Likely oscillatory motions of stochastic hyperelastic solids

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'

Likely equilibria of stochastic hyperelastic spherical shells and tubes

In large deformations, internally pressurised elastic spherical shells and tubes may undergo a limit-point, or inflation, instability manifested by a rapid transition in which their radii suddenly increase. The possible existence of such an instability depends on the material constitutive model. Here, we revisit this problem in the context of stochastic incompressible hyperelastic materials, and ask the question: what is the probability distribution of stable radially symmetric inflation, such that the internal pressure always increases as the radial stretch increases? For the classic elastic problem, involving isotropic incompressible materials, there is a critical parameter value that strictly separates the cases where inflation instability can occur or not. By contrast, for the stochastic problem, we show that the inherent variability of the probabilistic parameters implies that there is always competition between the two cases. To illustrate this, we draw on published experimental data for rubber, and derive the probability distribution of the corresponding random shear modulus to predict the inflation responses for a spherical shell and a cylindrical tube made of a material characterised by this parameter.Comment: arXiv admin note: text overlap with arXiv:1808.0126

Stochastic modelling and analysis of homogeneous hyperelastic solids

Combining finite elasticity and information theory, a stochastic method is devel oped in order to accurately predict and assess the behaviour of materials, and also to model experimental data. An explicit strategy to calibrate homogeneous isotropic hyperelastic models to mean values and the standard deviation of ei ther the stress-strain function or the nonlinear shear modulus is devised, and the technique of using Bayes Theorem to select the optimal model to represent the ma terial or data in question is presented, specifically here in relation to manufactured silicone specimens. An analysis of the behaviour of solid materials under various deformations, including necking instability, the inflation of cylindrical tubes and spheres, and the cavitation of spherical shells, when the material is stochastic, is demonstrated, before an extension to the dynamic finite deformations of stochastic hyperelastic solids, including the shear motion of a cuboid, the quasi-equilibrated radial-axial motion of a cylindrical tube, and the quasi-equilibrated radial motion of a spherical shell, is explored. Ultimately, it is determined that the amplitude and period of oscillation of stochastic bodies are characterised by probability dis tributions. Overall, the aim is to highlight the need for mathematical modelling to consider the variability obtained in experimental data, in the mechanical re sponses of materials, or in testing protocols, with a view to enhancing the accuracy of the mathematical modelling techniques employed, and, as a result, to provide an improved assessment or prediction of the behaviour of the materials in question

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

Motivated by the need to quantify uncertainties in the mechanical behaviour of solid materials, we perform simple uniaxial tensile tests on a manufactured rubber-like material that provide critical information regarding the variability in the constitutive responses between different specimens. Based on the experimental data, we construct stochastic homogeneous hyperelastic models where the parameters are described by spatially independent probability density functions at a macroscopic level. As more than one parametrised model is capable of capturing the observed material behaviour, we apply Baye theorem to select the model that is most likely to reproduce the data. Our analysis is fully tractable mathematically and builds directly on knowledge from deterministic finite elasticity. The proposed stochastic calibration and Bayesian model selection are generally applicable to more complex tests and materials

Radio Continuum and Star Formation in CO-rich Early Type Galaxies

In this paper we present new high resolution VLA 1.4 GHz radio continuum observations of five FIR bright CO-rich early-type galaxies and two dwarf early-type galaxies. The position on the radio-FIR correlation combined with striking agreements in morphology between high resolution CO and radio maps show that the radio continuum is associated with star formation in at least four of the eight galaxies. The average star formation rate for the sample galaxies detected in radio is approximately 2 solar masses per year. There is no evidence of a luminous AGN in any of our sample galaxies. We estimate Toomre Q values and find that the gas disks may well be gravitationally unstable, consistent with the above evidence for star formation activity. The radio continuum emission thus corroborates other recent suggestions that star formation in early type galaxies may not be uncommon.Comment: 21 pages, 7 figures, to be published in the Astronomical Journa