# Crack tip kinematics reveal the cohesive zone structure in brittle hydrogel
fracture

Chenzhuo Li  Xinyue Wei  Meng Wang  Mokhtar Adda-Bedia  John M. Kolinski

###### Abstract

When brittle hydrogels fail, several mechanisms conspire to alter the state of
stress near the tip of a crack, and it is challenging to identify which
mechanism is dominant. In the fracture of brittle solids, a sufficient far-
field stress results in the complete loss of structural strength as the
material ‘unzips’ at the tip of a crack, where stresses are concentrated.
Direct studies of the so-called small-scale yielding zone, where deformation
is large, are sparing. Using hydrogels as a model brittle solid, we probe the
small-scale yielding region with a combination of microscopy methods that
resolve the kinematics of the deformation. A zone over which most of the
energy is dissipated through the loss of cohesion is identified in the
immediate surroundings of the crack tip. With direct measurements, we
determine the scale and structure of this zone, and identify how the specific
loss mechanisms in this hydrogel material might generalize for brittle
material failure.

###### keywords:

Brittle Fracture , Fracture Cohesive Zone , High-resolution Deformation Fields
, Soft Materials

###### PACS:

0000 , 1111

###### MSC:

0000 , 1111

††journal: Journal of the Mechanics and Physics of Solids

[inst1]organization=Engineering Mechanics of Soft Interfaces, School of
Engineering, Ecole Polytechnique Fédérale de Lausanne,city=Lausanne,
postcode=1015, country=Switzerland

[inst2]organization=The Racah Institute of Physics, The Hebrew University of
Jerusalem,city=Jerusalem, postcode=91904, country=Israel

[inst3]organization=Université de Lyon, Ecole Normale Supérieure de Lyon,
CNRS, Laboratoire de Physique,city=Lyon, postcode=F-69342, country=France

Brittle hydrogels fail cohesively

Energy dissipation near the crack tip occurs both locally at the crack tip,
and in a distributed damage zone

##  1 Introduction

When brittle materials fail, they typically lose cohesion through the process
of fracture [1]. In the existing theory of brittle fracture, the loss of
cohesion is presumed to be confined to a small volume near the crack tip, as
asserted by the ‘small-scale yielding’ hypothesis [2, 1]. Outside of this
zone, the material is typically modelled as a linear elastic solid [3, 2],
where the stress diverges near the crack tip with the characteristic ‘inverse-
square-root’ $r^{-1/2}$ asymptotics. For soft materials such as hydrogels, the
diverging stress is typically accompanied by large deformation prior to
failure [4, 5] \- thus, several candidate mechanisms that regulate the stress
can become active, complicating the evaluation of the stress state near the
crack tip. In the absence of a clear failure mechanism at the crack tip,
significant open questions concerning the near crack tip fields remain
unanswered [6, 7]. Indeed, the large deformation in this region elicits
constitutive non-linearity, poro-elastic solvent flux for hydrogels, and
cohesive loss, rendering the nature of material failure at the crack tip
ambiguous. Measurement of deformation in this region is further complicated by
the singular nature of the crack tip field, which requires ever-increasing
resolution to characterize the deformation field with data that can be used to
identify the relevant failure mechanisms. Despite these challenges, direct
measurements of deformation and kinematics within the small-scale region near
the crack tip in a brittle solid can offer insight into how material failure
occurs at the smallest scales, and thus illuminate the physics of brittle
fracture generally.

To address the experimental challenges presented by the large deformation in
the diverging stress field at the crack tip, we probe the material kinematics
in the near-tip field of a quasi-static crack tip in a brittle polyacrylamide
hydrogel under remote tensile loading. Brittle hydrogels are used as a proxy
for the broad class of brittle solids, which have the merit of a lesser
material sound speed, facile preparation, and an expansive application domain
in their own right [8, 9, 10, 11, 12, 13, 14]. Polyacrylamide hydrogels with a
canonical composition that results in nearly ideally brittle material response
were used (see Materials and Methods for preparation protocol) [11]. Two
distinct, high-resolution optical methods are used in these experiments. In
the first method, we embed fluorescent dye in the hydrogel, and monitor the
crack tip opening displacement (CTOD) on a single plane extending a few
hundred microns from the crack tip using confocal microscopy. In the second
method, we embed particles in a hydrogel and measure the deformation within
the hydrogel near the crack tip. The particles are sufficiently small that
they do not alter the fracture dynamics [15]. These high-resolution
measurements enable us to probe the deformation kinematics to within
$10\text{\,}\mathrm{\SIUnitSymbolMicro m}$ of the crack tip.

Analysis of the near-tip kinematic data using known solutions for the crack
tip opening displacement [16], and direct evaluation of the $J$-integral [2]
for neo-Hookean solids [17, 18], show that the strain energy decreases
strongly within approximately $100\text{\,}\mathrm{\SIUnitSymbolMicro m}$ of
the crack tip, indicating dissipation in a cohesive zone. The functional form
of the cohesive law is evaluated from the decrease of the fracture energy near
the crack tip. By windowing the boundary of the $J$-integral, the structure of
the cohesive zone is measured. Careful characterization of the material, and
established estimates for solvent fluxes [19, 20, 16, 21] are used to evaluate
possible strain-stiffening and poro-elastic stress relaxation; we find that
alternative mechanisms that could modify the material response near the crack
tip are not active on the scale of the measured cohesive zone; instead, the
mode of failure is consistent with a cohesive zone reminiscent of the Irwin-
Orowon picture of network fracture [22]. For dynamic cracks in the same
hydrogel, a non-linear scale $\delta_{dyn}$ emerges near the crack tip [11];
this scale is attributed to the $1/r$ scaling that occurs at the crack tip of
strain-stiffening materials described by a generalized neo-Hookean material
model [17], or a weakly non-linear expansion of the displacement field[6, 4].
While we observe a similar crack tip feature in our data, the requisite strain
stiffening does not emerge at our strain rates, and thus a parabolic CTOD is
expected for the nearly-ideal neo-Hookean material model that describes our
hydrogel at the strain rates realized in our experiments [16]; thus, strain
stiffening is not responsible for the sharper CTOD 111At higher strain rates,
strain stiffening could emerge, but we cannot yet quantitatively analyze these
data as the material response at the strain rates typically achieved in
dynamic fracture remains to be characterized.. Finally, we discuss the
implications of the value of the cohesive zone stress at the crack tip, and
show that the small-scale but distributed damage in the cohesive zone
surrounds a more localized process zone on a smaller scale than the resolution
of our measurements, indicating a separation of scales between the cohesive
zone and the process zone where the material ultimately separates. While the
size of the cohesive zone should vary for different materials, the nature of
brittle material failure identified here may be universal for fracture in
brittle solids.

