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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. 

The response of a body described by a quasi-linear viscoelastic constitutive relation, whose material moduli depend on the mechanical pressure (that is one-third the trace of stress) is studied. The constitutive relation stems from a class of implicit relations between the histories of the stress and the relative deformation gradient. A-priori thresholding is enforced through the pressure that ensures that the displacement gradient remains small. The resulting mixed variational problem consists of an evolutionary equation with the Volterra convolution operator; this equation is studied for well-posedness within the theory of maximal monotone graphs. For isotropic extension or compression, a semi-analytic solution of the quasi-linear viscoelastic problem is constructed under stress control. The equations are studied numerically with respect to monotone loading both with and without thresholding. In the example, the thresholding procedure ensures that the solution does not blow-up in finite time. 

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. 

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). 

We consider the mathematical analysis and numerical approximation of a system of nonlinear partial differential equations that arises in models that have relevance to steady isochoric flows of colloidal suspensions. The symmetric velocity gradient is assumed to be a monotone nonlinear function of the deviatoric part of the Cauchy stress tensor. We prove the existence of a weak solution to the problem, and under the additional assumption that the nonlinearity involved in the constitutive relation is Lipschitz continuous we also prove uniqueness of the weak solution. We then construct mixed finite element approximations of the system using both conforming and nonconforming finite element spaces. For both of these we prove the convergence of the method to the unique weak solution of the problem, and in the case of the conforming method we provide a bound on the error between the analytical solution and its finite element approximation in terms of the best approximation error from the finite element spaces. We propose first a Lions–Mercier type iterative method and next a classical fixed-point algorithm to solve the finite-dimensional problems resulting from the finite element discretisation of the system of nonlinear partial differential equations under consideration and present numerical experiments that illustrate the practical performance of the proposed numerical method.