Modeling high strain-rate microcavitation in soft materials: the role of material behavior in bubble dynamics.

Inertial Microcavitation Rheometry (IMR) is a promising tool for characterizing the mechanical behavior of soft materials at high strain rates. In IMR, an isolated, spherical microbubble is generated inside a soft material, using either a spatially-focused pulsed laser or focused ultrasound, to probe the mechanical behavior of the soft material at high strain rates (>103 s-1). Then, a theoretical modeling framework for inertial microcavitation, incorporating all the dominant physics, is used to extract information regarding the mechanical behavior of the soft material by fitting model predictions to the experimentally measured bubble dynamics. To model the cavitation dynamics, approaches based on extensions of the Rayleigh-Plesset equation are commonly used; however, these approaches cannot consider bubble dynamics that involves appreciable compressible behavior and place a limit on the nonlinear viscoelastic constitutive models that may be employed to describe the soft material. To circumvent these limitations, in this work, we develop a finite-element-based numerical simulation capability for inertial microcavitation of spherical bubbles that enables appreciable compressibility to be accounted for and more complex viscoelastic constitutive laws to be incorporated. We first apply the numerical simulation capability to understanding the role of material compressibility during violent spherical bubble collapse, and based on finite-element simulations, we propose a Mach-number-based threshold of 0.08, beyond which bubble collapse is violent, and the bubble dynamics involves compressibility not accounted for in Rayleigh-Plesset-based approaches. Second, we consider more complex viscoelastic constitutive models for the surrounding material, including nonlinear elastic and power-law viscous behavior, and apply IMR by fitting computational results to experimental data from inertial microcavitation of polyacrylamide (PA) gels in order to determine material parameters for PA gels at high strain rates.

Size-dependence of the flow threshold in dense granular materials.

The flow threshold in dense granular materials is typically modeled by local, stress-based criteria. However, grain-scale cooperativity leads to size effects that cannot be captured with local conditions. In a widely studied example, flows of thin layers of grains down an inclined surface exhibit a size effect whereby thinner layers require more tilt to flow. In this paper, we consider the question of whether the size-dependence of the flow threshold observed in inclined plane flow is configurationally general. Specifically, we consider three different examples of inhomogeneous flow - planar shear flow with gravity, annular shear flow, and vertical chute flow - using two-dimensional discrete-element method calculations and show that the flow threshold is indeed size-dependent in these flow configurations, displaying additional strengthening as the system size is reduced. We then show that the nonlocal granular fluidity model - a nonlocal continuum model for dense granular flow - is capable of quantitatively capturing the observed size-dependent strengthening in all three flow configurations.

A predictive, size-dependent continuum model for dense granular flows.

Dense granular materials display a complicated set of flow properties, which differentiate them from ordinary fluids. Despite their ubiquity, no model has been developed that captures or predicts the complexities of granular flow, posing an obstacle in industrial and geophysical applications. Here we propose a 3D constitutive model for well-developed, dense granular flows aimed at filling this need. The key ingredient of the theory is a grain-size-dependent nonlocal rheology--inspired by efforts for emulsions--in which flow at a point is affected by the local stress as well as the flow in neighboring material. The microscopic physical basis for this approach borrows from recent principles in soft glassy rheology. The size-dependence is captured using a single material parameter, and the resulting model is able to quantitatively describe dense granular flows in an array of different geometries. Of particular importance, it passes the stringent test of capturing all aspects of the highly nontrivial flows observed in split-bottom cells--a geometry that has resisted modeling efforts for nearly a decade. A key benefit of the model is its simple-to-implement and highly predictive final form, as needed for many real-world applications.

Nonlocal modeling of granular flows down inclines.

Flows of granular media down a rough inclined plane demonstrate a number of nonlocal phenomena. We apply the recently proposed nonlocal granular fluidity model to this geometry and find that the model captures many of these effects. Utilizing the model's dynamical form, we obtain a formula for the critical stopping height of a layer of grains on an inclined surface. Using an existing parameter calibration for glass beads, the theoretical result compares quantitatively to existing experimental data for glass beads. This provides a stringent test of the model, whose previous validations focused on driven steady-flow problems. For layers thicker than the stopping height, the theoretical flow profiles display a thickness-dependent shape whose features are in agreement with previous discrete particle simulations. We also address the issue of the Froude number of the flows, which has been shown experimentally to collapse as a function of the ratio of layer thickness to stopping height. While the collapse is not obvious, two explanations emerge leading to a revisiting of the history of inertial rheology, which the nonlocal model references for its homogeneous flow response.

Continuum modeling of secondary rheology in dense granular materials.

Recent dense granular flow experiments have shown that shear deformation in one region of a granular medium fluidizes its entirety, including regions far from the sheared zone, effectively erasing the yield condition everywhere. This enables slow creep deformation to occur when an external force is applied to a probe in the nominally static regions of the material. The apparent change in rheology induced by far-away motion is termed the "secondary rheology," and a theoretical rationalization of this phenomenon is needed. Recently, a new nonlocal granular rheology was successfully used to predict steady granular flow fields, including grain-size-dependent shear-band widths in a wide variety of flow configurations. We show that the nonlocal fluidity model is also capable of capturing secondary rheology. Specifically, we explore creep of a circular intruder in a two-dimensional annular Couette cell and show that the model captures all salient features observed in experiments, including both the rate-independent nature of creep for sufficiently slow driving rates and the faster-than-linear increase in the creep speed with the force applied to the intruder.

Modeling of elasto-capillary phenomena.

