In the present work we derive an analytical expression for the pressure–deflection curve of circular membranes subjected to inflation. This problem has been studied mostly from a numerical point of ...view and there is still a lack of accurate closed-form solutions in nonlinear elasticity. The analytical formulation is developed with a semi-inverse method by setting a priori the kinematics of deformation of the membrane. A compressible Mooney–Rivlin material model is considered and a pressure–deflection relation is derived from the equilibrium. The kinematics is approximated and therefore the obtained solution is not exact. Consequently, the formulation is adjusted by introducing an additional polynomial function in the pressure–deflection equation. The polynomial is calibrated by fitting numerical solutions of the exact system of differential equilibrium equations. The calibration is done over a wide range of constitutive parameters that covers the response of all rubber materials for technological applications. As a result, a definitive and accurate expression of the applied pressure as a function of the deflection of the membrane is obtained. The formula is validated with finite element (FE) simulations and compared with other solutions available in the literature. The comparison shows that the present model is more accurate. In addition, unlike the other models, it can be applied to compressible materials. Experimental uniaxial and bulge tests are carried out on rubber materials and the model proposed is used to characterize the Mooney–Rivlin constitutive parameters. Since the pressure–deflection formula is accurate and easy-to-use, it is an innovative tool in engineering applications of inflated membranes.
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•Analytical pressure–deflection relation for circular inflated membranes.•For the first time a formula that is valid for compressible materials.•A comparison with other solutions shows that our formulation is more accurate.•Bulge tests are carried out and the model reproduces well the experimental data.
•A new set of experiments is conducted on the inflation of plane membranes.•The performance of Gent and Gent-Gent material models is evaluated, and it is shown that the Gent-Gent material model gives ...much better predictions than the original Gent material model.•The accuracy of using inflation of a plane membrane to mimic equibiaxial extension is quantified.
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The Gent material model is the simplest extension of the neo-Hookean material model that can describe the finite extensibility of the polymeric chains comprising the elastomer network. However, it is known that its fitting to experimental results of uniaxial tension is not satisfactory for moderate values of stretch, and the Gent-Gent model was proposed to remedy this deficiency. In this paper we provide further evidence on the good performance of the Gent-Gent model by using it to study the inflation of a circular plane membrane. For this problem, the deformation near the pole is equibiaxial and the associated nominal tension is a monotonic function of the stretch, but the pressure as a function of the stretch has both a maximum and a minimum. The Gent and Gent-Gent models are first fitted to our own experimental data for the nominal tension, and then used to predict variation of the pressure with respect to the stretch. By comparison with the experimental data, it is shown that the Gent-Gent model gives much better predictions than the Gent model.
The helical stability experienced by elastic cylinders is investigated using numerical methods. A doubly fiber-reinforced incompressible nonlinear elastic tube subject to axial loading, internal ...pressure and twist is examined using a numerical scheme based on the modified Riks (quasi-static) procedure. Under the application of such loadings (qualitatively physiological for arteries) vessels form tortuous shapes and a range of topologically and geometrically complex morphologies. These configurations can be highly unstable owing to the nonlinear interaction (geometry, material and self contact) among the multiple bifurcation modes. The present work attempts to model such complex configurations using a thick-walled cylindrical tube. These complex morphologies are most commonly termed as helical coiling, looping and winding in the biomechanics community. We show that these bifurcations are very sensitive to the applied pressure, axial stretch and the amount of twist. Illustration of the helical buckling is then provided by considering an anisotropic constitutive model that includes both fiber stretching and fiber shearing, as opposed to previous analyses that only consider fiber stretching. The numerical implementation is achieved via user routine in the finite element code Abaqus.
•Helical stability experienced by elastic cylinders is investigated.•Fiber-reinforced tubes are subject to axial loading, pressure, and twist.•The modified Riks (quasi-static) procedure is applied.•Under loading, vessels form tortuous shapes and complex morphologies.•Results show: curving, helical buckling, kinking, and looping.
•Arbitrary cross-section description for the ANCF beam elements is presented, which makes these elements more general to use.•Achilles sub-tendons’ finite element approximations with continuum ANCF ...beam elements.•Numerical tensile load experiments for Achilles sub-tendons.
