Smooth waves and shocks of finite amplitude in soft materials

Abstract

Recently developed soft materials exhibit nonlinear wave propagation with potential applications for energy trapping, shock mitigation and wave focusing. We address finitely deformed materials subjected to combined transverse and axial impacts, and study the resultant nonlinear waves. We determine the dependency of the induced motion on the impact, pre-deformation and the employed constitutive models. We analyze the neo-Hookean constitutive model and show it cannot capture shear shocks and tensile-induced shocks, in contrast with experimental results on soft materials. We find that the Gent constitutive model predicts that compressive impact may not be sufficient to induce a quasi-pressure shock-yet it may induce a quasi-shear shock, where tensile impact can trigger quasi-pressure shock-and may simultaneously trigger a quasi-shear shock, in agreement with experimental data. We show that the tensile impact must be greater than a calculated threshold value to induce shock, and demonstrate that this threshold is lowered by application of pre-shear.

Shear shock evolution in incompressible soft solids

“…While perfect incompressibility allows us to impose an arbitrary normal traction σ n = P 22 that will be instantaneously equilibrated throughout the body (as seen from (10b)), in a compressible material, coupling between the longitudinal and shearing deformations might become significant (see for example Ziv and Shmuel (2019)). Plugging (2) into (10a) gives us the nonlinear wave equation…”

Observation of vector solitary waves in soft laminates using a finite volume method

“…In this case, the slower velocity c − is a monotonically increasing function of the strains, and the corresponding wave is referred to as genuinely nonlinear. The faster velocity c + has local minima at certain strains (Ziv and Shmuel, 2019), and hence the corresponding wave is not genuinely nonlinear. In the following sections we focus on strains far from these local minima, such that this mode can be considered genuinely nonlinear.…”

“…The left half-space is subjected to a shock wave for which the pre-shock state is unstrained and at rest, and its post-shock state is 0.2, 2.3, 2.6, −32.4). These initial conditions were chosen to generate four shock waves, for which we obtain numerical solutions using Newton's method, as described by Ziv and Shmuel (2019) and in Appendix A. Fi g. 4(a) shows the distribution of ǫ 1 (upper panel) and ǫ 2 (lower panel) across −0.4 m < X 1 < 0.6 m at t = 10 ms. The black line corresponds to the numerical solution of the exact equations.…”

“…Highly deformable materials with architectured microstructure are nowadays accessible by virtue of the current manufacturing abilities (Bandyopadhyay et al, 2015, Truby and. These materials exhibit geometrical and constitutive nonlinearities that give rise to rich physics , and in particular diverse wave phenomena such as unidirectional transmission, shocks and self-reinforcing waves (Nadkarni et al, 2014, 2016, Samsonov et al, 2017, Ziv and Shmuel, 2019, Deng et al, 2019a, Katz and Givli, 2019. Waves in nonlinear solids have recently regained scientific and technological interest, owing to the realization that their features can be harnessed for various applications, such as signal transmission (Raney et al, 2016), impact mitigation (Yasuda et al, 2019), energy harvesting (Hwang and Arrieta, 2018) and mechanical diodes (Deng et al, 2018).…”

“…The model we use to describe the nonlinear response of the media in our test cases is the compressible Gent model (Gent, 1996). This model-originally developed for capturing the strain stiffening of rubber at large strains-was recently shown useful by Ziv and Shmuel (2019) for modeling shear shocks that were experimentally observed by Catheline et al (2003), Espíndola et al (2017), and modeling tensile-induced shocks that were experimentally observed by Niemczura and Ravi-Chandar (2011).…”