The Anelastic Ericksen Problem: Universal Deformations and Universal Eigenstrains in Incompressible Nonlinear Anelasticity

Ericksen’s problem consists of determining all equilibrium deformations that can be sustained solely by the application of boundary tractions for an arbitrary incompressible isotropic hyperelastic material whose stress-free configuration is geometrically flat. We generalize this by first, using a geometric formulation of this problem to show that all the known universal solutions are symmetric with respect to Lie subgroups of the special Euclidean group. Second, we extend this problem to its anelastic version, where the stress-free configuration of the body is a Riemannian manifold. Physically, this situation corresponds to the case where nontrivial finite eigenstrains are present. We characterize explicitly the universal eigenstrains that share the symmetries present in the classical problem, and show that in the presence of eigenstrains, the six known classical families of universal solutions merge into three distinct anelastic families, distinguished by their particular symmetry group. Some generic solutions of these families correspond to well-known cases of anelastic eigenstrains. Additionally, we show that some of these families possess a branch of anomalous solutions, and demonstrate the unique features of these solutions and the equilibrium stress they generate.

Fully-Coupled Transient Fluid–Solid Interaction Simulation of the pH-Sensitive Hydrogel-Based Microvalve

The pH-sensitive hydrogels are attractive candidates to act like a microvalve in microfluidic devices. In this study, we build a theory for the transient simulation of a pH-sensitive hydrogel-based microvalve. Three fields are involved in the theory, namely the electrochemical, mechanical and fluid fields. We utilize the Nernst–Planck equation to describe the ionic flux into the hydrogel through diffusion, electrical migration and convection. We model the hydrogel as a compressible isotropic hyperelastic material with the Gent model. Then we implement the theory in a nonlinear finite element framework to simulate the time-dependent fluid–solid interaction (FSI) behavior of the pH-sensitive microvalve. Our focus is on exploring the physics and phenomena involving in the simulation rather than simulating a complex geometry or presenting a new design. We manifest the significance of the FSI by comparing the transient FSI and non-FSI simulation of the microvalve. The most highlighted novelty of our study is accounting for time-dependent effects. The results demonstrate that the microvalve perfectly closes the channel much before it reaches its stationary state and the closing state, which is of high interest in the microvalve study is different from the stationary state.

Likely oscillatory motions of stochastic hyperelastic solids

Stochastic homogeneous hyperelastic solids are characterized by strain-energy densities where the parameters are random variables defined by probability density functions. These models allow for the propagation of uncertainties from input data to output quantities of interest. To investigate the effect of probabilistic parameters on predicted mechanical responses, we study radial oscillations of cylindrical and spherical shells of stochastic incompressible isotropic hyperelastic material, formulated as quasi-equilibrated motions where the system is in equilibrium at every time instant. Additionally, we study finite shear oscillations of a cuboid, which are not quasi-equilibrated. We find that, for hyperelastic bodies of stochastic neo-Hookean or Mooney–Rivlin material, the amplitude and period of the oscillations follow probability distributions that can be characterized. Further, for cylindrical tubes and spherical shells, when an impulse surface traction is applied, there is a parameter interval where the oscillatory and non-oscillatory motions compete, in the sense that both have a chance to occur with a given probability. We refer to the dynamic evolution of these elastic systems, which exhibit inherent uncertainties due to the material properties, as ‘likely oscillatory motions’.

Instability investigation on fluid-loaded pre-stretched cylindrical membranes

This paper discusses the evaluation of instabilities on the quasi-static equilibrium path of fluid-loaded pre-stretched cylindrical membranes and the switching to a secondary branch at a bifurcation point. The membrane is represented by only the in-plane stress components, for which an incompressible, isotropic hyperelastic material model is assumed. The free inflation problem yields governing equations and boundary conditions, which are discretized by finite differences and solved by a Newton–Raphson method. An incremental arclength-cubic extrapolation method is used to find generalized equilibrium paths, with different parametrizations. Limit points and bifurcation points are observed on the equilibrium path when fluid level is seen as the controlling parameter. An eigen-mode injection method is employed to switch to a secondary equilibrium branch at the bifurcation point. A limit point with respect to fluid level is observed for a partially filled membrane when the aspect ratio (length/radius) is high, whereas for smaller aspect ratios, the limit point with respect to fluid level is observed at over-filling. Pre-stretch is observed to have a stiffening effect in the pre-limit zone and a softening effect in the post-limit zone.

SIMULATION OF THE STATIC INFLATION OF THE PASSIVE LEFT VENTRICLE TO AN END-DIASTOLIC STATE

To initiate our simulations of canine left ventricle (LV) mechanics, we needed to specify an initial geometry and an initial wall stress distribution. Although there are sufficient measurements of LV geometries, there are no assessments of stresses under any conditions. To estimate a physiologically plausible range of stresses at end diastole, we have inflated an unloaded reference geometry using static pressure loads. The LV was modelled as a six-layered truncated prolate ellipsoid. The myocardium was defined as a slightly compressible, transversely isotropic, hyperelastic material. The reference LV was inflated statically by gradually increasing the pressure on its inner surface until an end-diastolic state was reached. The calculated dependence of normalized LV volume changes on the applied pressure was in good agreement with previous experimental results. Our calculated geometry was found to be comparable to previous measurements. The end-diastolic stresses were found to have complex variations, which cannot be determined by adopting an ad hoc stress-free, end-diastolic geometry. The calculated geometry and stress distribution are deemed to be suitable for use as initial states for cardiac cycle simulations.

Transversely isotropic cyclic stress-softening model for the Mullins effect

This paper models stress softening during cyclic loading and unloading of an elastomer. The paper begins by remodelling the primary loading curve to include a softening function and goes on to derive nonlinear transversely isotropic constitutive equations for the elastic response, stress relaxation, residual strain and creep of residual strain. These ideas are combined with a transversely isotropic version of the Arruda–Boyce eight-chain model to develop a constitutive relation that is capable of accurately representing the Mullins effect during cyclic stress softening for a transversely isotropic, hyperelastic material, in particular, a carbon-filled rubber vulcanizate.