# Cavitation Rheology of Model Yield Stress Fluids Based on Carbopol

Hadi Mohammadigoushki<EMAIL_ADDRESS>Department of Chemical and
Biomedical Engineering, FAMU-FSU College of Engineering, Florida State
University, Tallahassee, FL, 32310, USA  Kourosh Shoele  Department of
Mechanical Engineering, FAMU-FSU College of Engineering, Florida State
University, Tallahassee, FL, 32310, USA

###### Abstract

Measuring surface tension of yield stress fluids has remained a critical
challenge due to limitations of the traditional tensiometry techniques. Here,
we overcome those limits and successfully measure the surface tension and
mechanical properties of a model yield stress fluid based on Carbopol gels via
a needle-induced cavitation (NIC) technique. Our results indicate that the
surface tension is approximately 70$\pm$3 mN/m, and is independent of the
rheology of yield stress fluid over a wide range of yield stress values
$\sigma_{y}=0.5-120$ Pa. In addition, we demonstrate that a Young modulus
smaller than $E<$1 kPa can be successfully measured for Carbopol gels with NIC
method. Finally, we present a time-resolved flow structure around the cavity
in a host of yield stress fluids, and assess the impact of fluid rheology on
the detailed form of flow around the cavity. Interestingly, prior to the
critical point associated with cavitation, the yield stress fluid is weakly
deformed suggesting that the measured surface tension data reflect the near
equilibrium values. Beyond the critical point, the yield stress fluid
experiences a strong flow that is controlled by both the critical pressure and
the non-Newtonian rheology of the yield stress fluid.

##  I Introduction

Yield stress fluids are common in our daily life and have been the subject of
intense research in the past decadesfrigaard2019 ; balmforth2014 ; coussot2014
; malkin2017 . Prime examples include food stuff, paint, home care products,
drilling fluids, oil extraction products, concrete as well as other advanced
functional materials such as colloidal gelssaud2021 , emulsionsfoudazi2015 ,
soft glassy materialsovarlez2010  and jammed suspensionsgross2014 . In the
classical description, yield stress materials behave like a solid (or deform
in a finite way) below a critical stress threshold known as yield stress and
flow beyond this stress thresholdBing22 . Yield stress fluids commonly
interact with solid substratesnelson2020 . For example, application of yield
stress lotions to skin, painting the wall, 3D printing of polymer melts on
substrates. The performance of the yield stress material in these applications
is controlled by its wetting and surface tension. In addition, surface tension
of the yield stress fluids plays a critical role in environmentally important
applications such as oil-sand pond reclamationSMALL2015 ; pourzahedi2021  and
nuclear waste managementjohnson2017 ; kam2001 . For a simple liquid and at
equilibrium the viscous stresses are negligible and provided that the
gravitational forces are small, an equilibrium surface tension can be
quantifiedisraelachvili2011 . Unlike simple liquids, measuring the surface
tension of a yield stress fluid has remained a critical challenge mainly
because at equilibrium, the residual stresses in the yield stress fluids are
no longer negligible.

Carbopol gels provide a model yield stress fluid for surface tension analysis
and have been the subject of several studiesHU1991723 ; MANGLIK200155 ;
G_raud_2014 ; Bouj13 ; C5SM00569H ; Lopez18 . Carbopol gel is a transparent
system, and its rheological properties can be fine tuned by variation of
Carbopol concentration or solution pHgutowski2012 . This material is generally
considered as a non-thixotropic yield stress fluidbalmforth2014 ; jaworski2021
, making this system a model yield stress fluid for studies of surface
tension. The earliest attempt in measuring the surface tension in Carbopol
solutions goes back to the work of Hu et al.HU1991723 . These authors used a
maximum bubble pressure (MBP) method, and reported a surface tension of
approximately around the measured value for pure water ($\gamma$ $\approx$
72.5 mN/m) for their solutionsHU1991723 . A closer inspection of their
rheological data reveals that the Carbopol based fluids used by these authors
(Carbopol 934 at concentrations between 0.05-0.1 wt$\%$) did not exhibit a
yield stress rheologyHU1991723 . Manglik et al.MANGLIK200155  also used the
MBP method and measured the surface tension of a range of Carbopol solutions
with concentrations ranging from 0 to 2000 ppm, and showed that the surface
tension in this range of concentration is also close to the surface tension of
water MANGLIK200155 . However, the latter study did not report the details of
Carbopol neutralization and the rheology data, which makes it unclear if those
solutions were yield stress fluidsMANGLIK200155 . More recently, Geraud et al.
G_raud_2014  used a capillary rise experiment to measure the surface tension
of a yield stress fluid based on Carbopol gel. These authors noted a surface
tension of 51 mN/m for their systemG_raud_2014 . Boujlel and Coussot Bouj13
used a Wilhelmy plate and suggested that this device generates an equilibrium
surface tension around 66 mN/m at vanishingly small Capillary numbers Bouj13 .
The Capillary number is defined as the ratio of the yield stress to the
surface tension. Jørgensen et al. C5SM00569H  used a bridge tensiometer that
involved compression and extension of the yield stress fluids between two
walls. These authors showed that the surface tension data obtained from
compression experiments are smaller than those measured during extension and
this deviation increases as the yield stress of the material increases
C5SM00569H . More recently, Lopez and co-workers used a ring tensiometer and
reported a surface tension value of 73.4 mN/m for their Carbopol based yield
stress fluids Lopez18 . A summary of the above literature is reported in Table
(1).

Apparently, there is no consensus on the value of the surface tension in yield
stress fluids. The complications associated with the use of solid substrates
may play a significant role in this disparity. In particular, majority of
prior experiments have measured surface tension by using methods that involve
a contact between solid substrates and the yield stress fluid (e.g., Wilhemly
plate or Bridge tensiometer). It is very well known that yield stress fluids
are susceptible to wall-slip jalaal2015 ; walls2003 ; bertola2003 ; divoux2011
; abbasi2021 , and therefore, the fluid contact-line on such substrates may
not be pinned during surface tension measurements. In particular, Geraud et
al. G_raud_2014  briefly noted that in capillaries with smooth surfaces, the
yield stress fluid shows significant wall-slip. However, Jørgensen et al.
C5SM00569H  and Boujlel $\&$ Coussot Bouj13  neither specified the type of
surfaces used in their experiments, nor investigated the impact of wall-slip
on their surface tension data.