##  2 Materials and Methods

###  2.1 Confocal Microscopy for CTOD

Experiments are performed on thin ($200\text{\,}\mathrm{\SIUnitSymbolMicro
m}$) samples of polyacrylamide hydrogel whose solvent contains a fluorescent
dye as described in Sec. 2.3. The experimental setup for CTOD measurement is
schematically shown in Fig. 1(a), where a confocal microscopy is used to
capture the in-situ 3D information of the crack tip. The fluorescent sample
with an edge crack is placed under far-field mode I loading applied by
displacement controlled stages. Using a pinhole to block the out-of-focus
light, the detector only collects the emitted light from the focal point
inside the sample. We constructed the 3D crack by taking a volume scan of the
mode I crack during quasi-static propagation
($\sim$$10\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}\mathrm{/}\mathrm{s}$).
A representative reconstructed 3D CTOD volume is shown in Fig. 1(b). This 3D
reconstruction demonstrates the planar symmetry of the crack, and ensures that
the 2D CTOD is sufficiently representative of the fully 3D crack tip opening
profile for the clean cracks [23] at the focus of the present work.

Figure 1: Experimental setup for CTOD measurement near the crack tip with a
confocal microscope. (a) Schematic drawing of the confocal microscope system.
The pre-cracked $3\text{\,}\mathrm{cm}$ $\times$ $1\text{\,}\mathrm{cm}$
$\times$ $200\text{\,}\mathrm{\SIUnitSymbolMicro m}$ hydrogel sample is dyed
by Rhodamine 6G, immersed in water, and subjected to mode I loading via the
sample grips. The laser beam (wavelength 525 nm) is focused into the sample by
the objective (Nikon CFI Plan Fluor 10X W and CFI75 Apo 25XC W), exciting the
florescent dye. The emission light passes through the dichroic beam splitter,
and the out-of-focus light is blocked by the pinhole in front of the detector.
By rastering the beam in $x$ and $y$, and scanning the objective in $z$, the
confocal system collects the fully 3D geometry of the crack tip. (b) A typical
3D reconstruction of a quasi-static planar crack, imaged with the $25\times$
objective. The crack speed is
$\sim$$2.5\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}\mathrm{/}\mathrm{s}$.
The scale bar is $100\text{\,}\mathrm{\SIUnitSymbolMicro m}$. (c) A 2D planar
slice of the 3D reconstruction shown in (b). The scale bar is
$50\text{\,}\mathrm{\SIUnitSymbolMicro m}$. The CTOD is determined with a
custom image processing routine (using Python skimage packages [24]), and
shown in the red solid curve. A parabola is fitted to the CTOD curve and
plotted in the blue dashed line. The prefactor $a$ of the parabola $x=ay^{2}$
fitted to the CTOD data is related to the apparent fracture energy
$\Gamma_{\mathrm{app}}$ using energy balance, as shown in Eq. 13. At the crack
tip, the fitted parabola deviates from the CTOD as shown in the zoomed-in
view.

###  2.2 Microscopy of particle-laden hydrogel samples

Experiments are furthermore performed on hydrogel samples embedded with
particles, to confirm the near-tip energy dissipation by $J$-integral via
full-field deformation measurement. The particle-embedded hydrogel was
prepared as described in Sec. 2.3 and mounted on the testing apparatus shown
in Fig. 2(a). After carefully inserting an edge crack and loading the sample
in tension, the crack begins to propagate quasi-statically, as indicated by
the yellow dashed line in the time series in Fig. 2(b). Here, a small region
of interest, particles in the vicinity of the crack tip are seen to remain
bright, facilitating subsequent tracking within microns of the crack tip,
which allows direct probing of the near-tip fields and distinguishes our
analysis from the macroscale experiments [25]. Despite the large displacements
and diverging path lines near the crack tip, the particle tracking algorithm
remains robust. Example trajectories for two particles adjacent to the crack
tip are highlighted by red and blue circles in the time series shown in Fig.
2(b). Once the particle tracking is complete and the reference locations are
recovered by the procedure described in Fig. 4, components of the displacement
of each particle are directly recovered over the entire tracked region as
demonstrated by the displacement component along the $y-$axis, $u_{y}$, in
Fig. 2(c).

Figure 2:  Experimental setup for full-field deformation measurement near the
crack tip. (a) A thin ($200\text{\,}\mathrm{\SIUnitSymbolMicro m}$) sample of
hydrogel containing embedded particles is gripped within a microscale tensile
testing apparatus and immersed in a water bath. Before mounting, an edge crack
is introduced centered between the grips, orthogonal to the loading axis. A
Nikon TI eclipse microscope is configured for dark-field illumination with a
$10\times$ water-immersion objective and images the focal plane onto the
camera sensor. A sample image recorded with this configuration is pre-
processed with bandpass filter and shown bottom right; the scale bar is
$100\text{\,}\mathrm{\SIUnitSymbolMicro m}$. (b) A time series obtained from
the small region of interest indicated by the grey dashed line in (a) shows
both the progress of the crack and the displacement of the embedded particles;
two of approximately 4000 tracked particles are encircled with red and blue
throughout the time series. The crack position is indicated in each frame by
yellow dashed line. The scale bar shown in the first frame lower left is
$10\text{\,}\mathrm{\SIUnitSymbolMicro m}$. The crack direction and speed is
shown by the red arrow, bottom. (c) After tracking each particle, its
displacement from the initial position in the unstrained material is measured;
the $y$ component of displacement, $u_{y}$, is indicated at each particle by
color. In this way, material displacements are measured within
$10\text{\,}\mathrm{\SIUnitSymbolMicro m}$ of the crack tip. The scale bar is
$100\text{\,}\mathrm{\SIUnitSymbolMicro m}$.

###  2.3 Preparation of hydrogel samples

Two kinds of hydrogel samples, prepared with the same monomer chemistry, were
used in our fracture experiments - hydrogel samples dyed with fluorescence dye
and hydrogel samples embedded with micro particles.

To prepare the fluorescent samples, we used a precursor solution comprised of
13.8 wt.% monomer and 2.7 wt.% bis-acrylamide cross-linker, used in several
prior brittle fracture studies[9, 12, 26]. The stock solution is first
degassed in a vacuum chamber for 10 minutes, and then 0.2 % ammonium per-
sulfate (APS) initiator and 0.02 % tetramethylethylenediamine (TEMED) catalyst
is added to trigger the free-radical polymerization reaction. After mixing for
30 seconds, the solution is poured into a
$190\text{\,}\mathrm{\SIUnitSymbolMicro m}$ gap between glass plates. The
polymerization reaction runs for at least four hours. After polymerization,
the top glass plate was removed, and the hydrogel was cut into
$3\text{\,}\mathrm{cm}$ by $1\text{\,}\mathrm{cm}$ samples; these samples were
then placed into $2\times 10^{-4}\mathrm{mol/L}$ Rhodamine 6G solution to dye
the solvent. After 24 hours immersed in the dye solution, the hydrogel samples
were then ready to be mounted on the confocal microscope for the fracture
experiments.

