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Abstract In this paper, viscous incompressible hydromagnetic flow around a sphere has been investigated by considering the penetration of the magnetic field inside it. Earlier researchers have found it difficult because it not only adds an extra equation to the system of governing equations but also needs a proper matching of the components of the magnetic fields at the interface. However, using a higher‐order compact finite difference scheme, we have successfully solved the governing highly nonlinear and coupled system of partial differential equations and have obtained converged solutions throughout the domain of the parameter space. In this novel numerical investigation, we have calculated the magnetic field throughout the whole domain, that is, both inside the sphere and within the fluid, with a suitable matching at the interface–a feature that has allowed us to capture the actual interactions occurring between the fluid flow and the magnetic field and unfurl several new characteristics of scientific and technological value. In fact, we have found that the magnetic field penetrating inside the sphere can effectively cause the critical value of Reynolds number to increase and can help to suppress flow separation more effectively than otherwise. 

Abstract This paper investigates the propagation of longitudinal waves in some possible models of compressible Kelvin-Voigt viscoelastic solids. With respect to the elastic part two extensions of the classical neo-Hookean material model are proposed: the A-model, which incorporates a logarithmic volumetric function, and the B-model, based on the deviatoric invariants and a power-law volumetric function. For both models, we assume the same dissipative part given by the classical Navier–Stokes constitutive equation. These models are analyzed for their ability to describe a recovery phenomenon, ensuring conditions for monotonicity, boundedness, and uniqueness of solutions. The propagation of longitudinal traveling waves is proved. We show that the equation governing such motions is indeed a special case of the viscous p-system and a weakly nonlinear analysis demonstrates the emergence of Burgers’ equations. 

The quasi-static problem describes a nonlinear porous body with a non-penetrating Barenblatt’s crack driven by the fracturing fluid, and its propagation is under investigation. By this, a bulk modulus of the porous body depends linearly on the density, the fracture faces allow contact with cohesion, and leak-off of the fluid into reservoir is accounted by the model. The mathematical problem consists in finding time-continuous functions of a displacement and a mean fluid pressure in the fracture, which satisfy the coupled system of the variational inequality and the fluid mass balance, which is controlled by the volume of fracking fluid pumped into the fracture. Well-posedness of the governing relations is proved rigorously by applying the method of Lagrange multipliers and using optimality conditions for the constrained minimization problem. As anillustrative example, a numerical benchmark problem of the fluid-driven fracture is presented in one dimension and computed by a Newton-type algorithm. 

The circumferential shear of a nonlinear isotropic incompressible elastic annulus is studied using the neo-Hookean, Ogden constitutive relations in addition to a new constitutive relation for the Hencky strain in terms of the Cauchy stress. The predictions of the three constitutive relations to the specific boundary value problem are delineated. In view of the predictions being quite distinct between the new constitutive relation studied and that for the Ogden constitutive relation, it would be worthwhile to carry out an experiment to determine the efficacy of the models. 

Mechanics of a thin elastic coating is analysed using nonlinear constitutive relations based on the concept of a density-dependent Young’s modulus. In contrast to previous considerations on the subject, the proposed framework does not assume weak nonlinearity. Two-term asymptotic expansions are derived for four setups of boundary conditions along the upper face of the coating; in doing so, the lower face is supposed to be clamped. Explicit approximate formulae obtained in the paper have the potential to be implemented in diverse industrial applications related to thin porous coatings. 

We study the response of a class of transversely elastic bodies, wherein the Green–Saint Venant strain tensor is a function of the second Piola–Kirchhoff stress tensor, when the body is residually stressed. The notion of such non-Cauchy elastic bodies being transversely isotropic is defined in Rajagopal (Mech. Res. Commun. 64, 2015, 38–41), and by a body being residually stressed, we mean the interior of the body is not in a stress-free state although the boundary is free of traction as considered by Coleman and Noll (Arch. Ration. Mech. Anal. 15, 1964, 87–111) and by Hoger (Arch. Ration. Mech. Anal. 88, 1985, 271–289). 

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.