Carbopol Type | Concentration (wt$\%$) |  $\sigma_{y}$ [Pa] | Technique used |  $\gamma$ [mN/m] | Reference
---|---|---|---|---|---
934 | 0-0.2 | – | Maximum Bubble Pressure | 73 | MANGLIK200155
934 | 0.05-0.1 | – | Maximum Bubble Pressure | 72.5 | HU1991723
ETD 2050 | 0.5 | 4 | Capillary Rise | 52 | G_raud_2014
ETD 2050 | 0.25-2 | 0.3-38 | Bridge Tensiometer |  $\approx$15-100 | C5SM00569H
ETD 2050 | 0.25-0.5 | 0.3-1.75 | Ascending Bubble | 59-66 | C5SM00569H
U 10 | 0.1-0.5 | 9-80 | Wilhelmy Plate | 66 | Bouj13
980 | 0.09-0.2 | 3-28.5 | LAUDA Tensiometer | 73.4 | Lopez18
Table 1: Summary of previous studies that have measured the surface tension
for Carbopol gels. Here $\sigma_{y}$ and $\gamma$ denote the yield stress and
the surface tension of the fluid.

Alternatively, there exists other methods for surface tension measurements
that do not rely on fluid contact with a solid substrate. Examples include
drop weightyildirim2005 , pendant dropstauffer1965 ; lecacheux2022  and
maximum bubble pressure (MBP)mysels1990 ; fainerman2004 ; simon1851  methods.
Although drop weight and pendant drop methods have been well established for
simple fluids, these methods are not suitable for yield stress fluids not only
because the residual yield stresses affect the drop behavior compared to
simple liquids, but also the basic theories do not incorporate the impact of
residual yield stresses in the flow analysis for calculation of the surface
tension.

The MBP method was first introduced by Simon in 1851 simon1851 , and has been
widely used to measure the surface tension of a broad range of simple liquids
kuffner1961 ; austin1967 ; joos1981 ; fainerman2004 . In the MBP experiments a
capillary tube (or a needle) is immersed in the fluid, and subsequently a gas
is pumped through the needle into the liquid. As gas is injected into the
needle, the gas-liquid interface forms a curved shape and its curvature is
controlled by the pressure difference between the gas inside the needle and
the surrounding fluid. In principle, for any fluidic environment, the pressure
inside the needle (or cavity) $P$ is balanced by the outside pressure and can
be obtained as:

$P=P_{h}+\gamma(1/r_{1}+1/r_{2})+P_{out},$ (1)

where $P_{h}=\rho gz$ is the hydrostatic pressure at the needle tip, $\gamma$
is the surface tension, $r_{1}$ and $r_{2}$ are the principal radii of
curvature of the cavity and $P_{out}$ is the pressure of the surrounding
medium resisting against motion and growth of the cavity. Experimental
observations in simple liquids have shown that as the gas is injected into the
needle, the total pressure inside the needle increases and at some critical
point, it goes through a maximum and suddenly drops to small
valuesfainerman2004 ; mysels1990 ; mysels1986improvements . This maximum
pressure has been used to evaluate the surface tension of the
liquidsfainerman2004 ; mysels1990 ; mysels1986improvements ; mysels1989some .
However, to obtain the surface tension from the maximum pressure data, one has
to account for multiple factors (i.e., curvature of the cavity, Buoyancy as
well as hydrodynamic stresses in the liquid). Previous studies have shown that
in the limit of small needle sizes $R<1$mm, a spherical cavity forms at the
tip of the needlefainerman1998maximum . In particular, at the maximum pressure
($P_{c}$), the cavity forms a hemispherical shape with a radius equal to that
of the needle radius, and therefore, Eq. 1 can be expressed as:
$P_{c}-P_{h}=\frac{2\gamma}{R}+P_{out}$. Any gas injection beyond this point
causes the cavity to become unstable, and grow rapidly before detaching from
the needle. For needles with $R>1mm$, the cavity becomes non-
sphericalsugden1924determination . It has been shown that the impact of cavity
non-sphericity on the surface tension could be accounted by introducing a
correction factor $f$ into the above equation such that:

$P_{c}-P_{h}=f\frac{2\gamma}{R}+P_{out}.$ (2)

The correction factor can be obtained asfainerman1998maximum :
$f={\sum_{i=0}^{5}}a_{i}{(\frac{R}{a})}^{i},$ where capillary length
$a=\sqrt{\frac{2\gamma}{\Delta\rho g}}$. Note $a_{i}$ values are tabulated in
the literature fainerman1998maximum ; bendure1971dynamic . For viscous liquids
$P_{out}$ is associated with the viscous stresses. Previous studies have
investigated the impact of hydrodynamic stresses on surface tension data of a
range of viscous fluids fainerman2004correction . Fainerman and co-workers
showed that the viscous stresses can increase the surface tension by up to 3
mN/m over two orders of magnitude variation in the viscosity of a solution
with an equilibrium surface tension of 70 mN/m, thereby, suggesting a small
influence of the viscous stresses on equilibrium surface tension of viscous
liquids fainerman2004correction .

##  II Cavitation Rheology

Yield stress fluids are different from simple liquids in that below the yield
stress threshold, they behave like a solid and barely deform. As a result, the
gas pressure inside the capillary tube must overcome the residual stresses in
the solid before they can plastically deform the surrounding material. Over
the course of last decade, Crosby and co-workers have developed a needle-
induced cavitation (NIC) rheology technique based on the MBP method for a wide
range of synthetic hydrogels, rubbers and block co-polymers zimberlin2007 ;
kundu2009 ; zimberlin2010 ; barney2020 ; cui2011 ; delbos2012 ; hutchens2016 ;
barney2019 . It has been shown that for elastic materials, and in the limit of
small needle sizes, similar to simple liquids, the maximum pressure occurs at
the point where cavity forms a hemisphere at the tip of the needle and the
critical pressure $P_{c}$ is related to surface tension and Young modulus $E$
of the gel as zimberlin2007 ; gent1969 :

$P_{c}=5E/6+2\gamma/R,$ (3)

where, the elasticity of the surrounding material acts as an additional
pressure that cavity must overcome with $P_{out}=5E/6$. Therefore, the maximum
pressure measurement in the MBP method not only allows one to estimate the
surface tension of the gel, but also its elastic modulus, hence the term
cavitation rheology was used. Note that in Eq. (3) the surrounding material
behaves as a neo-Hookean solid with no plastic deformation. As a result, the
impact of local yielding of the materials on surface tension data is
neglected. The latter assumption may hold for stiff materials such as
synthetic hydrogels, rubbers and block co-polymers that are stiff and exhibit
strong elasticities. The yield stress fluids based on Carbopol gels are
expected to be much softer than synthetic hydrogels, acrylic triblock gels or
copolymers used in previous studies zimberlin2007 ; barney2019 . For yield
stress materials based on Carbopol, the $P_{out}$ resistance may be related to
both the elastic response of the yield stress fluid before reaching plasticity
as well as the plastic response of the medium after it passes the yield limit
in the vicinity of a growing bubble. Consequently, Eq. (3) should be modified
for soft yield stress fluids prepared from Carbopol solutions.