To prepare particle-embedded hydrogel, we mixed polystyrene particles
(diameter $1.1\text{\,}\mathrm{\SIUnitSymbolMicro m}$) with the degassed
monomer/cross-linker solution at a concentration 0.005 wt.%, and sonicated for
5 minutes to disperse the aggregated particles and ensure their random
distribution. The solution was was polymerized with APS and TEMED as described
above, and poured into the same mold geometry. The polymerized hydrogel, now
containing the polystyrene micro-particles, was then cut into samples of the
same $3\text{\,}\mathrm{cm}$ by $1\text{\,}\mathrm{cm}$ size, and immersed in
water for 24 hours to swell to an equilibrium state for fracture experiments
in a waterbath.

As prepared, neither the Rhodamine 6G dye or the micro-particles are supposed
not to affect the elastic properties or the fracture properties of the
hydrogel. Therefore, the prepared hydrogels can be well modelled as
incompressible neo-Hookean materials. The Young’s modulus, fracture energy,
and Rayleigh wave speed can be estimated as those of the pure hydrogel [9, 23,
26], i.e., approximately $90\text{\,}\mathrm{kPa}$,
$5.28\text{\,}\mathrm{J}\mathrm{/}\mathrm{m}^{2}$, and
$5.5\text{\,}\mathrm{m}\mathrm{/}\mathrm{s}$, respectively.

###  2.4 Derivation of apparent fracture energy and constitutive response from
CTOD

The evaluation of apparent fracture energy from the CTOD is based on linear
elastic fracture mechanics (LEFM) theory. From LEFM, the mode I displacement
field [1] is

$\begin{split}&u_{x}=\frac{K_{I}}{2\mu}\sqrt{\frac{r}{2\pi}}\cos{\frac{\theta}{2}}\left(\kappa-1+2\sin^{2}\frac{\theta}{2}\right),\\\
&u_{y}=\frac{K_{I}}{2\mu}\sqrt{\frac{r}{2\pi}}\sin{\frac{\theta}{2}}\left(\kappa+1-2\cos^{2}\frac{\theta}{2}\right),\end{split}$
(1)

where $r$ and $\theta$ are polar coordinates in the undeformed states,
$\kappa=\frac{3-\nu}{1+\nu}$ for plane stress, and the Poisson’s ratio
$\nu=0.5$ due to incompressibility. On the fracture surfaces $\theta=\pm\pi$,
the displacement field in equation 1 becomes

$\begin{split}&u_{x}=0,\\\
&u_{y}=\pm\frac{K_{I}}{2\mu}\sqrt{\frac{r}{2\pi}}(\kappa+1).\end{split}$ (2)

Given $u_{x}=x-X$ and $u_{y}=y-Y$, the CTOD on the fracture surface is

$x=-r=-\frac{9\pi\mu^{2}}{8K_{I}^{2}}y^{2}=-ay^{2}.$ (3)

The stress intensity factor $K_{I}$ is related to the strain energy release
rate by $G=K_{I}^{2}/E=3K_{I}^{2}/\mu$ ($\mu=3E$ due to incompressibility);
thus, $G$ can be derived from the prefactor $a$ of the parabolic CTOD
predicted from LEFM. At the critically loaded state, this is equal to the
fracture energy of the material by energy balance. Thus, the apparent fracture
energy $\Gamma_{\text{app}}$ from LEFM is

$\Gamma_{\text{app}}=G_{c}=\frac{3\pi\mu}{8a}.$ (4)

The total measured fracture energy $\Gamma_{\mathrm{app}}$ consists of the
energy dissipation in the cohesive zone $\Gamma_{c}$ and the constant energy
dissipation $\Gamma^{*}$ at the crack tip, which is

$\Gamma_{\mathrm{app}}=\Gamma_{\mathrm{c}}+\Gamma^{*}.$ (5)

While $\Gamma^{*}$ remains constant, as it is localized at the crack,
$\Gamma_{c}$ represents the work required per unit surface area in order to
separate the crack faces to a separation distance $\delta_{c}$ by overcoming
the cohesive stress $\sigma_{c}$; thus, $\Gamma_{c}$ is given by [27]

$\Gamma_{c}=\int_{0}^{\delta_{c}}\sigma_{c}(\delta)d\delta.$ (6)

The constitutive response within the cohesive zone is thus given by

$\sigma_{c}=\frac{d\Gamma_{c}}{d\delta}=\frac{d(\Gamma_{\mathrm{app}}-\Gamma^{*})}{d\delta}=\frac{d\Gamma_{\mathrm{app}}}{d\delta}.$
(7)

###  2.5 Experimental measurement of deformation fields and calculation of
$J$-integral

We carried out the full-field deformation measurement near the crack tip using
the experimental setup shown in Fig. 2 (a). The hydrogel was dispersed with
$1.1\text{\,}\mathrm{\SIUnitSymbolMicro m}$ diameter polystyrene particles and
was cut into $3\text{\,}\mathrm{cm}$ by $1\text{\,}\mathrm{cm}$ rectangular
sheets after cross-linking. The sample was placed onto one surface of a pair
of magnetically actuated grips and a displacement-controlled loading frame.
Tension is applied via a servo motor in the $y$ direction, such that the
displacement between grips are increased symmetrically; in this way, the
center of the sample remain stationary in the laboratory frame of reference. A
pre-crack is introduced along $x$ direction on an edge of the sample centered
between the grips. During the experiment, the sample was immersed in water and
illuminated with a dark-field configuration. A $10\times$ water-immersion
objective (Nikon Plan Fluor $10\times$/0.30W) was mounted on microscope (Nikon
eclipse Ti, tube lens magnification: $1.5\times$) to focus the scattered light
from the particles onto the sensor of a high-resolution camera (Hamamatsu
C13440, resolution: $2048\times 2048$ pixels, bit depth: 16 bit), and imaged
at up to 100 frames per second. As a consequence of the water-immersion
objective and the water bath, the refractive index of the hydrogel (1.365) is
well-matched to that of the surrounding water (1.333), and thus lensing
effects near the crack tip were effectively eliminated, as can be seen from an
sample image shown on the bottom right inset in Fig. 2(a). By carefully
increasing mode I loading, the initial crack starts to propagate quasi-
statically, at a velocity of approximately
$10\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}\mathrm{/}\mathrm{s}$. At
this rate of crack propagation, particles remain approximately stationary for
the duration of the camera’s exposure, eliminating motion blur. The microscope
was adjusted to focus on the mid-plane of the sample, and a region of interest
about $1\text{\,}\mathrm{mm}$ ahead of the current crack tip was recorded by
the camera at 3 frames per second.

The recorded images of crack, only if the crack travels in a straight path
(ensuring pure mode I loading), were first rotated to ensure that the crack
travels horizontally and the loading is along vertical direction. A bandpass
filter is then applied to all images to eliminate both image noise and large
undesired structures appearing on images, e.g., floating dust.

