Likely chirality of stochastic anisotropic hyperelastic tubes

Abstract

When an elastic tube reinforced with helical fibres is inflated, its ends rotate. In large deformations, the amount and chirality of rotation is highly non-trivial, as it depends on the choice of strain-energy density and the arrangements of the fibres. For anisotropic hyperelastic tubes where the material parameters are single-valued constants, the problem has been satisfactorily addressed. However, in many systems, the material parameters are not precisely known, and it is therefore more appropriate to treat them as random variables. The problem is then to understand chirality in a probabilistic framework. Here, we develop a procedure for examining the elastic responses of a hyperelastic cylindrical tube of stochastic anisotropic material, where the material parameters are spatially-independent random variables defined by probability density functions. The tube is subjected to uniform dead loading consisting of internal pressure, axial tension and torque. Assuming that the tube wall is thin and that the resulting deformation is the combined inflation, extension and torsion from the reference circular cylindrical configuration to a deformed circular cylindrical state, we derive the probabilities of radial expansion or contraction, and of right-handed or lefthanded torsion. We refer to these stochastic behaviours as 'likely inflation' and 'likely chirality', respectively.

Uncertainty quantification of elastic material responses: testing, stochastic calibration and Bayesian model selection

“…Our analysis is fully tractable mathematically and builds directly on knowledge from deterministic finite elasticity. This study highlights the need for continuum models to consider the variability in the elastic behaviour of materials at large strains, and complements our previous theoretical investigations of how elastic solutions of fundamental problems in nonlinear elasticity can be extended to stochastic hyperelastic models [33][34][35][38][39][40].…”

“…Presently, theoretical approaches have been able to successfully contend with cases of simple geometry such as cubes, spheres, shells and tubes. Specifically, within the stochastic framework, the classic problem of the Rivlin cube was considered in [38], the static and dynamic inflation of cylindrical and spherical shells was treated in [34,35], the static and dynamic cavitation of a sphere was analysed in [33,40], and the stretch and twist of anisotropic cylindrical tubes was examined in [39]. These problems were drawn from the analytical finite elasticity literature and incorporate random variables as basic concepts along with mechanical stresses and strains.…”

Likely oscillatory motions of stochastic hyperelastic solids

“…To study the effect of probabilistic model parameters on predicted mechanical responses, in [64,65,67,68], for different bodies with simple geometries at finite strain deformations, it was shown explicitly that, in contrast to the deterministic elastic problem where a single critical value strictly separates the stable and unstable cases, for the stochastic problem, there is a probabilistic interval where the stable and unstable states always compete, in the sense that both have a quantifiable chance to be found. In addition, revisiting these problems from a novel perspective offered fresh opportunities for gaining new insights into the fundamental elastic solutions, and correcting some inconsistencies found in the previous works.…”

“…In this paper, we extend the stochastic framework developed in [64,65,67,68] to study radial oscillations of cylindrical and spherical shells of stochastic incompressible isotropic hyperelastic material formulated as quasi-equilibrated motions. For these motions, the system is in equilibrium at every time instant.…”

Likely striping in stochastic nematic elastomers

“…To attain this, we start with a generalised nematic strain energy function similar to that proposed by Fried and Sellers [24,25], where we take as reference configuration the stress-free state of the isotropic phase at high temperature [28][29][30][31][32], instead of the nematic phase in which the cross-linking was produced [10,24,25,27,[33][34][35]. Recognising the fact that some uncertainties may arise in the mechanical responses of liquid crystal elastomers, and inspired by recent developments in stochastic finite elasticity [36][37][38][39][40][41][42][43][44][45][46][47][48], we then design stochastic elastic-nematic strain energy functions to capture the variability in physical responses of nematic solid materials at a macroscopic level.…”