To the best of our knowledge, the theoretical analysis of the cavitation
phenomenon in soft yield stress fluids has not been considered before.
Additionally, there are currently no studies that employ the NIC technique to
measure the surface tension of the yield stress fluids based on Carbopol gels.
Although previous studies have used the NIC method to measure the Young
modulus of a wide range of stiff materials with a modulus in the range of 1
kPa $<E<$ 60 kPabarney2019 , the yield stress fluids based on Carbopol gels
are expected to be much softer than stiff gels and co-polymers barney2019 .
Hence, it is still unclear if the NIC technique is sensitive enough to measure
a Young modulus in the range of E $<$ 1 kPa that is relevant for Carbopol
gels. The main goal of the first part of this paper is to develop a
theoretical framework for cavitation in soft yield stress fluids and its
subsequent application in experiments to evaluate the surface tension and the
mechanical properties of the yield stress fluids based on Carbopol gels.

On a relevant note and from a biological perspective, a very important health
issue, traumatic brain injury (TBI), has been associated with the cavitation
of a bubble in biological tissues that exhibit yield stress properties
barney2020 . The leading hypothesis is that the cavitation introduces a strong
flow, which consequently deforms and damages the surrounding tissue, and that
causes TBIcanchi2017 . To test this hypothesis, one must first evaluate the
flow field generated by a cavitation process in a yield stress material. A
direct and in situ measurement of the flow profile around a cavity in yield
stress materials do not exist. Imaging the detailed form of flow structure
around a cavity in a model yield stress fluid provides important insights that
will significantly advance our understanding of the origin of the TBI. The
main goal of the second part of this paper is to provide the first temporally-
resolved form of flow structure around a cavity in model yield stress fluid
based on Carbopol gels.

##  III Materials and Methods

Yield stress is observed in a host of materials. The most popular yield stress
fluid is a polymeric gel based on an aqueous solution of Carbopolbalmforth2014
. Different models of Carbopol (940, 980, Ultrez-10 and etc) are commercially
available. In this study we use Carbopol 940 and Ultrez-10 which are known as
non-thixotropic model yield stress fluids balmforth2014 . The concentration of
the Carbopol is varied from 0.02 wt$\%$ to 0.5 wt$\%$. Yield stress fluids are
made by gently mixing Carbopol with de-ionized water and neutralized by adding
1.5 mass fraction of triethanolamine to the Carbopol solution. In addition to
Carbopol based fluids, we use a Newtonian fluid (corn syrup from Golden
Barrel) for the purpose of comparison with the yield stress fluids (see
rheological properties in Table S(1) of the supplementary materials).

Yield stress fluids were characterized using a commercial rheometer (TA
Discovery HR 10) and a standard concentric cylinders geometry with Ri =14.01
mm and Ro = 15.185 mm, where Ri and Ro are the radii of the inner and outer
cylinders. Because wall-slip is significant and can affect the rheological
characterization of the yield stress fluids, we have roughened the concentric
cylinders geometry using a sand blasting protocol. As in our previous studies
Rassolov2020 ; Rassolov2022 ; nazari2023helical , two types of measurements
are performed: Small Amplitude Oscillatory Shear (SAOS) was used to obtain the
linear viscoelastic responses. In particular, the storage and loss moduli are
measured as a function of angular frequency. In addition, the flow curves are
measured using a steady applied shear experiment.

To measure the surface tension and the Young modulus of the yield stress
fluids, we use an in-house custom-made NIC method. Fig. 1 shows a schematic of
the custom made NIC technique, which consists of a programmable syringe pump
(model NE-1000 from New Era), blunt needle (from McMaster-Carr), tubing,
wiring, differential pressure sensor (model PX-26 series from OMEGA), high-
speed camera (Phantom miro M310), data acquisition device; DAQ (from National
Instruments), and a computer. The LabVIEW software and the DAQ allow the
differential pressure sensor, programmable syringe pump, and high-speed camera
to work in a synchronized manner. Using this setup, we are able to record, and
capture the temporal evolution of cavity growth and the associated pressure
changes. The inner radius of the needle (or the capillary tube) used in these
experiments is varied from $R$ = 76 $\mu$m - 850 $\mu$m, which allows us to
access a wide range of critical pressures. In each experiment, the needle is
gently placed in the yield stress fluid and the air is injected in the
capillary tube at a constant rate of $Q=0.3~{}\mu$L/hr. Fluids are placed in a
cubical container with flat side walls to minimize optical distortions. In
addition to NIC method, we use a pendant drop technique to measure the surface
tension of primarily non-yield stress fluids (see more details about this
technique in the supplementary materials). The surface tension data obtained
from the pendant drop experiments (for simple liquids) are compared with the
results obtained by the NIC method to ensure the accuracy of the NIC method.

We have also performed particle image velocimetry (PIV) to temporally resolve
the flow field around the cavity in Carbopol gels. For PIV analysis, we
generate a sheet of laser light (with a wavelength of 532 nm) that passes
through the cavity. The fluids are seeded by glass microspheres (model 110P8
from Potters with a mean diameter $\approx 8~{}\mu$m). The small amount of
these seeding particles (40 ppm by mass) does not affect the rheological
properties of the fluids.

Gas

Syringe Pump

Pressure Sensor

Computer & Control Unit

Data Acquisition Unit

High Speed Camera

Figure 1: A schematic of the NIC device along with the necessary parts used in
this paper.

##  IV Results and Discussion

###  IV.1 Bulk rheology

Fig. 2(a) shows the flow curves of sample representative yield stress fluids
based on Ultrez-10 along with the best fit to the Herschel-Bulkley fluid model
(dashed curves). The Herschel-Bulkley model is defined as:
$\sigma=\sigma_{y}+K{\dot{\gamma}}^{n}$, where $\sigma$, $\sigma_{y}$, $K$,
$\dot{\gamma}$ and $n$ denote the shear stress, yield stress, consistency
factor, rate of deformation and the shear-thinning index, respectively. As
expected, the increase in the Carbopol concentration gives rise to a stronger
yield stress fluid. In addition, Fig. 2(b) shows the measured storage and loss
modulii as a function of angular frequency for sample yield stress fluids
based on Ultrez-10. As the Carbopol concentration increases both the storage
and loss moduli increase, which is again expected and consistent with the
reported values in the literaturegutowski2012 . Moreover, the storage modulus
approaches an asymptotic value (called shear modulus hereafter) as the angular
frequency decreases. Similar rheological properties have been reported for the
yield stress fluids based on CBP-940 and a summary of the rheological
properties of the Carbopol solutions are included in the table (S1) and Fig.
S1 of the supplementary materials. In addition, the shear viscosity of these
fluids is not sensitive to ramp-up or ramp-down in shear rate (see Fig. S2 in
the supplementary materials), thereby confirming that these yield stress
fluids are not thixotropic.