Naturally, one wonders whether the embedded particles used to track material
displacements might alter the mechanical response, and furthermore the
fracture mechanics, of the hydrogel. To evaluate whether this might be the
case, we applied the CTOD analysis to a sample with embedded particles, and we
find that neither the fracture energy, nor its functional form are altered in
the particle embedded sample, as can be seen in Fig. 3. This underscores our
prior study of similar particle-embedded samples in the dynamic fracture
context [15]. A plausible explanation for why the particles are truly passive,
even in the context of fracture mechanics, is that they are significantly
smaller than the typical yield scale in the material, where fracture processes
occur; such scales have been evaluated in detailed studies of liquid-liquid
phase separation in elastomer networks such as PDMA [28].

Figure 3: CTOD measurement of a typical sample with particles. (a) Extracted
CTOD from one of the typical samples with embedded particles used in the
J-integral measurements. The CTOD is highlighted by the red curve. The scale
bar is $50\text{\,}\mathrm{\SIUnitSymbolMicro m}$.(b) The apparent fracture
energy $\Gamma_{\mathrm{app}}$ of the particle sample in (a) measured by the
CTOD method. $\Gamma_{\mathrm{app}}$ asymptotes to
$\Gamma^{*}=$$1.94\text{\,}\mathrm{J}\mathrm{/}\mathrm{m}\mathrm{{}^{2}}$ at
the crack tip, and converges to a background fracture energy as the window
extent moves away from the crack tip. The curve is consistent as in the main
text. This result shows that the CTOD measurement agrees with the J-integral
measurements using particle tracking.

To track the particles from the pre-processed images, an open source particle
tracking algorithm, TrackPy [29, 30], is used. Two steps are necessary to
successfully track particles - first, the particles must be located in each
image, and second, the particles must be linked to their subsequent position
in each succeeding image. In the first step, an estimated particle diameter is
provided to construct a spatial filter; the particles identified by their
intensity are then filtered by their mass (integrated brightness), percentile
and eccentricity. To ensure sub-pixel resolution of particle positions, the
uniformity of the distribution of the residual of each particles $x$ and $y$
coordinates is evaluated. For a typical particle image in our experiment,
approximately 4000 particles are located with the field-of-view of
$800\text{\,}\mathrm{\SIUnitSymbolMicro m}$ by
$800\text{\,}\mathrm{\SIUnitSymbolMicro m}$.

The particles thus located are then linked between frames based on the
particles’ most recent velocity. Despite the large deformation and large
rotation near the crack tip, we employ a sufficient frame rate to ensure the
robustness of the velocity-based prediction of particle locations in
subsequent frame. With accurate particle locations and reliable particle
linking, the trajectories of particles are successfully tracked.

Figure 4: Estimation of particles’ location at undeformed state. The
estimation is based on two frames of the recorded movie of crack propagation -
the first frame (a) and the last frame (b). In the first frame, since the
crack is still hundred of micrometers from the field-of-view (FOV), the
hydrogel sample in the FOV is uniformly stretched. In the last frame, the
crack has propagated through, and on the crack surface, the boundary is
traction-free. For a pair of particles that is perpendicular to the crack
surface in the last frame and vertically aligned in the first frame (indicated
by red arrows in (a) and (b)), the distance between the two particles can be
measured as $l$ in the first frame under uniform stretch and $L$ in the last
frame near the traction-free boundary. By comparing the distance change of the
particle pair, the uniform stretch in the first frame can be calculated as
$\lambda=l/L$. (c) The uniform stretch is evaluated for 10 pairs of particles
across the image from the left to the right, 5 on the upper crack surface and
5 on the lower crack surface. The average uniform stretch is measured as 1.09
with a standard deviation of 0.01. (d) The tracked particles (red dot) in the
frame and their estimated locations at undeformed state (blue cross). With the
measured uniform stretch $\bar{\lambda}$ from (c) and the tracked particle
location $(x,y)$ in the first frame, the particle locations at undeformed
state can be retrieved as
$(X,Y)=(x/(1-\nu(\bar{\lambda}-1)),y/\bar{\lambda}))$, with any arbitrary
selection of the laboratory frame of reference, where $\nu$ is the Poisson’s
ratio and equals to 0.5 for our incompressible hydrogel. Note that this is
only an example of one experiment. The same procedure is carried out on all
three experiments and the measured uniform stretch values are all around 1.09.

With the particle locations tracked at current states $\mathbf{x}$ and their
locations at reference state $\mathbf{X}$ estimated as shown in Fig. 4,
displacement vectors for each particle can be directly calculated as
$\mathbf{u}=\mathbf{x}-\mathbf{X}$. By performing finite difference on the
displacement components of neighboring particles, the deformation gradient
tensor $\mathbf{F}$ of each particle is estimated by

$\mathbf{F}=\mathbf{I}+\mathbf{\nabla_{X}u}.$ (8)

Then, it is straightforward to obtain the stretch and rotation tensor by polar
decomposition, and thus to obtain the Lagrange finite strain tensor via the
right Cauchy-Green deformation tensor. With an appropriate constitutive law,
the displacements and deformation gradient tensor can be used to calculate the
$J$-integral around the crack tip.

The $J$-integral is explicitly calculated from deformation fields by [2, 17,
18]

$J=\int_{c}\left(Wn_{1}-\sigma_{ij}n_{j}\frac{\partial u_{i}}{\partial
x_{1}}\right)ds,i,j=1,2,$ (9)

where $c$ is an arbitrary counterclockwise integral path with an outward unit
vector $\mathbf{n}$, $W$ is the strain energy density, $\sigma_{ij}$ is the
first Piola-Kirchhoff stress tensor, ${\partial u_{i}}/{\partial x_{1}}$ are
the components of the displacement gradient tensor. To ensure the correct
representation of energy release rate, $J$ must be calculated with strain
energy density and stress tensor that are derived based on proper material
model - in our case, the neo-Hookean material model.

The hydrogel used in our fracture experiments is known as an incompressible
neo-Hookean material, whose strain energy density $W$ is defined as

$W=\frac{1}{2}\mu(I_{1}-3),$ (10)

where $\mu$ is the shear modulus of the hydrogel, $I_{1}$ is the first
invariable of the right Cauchy-green deformation tensor $\mathbf{C}$, i.e.,
$I_{1}=\text{tr}\mathbf{C}=\text{tr}(\mathbf{\tilde{F}}^{\text{T}}\mathbf{\tilde{F}})$.
It it noted that the deformation gradient tensor $\mathbf{\tilde{F}}$ used
here is for 3D deformation; however, deformation gradient tensor $\mathbf{F}$
measured from our experiments is 2D. Considering the incompressibility of the
hydrogel, i.e., $\det(\mathbf{\tilde{F}})=1$, and our focal plane coincides
with the mid-plane of the sample, Eq. 10 can be simplified as

$W=\frac{1}{2}\mu(\text{tr}(\mathbf{F}^{\text{T}}\mathbf{F})+\det(\mathbf{F})^{-2}-3),$
(11)

where $W$ can be directly calculated from the measured $\mathbf{F}$. In this
case, the in-plane Piola-Kirchoff stress is given by:

$\mathbf{\sigma}=\mu\mathbf{F}-\mu\det(\mathbf{F})^{-2}\mathbf{F}^{-\text{T}}.$
(12)

Once we obtain the deformation gradient evaluated at each particle, we
linearly interpolate the discrete data ($\mathbf{F},W,\mathbf{\sigma}$) to
construct the field data near the crack tip. Over these deformation fields,
$J$-integrals are calculated according to Eq. 9 along rectangular integral
paths.

