Abstract
Abstract 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’.
Highlights
Motivated by numerous long-standing and modern engineering problems, oscillatory motions of cylindrical and spherical shells made of linear elastic material (Krauss, 1967; Love, 1888, 1944; Reissner, 1941) have generated a wide range of experimental, theoretical and computational studies (Alijani & Amabili, 2014; Amabili, 2008; Amabili & Païdoussis, 2003; Breslavsky & Amabili, 2018; Dong et al, 2018)
We extend the stochastic framework developed in Mihai et al (2018a,b, 2019a,b) to study radial oscillations of cylindrical and spherical shells of stochastic incompressible isotropic hyperelastic material formulated as quasi-equilibrated motions
We provided here a synthesis on the analysis of finite amplitude oscillations resulting from dynamic finite deformations of given isotropic incompressible nonlinear hyperelastic solids and extended this to non-deterministic oscillatory motions of stochastic isotropic incompressible hyperelastic solids with similar geometries
Summary
Motivated by numerous long-standing and modern engineering problems, oscillatory motions of cylindrical and spherical shells made of linear elastic material (Krauss, 1967; Love, 1888, 1944; Reissner, 1941) have generated a wide range of experimental, theoretical and computational studies (Alijani & Amabili, 2014; Amabili, 2008; Amabili & Païdoussis, 2003; Breslavsky & Amabili, 2018; Dong et al, 2018). The governing equations for large amplitude oscillations of cylindrical tubes and spherical shells of homogeneous isotropic incompressible nonlinear hyperelastic material, formulated as special cases of quasi-equilibrated motions (Truesdell, 1962), were reviewed in Truesdell & Noll (2004). To study the effect of probabilistic model parameters on predicted mechanical responses, in Mihai et al (2018a,b, 2019a,b), 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.
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