Figure 2: (a) steady shear stress as a function of applied shear rate for
yield stress fluids based on Carbopol Ultrez-10. (b) Storage (filled symbols)
and loss moduli (open symbols) as a function of angular frequency. Different
symbols correspond to various Carbopol concentration as 0.06 wt$\%$ ($\circ$),
0.1 wt$\%$ ($\square$), 0.3 wt$\%$ ($\diamond$) and 0.5 wt$\%$ ($\triangle$).

Fig. 3 shows a summary of the yield stress and the shear modulus of these
yield stress fluids as a function of Carbopol concentration. Interestingly, at
low Carbopol concentrations, the yield stress and the shear modulus increase
rather sharply as the Carbopol concentration increases. However, beyond a
critical concentration (between 0.1-0.15 wt$\%$), the increase in yield stress
and the shear modulus becomes more gradual. This trend can be rationalized as
follows. Carbopol is made up of high molecular weight polymers that swell upon
mixing and neutralization in aqueous solutionsOel22 . At low Carbopol
concentrations, the swollen polymer particles form a percolated structure,
which at higher concentrations, the polymer swelling increases, thereby,
decreasing the distance between polymer particles in the solution. The
decrease in particle spacing increases the yield stress and the shear modulus.
Beyond a critical point, where particles create a jammed structure, further
increase of the Carbopol content does not dramatically affect the jammed
nature of the solution and therefore, the yield stress and the elastic modulus
increase more gradually.

Figure 3: (a) Yield stress and (b) shear (elastic) modulus as a function of
Carbopol concentration for the two yield stress fluid systems. The continuous
and dashed lines are guide to the eye.

###  IV.2 Needle induced cavitation

As noted above, the maximum pressure inside the capillary tube in soft yield
stress fluids may be related to surface tension, the yield stress and the
elastic modulus of the material. Therefore, in the first step, we developed a
mechanical analysis of the cavitation in yield stress fluids. In the limit of
small capillary size ($R<1mm$ or for a nearly spherical bubble), in addition
to the surface tension pressure $2\gamma/R$, the effective pressure acting on
the yield materials surrounding the cavity $P_{out}=P_{out}(\sigma_{y},E)$. In
the near field, adjacent to the bubble, the material may yield and far away
from the bubble, the yield stress material behaves as an elastic solid.
Considering the yield stress material as a linear elastic perfectly plastic
material, the total pressure acting on the cavity $P_{out}$ = $P_{out}|_{1}$
\+ $P_{out}|_{2}$. Here $P_{out}|_{1}$ and $P_{out}|_{2}$ are pressures
associated with plastically deformed response of the surrounding medium near
cavity and the confinement induced by the elastic response of the yield stress
material, respectively. Based on our analysis, the total pressure contribution
from the surrounding yield stress material on the cavity can be written as
(see appendix for more details on the derivation):

$P_{out}\,=\frac{2\sigma_{y}}{3}\left\\{1+\ln\left(\frac{2E}{3\sigma_{y}}\right)\right\\}+\frac{2\pi^{2}}{27}E.$
(4)

Note the latter term in total pressure ($P_{out}|_{2}=\frac{2\pi^{2}}{27}E$)
is analogous to the Eq. 3 derived for elastic networkszimberlin2007 ; gent1969
. Therefore, in the limit of small capillary tube diameters, the maximum
pressure inside the growing bubble in a soft yield stress material can be
given as:

$P_{c}\,=\frac{2\gamma}{R}+\frac{2\sigma_{y}}{3}\left\\{1+\ln\left(\frac{2E}{3\sigma_{y}}\right)+\left(\frac{\pi}{3}\right)^{2}\frac{E}{\sigma_{y}}\right\\}.$
(5)

The above Eq. 5 will be used to assess the surface tension as well as the
elastic modulus of the yield stress fluids based on Carbopol. Subsequent to
the above theoretical analysis, we performed needle-induced cavitation
experiments in a Newtonian corn syrup as well as yield stress fluids. Fig. 4
shows the temporal evolution of the pressure in the Newtonian solution based
on corn syrup (Fig. 4(a)) and a sample yield stress fluid (0.5wt$\%$
Ultrez-10; Fig. 4(b)). In these experiments, the volume of the air left in the
syringe at any point in time can be given as: $V(t)=V_{0}-Qt+V_{b}$, where
$V_{0}$, $Q$ and $V_{b}$ are the initial air volume (which is 10mL in this
work), flow rate by which the air is pumped and the volume of the cavity. If
we assume that air is an ideal gas and experiments are performed in isothermal
conditions, $PV$ = $P_{0}V_{0}$ and therefore, we will have:

${\Delta
P}/{P_{0}}={(P-P_{0})}/{P_{0}}={1}/({1-\frac{Q}{V_{0}}t+\frac{V_{b}}{V_{0}}})\,-\,1$
(6)

In the limit of $V_{b}<<V_{0}$, and $Qt<V_{0}$ (which are satisfied in our
experiments) pressure should increase linearly with time regardless of the
type of fluid. Indeed, our results of Fig. 4(a,b) show that for both Newtonian
and the yield stress fluids, the pressure increases linearly up to a critical
value before it drops quickly. Additionally, the critical pressure increases
as the needle diameter decreases, which is consistent with the predictions of
Eq. (2) for the Newtonian fluid and Eq. (5) for the yield stress fluid.

Figure 4: Temporal evolution of the pressure in (a) corn syrup and (b)
0.5wt$\%$ yield stress fluid based on Ultrez-10. The transient pressure is
shown for various needle tube diameters $d$. The lower snapshots show the
simultaneous temporal evolution of cavity in corn syrup (top row) and in
0.5wt$\%$ yield stress fluid based on Ultrez-10 (lower row). Note that (i-v)
in snapshots refer to the time instances at which pressure is measured before
or after the critical point in part (a) and part (b). The scale bar in
snapshot is 0.8 mm.

The corresponding temporal evolution of the cavity size are shown for the corn
syrup (top row in Fig. 4) and the yield stress fluid (bottom row in Fig. 4).
As the pressure increases towards the critical point, the cavity gradually
protrudes out of the needle tip and attains a smaller radius of curvature. At
the critical point, the cavity forms a hemisphere with a diameter that is
approximately equal to the diameter of the needle for both Newtonian and the
yield stress fluids. This is unsurprising because the correction factor $f$ in
these experiments is very close to unity ($f=0.9968$). Beyond the critical
pressure, the explosive growth of the cavity radius gives rise to the pressure
drop. The experiments of Fig. 4 are performed in all solutions and the
measured critical pressure values along with the best fit to Eq. 5 could be
used to assess the surface tension and the Young modulus of the yield stress
fluids. Prior to assessment of the yield stress fluid properties, and in order
to ensure the accuracy and precision of our NIC device, we first report the
results for a broad range of non-yield stress fluids.