##  3 Results

###  3.1 Measurement of the apparent fracture energy in the near-tip region
via CTOD

LEFM predicts a parabolic CTOD for mode I cracks [1], where the pre-factor of
the parabola, $a$, is governed by the strain energy release rate $G_{c}$. By
Griffith’s energy balance criterion, $G_{c}$ is equal to the material fracture
energy when the crack is critically loaded and propagating. In our
experiments, the quasi-static propagation ensures that the crack is observed
in a critical state, and thus satisfies energy balance. Under this condition,
a parabola fitted to the measured CTOD (measured as described in section 2.1)
can be used to directly compute the apparent fracture energy
$\Gamma_{\text{app}}$ (See detailed derivation in section 2). For a static
mode I crack, $\Gamma_{\text{app}}$ is given by

$\Gamma_{\text{app}}=G_{c}=\frac{3\pi\mu}{8a},$ (13)

where $\mu$ is the shear modulus of the material. The measured CTOD is
perfectly parabolic in the region away from the crack tip, where the fracture
behavior is known as ‘$K$-dominant’, as shown in Fig. 1(c). In the small
region near the crack tip, however, the CTOD deviates from the fitted parabola
in a manner similar to $\delta_{dyn}$[11], demanding further investigation.

$\Gamma_{\text{app}}$ is experimentally determined from the CTOD data by
evaluating the best fit parabola over different window extents using a non-
linear least squares routine and Eq. 13. As the window extent changes for the
evaluation of the best fit, the value of $a$ can change; thus, the value of
$\Gamma_{\text{app}}$ is a function of window extent used to fit the parabolic
geometry to the CTOD data. Analysis of the CTOD data in this manner shows that
$\Gamma_{\text{app}}$ decreases near the crack tip, as shown in Fig. 5(a).
Further from the crack tip, $\Gamma_{\text{{app}}}$ plateaus to a background
fracture energy $\Gamma_{\text{bg}}$. The variation of $\Gamma_{\text{app}}$
represents the loss of cohesion in the near-tip region. At the crack tip,
$\Gamma_{\mathrm{app}}$ asymptotes to the value
$\Gamma^{*}_{\mathrm{le}}=1.77\pm 0.44\mathrm{J/m^{2}}$ by linear
extrapolation of the data within a $25\text{\,}\mathrm{\SIUnitSymbolMicro m}$
range near the crack tip (distribution of $\Gamma^{*}_{\mathrm{le}}$ for 60
measurements is shown in Fig. 5(c)). $\Gamma^{*}_{\mathrm{le}}$ is the energy
dissipation at the crack tip, and is distinct from the cohesive loss occurring
within a ca. $100\text{\,}\mathrm{\SIUnitSymbolMicro m}$ neighborhood of the
crack. These measurements suggest two distinct regions where energy is
dissipated - a distributed damage region, and a highly-localized process
region at the crack tip.

Figure 5: CTOD evaluation of the apparent fracture energy near the crack tip.
(a) Apparent fracture energy is plotted as a function of window extent for the
parabolic fit. The mean and standard deviation are measured from 6 quasi-
static crack samples consisting of 10 frames each. The mean is shown in the
red curve and the standard deviation is indicated by the blue region for each
value of the window extent. $\Gamma_{\mathrm{{app}}}$ approaches the
background fracture energy $\Gamma_{\mathrm{{bg}}}$ as the window extent
enters the K-dominant region. Using linear extrapolation of the data within
$25\text{\,}\mathrm{\SIUnitSymbolMicro m}$ distance to the crack tip,
$\Gamma_{\mathrm{{app}}}$ asymptotes to the value of
$\Gamma^{*}_{\mathrm{le}}=1.77\pm 0.44\mathrm{J/m^{2}}$ as the window extent
shrinks to zero at the crack tip. Inset: different window extents result in
different values of $\Gamma_{\mathrm{app}}$ by altering the range of the data
used for the fit. The left boundary of the window (green dashed rectangle)
extends behind the crack, defining the window extent. (b) The constitutive law
of the cohesive zone is defined by the tensile stress as a function of the
separation of the crack faces, $\sigma_{c}(\delta)$, for the brittle hydrogel
network. $\sigma_{c}$ is determined from $\Gamma_{\mathrm{{app}}}(\delta)$,
shown inset, as described in the main text.(c) Distribution of $\Gamma^{*}$
from the CTOD measurements, where $\Gamma^{*}$ is the asymptotic apparent
fracture energy at the crack tip. The distribution is from 60 measurements
obtained with 6 independent samples, which each frame treated individually.
The mean is $1.77\mathrm{J/m^{2}}$ and the standard deviation is
$0.44\mathrm{J/m^{2}}$.

The constitutive behavior within the cohesive zone [27] is defined by the
variation of the cohesive stress $\sigma_{c}$ in this region; thus, one can
evaluate $\sigma_{c}$ by the derivative of $\Gamma_{\mathrm{app}}$ with
respect to the crack face separation distance $\delta$ [27], as

$\sigma_{c}(\delta)=\frac{d\Gamma_{\mathrm{app}}(\delta)}{d\delta};$ (14)

a derivation of this expression presented in Materials and Methods. The
constitutive law of the cohesive zone is shown in Fig. 5(b).

###  3.2 Measurement of the J-integral in the near-tip region via particle
tracking

The displacement vector field is determined using the microscopy method
described in section 2.2. Once the displacement vector field is determined via
particle tracking for each particle, the deformation gradient tensor,
$\mathbf{F}$, is estimated with finite-difference applied to the displacement
components of the neighboring particles [31]. Using the estimate of the
deformation gradient tensor, we calculate the strain energy density $W$ and
the first Piola-Kirchoff stress tensor fields $\mathbf{\sigma}$ using the neo-
Hookean material model. Expressions for $W$ and $\mathbf{\sigma}$ are provided
in section 2.

Using these deformation fields, the $J$-integral is evaluated around the crack
tip along a series of counterclockwise, rectangular paths as shown in Fig.
6(a). The left boundaries of the integral paths vary from the crack tip to the
far crack tail (indicated by the $x$ coordinate of its midpoint $A$, $X_{A}$),
while the other boundaries remain fixed at
$120\text{\,}\mathrm{\SIUnitSymbolMicro m}$ from the crack tip. Values of $J$
from integral paths with different left boundary locations are shown in Fig.
6(b). When the left boundary of the integral path is far away from the crack
tip, i.e., $|X_{A}|>150$\mathrm{\SIUnitSymbolMicro m}$$, the value of $J$ is
nearly constant. In this region, the $J$-integral remains path-independent
with a value of approximately $5\text{\,}\mathrm{J}\mathrm{/}\mathrm{m}^{2}$
in agreement with the reported values of the fracture energy of this material
[23].

