deformation and stress during the accretion-ablation process, and the residual stretch and stress after the completion of the accretion-ablation process, are computed. This article is part of the theme issue 'Foundational issues, analysis and geometry in continuum mechanics'.

Geometric charges and nonlinear elasticity of two-dimensional elastic metamaterials.

Problems of flexible mechanical metamaterials, and highly deformable porous solids in general, are rich and complex due to their nonlinear mechanics and the presence of nontrivial geometrical effects. While numeric approaches are successful, analytic tools and conceptual frameworks are largely lacking. Using an analogy with electrostatics, and building on recent developments in a nonlinear geometric formulation of elasticity, we develop a formalism that maps the two-dimensional (2D) elastic problem into that of nonlinear interaction of elastic charges. This approach offers an intuitive conceptual framework, qualitatively explaining the linear response, the onset of mechanical instability, and aspects of the postinstability state. Apart from intuition, the formalism also quantitatively reproduces full numeric simulations of several prototypical 2D structures. Possible applications of the tools developed in this work for the study of ordered and disordered 2D porous elastic metamaterials are discussed.

Strain energy density as a Gaussian process and its utilization in stochastic finite element analysis: application to planar soft tissues.

Data-based approaches are promising alternatives to the traditional analytical constitutive models for solid mechanics. Herein, we propose a Gaussian process (GP) based constitutive modeling framework, specifically focusing on planar, hyperelastic and incompressible soft tissues. The strain energy density of soft tissues is modeled as a GP, which can be regressed to experimental stress-strain data obtained from biaxial experiments. Moreover, the GP model can be weakly constrained to be convex. A key advantage of a GP-based model is that, in addition to the mean value, it provides a probability density (i.e. associated uncertainty) for the strain energy density. To simulate the effect of this uncertainty, a non-intrusive stochastic finite element analysis (SFEA) framework is proposed. The proposed framework is verified against an artificial dataset based on the Gasser-Ogden-Holzapfel model and applied to a real experimental dataset of a porcine aortic valve leaflet tissue. Results show that the proposed framework can be trained with limited experimental data and fits the data better than several existing models. The SFEA framework provides a straightforward way of using the experimental data and quantifying the resulting uncertainty in simulation-based predictions.

Exponents of the one-term Ogden model: insights from simple shear.

Isotropic one-term Ogden models are widely used to predict the mechanical response of both incompressible elastomers and soft tissue. Even though the exponent might be chosen to yield excellent agreement with some aspects of mechanical response, there is no guarantee that these models will be physically realistic in all situations. We show here that, in particular, the predictions of models with either negative or large positive exponents do not seem physically realistic in simple shear. The mechanical response of materials in shear should be physically realistic to ensure rational and reliable predictions for complex geometries and boundary conditions. We suggest that for problematic values of exponents of one-term models that extra Ogden invariants should necessarily be included in the model. This article is part of the theme issue 'The Ogden model of rubber mechanics: Fifty years of impact on nonlinear elasticity'.

Linear, weakly nonlinear and fully nonlinear models for soft tissues: which ones provide the most reliable estimations of the stiffness?

Benign and malignant lesions in tissues or organs can be detected by elastographic investigations in which pathological regions are spotted from local alterations of the stiffness. As is known, the shear modulus provides a measure of the stiffness of an elastic material. Based on the classical theory of linear elasticity, an elastogram yields estimations of the linear shear modulus from measurements of the speed of small-amplitude transverse waves propagating in the medium tested. In this paper, we show that the estimation of the shear modulus can be improved significantly by employing the fourth-order weakly nonlinear theory of elasticity (FOE), and indicate how the stiffness can be assessed more precisely with the use of FOE. We discuss also why FOE provides more reliable results than the fully nonlinear theory of elasticity. This article is part of the theme issue 'The Ogden model of rubber mechanics: Fifty years of impact on nonlinear elasticity'.

Cell contraction induces long-ranged stress stiffening in the extracellular matrix.