Surface energy is an important factor in the deformation of fluids but is typically a minimal or negligible effect in solids. However, when a solid is soft and its characteristic dimension is small, forces due to surface energy can become important and induce significant elastic deformation. The interplay between surface energy and elasticity can lead to interesting elasto-capillary phenomena. We present a finite-element-based numerical simulation capability for modeling these effects in a static, implicit framework. We demonstrate the capacity of the simulation capability by examining three elasto-capillary problems: (i) wetting of an elastic hemispherical droplet on a substrate, (ii) cavitation of an elastomer, and (iii) the Rayleigh-Plateau instability in soft elastic filaments.

3D Viscoelastic traction force microscopy.

Native cell-material interactions occur on materials differing in their structural composition, chemistry, and physical compliance. While the last two decades have shown the importance of traction forces during cell-material interactions, they have been almost exclusively presented on purely elastic in vitro materials. Yet, most bodily tissue materials exhibit some level of viscoelasticity, which could play an important role in how cells sense and transduce tractions. To expand the realm of cell traction measurements and to encompass all materials from elastic to viscoelastic, this paper presents a general, and comprehensive approach for quantifying 3D cell tractions in viscoelastic materials. This methodology includes the experimental characterization of the time-dependent material properties for any viscoelastic material with the subsequent mathematical implementation of the determined material model into a 3D traction force microscopy (3D TFM) framework. Utilizing this new 3D viscoelastic TFM (3D VTFM) approach, we quantify the influence of viscosity on the overall material traction calculations and quantify the error associated with omitting time-dependent material effects, as is the case for all other TFM formulations. We anticipate that the 3D VTFM technique will open up new avenues of cell-material investigations on even more physiologically relevant time-dependent materials including collagen and fibrin gels.

Predicting complex nonspherical instability shapes of inertial cavitation bubbles in viscoelastic soft matter.

Inertial cavitation in soft matter is an important phenomenon featured in a wide array of biological and engineering processes. Recent advances in experimental, theoretical, and numerical techniques have provided access to a world full of nonlinear physics, yet most of our quantitative understanding to date has been centered on a spherically symmetric description of the cavitation process in water. However, cavitation bubble growth and collapse rarely occur in a perfectly symmetrical fashion, particularly in soft materials. Predicting the onset of dynamically arising, nonspherical instabilities in soft matter has remained a significant, unresolved challenge, in part due to the additional constitutive complexities introduced by the surrounding nonlinear viscoelastic solid. Here, we provide a new theoretical framework capable of accurately predicting the onset of nonspherical instability shapes of a bubble in a soft material by explicitly accounting for all pertinent nonlinear interactions between the cavitation bubble and the solid surroundings. Comparison with high-resolution experimental images from laser-induced cavitation events in a polyacrylamide hydrogel show excellent agreement. Interestingly, and consistent with experimental findings, our model predicts the emergence of various dynamic instability shapes for circumferential bubble stretch ratios greater than 1, in contrast to most quasistatic investigations. Our new theoretical framework not only provides unprecedented insight into the cavitation dynamics in a soft, nonlinear solid, but also provides a quantitative means of interpreting bubble dynamics relevant to a wide array of engineering and medical applications as well as natural phenomena.

Nonlocal continuum modeling of dense granular flow in a split-bottom cell with a vane-shaped intruder.

Shear flow in one spatial region of a dense granular material-induced, for example, through the motion of a boundary-fluidizes the entire granular material. One consequence is that the yield condition vanishes throughout the granular material-even in regions that are very far from the "primary," boundary-driven shear flow. This phenomenon may be characterized through the mechanics of intruders embedded in the granular medium. When there is no primary flow, a critical load must be exceeded to move the intruder; however, in the presence of a primary flow, intruder motion occurs even when an arbitrarily small external load is applied to an intruder embedded in a region far from the sheared zone. In this paper, we apply the nonlocal granular fluidity (NGF) model-a continuum model that involves higher-order flow gradients-to simulate the specific case of dense flow in a split-bottom cell with a vane-shape intruder. Our simulations quantitatively capture the key features of the experimentally observed phenomena: (1) the vanishing of the yield condition, (2) an exponential-type relationship between the applied torque and the rotation rate, (3) the effect of the distance between the intruder and the primary flow zone, and (4) the direction-dependence of the torque/rotation-rate relation, in which the observed relation changes depending on whether the intruder is forced to rotate along with or counter to the primary flow. Importantly, this represents the first fully three-dimensional validation test for a nonlocal model for dense granular flow in general and for the NGF model in particular.

Comparative study of the dynamics of laser and acoustically generated bubbles in viscoelastic media.

Experimental observations of the growth and collapse of acoustically and laser-nucleated single bubbles in water and agarose gels of varying stiffness are presented. The maximum radii of generated bubbles decreased as the stiffness of the media increased for both nucleation modalities, but the maximum radii of laser-nucleated bubbles decreased more rapidly than acoustically nucleated bubbles as the gel stiffness increased. For water and low stiffness gels, the collapse times were well predicted by a Rayleigh cavity, but bubbles collapsed faster than predicted in the higher stiffness gels. The growth and collapse phases occurred symmetrically (in time) about the maximum radius in water but not in gels, where the duration of the growth phase decreased more than the collapse phase as gel stiffness increased. Numerical simulations of the bubble dynamics in viscoelastic media showed varying degrees of success in accurately predicting the observations.

Small-amplitude acoustics in bulk granular media.

We propose and validate a three-dimensional continuum modeling approach that predicts small-amplitude acoustic behavior of dense-packed granular media. The model is obtained through a joint experimental and finite-element study focused on the benchmark example of a vibrated container of grains. Using a three-parameter linear viscoelastic constitutive relation, our continuum model is shown to quantitatively predict the effective mass spectra in this geometry, even as geometric parameters for the environment are varied. Further, the model's predictions for the surface displacement field are validated mode-by-mode against experiment. A primary observation is the importance of the boundary condition between grains and the quasirigid walls.