Predicting the pressure–volume curve of an elastic microsphere composite

Abstract

The effective macroscopic response of nonlinear elastomeric inhomogeneous materials is of great interest in many applications including nonlinear composite materials and soft biological tissues. The interest of the present work is associated with a microsphere composite material, which is modelled as a matrix-inclusion composite. The matrix phase is a homogeneous isotropic nonlinear rubber-like material and the inclusion phase is more complex, consisting of a distribution of sizes of stiff thin spherical shells filled with gas. Experimentally, such materials have been shown to undergo complex deformation under cyclic loading. Here, we consider microspheres embedded in an unbounded host material and assume that a hydrostatic pressure is applied in the 'far-field'. Taking into account a variety of effects including buckling of the spherical shells, large deformation of the host phase and evolving microstructure, we derive a model predicting the pressure-relative volume change load curves. Nonlinear constitutive behaviour of the matrix medium is accounted for by employing neo-Hookean and Mooney-Rivlin incompressible models. Moreover a nearly-incompressible solution is derived via asymptotic analysis for a spherical cavity embedded in un unbounded isotropic homogeneous hyperelastic medium loaded hydrostatically. The load-curve predictions reveal a strong dependence on the microstructure of the composite, including distribution of microspheres, the stiffness of the shells, and on the initial volume fraction of the inclusions, whereas there is only a modest dependence on the characteristic properties of the nonlinear elastic model used for the rubber host.

The inflation of viscoelastic balloons and hollow viscera

“…Although inhomogeneous, the purely radial nonlinear elastic deformation associated with the inflation and deflation of an incompressible hollow sphere subjected to hydrostatic pressure is a universal solution belonging to 'Family 4' (see [58,59]). The problem is straightforward and its solution has long been known (see, for example, [28,58] and references therein). The equilibrium equation can be integrated exactly to yield a nonlinear equation for the internal deformed void radius a (or equivalently for the external radius b) in terms of the imposed pressure difference and the undeformed radius A (or the external undeformed radius B, respectively), in the form…”

“…B, b → ∞, on r = ∞) (figure 2). Given a strain energy function, the integral on the right-hand side in (1.1) is determined in a straightforward manner and the deformed radius a is determined numerically [28]. The corresponding linear viscoelasticity problem is also straightforward, and since the associated constitutive law in that case may be inverted without difficulty, both imposed displacement and pressure conditions can be derived.…”

A nonlinear elasticity approach to modelling the collapse of a shelled microbubble

“…Experimental studies (Falou et al, 2010;Tu et al, 2009) have shown that the Poisson ratio of a typical ultrasound contrast agent is between 0.48 and 0.49 which is similar to rubber. The common approach to modelling rubber for moderate strains is to utilize nonlinear strain energy density functions which are either neo-Hookean or Mooney-Rivlin (De Pascalis et al, 2013). The key reason for introducing a compressible, third invariant is to account for the material parameter described by Poisson's ratio and to identify how this parameter influences the collapse time of the shell whereas a neo-Hookean or Mooney-Rivlin strain energy density function only utilizes the shear modulus (De Pascalis et al, 2013).…”

“…The common approach to modelling rubber for moderate strains is to utilize nonlinear strain energy density functions which are either neo-Hookean or Mooney-Rivlin (De Pascalis et al, 2013). The key reason for introducing a compressible, third invariant is to account for the material parameter described by Poisson's ratio and to identify how this parameter influences the collapse time of the shell whereas a neo-Hookean or Mooney-Rivlin strain energy density function only utilizes the shear modulus (De Pascalis et al, 2013). The compressible neo-Hookean strain energy density function is (Gorb & Walton, 2010)…”

Syntactic foam under compressive stress: Comparison of modeling predictions and experimental measurements

“…De Pascalis et al. 18 have developed a mathematical model predicting the pressure/volume relationship of syntactic foam using Shorter’s work. He proposed the idea of modeling the compression as a two-step process: linear and non-linear, using the critical pressure defined by Fok and Allwright to set the transition between them.…”