Keywords

13.1 Introduction

Polymeric foams have been extensively used in shock isolation applications because of their superior shock or impact energy absorption capability. Experimentally, it is challenging when characterizing very soft polymeric foams with a Kolsky compression bar because radial-inertia-induced stress can overshadow the intrinsic material response. Currently existing analyses on radial inertia effect are based on the assumption of constant Poisson’s ratio [1,2,3,4,5], which is not applicable to foam materials. Foam materials usually possess very small Poisson’s ratio before densification and can approach a nearly rubbery state after densification. This drastic change in Poisson’s ratio may result in a sudden radial confinement in the specimen during densification process. Therefore, it is desirable to understand the radial inertia effect of polymeric foams that are subjected to large deformation (passing densification) at high strain rates.

In this paper, we develop an analytical method to calculate the additional stress induced by radial inertia in a polymeric foam specimen. The effect of changing profile of Poisson’s ratio during deformation on radial inertia is investigated. The analytical results are also compared with experimental results obtained from Kolsky compression bar tests on a silicone foam.

13.2 Radial Inertia in a Compressible Cylindrical Solid

In this study, a cylindrical specimen configuration, shown in Fig. 13.1, with initial dimensions of radius, a 0, and length, l 0, for radial inertia analysis. An axial compression at velocity V x = V x(t) generates a radial expansion with the velocity, V r = V r(r, t). From mass and momentum conservations, the additional axial stress induced by radial inertia is calculated as

$$ {\sigma}_r\left(r,t\right)=\rho (t)\cdot \frac{a^2(t)-{r}^2(t)}{2\left(1-{e}_x(t)\right)}\cdot \left[\nu (t){\ddot{e}}_x(t)+\nu (t)\cdot \left(\nu (t)+1\right)\frac{{\dot{e}}_x^2(t)}{1-{e}_x(t)}+\frac{d\nu \left({e}_x\right)}{{d e}_x}{\dot{e}}_x^2(t)\right] $$
(13.1)

where ρ and ν are instantaneous density and Poisson’s ratio of the specimen; e x is engineering axial strain subjected to the specimen. Equation (13.1) indicates that the radial inertia effect includes three parts induced by (1) strain acceleration (the first term in the bracket in Eq. (13.1)); (2) large strain and high strain rate (the second term in the bracket in Eq. (13.1)); and (3) change of Poisson’s ratio and high strain rate (the third term in the bracket in Eq. (13.1). The first two parts are also dependent on instantaneous Poisson’s ratio.

Fig. 13.1
figure 1

Cylindrical specimen configuration

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For a hyperelastic foam, i.e. silicone foam, the Poisson’s ratio follows a Boltzmann sigmoidal function of engineering axial strain,

$$ \nu \left({e}_x\right)=\frac{\nu_1-{\nu}_2}{1+\exp \left(\frac{e_x-{e}_{x0}}{\delta}\right)}+{\nu}_2 $$
(13.2)

where ν 1, ν 2, e x0, and δ are constants. Applying Eq. (13.2) into Eq. (13.1) yields an average radial-inertia-induced stress along axial direction,

$$ {\overline{\sigma}}_r(t)=\frac{\rho_0\cdot {a}_0^2}{4{\left(1-{e}_x(t)\right)}^2}\cdot \left[\begin{array}{l}{\ddot{e}}_x(t)\cdot \left({\nu}_2+\frac{\nu_1-{\nu}_2}{1+\exp \left(\frac{e_x(t)-{e}_{x0}}{\delta}\right)}\right)\\ {}+\left({\nu}_2+\frac{\nu_1-{\nu}_2}{1+\exp \left(\frac{e_x(t)-{e}_{x0}}{\delta}\right)}\right)\cdot \left(1+{\nu}_2+\frac{\nu_1-{\nu}_2}{1+\exp \left(\frac{e_x(t)-{e}_{x0}}{\delta}\right)}\right)\cdot \frac{{\dot{e}}_x^2(t)}{1-{e}_x(t)}\\ {}+{\dot{e}}_x^2(t)\cdot \frac{\nu_2-{\nu}_1}{\delta}\cdot \frac{\exp \left(\frac{e_x(t)-{e}_{x0}}{\delta}\right)}{{\left[1+\exp \left(\frac{e_x(t)-{e}_{x0}}{\delta}\right)\right]}^2}\end{array}\right] $$
(13.3)

For a silicone foam specimen with a density of ρ 0 = 600 kg/m3, a porosity of 50%, a diameter of 15 mm (a 0 = 7.5 mm), the constants of Poisson’s ratio are ν 1 = 0.21, ν 2 = 0.43, ε x0 = 0.525, δ = 0.01. The average radial-inertia-induced stress of a silicone foam specimen subjected to a constant strain rate of 4000 s−1 with a linear ramping time of 40 μs was calculated with Eq. (13.3) and is plotted in Fig. 13.2.

Fig. 13.2
figure 2

Additional axial stress induced by radial inertia

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13.3 Conclusions

A general radial inertia analysis is conducted in this study to better understand the dynamic compressive response of compressible materials, particularly polymeric foam materials with drastic change of Poisson’s ratio before and after densification. This analysis is applicable to any compressible or incompressible material, with a constant or varied Poisson’s ratio, subjected to high-strain-rate compression. Strain acceleration, large strain, high strain rate, and the change of Poisson’s ratio all contribute, but not necessarily equally, to radial inertia, which results in additional axial stress in dynamic specimen stress measurements.