Animal cells in tissues are supported by biopolymer matrices, which typically exhibit highly nonlinear mechanical properties. While the linear elasticity of the matrix can significantly impact cell mechanics and functionality, it remains largely unknown how cells, in turn, affect the nonlinear mechanics of their surrounding matrix. Here, we show that living contractile cells are able to generate a massive stiffness gradient in three distinct 3D extracellular matrix model systems: collagen, fibrin, and Matrigel. We decipher this remarkable behavior by introducing nonlinear stress inference microscopy (NSIM), a technique to infer stress fields in a 3D matrix from nonlinear microrheology measurements with optical tweezers. Using NSIM and simulations, we reveal large long-ranged cell-generated stresses capable of buckling filaments in the matrix. These stresses give rise to the large spatial extent of the observed cell-induced matrix stiffness gradient, which can provide a mechanism for mechanical communication between cells.

Strongly anisotropic surface elasticity and antiplane surface waves.

Within the new model of surface elasticity, the propagation of anti-plane surface waves is discussed. For the proposed model, the surface strain energy depends on surface stretching and on changing of curvature along a preferred direction. From the continuum mechanics point of view, the model describes finite deformations of an elastic solid with an elastic membrane attached on its boundary reinforced by a family of aligned elastic long flexible beams. Physically, the model was motivated by deformations of surface coatings consisting of aligned bar-like elements as in the case of hyperbolic metasurfaces. Using the least action variational principle, we derive the dynamic boundary conditions. The linearized boundary-value problem is also presented. In order to demonstrate the peculiarities of the problem, the dispersion relations for surface anti-plane waves are analysed. We have shown that the bending stiffness changes essentially the dispersion relation and conditions of anti-plane surface wave propagation. This article is part of the theme issue 'Modelling of dynamic phenomena and localization in structured media (part 2)'.

Nanoparticle amount, and not size, determines chain alignment and nonlinear hardening in polymer nanocomposites.

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.

Revisiting the wrinkling of elastic bilayers I: linear analysis.

Wrinkling is a universal instability occurring in a wide variety of engineering and biological materials. It has been studied extensively for many different systems but a full description is still lacking. Here, we provide a systematic analysis of the wrinkling of a thin hyperelastic film over a substrate in plane strain using stream functions. For comparison, we assume that wrinkling is generated either by the isotropic growth of the film or by the lateral compression of the entire system. We perform an exhaustive linear analysis of the wrinkling problem for all stiffness ratios and under a variety of additional boundary and material effects. Namely, we consider the effect of added pressure, surface tension, an upper substrate and fibres. We obtain analytical estimates of the instability in the two asymptotic regimes of long and short wavelengths. This article is part of the theme issue 'Rivlin's legacy in continuum mechanics and applied mathematics'.

Are Homeostatic States Stable? Dynamical Stability in Morphoelasticity.

Biological growth is often driven by mechanical cues, such as changes in external pressure or tensile loading. Moreover, it is well known that many living tissues actively maintain a preferred level of mechanical internal stress, called the mechanical homeostasis. The tissue-level feedback mechanism by which changes in the local mechanical stresses affect growth is called a growth law within the theory of morphoelasticity, a theory for understanding the coupling between mechanics and geometry in growing and evolving biological materials. This coupling between growth and mechanics occurs naturally in macroscopic tubular structures, which are common in biology (e.g., arteries, plant stems, airways). We study a continuous tubular system with spatially heterogeneous residual stress via a novel discretization approach which allows us to obtain precise results about the stability of equilibrium states of the homeostasis-driven growing dynamical system. This method allows us to show explicitly that the stability of the homeostatic state depends nontrivially on the anisotropy of the growth response. The key role of anisotropy may provide a foundation for experimental testing of homeostasis-driven growth laws.

Fibrous nonlinear elasticity enables positive mechanical feedback between cells and ECMs.