When $|X_{A}|<150$\mathrm{\SIUnitSymbolMicro m}$$, a decrease of $J$ is
observed, similar to the drop of $\Gamma_{\mathrm{{app}}}$ measured by the
CTOD, as shown in Fig. 5(a). We measured a background $J_{\mathrm{{bg}}}$ from
a far-field ($400\text{\,}\mathrm{\SIUnitSymbolMicro m}$ to the crack tip)
integral path, and quantified the fractional decrease of the near-crack-tip
$J$ to the background $J_{\mathrm{{bg}}}$. $J$ decreased by $\approx 10$% from
$J_{\mathrm{{bg}}}$ for an integration window extending between
$150\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and
$70\text{\,}\mathrm{\SIUnitSymbolMicro m}$ from the crack tip. A sharper
decrease to approximately 20% of $J_{\mathrm{{bg}}}$ is observed when
$|X_{A}|<70$\mathrm{\SIUnitSymbolMicro m}$$. This suggests, on the one hand,
that a cohesive region with a finite size of approximately
$150\text{\,}\mathrm{\SIUnitSymbolMicro m}$ exists behind the crack tip; on
the other hand, it also suggests that most of the energy dissipates in this
cohesive region rather than ahead of the crack, despite the dominant stresses
ahead of the crack tip.

Figure 6: $J$-integral evaluation with varying paths near the crack tip. (a)
The stress concentration at the crack tip can be readily observed from the
plot of the $\sigma_{YY}$ component of the first Piola-Kirchhoff stress, shown
in the material frame of reference. The crack is indicated and labelled in
red, extending from $(0,0)$ in the $-X$ direction. Several integral paths,
used in the evaluation of the $J$-integral, are depicted by the dashed lines;
only the left boundary location, indicated by the $x$ coordinate of its
midpoint $A$, is varied upon changing the integral path. (b) The $J$-integral
was evaluated for 13 separate measurements during crack propagation in 3
different gel samples. The left boundary is systematically varied in
$10\text{\,}\mathrm{\SIUnitSymbolMicro m}$ steps. Far from the crack tip, the
change of integral window/contour does not affect the calculated value of $J$,
consistent with its path-independence. When the window extent shrinks to less
than $150\text{\,}\mathrm{\SIUnitSymbolMicro m}$, $J$ starts to drop, with a
similar decreasing trend to what is observed from the CTOD measurement shown
in Fig. 5(a). The color gradient in the background represents the fractional
deviation of the measured $J$ to a background $J_{\mathrm{bg}}$. The error
bars indicate the standard deviation among the 13 measurements.

To probe the local structure of energy dissipation, we evaluate the
$J$-integral along the perimeter of $100\text{\,}{\mathrm{\SIUnitSymbolMicro
m}}^{2}$ square windows. A map of the dissipation emerges as these windows are
rastered around the crack tip with a step of
$5\text{\,}\mathrm{\SIUnitSymbolMicro m}$. This process is repeated for 13
individual recordings of the crack tip field, which are then averaged as shown
Fig. 7(a). We exploit the path independence of $J$ by recognizing that any
finite value returned from the evaluation of $J$ represents the local
dissipation in the square window region. The local energy dissipation is
primarily distributed along $-X$, where the material has just fractured, as
shown in Fig. 7. Away from the crack tip, the localized value of $J$ decreases
to zero, indicating that no dissipation takes place in these regions, as is
expected for $K$-dominance.

The structure of the cohesive zone can be characterized by taking slices of
the $J$ map along the crack path $Y=0$ and perpendicular to it along $X=0$.
The energy dissipation represented by the value of $J$ is symmetric and
localized in a $50\text{\,}\mathrm{\SIUnitSymbolMicro m}$ region on either
side of the crack face, as shown in Fig. 7(b). In the direction of crack
advancement, the value of $J$ is non-zero only
$20\text{\,}\mathrm{\SIUnitSymbolMicro m}$ ahead of the crack tip, but extends
approximately $150\text{\,}\mathrm{\SIUnitSymbolMicro m}$ behind the crack, as
can be seen in Fig. 7(c). This asymmetry of energy dissipation is consistent
with our observation from CTOD measurement shown in Fig. 5(a) and the windowed
$J$ evaluation in Fig. 6(b). From the color-plot of $J$ in Fig. 7(a), the
cohesive zone for a quasi-static crack in this hydrogel looks like a droplet
moving along a surface in the direction of the crack with dimensions of the
order of $100\text{\,}\mathrm{\SIUnitSymbolMicro m}$ along each axis.

Figure 7: Structure of the cohesive zone. (a) The field of $J$ (averaged over
13 measurements) calculated locally by rastering the integral path which is
defined by the perimeter of small square windows (top left) with side length
of $10\text{\,}\mathrm{\SIUnitSymbolMicro m}$. Far from the crack tip, $J$ is
approximately zero, suggesting that no energy is dissipated in these regions;
however, close to the crack tip, $J$ increases, quantitatively indicating the
energy dissipated locally within the window through the material decohesion.
To evaluate the structure of this cohesive zone, the values of $J$ along $X=0$
(green dashed line) and $Y=0$ (blue dashed line) are extracted and presented
in (b) and (c), respectively. $X$ and $Y$ stand for the coordinates of the
center of the window. (b) Localized J values increase near the crack tip and
are symmetric about the crack along $Y$. The energy dissipation is localized
in a $50\text{\,}\mathrm{\SIUnitSymbolMicro m}$ region on either side of the
crack. (c) Along $X$, the localized values of $J$ are not symmetric about the
crack tip; Energy dissipation is observed only in a very limited area ahead of
the crack tip ($20\text{\,}\mathrm{\SIUnitSymbolMicro m}$) but extends
approximately $150\text{\,}\mathrm{\SIUnitSymbolMicro m}$ into the crack tail,
consistent with our observation from the CTOD measurement shown in Fig. 5(a)
and the windowed $J$ evaluation in Fig. 6(b).

###  3.3 Evaluation of strain stiffening

Among the various mechanisms that might alter the constitutive behavior under
large stretch near the crack tip, strain stiffening is the primary means by
which the observed sharper CTOD could emerge [16]. In order to evaluate
whether strain stiffening occurs for the hydrogel material used in our
experiments, we evaluate the constitutive response at the largest values of
stretch observed near the crack tip with the particle tracking measurements.