Fig. 5 shows the critical pressure as a function of needle diameter for corn
syrup along with Carbopol based solutions that are in the dilute regime and
show non-yield stress rheology (see Table (S1) in the supplementary materials
for rheological properties). Subsequently, the results were fitted to Eq. (2).
In experiments performed in this study, the maximum needle radius used is
$0.85$mm and therefore, the maximum correction factor due to non-sphericity of
the cavity is about $f=0.97$. Additionally, unlike previous MBP studies that
have mainly used one needle size to assess the surface tension of liquids, we
fit the critical pressure data to a wide range of needle sizes and that
reduces the error associated with using only one measurement point. The
resulting intercepts and slopes are summarized in Table (2).

The first notable observation is that $P_{out}$ contribution from viscous
stresses is very small ($\leq$ 8 Pa) both for the corn syrup as well as other
non-yield stress fluids based on Carbopol. Note that the maximum $P_{out}$
measured in our experiments is smaller than the pressure associated with the
surface tension of the liquid by orders of magnitude. For example, for the
largest needle used, $R=0.85$mm, the surface tension pressure is $\approx$
0.17 kPa while $P_{out}=0.0004-0.008$ kPa indicating a negligible impact of
the viscous stresses on surface tension measurements. In addition, although
some of these viscous liquids (cf. Carbopol 940 0.025 wt $\%$ and 0.05 wt $\%$
in Table S(1) of the supplementary information) have significantly different
shear viscosity, the resulting $P_{out}$ is still negligible and similar in
magnitude. We conclude from these results that the impact of viscous stresses
on surface tension data is negligible. This conclusion is consistent with
previous findings on viscous fluids fainerman2004correction .

Secondly, the slope of each graph, which represents the estimated surface
tension of these fluids are listed in Table 2. In particular, for corn syrup,
the measured surface tension is consistent with the value reported in the
literature llewellin2002 ; borhan1999 . In addition, for dilute Carbopol
solutions the surface tension is very close to that measured for pure water,
independent of the type and concentration of the Carbopol. The latter data are
consistent with measurements of Manglik et al MANGLIK200155  and Hu et al.
HU1991723  who have reported a surface tension of $\approx$ 72.5-73 mN/m for
dilute Carbpol solutions based on Carbopol 934. Finally, included in the Table
2 are the measured surface tension data of these fluids by a pendant drop
method (see details in Fig. S3 of the supplementary materials). The resulting
surface tension values obtained by the pendant drop method are consistent with
the NIC results, thereby, confirming the accuracy of our NIC method for
evaluating the surface tension of simple liquids.

Figure 5: Critical pressure as a function of capillary tube size for non-yield stress fluids. The dashed lines indicate the best linear fit to the experimental data.  Fluid | [wt$\%$] |  $P_{out}$ [kPa] |  $\gamma$ [mN/m] |  $\gamma^{*}$ [mN/m] |  $\gamma^{**}$ [mN/m]
---|---|---|---|---|---
corn syrup |  | -0.008 | 83.2 | 81 borhan1999  | 80.5$\pm$2
940 | 0.025 | -0.006 | 70 | $--$ | 71.0$\pm$2
0.05 | -0.005 | 72 | $--$ | 72.2$\pm$1
Ultrez$-$10 | 0.04 | -0.004 | 71.1 | $--$ | 70.9$\pm$2
0.05 | 0.004 | 72.1 | $--$ | 72.2$\pm$3
Table 2: A summary of the surface tension and fitting to the Newtonian as well
as dilute Carbpol based solutions. ∗ denotes the surface tension data reported
in the literature and ∗∗ are the surface tension values obtained by a pendant
drop method.  Figure 6: Critical pressure as a function of inverse capillary
size for yield stress fluids based on (a) Ultrez-10 and (b) 940. The dashed
line is the best fit of Eq. 5 to the experimental data.

Subsequent to the above experiments, the needle induced cavitation experiments
are performed on all of the yield stress fluids and the critical pressure for
a wide range of needle sizes is measured. Fig. 6 shows the critical pressure
as a function of capillary radius for the two yield stress systems of this
paper. Included in these figures is also the best fit of Eq. 5 to the
experimental data. First, the slope of the data in all experiments is similar
hinting at a similar surface tension data for all of these systems over a
broad range of Carbopol concentrations. Additionally, the intercept of the
fitted line to these experimental data is no longer negligible and increases
as the Carbopol concentration increases. Finally, the measured critical
pressure data are independent of the imposed flow rate ($Q=0.1-10~{}\mu$L/hr;
see Fig. S4 in the supplementary materials) and the needle insertion procedure
(whether needle is gently inserted into the sample or retracted following a
protocol suggested in barney2019 ).

The resulting surface tension data for the two yield stress fluids are shown
in Fig. 7(a). Interestingly, the surface tension does not change substantially
as a function of yield stress. The average surface tension data amongst all
these yield stress fluids is approximately 70$\pm$3 mN/m. These results are
consistent with the assumption made by Lopez et al Lopez18  that the surface
tension of the yield stress fluids does not change as the yield stress
increases. Additionally, Boujlel and Coussot Bouj13  reported the equilibrium
surface tension of the yield stress fluids to be approximately 10$\%$ less
than the surface tension of the pure water, which are close to our measured
values as well. The difference between the data provided by Boujlel and
Coussot Bouj13  and our measured values is presumably due to potential role of
wall-slip in experiments of Boujlel and Coussot Bouj13 .

Figure 7: (a) Surface tension as a function of yield stress for yield stress
fluids of this study. (b) Young modulus as a function of shear modulus for the
yield stress fluids. The line in part (b) is the equivalent point.

In addition to surface tension, we have assessed the Young modulus, $E$ of the
yield stress fluids by fitting the intercepts to the Eq. 5. Fig. 7(b) shows
the Young modulus as a function of shear modulus for all of the yield stress
fluids considered in this study. For both systems, the Young modulus increases
linearly with the shear modulus.