In native states, animal cells of many types are supported by a fibrous network that forms the main structural component of the ECM. Mechanical interactions between cells and the 3D ECM critically regulate cell function, including growth and migration. However, the physical mechanism that governs the cell interaction with fibrous 3D ECM is still not known. In this article, we present single-cell traction force measurements using breast tumor cells embedded within 3D collagen matrices. We recreate the breast tumor mechanical environment by controlling the microstructure and density of type I collagen matrices. Our results reveal a positive mechanical feedback loop: cells pulling on collagen locally align and stiffen the matrix, and stiffer matrices, in return, promote greater cell force generation and a stiffer cell body. Furthermore, cell force transmission distance increases with the degree of strain-induced fiber alignment and stiffening of the collagen matrices. These findings highlight the importance of the nonlinear elasticity of fibrous matrices in regulating cell-ECM interactions within a 3D context, and the cell force regulation principle that we uncover may contribute to the rapid mechanical tissue stiffening occurring in many diseases, including cancer and fibrosis.

Stress controls the mechanics of collagen networks.

Collagen is the main structural and load-bearing element of various connective tissues, where it forms the extracellular matrix that supports cells. It has long been known that collagenous tissues exhibit a highly nonlinear stress-strain relationship, although the origins of this nonlinearity remain unknown. Here, we show that the nonlinear stiffening of reconstituted type I collagen networks is controlled by the applied stress and that the network stiffness becomes surprisingly insensitive to network concentration. We demonstrate how a simple model for networks of elastic fibers can quantitatively account for the mechanics of reconstituted collagen networks. Our model points to the important role of normal stresses in determining the nonlinear shear elastic response, which can explain the approximate exponential relationship between stress and strain reported for collagenous tissues. This further suggests principles for the design of synthetic fiber networks with collagen-like properties, as well as a mechanism for the control of the mechanics of such networks.

Nonlinear elasticity with limiting small strain for cracks subject to non-penetration.

A major drawback of the study of cracks within the context of the linearized theory of elasticity is the inconsistency that one obtains with regard to the strain at a crack tip, namely it becoming infinite. In this paper we consider the problem within the context of an elastic body that exhibits limiting small strain wherein we are not faced with such an inconsistency. We introduce the concept of a non-smooth viscosity solution which is described by generalized variational inequalities and coincides with the weak solution in the smooth case. The well-posedness is proved by the construction of an approximation problem using elliptic regularization and penalization techniques.

Geophysics-Inspired Nonlinear Stress-Strain Law for Biological Tissues and Its Applications in Compression Optical Coherence Elastography.

We propose a nonlinear stress-strain law to describe nonlinear elastic properties of biological tissues using an analogy with the derivation of nonlinear constitutive laws for cracked rocks. The derivation of such a constitutive equation has been stimulated by the recently developed experimental technique-quasistatic Compression Optical Coherence Elastography (C-OCE). C-OCE enables obtaining nonlinear stress-strain dependences relating the applied uniaxial compressive stress and the axial component of the resultant strain in the tissue. To adequately describe nonlinear stress-strain dependences obtained with C-OCE for various tissues, the central idea is that, by analogy with geophysics, nonlinear elastic response of tissues is mostly determined by the histologically confirmed presence of interstitial gaps/pores resembling cracks in rocks. For the latter, the nonlinear elastic response is mostly determined by elastic properties of narrow cracks that are highly compliant and can easily be closed by applied compressing stress. The smaller the aspect ratio of such a gap/crack, the smaller the stress required to close it. Upon reaching sufficiently high compressive stress, almost all such gaps become closed, so that with further increase in the compressive stress, the elastic response of the tissue becomes nearly linear and is determined by the Young's modulus of the host tissue. The form of such a nonlinear dependence is determined by the distribution of the cracks/gaps over closing pressures; for describing this process, an analogy with geophysics is also used. After presenting the derivation of the proposed nonlinear law, we demonstrate that it enables surprisingly good fitting of experimental stress-strain curves obtained with C-OCE for a broad range of various tissues. Unlike empirical fitting, each of the fitting parameters in the proposed law has a clear physical meaning. The linear and nonlinear elastic parameters extracted using this law have already demonstrated high diagnostic value, e.g., for differentiating various types of cancerous and noncancerous tissues.

Symmetries and symmetry-generated averages of elastic constants up to the sixth order of nonlinearity for all crystal classes, isotropy and transverse isotropy.

