High-Speed, Full-Field Deformation Measurements Near Inertial Microcavitation Bubbles Inside Viscoelastic Hydrogels

Abstract

Background

Inertial microcavitation is a well-known phenomenon that generates large stresses and deformations at extremely high loading rates in various soft materials, ranging from commercial polymer coatings to biological tissues. Recent advances in soft material characterization have taken advantage of inertial cavitation as a means towards a high-rate, minimally invasive soft material rheology approach. Yet, most of these studies rely on idealizations to infer the full deformation fields around the bubble based only on the experimentally measured temporal evolution of the bubble radius (akin to relying on crosshead strain data in a traditional materials test).

Objective

Here, we develop an experimental method to quantitatively measure full-field deformation and associated strains due to laser-induced inertial cavitation (LIC) in gelatin hydrogels, where the surrounding material is subjected to ultra-high strain rates ($10^3$ $\sim$ $10^6$ s$^{-1}$).

Methods

Our method combines two broad experimental techniques: the embedded speckle plane patterning (ESP) method and spatiotemporally adaptive quadtree mesh digital image correlation (STAQ-DIC).

Results

We illustrate the powerful capability of our approach by testing three concentrations of gelatin hydrogels 6%, 10%, and 14% as benchmark cases and quantitatively capture their kinematics during LIC.

Conclusions

These full-field, quantitative investigations are of significant interest in many cavitation-related applications including high strain-rate material characterization, guided advanced laser & ultrasound therapies, tissue engineering, and advanced manufacturing.

Appendices

Appendix 1

Theoretical Kinematic Fields in LIC

Consider a spherical bubble (see Fig. 12) with reference undeformed configuration $\mathcal {B}_0(r_0,\varphi _0,\theta _0)$, $\lbrace R_{0} \leqslant r_0 < \infty , \ 0 \leqslant \varphi _0 \leqslant \pi , \ 0 \leqslant \theta _0 \leqslant 2 \pi \rbrace$, and current deformed configuration $\mathcal {B}(r,\varphi ,\theta )$, $\lbrace R \leqslant r < \infty , \ 0 \leqslant \varphi \leqslant \pi , \ 0 \leqslant \theta \leqslant 2 \pi \rbrace$, where $\lbrace r_0, r \rbrace$ represent referential and current radial coordinates, $\lbrace \varphi _0, \varphi \rbrace$ are referential and current azimuthal angular coordinates, and $\lbrace \theta _0, \theta \rbrace$ are referential and current polar angular coordinates. The time-dependent bubble radius is R(t), and $R_0$ denotes the undeformed bubble radius. We assume a spherically symmetric motion, in which $r = r(r_0,t)$, $\varphi =\varphi _0$, and $\theta =\theta _0$, and the components of the deformation gradient tensor $\mathbf {F}$ in the spherical coordinate system are

$$\begin{aligned} \mathbf {F} = \begin{bmatrix} \frac{\partial r}{\partial r_0} &{} 0 &{} 0 \\ 0 &{} \frac{r}{r_0} &{} 0 \\ 0 &{} 0 &{} \frac{r}{r_0} \end{bmatrix}. \end{aligned}$$

(4)

Fig. 12

Spherical coordinate system $\lbrace r, \varphi , \theta \rbrace$

We assume that the surrounding material is incompressible, so that det($\mathbf {F}$) = 1, and the spherically symmetric motion is described by

$$\begin{aligned} r = \left( r_0^3 + R(t)^3 - R_0^3 \right) ^{1/3}. \end{aligned}$$

(5)

Equation (5) may be inverted to obtain the reference map $r_0 = (r^3 + R_0^3 - R(t)^3)^{1/3}$. For a spherically symmetric, incompressible motion, the only non-zero components of the displacement and the velocity vectors are the radial components, and their spatial descriptions are given by

$$\begin{aligned} u_r(r,t) = r - r_0(r,t), \end{aligned}$$

(6)

and

$$\begin{aligned} v_r(r,t) = \frac{R^2 \dot{R}}{r^2}, \end{aligned}$$

(7)

where the superposed dot denotes the derivative with respect to time t. The spatial description of the radial component of the acceleration vector is

$$\begin{aligned} a_r(r,t) = \frac{R^2 \ddot{R} + 2 R \dot{R}^2 }{r^2} - \frac{2 R^4 \dot{R}^2 }{r^5}. \end{aligned}$$

(8)

Finally, the Hencky (logarithmic) strain tensor is defined as $\mathbf {E} = (1/2) \text {ln} (\mathbf {F}^{{\scriptscriptstyle \mathsf {T}}} \mathbf {F})$. For a spherically symmetric, incompressible motion, the components of the logarithmic strain tensor in the spherical coordinate system are

$$\begin{aligned} \mathbf {E} = \begin{bmatrix} E_{rr} &{} 0 &{} 0 \\ 0 &{} E_{\varphi \varphi } &{} 0 \\ 0 &{} 0 &{} E_{\theta \theta } \end{bmatrix}, \end{aligned}$$

(9)

where $E_{\varphi \varphi } = E_{\theta \theta }$, and the spatial descriptions of the radial and circumferential logarithmic strain components are

$$\begin{aligned} E_{rr} = -2\ln \left( \dfrac{r}{r_0(r,t)}\right) \quad \text {and}\quad E_{\theta \theta } = \ln \left( \dfrac{r}{r_0(r,t)}\right) . \end{aligned}$$

(10)

Appendix 2

Other DIC Measured Displacement and Strain Results

Fig. 13

Kymographs of kinematic fields in three concentrations, (a) 6%, (b) 10%, and (c) 14%, of gelatin due to a single laser-induced cavitation bubble. The kymographs are created by taking a 10 pixel-wide vertical slice of the full-field data symmetric about the center of the cavitation bubble for each frame over 250 frames at a camera frame rate of 2 million frames/sec. The resulting (i) circumferential displacement field $u_{\theta }$ and (ii) shear logarithmic strain $E_{r\theta }$ are plotted against time for all three gelatin concentrations