Stability of an inflated hyperelastic membrane tube with localized wall thinning

Abstract

It is now well-known that when an infinitely long hyperelastic membrane tube free from any imperfections is inflated, a transcritical-type bifurcation may take place that corresponds to the sudden formation of a localized bulge. When the membrane tube is subjected to localized wall-thinning, the bifurcation curve would "unfold" into the turning-point type with the lower branch corresponding to uniform inflation in the absence of imperfections, and the upper branch to bifurcated states with larger amplitude. In this paper stability of bulged configurations corresponding to both branches is investigated with the use of the spectral method. It is shown that under pressure control and with respect to axi-symmetric perturbations, configurations corresponding to the lower branch are stable but those corresponding to the upper branch are unstable. Stability or instability is established by demonstrating the non-existence or existence of an unstable eigenvalue (an eigenvalue with a positive real part). This is achieved by constructing the Evans function that depends only on the spectral parameter. This function is analytic in the right half of the complex plane where its zeroes correspond to the unstable eigenvalues of the generalized spectral problem governing spectral instability. We show that due to the fact that the skew-symmetric operator J involved in the Hamiltonian formulation of the basic equations is onto, the zeroes of the Evans function can only be located on the real axis of the complex plane. We also comment on the connection between spectral (linear) stability and nonlinear (Lyapunov) stability.

Likely oscillatory motions of stochastic hyperelastic solids

“…Thus, the accelerationr can be computed in terms of the accelerationä on the inner surface. By the governing equations (46), the condition (4) is valid for x = (r, θ, z) T , since…”

“…In this case, the integrand is negative for 0 < r 2 /R 2 < 1/α and positive for r 2 /R 2 > 1/α. Using the first equation in (46), it is straightforward to show that 0 < r 2 /R 2 < 1/α (respectively, r 2 /R 2 > 1/α) is equivalent to 0 < a 2 /A 2 < 1/α (respectively, a 2 /A 2 > 1/α). When α = 1, the modulus defined by (58) coincides with the generalised shear modulus defined in [104, p. 174], and also in [15].…”

“…In this case, as R 2 /r 2 → 1, the three stress components defined by (57) are equal. Next, for the cylindrical tube deforming by (46), we set the inner and outer radial pressures acting on the curvilinear surfaces r = a(t) and r = b(t) at time t (measured per unit area in the present configuration), as T 1 (t) and T 2 (t), respectively [104, pp. 214-217].…”

“…Motivated by numerous long-standing and modern engineering problems, oscillatory motions of cylindrical and spherical shells made of linear elastic material [55,57,58,78] have generated a wide range of experimental, theoretical, and computational studies [5-7, 18, 29]. In contrast, time-dependent finite oscillations of cylindrical tubes and spherical shells of nonlinear hyperelastic material, relevant to the modelling of physical responses in many biological and synthetic systems [3,9,28,[40][41][42]56], have been less investigated, and much of the work in finite nonlinear elasticity has focused on the static stability of pressurised shells [2,17,21,22,24,31,34,36,38,59,70,81,90,111], or on wave-type solutions in infinite media [46,77].…”

Characterization and dynamical stability of fully nonlinear strain solitary waves in a fluid-filled hyperelastic membrane tube

“…In [17] a stability analysis of the aneurysm solutions in the presence of a mean flow was undertaken and it was found that if the speed of the mean flow is large enough, then the aneurysm solutions may be spectrally stable. It was found in [18] that for membrane tubes with localized wall thinning there exist two families of bulging solitary waves, and the lower family with amplitudes increasing with the inflation pressure is spectrally stable. In the latter case, we can speak about the standing wave (not orbital) stability, because the problem has no translational invariance any more.…”