Likely oscillatory motions of stochastic hyperelastic solids

Abstract

Stochastic homogeneous hyperelastic solids are characterised 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 characterised. 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".Key words: stochastic hyperelastic models, dynamic finite strain deformation, quasi-equilibrated motion, finite amplitude oscillations, incompressibility, applied probability."Denominetur motus talis, qualis omni momento temporis t praebet configurationem capacem aequilibrii corporis iisdem viribus massalibus sollicitati, 'motus quasi aequilibratus'. Generatim motus quasi aequilibratus non congruet legibus dynamicis et proinde motus verus corporis fieri non potest, manentibus iisdem viribus masalibus." -C. Truesdell (1962) [103]

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 chirality of stochastic anisotropic hyperelastic tubes

“…The present study is part of an ongoing investigation where we aim to illustrate explicitly how the elastic solution of fundamental nonlinear elasticity problems can be extended to the stochastic case [30][31][32]34]. For numerical methods applied to problems that are intractable analytically, we refer to [51,52].…”

“…Then, the non-zero components of the associated stress tensor, given by (32), with B defined by (66), take the form,…”

“…Such problems can offer significant insight into how the stochastic framework builds on the finite elasticity theory, and revisiting them from the stochastic perspective can offer also opportunities for gaining new insights into the elastic solutions. More complex, but mathematically tractable problems can then be treated in a similar manner [32]. To study the effect of probabilistic parameters in the case of more complex geometries and loading conditions, suitable numerical approaches built on rigorous analysis have started to be developed in [51,52].…”