To characterize the mechanical response of our hydrogel material, we load it
in uniaxial tension and evaluate the stress as a function of stretch, as shown
in Fig. 8(a). The $\sigma_{\mathrm{eng}}-\lambda$ curves are fitted to both
the incompressible neo-Hookean model and the generalized neo-Hookean model
respectively222In the generalized neo-Hookean material model, the strain
energy density function is given by
$W=\frac{\mu}{2b}\left[\left(1+\frac{b}{n}(I-3)\right)^{n}-1\right]$, where
$\mu$ is the small-strain shear modulus, and $b$ and $n$ are material
parameters [32]. . The generalized neo-Hookean model can account for strain
stiffening and material yield through two parameters $b$ and $n$, where $b$ is
a parameter corresponding to yield and $n$ is a stiffening exponent [32]. When
$n=1$, the generalized neo-Hookean model reduces to the neo-Hookean model with
the strain energy density given by $W=\frac{\mu}{2}\left(I-3\right)$. A fit to
the generalized neo-Hookean model can be sensitive, particularly for data that
are well-described by the neo-Hookean material model; nevertheless, using non-
linear least squares we obtain a good fit with $n=1.07\pm 0.02$. Notably, this
value is well below the threshold above which the strain stiffening leads to a
sharper CTOD at $n=3/2$ [16].

Mild strain stiffening emerges in comparison to the classical neo-Hookean
material response; we evaluate the relative difference between our material
and the neo-Hookean response via the relative percentage difference of the
stress, the tangent modulus and the relative percentage difference of the
tangent modulus, as shown in Fig. 8(b)-(d). Under small stretches $\lambda$
and even moderate stretches ($\lambda<2$), the deviation generally remains
within $5\%$. The tangent modulus $E_{\mathrm{t}}$ decreases with the
incremental stretch. For $\lambda<1.6$, the theoretical tangent modulus,
$E_{\mathrm{t,NH}}$, agrees well with the experimental tangent modulus,
$E_{\mathrm{t,exp}}$. For a larger stretch, $E_{\mathrm{t,exp}}$ starts to
deviate from the neo-Hookean prediction $E_{\mathrm{t,NH}}$ and shows a larger
modulus, implying that the hydrogel is stiffer than neo-Hookean prediction in
this small region. However, the region where stretch is larger than 1.8 is
only within $20\text{\,}\mathrm{\SIUnitSymbolMicro m}$ around the crack tip,
as shown in Fig. 8(e). Since our observation of apparent fracture energy drop
and J-integral decrease has a characteristic length of around
$100\text{\,}\mathrm{\SIUnitSymbolMicro m}$, strain stiffening is excluded
from the possible reasons for the energy and $J$ drop. We then conclude that
the energy drop is a result of energy dissipation within the cohesive zone.

Figure 8: (a) Stress-stretch ($\sigma_{\mathrm{eng}}-\lambda$) curves of two
uniaxial tensile tests of intact hydrogel samples - a monotonic test until
sample fracture ($\lambda_{\mathrm{max}}\approx 2.24$), and a cyclic test with
stretch range from 1 to 1.9. The experimental curves are fitted to the
incompressible neo-Hookean model and the generalized neo-Hookean model [16],
as shown in the red and purple dashed curves respectively. The experimental
data starts to visibly deviate from the theoretical prediction at stretches
larger than 1.8. (b) The percentile deviation of the experimentally measured
stress to the neo-Hookean prediction. The data with $\lambda<1.1$ is truncated
due to the huge numerical fluctuation of the deviation value as the stress
approach zero at $\lambda=1$. (c) The tangent moduli ($E_{\mathrm{t}}$) for
both experimental data and neo-Hookean model. (d) The percentile deviation of
tangent moduli measured from experiments to neo-Hookean model. (e) The maximum
principle stretch field near the crack tip measured by particle tracking. The
stretch remains slightly larger than 1 until very close to the crack tip.
Inset: a magnified view of the stretch field in the immediate vicinity of the
crack tip (crack indicated by yellow dashed line). The region with stretch
larger than 1.8 is roughly characterized by a circle with radius of
$20\text{\,}\mathrm{\SIUnitSymbolMicro m}$. The scale bar is
$100\text{\,}\mathrm{\SIUnitSymbolMicro m}$ in the main plot and
$20\text{\,}\mathrm{\SIUnitSymbolMicro m}$ in the inset.

##  4 Discussion and Conclusions

Our micro-scale experiments demonstrate how a canonical brittle material
converts strain energy into both distributed, and much more localized,
dissipation at a crack tip. Within this material, there is an enormous
separation of scales - the typical scale of the sample is centimetric, whereas
the near-tip dissipation region extends no further than a couple of hundred
microns away from the very tip of the crack. It is interesting to note that
the CTOD data show two regions where dissipation occurs - a finite-sized
cohesive region that extends approximately
$100\text{\,}\mathrm{\SIUnitSymbolMicro m}$ from the crack tip and an
optically unresolved region, in which the remainder of the strain energy,
$\Gamma^{*}_{\mathrm{le}}$, is dissipated. The cohesive region reveals the
delocalized energy dissipation due to the distributed chain scission in the
hydrogel network. The size of the cohesive region is consistent with the
molecular damage zone directly visualized by fluorescence mechanophores in
elastomer [33], and our observation confirms the imperfection of polymer
networks in the Irwin-Orowan model [22]. We discuss alternative mechanisms for
the observed CTOD data, and provide numerical simulations for these
mechanisms, in Appendix A.

Having eliminated poro-elastic stress relaxation and strain stiffening from
the possible mechanisms responsible for modifying the stress state at the
crack tip, we are left with cohesive loss as the primary remaining mechanism
that can explain our observations. In this case, we ought to have a
quantitative estimate of the cohesive zone scale that should agree with the
observed scale of the measured cohesive region. Indeed, we can recover an
estimate for this scale by comparing the measured fracture energy to the
highly repeatable work of fracture (integrated work on the material prior to
failure in a defect-insensitive sample) [22, 34]. Here the fracture energy is
$\Gamma\approx$ $5\text{\,}\mathrm{J}\text{\,}{\mathrm{m}}^{-2}$ and the work
of fracture is calculated by numerically integrating the loading curve as
$\mathcal{W}=$ $54\text{\,}\mathrm{kJ}\text{\,}{\mathrm{m}}^{-3}$; thus, the
cohesive lengthscale is $\ell_{coh}\approx$
$93\text{\,}\mathrm{\SIUnitSymbolMicro m}$, a value in almost perfect
agreement with the scale of the cohesive zone we measure, identifying the
importance of cohesive loss as the primary mechanism responsible for
determining the state of stress at the crack tip in this brittle hydrogel.
Notably, similar analysis yields a similar scale for the cohesive zone in
brittle high-tensile steels [35], highlighting the universality of this
analysis for different materials.