###  IV.3 Flow visualization around cavity

Fig. 4 indicates that the cavitation experiments comprise of two steps. In the
first stage, as the pressure increases linearly with time, a cavity forms at
the tip of the needle and gradually decreases its radius of curvature. The
second step is associated with the post critical pressure and consists of an
instability that is characterized by sudden expansion of the cavity. Fig. 8(a)
shows the temporal evolution of the pressure for the Newtonian corn syrup
along with two yield stress fluids. The time-resolved velocity profiles for
these three sample experiments are shown in Fig. 8(c-e) at different points
along the pressure curves of the Fig. 8(a). At times where the pressure values
are below the critical pressure and even at the critical pressure, where the
radius of the cavity is the same as the needle inner radius, the flow field
around the cavity is weak (see (i) Fig. 8(c-e)). Therefore, these fluids,
regardless of their rheology, do not experience a strong flow at the critical
point. The latter observation gives further credence to the aforementioned
assumption that at the critical point, the viscous stresses are weak and their
impact on surface tension is negligible.

However, post-instability (i.e., $t-t_{c}>0$) sudden expansion of cavity
introduces a strong flow to the surrounding fluid shortly after the critical
point (see the velocity field in (ii) Fig. 8(c-e)). Subsequently, at some
point during rapid pressure drop, the flow reaches its maximum strength (see
(iii) in the velocity field of Fig. 8(c-e)). Eventually, the flow field
subsides to equilibrium at longer times (see (iv) in Fig. 8(c-e)). The above
trend is observed over the range of the needle sizes used in this study both
for the viscous and the yield stress fluids. The most dominant component of
the fluid flow around a cavity is the compression. Therefore, to characterize
the strength of the compression around the cavity, we use the maximum
extension rate around the cavity as a function of time. Therefore, the
volumetric extension rate in spherical coordinate, at each point in time, is
defined as:

$\dot{\varepsilon}=\mathbf{\nabla}\cdot\mathbf{U}=\frac{\partial
u_{r}}{\partial r}+\frac{1}{r}\frac{\partial
u_{\theta}}{\partial\theta}+\frac{u_{r}}{r}+\frac{1}{r\sin(\theta)}\frac{\partial
u_{\phi}}{\partial\phi}+\frac{u_{r}+u_{\theta}\cot(\theta)}{r}.$ (7)

Here $\mathbf{U}$ is the fluid velocity and $r$ is the instantaneous distance
from the center of the bubble. Although flow around the cavity is three
dimensional, our PIV method allows us to map the 2D velocity profiles around
the cavity. Additionally, because of symmetry of the flow around the cavity,
$u_{\theta}=0$ and $u_{\phi}=0$. Hence, $\dot{\varepsilon}=\frac{\partial
u_{r}}{\partial r}+\frac{2u_{r}}{r}.$ The extension rate is obtained by first
fitting a smooth function to the experimentally measured velocity profiles and
then, differentiating from the smooth velocity function. At each point in
time, the fluid around the cavity experiences a maximum extension rate, which
typically occurs close to the cavity boundaries. We assess the flow strength
in each of the above experiments by comparing the maximum extension rate that
each fluid experiences during cavitation process. Fig. 8(b) shows the temporal
evolution of the maximum extension rate that each fluid experiences around the
cavity.

Interestingly, a careful comparison between the flow field of the Newtonian
and the yield stress fluids of Fig. 8(c-e), reveals a significant difference
between these cases. Despite the critical pressure being the same for the
Newtonian fluid and 0.2wt$\%$ Carbopol solution, the velocity fields and the
maximum strain rate in the yield stress fluid are stronger than the Newtonian
counterpart. This difference is presumably connected to the non-Newtonian
rheology of the yield stress fluid. On the other hand, as the critical
pressure for the onset of cavitation increases, the flow strength (velocity
field and/or the characteristic maximum strain rate) before and at the onset
of cavitation (e.g., (i) in Fig. 8(e)) is very negligible and similar to those
shown for other systems in Fig. 8(c-d). However, for post-instability, the
flow field is much stronger in Fig. 8(e) than those shown for other systems in
Fig. 8(c,d). Taken together, these results suggest that in addition to the
critical pressure, the non-Newtonian rheology of the yield stress fluids
control the detailed form of flow structure around a cavity.

###  IV.4 Dimensionless analysis

Finally, we come back to analyzing the stresses involved in cavitation
experiments. In principle, several forces can be involved in cavitation
experiments; inertia, viscous, gravitational, elastic, yield stress and the
surface tension. To assess the importance of these stresses, we use a range of
dimensionless numbers. Note that the surface tension values are evaluated at
the critical point, where pressure shows a maximum. Hence, we will assess the
dimensionless numbers at the critical point.

Figure 8: (a) Temporal evolution of pressure in NIC experiments measured for
various fluids and needle sizes. Here $t_{c}$ refers to the time that pressure
reaches the critical value. (b) The temporal evolution of the maximum strain
rate around the cavity for various fluids. Here $t_{max}$ refers to the time
that strain rate reaches the maximum value. The maximum strain rates that
fluids experience over the course of cavitation are 18 [1/s] for corn syrup,
75 [1/s] for 0.2 wt$\%$ and 115 [1/s] for 0.5 wt$\%$ Carbopol gels. The
temporal evolution of the velocity profiles are shown at different instances
during cavity formation and growth. Each of velocity profiles corresponds to
corn syrup (c), 0.2wt$\%$ Ultrez-10 (d) and 0.5wt$\%$ Ultrez-10 (e).

Prior and at the critical point, the deformation of the fluid, although small,
is controlled by the gas injection rate. As noted before, the injection rate
used in these experiments is very small and fixed for all needle sizes to
$Q=0.3~{}\mu$L/hr. We can estimate a characteristic deformation rate (at the
walls of the capillary tube) associated with this flow rate as
$\dot{\gamma}\approx{4Q}/{\pi R^{3}}\approx 10^{-4}-10^{-2}$ [1/s]. The
importance of inertia can be assessed using a Reynolds number defined as:
$Re=\rho\dot{\gamma}d^{2}/\eta(\dot{\gamma})$. Here, $\rho$, $\dot{\gamma}$,
$d$ and $\eta$ represent the density of the fluid, characteristic deformation
rate, the diameter of the capillary tube and the shear dependent viscosity of
the surrounding fluid. Although inertia depends on the choice of the needle
and fluid, the maximum Reynolds number for all experiments is very small
($Re\approx 10^{-6}$), rendering the effects of inertia negligible.

