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Abstract This study proposes a simple and novel class of stretch-limiting constitutive relations for perfectly flexible elastic strings drawn from modern advances in constitutive theory for elastic bodies. We investigate strings governed by constitutive relations where stretch is a bounded, piecewise linear function of tension, extending beyond the traditional Cauchy elasticity framework. Our analysis includes explicit solutions for catenaries and longitudinal, piecewise constant stretched motions. 

Many viscous liquids behave effectively as incompressible under high pressures but display a pronounced dependence of viscosity on pressure. The classical incompressible Navier-Stokes model cannot account for both features, and a simple pressure-dependent modification introduces questions about the well-posedness of the resulting equations. This paper presents the first study of a second-gradient extension of the incompressible Navier-Stokes model, recently introduced by the authors, which includes higher-order spatial derivatives, pressure-sensitive viscosities, and complementary boundary conditions. Focusing on steady flow down an inclined plane, we adopt Barus' exponential law and impose weak adherence at the lower boundary and a prescribed ambient pressure at the free surface. Through numerical simulations, we examine how the flow profile varies with the angle of inclination, ambient pressure, viscosity sensitivity to pressure, and internal length scale. 

The work reported in ``Granular micromechanics-based identification of isotropic strain gradient parameters for elastic geometrically nonlinear deformations" misidentified key terms in the grain-pair objective relative displacement when accounting for the second gradient of placement. In this paper, we correct that oversight by deriving a revised expression for the grain-pair objective relative displacement within the granular micromechanics framework. The amended terms, which resemble Christoffel symbols expressed in terms of strain gradients, modify the contributions of both the normal and tangential components to the strain energy and, consequently, alter the identified strain gradient elastic parameters. Importantly, the identification of the standard (first gradient) elastic tensor remains unchanged. This brief paper presents the corrected derivation, the resulting stiffness tensors for anisotropic strain gradient elasticity, and updated analytical expressions for the material parameters in both 2D and 3D isotropic settings. 

We introduce second-gradient models for incompressible viscous fluids, building on the framework introduced by Fried and Gurtin. We propose a new and simple constitutive relation for the hyperpressure to ensure that the models are both physically meaningful and mathematically well-posed. The framework is further extended to incorporate pressure-dependent viscosities. We show that for the pressure-dependent viscosity model, the inclusion of second-gradient effects guarantees the ellipticity of the governing pressure equation, in contrast to previous models rooted in classical continuum mechanics. The constant viscosity model is applied to steady cylindrical flows, where explicit solutions are derived under both strong and weak adherence boundary conditions. In each case, we establish convergence of the velocity profiles to the classical Navier-Stokes solutions as the model's characteristic length scales tend to zero. 

The concept of ‘effective mass’ is frequently used for the simplification of complex lumped parameter systems (discrete dynamical systems) as well as materials that have complicated microstructural features. From the perspective of wave propagation, it is claimed that for some bodies described as metamaterials, the corresponding ‘effective mass’ can be frequency dependent, negative or it may not even be a scalar quantity. The procedure has even led some authors to suggest that Newton’s second law needs to be modified within the context of classical continuum mechanics. Such absurd physical conclusions are a consequence of appealing to the notion of ‘effective mass’ with a preconception for the constitutive structure of the metamaterial and using a correct mathematical procedure. We show that such unreasonable physical conclusions would not arise if we were to use the appropriate ‘effective constitutive relation’ for the metamaterial, rather than use the concept of ‘effective mass’ with an incorrect predetermined constitutive relation. 

We propose a thermodynamically based approach for constructing effective rate-type constitutive relations describing finite deformations of metamaterials. The effective constitutive relations are formulated as second-order in time rate-type Eulerian constitutive relations between only the Cauchy stress tensor, the Hencky strain tensor and objective time derivatives thereof. In particular, there is no need to introduce additional quantities or concepts such as “micro-level deformation”,“micromorphic continua”, or elastic solids with frequency dependent material properties. Moreover, the linearisation of the proposed fully nonlinear (finite deformations) constitutive relations leads, in Fourier/frequency space, to the same constitutive relations as those commonly used in theories based on the concepts of frequency dependent density and/or stiffness. From this perspective the proposed constitutive relations reproduce the behaviour predicted by the frequency dependent density and/or stiffness models, but yet they work with constant—that is motion independent—material properties. This is clearly more convenient from the physical point of view. Furthermore, the linearised version of the proposed constitutive relations leads to the governing partial differential equations that are particularly simple both in Fourier space as well as in physical space. Finally, we argue that the proposed fully nonlinear (finite deformations) second-order in time rate-type constitutive relations do not fall into traditional classes of models for elastic solids (hyperelastic solids/Green elastic solids, first-order in time hypoelastic solids), and that the proposed constitutive relations embody a new class of constitutive relations characterising elastic solids. 