Achilles sub-tendons are materially and geometrically challenging structures that can nearly undergo around 15% elongation from their pre-twisted initial states during physical activities. Sub-tendons’ cross-sectional shapes are subject-specific, varying from simple to complicated. Therefore, the Achilles sub-tendons are often described by three-dimensional elements that lead to a remarkable number of degrees of freedom. On the other hand, the continuum-based beam elements in the framework of the absolute nodal coordinate formulation have already been shown to be a reliable and efficient replacement for the three-dimensional continuum elements in some special problems. So far, that element type has been applied only to structures with a simple cross-section geometry. To computationally efficiently describe a pre-twisted Achilles sub-tendon with a complicated cross-section shape, this study will develop a continuum-based beam element based on the absolute nodal coordinate formulation with an arbitrary cross-section description. To demonstrate the applicability of the developed beam element to the Achilles sub-tendons, 16 numerical examples are considered. During these numerical tests, the implemented cross-section descriptions agreed well with the reference solutions and led to faster convergence rates in comparison with the solutions provided by commercial finite element codes. Furthermore, it is demonstrated that in the cases of very complicated cross-sectional forms, the commercial software ANSYS provides inflated values for the elongation deformation in comparison with ABAQUS (about 6.2%) and ANCF (about 9.4%). Additionally, the numerical results reveal a possibility to model the whole sub-tendons via coarse discretization with high accuracy under uniaxial loading. This demonstrates the huge potential for use in biomechanics and also in multibody applications, where the arbitrary cross-section of beam-like structures needs to be taken into account.
We derive a one-dimensional (1d) model for the analysis of bulging or necking in an inflated hyperelastic tube of finite wall thickness from the three-dimensional (3d) finite elasticity theory by ...applying the dimension reduction methodology proposed by Audoly and Hutchinson (2016). The 1d model makes it much easier to characterize fully nonlinear axisymmetric deformations of a thick-walled tube using simple numerical schemes such as the finite difference method. The new model recovers the diffuse interface model for analyzing bulging in a membrane tube and the 1d model for investigating necking in a stretched solid cylinder as two limiting cases. It is consistent with, but significantly refines, the exact linear and weakly nonlinear bifurcation analyses. Comparisons with finite element simulations show that for the bulging problem, the 1d model is capable of describing the entire bulging process accurately, from initiation, growth, to propagation. The 1d model provides a stepping stone from which similar 1d models can be derived and used to study other effects such as anisotropy and electric loading, and other phenomena such as rupture.
Durotaxis of cells anchored to the extracellular matrix through focal adhesions has been systematically studied through both analytical and computational approaches. However, recent experiments have ...revealed the attitude of certain cells to unexpectedly migrate towards comparatively softer substrates, thus suggesting the possibility for negative durotaxis to manifest. Cell migration is possible because focal adhesions grow and disrupt, thus operating like adhesive structures undergoing a chemo-physical degradation process. In the present contribution, this degradation process is described through an elastic-damaging cohesive law deduced from a convex–concave pseudo-elastic potential, which confers a variational structure to the mechanical model of the adhesion structure and makes the derivation of analytical solutions possible. Furthermore, the obtained traction-separation cohesive law is amenable to a straightforward implementation into finite element codes. Finite elasticity of the cell body is considered while durotaxis is triggered by applying a contractile pre-stretch to the cell. It is shown that displacement- or force-driven degradation processes may lead to different kinds of durotaxis. The consistency and effectiveness of the proposed approach are showcased in one- and three-dimensional examples of cell–substrate systems.
The theory of superimposed incremental elastic deformations is applied to a deformed configuration of a residually-stressed circular cylindrical tube. The tube is subject to internal or external ...pressure and an axial load that maintain its circular cylindrical shape, and then bifurcations from this shape are analysed. Detailed governing equations and boundary conditions are provided for axisymmetric, prismatic and asymmetric bifurcations for a general form of strain–energy function that incorporates radial and circumferential residual stress components. The theory is applied to a simple model strain–energy function with two material parameters and a parameter that reflects the magnitude of the residual stress. Numerical computations are used to illustrate the dependence of the results of the bifurcation analysis on these parameters and the axial and radial underlying deformation. It is shown that in general the presence of residual stress has a significant effect compared with the corresponding results without residual stress.
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