The impact of gravitational forces can be estimated by a Bond number defined
as $Bo={\Delta\rho ga^{2}}/{\gamma}$. At the critical point, the cavity forms
a hemisphere with a radius equal to the needle size. Therefore, at the
critical point, the Bond number is controlled by the needle size and varies as
$Bo\approx 3\times 10^{-3}-0.1$ for all needle sizes used in this paper.
Therefore, the effect of gravity on surface tension data is negligible. The
latter finding is consistent with nearly spherical bubble shapes observed in
all of our experiments. To assess the impact of viscous stresses, we first
start with cavitation experiments on Newtonian and the non-yield stress fluids
that suggest the viscous stresses are negligible compared to the surface
tension effects. In experiments with the yield stress fluids, the viscous
stresses, if strong enough, may plastically deform the material near the
cavity. To assess the relative importance of viscous and yield stresses at the
critical point, we employ a Bingham number defined as
$Bi=\sigma_{y}/{\tau_{v}}$, where viscous stress
$\tau_{v}=\eta({\dot{\gamma})}\dot{\gamma}$. At the critical point, the
Bingham number varies as $Bi\approx 0.9-0.97$ indicating that viscous stresses
can barely deform the yield stress fluid around the cavity at the critical
point. This result suggests that in experiments with the yield stress fluids,
the medium around the cavity should experience a very weak deformation in the
vicinity of the cavity and the elastic response of the surrounding material
should be dominant. In fact, in fitting the experimental data of Fig. 6(a,b)
to Eq. (5), we noticed that the pressure contribution from the plastically
deformed zone near the surface of the cavity (indicated by $P_{out}|_{1}$) is
much smaller than the elastic resistance of the surrounding fluid
($P_{out}|_{1}<<P_{out}|_{2}$). The latter result is consistent with the above
hypothesis that the viscous stresses are not strong enough to significantly
deform the yield stress fluids at the critical pressure and the elastic
response of the yield stress material is dominant (i.e., ${\tau_{v}}/{E}<<1$
or ${\sigma_{y}}/{E}<<1$).

##  V Conclusion

In summary, we have performed needle induced cavitation experiments to assess
the surface tension, Young modulus and the detailed form of flow structure
around the cavity in a broad range of yield stress fluids. The findings of
this study can be summarized as follows: First, the measured surface tension
values for Carbopol based yield stress fluids is close to surface tension of
pure water over a wide range of yield stress values ($\sigma_{y}=0.5-120$ Pa).
Secondly, we demonstrated that NIC technique can be successfully used to
measure the Young modulus as small as 10 Pa for yield stress gels. Thirdly,
our flow visualization experiments revealed that for $P\leq P_{c}$, the fluid
is barely perturbed by viscous stresses. However, post-instability the
strength of the flow increases up to a local maximum in strain rate (or
deformation rate) before it subsides gradually towards equilibrium at longer
times. Finally, our results show that the flow strength post-instability is
controlled by the critical pressure as well as the non-Newtonian rheology of
the yield stress fluids.

Although we performed NIC experiments in a range of yield stress fluids with
$\sigma_{y}=0.5-120$ Pa, by no means this method (and assessment of surface
tension) is limited to this range of yield stress values and may be adopted
for stiffer gels. Moreover, the impact of this work goes beyond the evaluation
of the surface tension for yield stress materials. In fact, a broad range of
biological tissues and cells are soft and assessing their Young modulus
requires highly sensitive and sophisticated methods such as AFM or nano-
indentation methods that often generates complex and ambiguity in the obtained
resultsgarcia2018 ; guz2014if . The results of this work lend credence to NIC
method as a fast and particularly straightforward technique that can be used
for measuring the mechanical properties of biological samplespavlovsky2014 .
This will be the focus of our future work.

##  VI Supporting Information

Illustrates further information about the rheological properties of the fluids
and the cavitation rheology experiments.

##  VII Acknowledgement

The authors are grateful to Anas Al-Humiari, Scott Hannahs and Richard Crisler
for their help in implementing the NIC device. HM is grateful to Philipe
Coussot, Randy Ewoldt and David Venerus for several helpful discussions.

##  VIII Appendix: Plastic response of expanding bubble in a yield stress
environment

The calculation of the plastic response of an expanding bubble is based on the
theoretical formulation by Bishop et al. bishop_theory_1945 . We assume the
surrounding material is extended between the internal radius of $R_{0}$ (here
is the effective radius of the bubble as it starts to penetrate the
surrounding elastic medium) and a large external radius of $R_{\infty}$ as
shown in Fig. 9a. The inner surface is under uniform pressure of $P_{out}$,
while for a very large medium, we assume that there is no pressure on the
external surface. Because of the spherical symmetry of the problem, a stress-
based solution is used to formulate the problem when the material remains
elastic. The nonzero stress components are the radial stress $\sigma_{r}$ and
the hoop stresses $\sigma_{\theta}=\sigma_{\phi}$. In the absence of body
forces, the equilibrium equation in the radial direction is

$\frac{d\sigma_{r}}{dr}+\frac{2}{r}\left(\sigma_{r}-\sigma_{\theta}\right)\,=\,0.$
(8)

In addition, the only displacement component is in the radial direction,
$u=u(r)$ and the corresponding stain components in spherical polar coordinates
are

$\epsilon_{rr}=\frac{du}{dr},\hskip
28.45274pt\epsilon_{\theta\theta}\,=\,\epsilon_{\phi\phi}\,=\,\frac{u}{r}.$
(9)

For a linearly elastic medium with Young Modulus of $E$ and Poisson ratio
of$\nu$, the stress-strain relations can be written as,

$\epsilon_{rr}=\frac{1}{E}(\sigma_{r}-\nu\sigma_{\theta}),\hskip
28.45274pt\epsilon_{\theta\theta}=\frac{1}{E}\left[\sigma_{\theta}-\nu(\sigma_{r}+\sigma_{\theta})\right],$
(10)

and to form Beltrami-Michell compatibility relationship of

$\frac{d}{dr}\left(\sigma_{r}+2\sigma_{\theta}\right)\,=\,0.$ (11)

Eqs. 8 and 11 along with the boundary conditions can be solved to find the
Lame solution in spherical polar coordinate as asaro2006mechanics ,

$\sigma_{r}\,=\,-P_{out}\left(\frac{R_{\infty}^{3}}{r^{3}}-1\right)/\left(\frac{R_{\infty}^{3}}{R_{0}^{3}}-1\right)$
(12)
$\sigma_{\theta}\,=\,\sigma_{\phi}\,=\,P_{out}\left(\frac{R_{\infty}^{3}}{2r^{3}}+1\right)/\left(\frac{R_{\infty}^{3}}{R_{0}^{3}}-1\right)$
(13)

where $r$ is the radial distance. Here, $\sigma_{r}$ is compressible stress
and $\sigma_{\theta}$ is tensile stress and $|\sigma_{r}|>|\sigma_{\theta}|$.
Equivalently, we can represent the spherical stress field as the summation of
the hydrostatic isotropic pressure of $-\sigma_{\theta}$ (equivalently a
hydrostatic tension of $\sigma_{\theta}$) and uni-axial compressive stress in
the radial direction and zero in other directions of
$(\sigma_{\theta}-\sigma_{r},0,0)$. This stress condition simplifies the von
Mises yield condition of the elastic materials as hill1998mathematical ,

$\sigma_{\theta}-\sigma_{r}\,=\,\sigma_{y}$ (14)

Figure 9: (a) Elastic region around a spherical bubble expanding by uniform
distributed pressure $P_{out}$ at the initial stage. (b) The elastic and
plastic regions around the expanded spherical bubble at a later stage.