Algebraic expressions for averaging linear and nonlinear stiffness tensors from general anisotropy to different effective symmetries (11 Laue classes elastically representing all 32 crystal classes, and two non-crystalline symmetries: isotropic and cylindrical) have been derived by automatic symbolic computations of the arithmetic mean over the set of rotational transforms determining a given symmetry. This approach generalizes the Voigt average to nonlinear constants and desired approximate symmetries other than isotropic, which can be useful for a description of textured polycrystals and rocks preserving some symmetry aspects. Low-symmetry averages have been used to derive averages of higher symmetry to speed up computations. Relationships between the elastic constants of each symmetry have been deduced from their corresponding averages by resolving the rank-deficient system of linear equations. Isotropy has also been considered in terms of generalized Lamé constants. The results are published in the form of appendices in the supporting information for this article and have been deposited in the Mendeley database.

Nonlinear Elasticity of Blood Vessels and Vascular Grafts.

The transplantation of vascular grafts has emerged as a prevailing approach to address vascular disorders. However, the development of small-diameter vascular grafts is still in progress, as they serve in a more complicated mechanical environment than their counterparts with larger diameters. The biocompatibility and functional characteristics of small-diameter vascular grafts have been well developed; however, mismatch in mechanical properties between the vascular grafts and native arteries has not been accomplished, which might facilitate the long-term patency of small-diameter vascular grafts. From a point of view in mechanics, mimicking the nonlinear elastic mechanical behavior exhibited by natural blood vessels might be the state-of-the-art in designing vascular grafts. This review centers on elucidating the nonlinear elastic behavior of natural blood vessels and vascular grafts. The biological functionality and limitations associated with as-reported vascular grafts are meticulously reviewed and the future trajectory for fabricating biomimetic small-diameter grafts is discussed. This review might provide a different insight from the traditional design and fabrication of artificial vascular grafts.

A viscoelastic constitutive framework for aging muscular and elastic arteries.

The evolution of arterial biomechanics and microstructure with age and disease plays a critical role in understanding the health and function of the cardiovascular system. Accurately capturing these adaptative processes and their effects on the mechanical environment is critical for predicting arterial responses. This challenge is exacerbated by the significant differences between elastic and muscular arteries, which have different structural organizations and functional demands. In this study, we aim to shed light to these adaptive processes by comparing the viscoelastic mechanics of autologous thoracic aortas (TA) and femoropopliteal arteries (FPA) in different age groups. We have extended our fractional viscoelastic framework, originally developed for FPA, to both types of arteries. To evaluate this framework, we analyzed experimental mechanical data from TA and FPA specimens from 21 individuals aged 13 to 73 years. Each specimen was subjected to a multi-ratio biaxial mechanical extension and relaxation test complemented by bidirectional histology to quantify the structural density and microstructural orientations. Our new constitutive model accurately captured the mechanical responses and microstructural differences of the tissues and closely matched the experimentally measured densities. It was found that the viscoelastic properties of collagen and smooth muscle cells (SMCs) in both the FPA and TA remained consistent with age, but the viscoelasticity of the SMCs in the FPA was twice that of the TA. Additionally, changes in collagen nonlinearity with age were similar in both TA and FPA. This model provides valuable insights into arterial mechanophysiology and the effects of pathological conditions on vascular biomechanics. STATEMENT OF SIGNIFICANCE: Developing durable treatments for arterial diseases necessitates a deeper understanding of how mechanical properties evolve with age in response to mechanical environments. In this work, we developed a generalized viscoelastic constitutive model for both elastic and muscular arteries and analyzed both the thoracic aorta (TA) and the femoropopliteal artery (FPA) from 21 donors aged 13 to 73. The derived parameters correlate well with histology, allowing further examination of how viscoelasticity evolves with age. Correlation between the TA and FPA of the same donors suggest that the viscoelasticity of the FPA may be influenced by the TA, necessitating more detailed analysis. In summary, our new model proves to be a valuable tool for studying arterial mechanophysiology and exploring pathological impacts.