We present an asymptotic framework that rigorously generates nonlinear constitutive relations between stress and linearized strain for elastic bodies. Each of these relations arises as the leading-order relationship satisfied by a oneparameter family of nonlinear constitutive relations between stress and nonlinear strain. The asymptotic parameter limits the overall range of strains that satisfy the corresponding constitutive relation in the one-parameter family, while the stresses can remain large (relative to a fixed stress scale). This differs from classical linearized elasticity where a fixed constitutive relation is assumed, and the magnitude of the displacement gradient serves as the asymptotic parameter. Also unlike classical approaches, the constitutive relations in our framework are expressed as implicit relationships between stress and strain rather than requiring stress explicitly expressed as a function of strain, adding conceptual simplicity and versatility. We demonstrate that our framework rigorously justifies nonlinear constitutive relations between stress and linearized strain including those with density-dependent Young’s moduli or derived from strain energies beyond quadratic forms. 

Introduction: Dissection or rupture of the aorta is accompanied by high mortality rates, and there is a pressing need for better prediction of these events for improved patient management and clinical outcomes. Biomechanically, these events represent a situation wherein the locally acting wall stress exceed the local tissue strength. Based on recent reports for polymers, we hypothesized that aortic tissue failure strength and stiffness are directly associated with tissue mass density. The objective of this work was to test this novel hypothesis for porcine thoracic aorta. Methods: Three tissue specimens from freshly harvested porcine thoracic aorta were treated with either collagenase or elastase to selectively degrade structural proteins in the tissue, or with phosphate buffer saline (control). The tissue mass and volume of each specimen were measured before and after treatment to allow for density calculation, then mechanically tested to failure under uniaxial extension. Results: Protease treatments resulted in statistically significant tissue density reduction (sham vs. collagenase p = 0.02 and sham vs elastase p = 0.003), which in turn was significantly and directly correlated with both ultimate tensile strength (sham vs. collagenase p = 0.02 and sham vs elastase p = 0.03) and tangent modulus (sham vs. collagenase p = 0.007 and sham vs elastase p = 0.03). Conclusions: This work demonstrates for the first time that tissue stiffness and tensile strength are directly correlated with tissue density in proteolytically-treated aorta. These findings constitute an important step towards understanding aortic tissue failure mechanisms and could potentially be leveraged for non-invasive aortic strength assessment through density measurements, which could have implications to clinical care. 

Recent works have shown that in contrast to classical linear elastic fracture mechanics, endowing crack fronts in a brittle Green-elastic solid with Steigmann-Ogden surface elasticity yields a model that predicts bounded stresses and strains at the crack tips for plane-strain problems. However, singularities persist for anti-plane shear (mode-III fracture) under far field loading, even when Steigmann-Ogden surface elasticity is incorporated. This work is motivated by obtaining a model of brittle fracture capable of predicting bounded stresses and strains for all modes of loading. We formulate an exact general theory of a three-dimensional solid containing a boundary surface with strain-gradient surface elasticity. For planar reference surfaces parameterized by flat coordinates, the form of surface elasticity reduces to that introduced by Hilgers and Pipkin, and when the surface energy is independent of the surface covariant derivative of the stretching, the theory reduces to that of Steigmann and Ogden. We discuss material symmetry using Murdoch and Cohen’s extension of Noll’s theory. We present a model small-strain surface energy that incorporates resistance to geodesic distortion, satisfies strong ellipticity, and requires the same material constants found in the Steigmann-Ogden theory. Finally, we derive and apply the linearized theory to mode-III fracture in an infinite plate under far-field loading. We prove that there always exists a unique classical solution to the governing integro-differential equation, and in contrast to using Steigmann-Ogden surface elasticity, our model is consistent with the linearization assumption in predicting finite stresses and strains at the crack tips. 

We present an asymptotic framework that rigorously generates nonlinear constitutive relations between stress and linearized strain for elastic bodies. Each of these relations arises as the leading order relationship satisfied by a one-parameter family of nonlinear constitutive relations between stress and nonlinear strain. The asymptotic parameter limits the overall range of strains that satisfy the corresponding constitutive relation in the one parameter family while the stresses can remain large (relative to a fixed stress scale). This differs from classical linearized elasticity where a fixed constitutive relation is assumed, and the magnitude of the displacement gradient serves as the asymptotic parameter. Also unlike classical approaches, the constitutive relations in our framework are expressed as implicit relationships between stress and strain rather than requiring stress explicitly expressed as a function of strain, adding conceptual simplicity and versatility. We demonstrate that our framework rigorously justifies nonlinear constitutive relations between stress and linearized strain including those with density-dependent Young’s moduli or derived from strain energies beyond quadratic forms.