From Equations 12,13 and 14, we find that the corresponding pressure to the
onset of the yield condition in linear elastic perfectly plastic materials is,

$P_{out,\sigma_{y}}\,=\,\frac{2\sigma_{y}}{3}\left(1-\frac{R_{0}^{3}}{R_{\infty}^{3}}\right)$
(15)

For an infinite domain, we have $\frac{R_{0}}{R_{\infty}}\rightarrow 0$ and
the inner layer of medium near the surface yields at a pressure equal to two-
thirds of the yield stress. Given the typical $\sigma_{y}$ values of Carbopol
gels, one can find that in almost all cases the material passes its purely
elastic response regime very fast at the beginning of the bubble creation.
Afterward, we have an elastic-perfectly plastic response. In this case, Eqs. 8
and 11 are used to find the stress components in the elastic region, while in
the plastic region, the modified system of equations of Eqs. 8 and 14 are
employed. To find the final solution, the continuation of the radial stress is
used, and with the assumption that the yield surface is placed at a radius
$R_{c}$ (Fig. 9b), we can find the stress fields inside and outside the
plastic regions as

$\left.\begin{array}[]{ll}\sigma_{r}=-\frac{2\sigma_{y}}{3}\left(\frac{R_{c}^{3}}{r^{3}}\right)\\\
\sigma_{\theta}=\frac{2\sigma_{y}}{3}\left(\frac{R_{c}^{3}}{2r^{3}}\right)\end{array}\right\\}\,\,\,\,c\leq
r$ (16)

$\left.\begin{array}[]{ll}\sigma_{r}=-2\sigma_{y}\ln\left(\frac{R_{c}}{r}\right)-\frac{2\sigma_{y}}{3}\\\
\sigma_{\theta}=\sigma_{y}-2\sigma_{y}\ln\left(\frac{R_{c}}{r}\right)-\frac{2\sigma_{y}}{3}\end{array}\right\\}\,\,\,\,c\geq
r$ (17)

If the bubble is at the radius $R$, one can use the above relation and relate
$R_{c}$ and $P_{out}$ as,

$P_{out}\,=\,2\sigma_{y}\ln\left(\frac{R_{c}}{R}\right)+\frac{2\sigma_{y}}{3}$
(18)

Unfortunately, the above relation is undetermined to provide a $P_{out}-R$
relation, as it involves an unknown $R_{c}$ value. To close the problem, a
much more complicated solution procedure should be followed to calculate the
elastic and plastic strain field based on $R$ and $R_{c}$ and to determine
$R_{c}$ value. A simplifying assumption can be made for the medium response by
assuming that at a later time of bubble growth, the bubble behaves similarly
to a spherical bubble that is expanded from zero radii. In this case, we can
show that $R_{c}/R$ should remain constant based on the self-similarity and
takes the form of hill1998mathematical

$\frac{R_{c}}{R}\,=\left(\frac{E}{3(1-\nu)\sigma_{y}}\right)^{\frac{1}{3}}$
(19)

and therefore we can rewrite the equation 18 as

$P_{out}\,=\frac{2\sigma_{y}}{3}\left\\{1+\ln\left(\frac{E}{3(1-\nu)\sigma_{y}}\right)\right\\}$
(20)

Assuming the $\nu$ of the incompressible medium is 0.5, we can find that
$P_{out}$ is a constant independent of the bubble radius and is equal to

$P_{out}\,=\frac{2\sigma_{y}}{3}\left\\{1+\ln\left(\frac{2E}{3\sigma_{y}}\right)\right\\}$
(21)

Given that we assume here the bubble expands from a zero initial radius, the
above solution should be regarded as an upper limit of a bubble expansion from
a finite-size needle in a perfect plastic regime. In the above model, we
assumed that the stress level could not exceed a fixed yield limit. However,
for many polymeric media, there is still elastic stress even after the yield
condition n2019elastoplastic , especially if the material is confined and can
not undergo large permanent deformation. Equivalently, it is possible to
define the terminal steady-state stress in the post-yield condition as
hill1998mathematical

$\sigma\,=\,\sigma_{y}+h_{\sigma}(\epsilon)$ (22)

where $h_{\sigma}$ is a function expressing the change of total stress as a
function of the logarithmic total strain $\epsilon$. Here, the material
undergoes strain-hardening or work-hardening and the von-Mises yield condition
of Eq. 14 is modified to hill1998mathematical ,

$\sigma_{\theta}-\sigma_{r}\,=\,\sigma_{y}\,+\,h\left\\{2\ln\left(\frac{r}{r_{0}}\right)-\frac{1-2\nu}{E}\left(\sigma_{r}+\sigma_{\theta}\right)\right\\}.$
(23)

where $r_{0}$ is the initial radial position of the element. If the medium is
incompressible ($\nu=\frac{1}{2}$), the equilibrium condition of Eq. 8 can be
simplified as

$\frac{d\sigma_{r}}{dr}\,=\,\frac{2}{r}\left(\sigma_{\theta}-\sigma_{r}\right)\,=\,\frac{2\sigma_{y}}{r}+\frac{2h_{\sigma}[2\ln(r/r_{0})]}{r}$
(24)

If we assume that the bubble is expanded from zero radius in the infinite
medium, we can use the self-similarity property of the spherical shape and
scale spatial lengths based on the current size of the bubble $R$. By adopting
this condition, and solving Eq. 24, a a revised relation for $P_{out}$ can be
found in the form of

$P_{out}\,=\frac{2\sigma_{y}}{3}\left\\{1+\ln\left(\frac{2E}{3\sigma_{y}}\right)\right\\}+2\int_{1}^{\left(\frac{2E}{3\sigma_{y}}\right)^{\frac{1}{3}}}\,h_{\sigma}\left\\{\frac{2}{3}\ln\left(\frac{t^{3}}{t^{3}-1}\right)\right\\}\frac{dt}{t}$
(25)

where for the linear $h_{\sigma}(\epsilon)=E\epsilon$ function (when the
stress-strain curve is represented with a slope of $E$.) bishop_theory_1945 ,
the relation can be simplified as

$P_{out}\,=\frac{2\sigma_{y}}{3}\left\\{1+\ln\left(\frac{2E}{3\sigma_{y}}\right)\right\\}+\frac{2\pi^{2}}{27}E$
(26)

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