Buckling of a spherical shell embedded in an elastic medium loaded by a far-field hydrostatic pressure

Abstract

Elastic buckling of a spherical shell, embedded in an elastic material and loaded by a far-field hydrostatic pressure is analysed using the energy method together with a Rayleigh—Ritz trial function. For simplicity, only axisymmetric deformations are considered and inextensional buckling is assumed. The strains within the structure that are pre-critical are assumed to be small for the linear theory to be applicable. An expression is derived relating the pressure load to the buckling mode number, from which the upper-bound critical load can be determined. It is found that the presence of the surrounding elastic medium increases the critical load of the shell and the corresponding buckling mode number. However, the results also show that the strain of the shell at the point of instability may not be small for typical values of material and geometric constants.

Influence of shell properties on high-frequency ultrasound imaging and drug delivery using polymer-shelled microbubbles

“…The linear elastic solution for compression-induced stress on a perfectly spherical thin shell can be expressed as $${\sigma }_{\text{crit}}=\frac{E}{\sqrt{3(1-{\nu }^2)}}true(\frac{2h}{d}true),$$ and the corresponding critical buckling pressure ( P crit = 4 h σ crit / d ) is given as $$P_{\text{crit}}=\frac{2E}{\sqrt{3(1-{\nu }^2)}}{\left(\frac{2h}{d}\right)}^2=A{\chi }^2,$$ where E is the Young's modulus, ν is the Poisson's ratio, h is the shell thickness, d is the diameter, A is a lumped constant describing the material properties, and χ is the STRR [37]. A Poisson's ratio of 0.3 was considered an appropriate approximation for most common materials, resulting in A = 1.21 E [38] (a positive coefficient of A implies compression of the sphere).…”

Predicting the pressure–volume curve of an elastic microsphere composite

“…In this section we discuss the buckling of a spherical shell inside an unbounded elastic matrix medium, loaded by a far-field hydrostatic pressure p. We employ a buckling model introduced by Fok and Allwright (2001). Given a distribution of sizes of microspheres inside the material, our aim is to determine which of them, for a given imposed pressure p, have buckled and which remain unbuckled.…”

“…Initial work into the buckling pressure of a spherical shell embedded in an unbounded uniform elastic medium has been carried out by Fok and Allwright (2001) and Jones et al (2008). We shall discuss these models and their assumptions later on in the paper, particularly that of Fok and Allwright (2001) which is the model that we shall adopt for buckling here. As described above, Shorter et al (2008) also carried out some experimental work related to this problem.…”

An investigation of contrast-agent shell-rupture threshold in response to overpressure

“…where E is the Young's modulus, ν is the Poisson's ratio, h is the shell thickness, d is the diameter, A is a lumped constant describing the material properties, and χ is the shell-thicknessto-radius ratio (STRR) [10].…”