Likely equilibria of stochastic hyperelastic spherical shells and tubes

Abstract

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.

Likely oscillatory motions of stochastic hyperelastic solids

“…Note that the random variables µ and R 1 are independent, depending on parameters (ρ 1 , ρ 2 ) and (ζ 1 , ζ 2 ), respectively, whereas µ 1 and µ 2 are codependent variables as they both require (µ, R 1 ) to be defined. For the numerical illustration of our subsequent results, throughout this paper, we assume that the random shear modulus µ follows the Gamma distribution represented in Figure 1, where the shape and scale parameters are ρ 1 = 405 and ρ 2 = 0.01, respectively [65]. Different simulations were then created by fixing the parameters (given in each figure caption), and repeatedly drawing random samples from the underlying distribution.…”

Likely chirality of stochastic anisotropic hyperelastic tubes

“…However, the physical behaviour of stochastic anisotropic materials deserves further attention, and we hope that our theoretical analysis may serve as a motivation for future experimental work. For the stochastic model described by (23), the following mathematical constraints guarantee that the random shear modulus of the matrix material under infinitesimal deformations, µ, and its inverse, 1/µ, are second-order random variables, i.e., they have finite mean value and finite variance [30,31,33,34,[48][49][50],…”

“…This question has begun to be addressed analytically in [30,31,34], with the special cases of stochastic hyperelastic bodies with simple geometries for which the finite deformation is known. Examples include the cavitation of a sphere under uniform tensile dead load [30], the inflation of pressurised spherical and cylindrical shells [31], and the classical problem of the Rivlin cube [34]. For these fundamental problems, to which the elastic solution is known explicitly, the sensitive dependence on parameter probabilities was demonstrated by showing that, in contrast to the deterministic problem, where a single critical value strictly separates the cases where instability can or cannot occur, for the stochastic problem, there is a probabilistic interval where the two cases compete, in the sense that both have a quantifiable chance to be found.…”

Likely Cavitation in Stochastic Elasticity

“…For stochastic hyperelastic models, the immediate question is: what is the influence of the random model parameters on the predicted nonlinear elastic responses? This question was previously considered by us in [36], for the stochastic Rivlin cube, and in [37], for the symmetric inflation of internally pressurised stochastic spherical shells and tubes. These idealised problems illustrate some important effects on the likely elastic responses of stochastic hyperelastic materials under large strains.…”

“…Here, we address this question by employing a similar approach as in [36,37] to revisit, in the context of stochastic elasticity, the cavitation problem of incompressible spheres of stochastic homogeneous isotropic hyperelastic materials under uniform radial tensile dead loads. Moreover, for all homogeneous isotropic hyperelastic models considered so far in the literature, cavitation appears as a supercritical bifurcation, where typically, after bifurcation, the cavity radius monotonically increases as the applied load increases (see, e.g., [8]).…”