How might our dissipation scale and sharp feature in the cohesive region near
the crack tip relate to the length scale $\delta_{dyn}$ observed in extensive
experiments on dynamic fracture [11, 12, 26, 4]? This point is less clear, as
that feature has long been ascribed to the $1/r$ scaling predictions of prior
theoretical works [17, 6]. However, it is known that this scale grows with
crack speed [11, 26], as does the fracture energy [26]; thus, if
$\delta_{dyn}$ somehow relates to the scale of the cohesive zone, it might
account for the velocity dependence of the fracture energy observed in dynamic
fracture; however, we cannot rule out strain stiffening at the strain rates
realized in dynamic fracture. Direct evaluation of the material constitutive
response must be evaluated at the rates typical of dynamic fracture to
determine whether the critical 3/2 strain stiffening exponent of the
generalized neo-Hookean model is exceeded at these strain rates; given the
amplitude of the stretch realized near the crack tip, this is a significant
experimental challenge.

The chemistry of the polyacrylamide hydrogel used in this study was selected
for its brittle material response; however, polyacrylamide gels, among other
hydrogels, have an incredible spectrum of mechanical responses [36, 22, 37,
38, 39, 40, 41, 14]. It is likely that hydrogels with a visco- or poro-elastic
constitutive response might have a different cohesive zone structure. In this
work, we employ a general approach for directly measuring the cohesive zone
law, and quantitatively evaluate effects such as strain stiffening and solvent
transport. When a material has an alternative dominant dissipation mechanism
through, e.g., plasticity as in the case of double-network hydrogels [42, 5],
this method must be considered carefully. Indeed, cracks in brittle materials
and cracks in plastic materials have different crack tip fields [17, 1, 43,
44].

Interestingly, our measurement of the apparent fracture energy identifies a
baseline value of the fracture energy _within_ the process zone, which we call
$\Gamma^{*}_{\mathrm{le}}$; this is the energy required to cause the rupture
of bonds at the crack tip, and there is a separation of scales between this
process and the cohesive zone. This quantity can be estimated from the mass
fraction of the monomer and the bond energy using a characteristic molecular
scale of the order of $2\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ \- for
polymerized acrylamide, the C-C bond energy is about
$7\text{\times}{10}^{-19}\text{\,}\mathrm{J}$; the cross sectional area of the
molecule is approximately $4\text{\times}{10}^{-20}\text{\,}\mathrm{m}$2, and
the mass ratio is about 10%, yielding an estimated
$1.7\text{\,}\mathrm{J}\text{\,}{\mathrm{m}}^{-2}$, in rough agreement with
the measured value of $\Gamma^{*}_{\mathrm{le}}$. Notably, the CTOD
measurements and the $J$-integral measurements indicate different values for
$\Gamma^{*}$. We attribute this to inaccuracies arising from linear
extrapolation of $\Gamma_{\mathrm{app}}$. The linear extrapolation likely
overestimates the value of $\Gamma^{*}$, as the _curvature_ of
$\Gamma_{\mathrm{app}}$ is negative as the window extent asymptotically
vanishes.

We have evaluated the energy loss occurring within the cohesive region of a
brittle hydrogel. This material is well-described by a neo-Hookean
constitutive law over all values of the stretches measured here, with
negligible strain stiffening. Poroelastic modifications to the stress might
occur, albeit at scales significantly smaller than the observed cohesive zone,
at the limit of the spatial resolution of our measurements. The loss of
cohesion will likely vary with different materials. Our measurements can
motivate the use, and the spatial structure, of the phase-field in numerical
calculation of brittle fracture [45], even in dynamic, unstable cases [46].
Notably, our observations confirm the long-standing idea of scale separation
between the material’s elastic response and the small scale region at the tip
of a brittle crack - the so-called autonomy of the crack tip[2], as there are
4 orders of magnitude separating the sample width and the extent of the
cohesive zone, ensuring that effects from the boundaries of the sample in the
crack plane do not alter the structure of the cohesive zone. In this work, we
have characterized and measured the local dissipation during failure of a
brittle solid using high-precision, high magnification measurements of the
kinematics of near-crack-tip fields, opening a door toward future studies and
extensions of this approach to other material systems.

##  Appendix A Alternative Constitutive Response at the Crack Tip

While it is impossible to determine the constitutive response of the material
near the crack tip from purely kinematic data, we can use estimates of
different possible mechanisms to evaluate whether they play a dominant role in
altering crack tip stress. One primary mechanism at play in hydrogels is that
of stress relaxation due to solvent transport. A typical lengthscale used to
evaluate the scale over which pore fluid migrates to relieve stress is derived
from the ratio of crack speed, $v$, to the solvent diffusion constant, $D$
[21]; this can readily be compared to the scale of the cohesive zone measured
in our work. For the cracks in our hydrogel, $v=$
$10\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{\,}{\mathrm{s}}^{-1}$ and $D=$
$1\text{\times}{10}^{-10}\text{\,}{\mathrm{m}}^{2}\text{\,}{\mathrm{s}}^{-1}$[47];
thus the length scale over which stress is relaxed by solvent migration is
less than $10\text{\,}\mathrm{\SIUnitSymbolMicro m}$; this is significantly
smaller than the scale of the cohesive region we have identified.

An alternative mechanism that might be at play, and has indeed been invoked to
explain dynamic fracture response [6, 4], is that of strain stiffening in the
context of a generalized neo-Hookean constitutive response. Indeed, it is
known that below a certain threshold of strain stiffening, neo-Hookean solids
have a parabolic CTOD, and that a strain stiffening parameter exceeding 3/2
[16] must be present to observe the characteristic $1/r$ scaling[17]. For our
material, we directly measured the material response subject to uniaxial
tension. We find that the hydrogel we use is nearly ideally neo-Hookean at the
strain rates we tested (and those realized in our fracture experiments); thus,
we anticipate a parabolic CTOD, and any ‘sharpness’ in our measured CTOD is
due to an alternative mechanism rather than strain stiffening of a generalized
neo-Hookean material model.

This is confirmed with numerical calculations carried out with COMSOL
Multiphysics 6.0, where the CTOD remains parabolic to within the grid
resolution (ca. $1\text{\,}\mathrm{\SIUnitSymbolMicro m}$ to scale) for an
ideally neo-Hookean solid, as shown in Fig. 9(b). Further calculations show
that a region with lesser modulus leads to a _blunter_ CTOD, as shown in Fig.
9(d).

Figure 9: Comparison of experimental and numerical CTODs from finite element
simulation of half the sample and their parabolic fittings. (a) 2D finite
element simulation setup for plane stress. The
$10\text{\,}\mathrm{m}\mathrm{m}$ $\times$ $15\text{\,}\mathrm{m}\mathrm{m}$
rectangular geometry takes half the size of the experimental sample by
symmetry. The displacement conditions to simulate the crack under mode I
loading are shown on each boundary. The material uses the neo-Hookean
constitutive model with the shear modulus of
$35\text{\,}\mathrm{k}\mathrm{P}\mathrm{a}$. The mesh is built with 166033
vertices after adaptive mesh refinement. (b) CTOD and parabolic fittings from
simulation and experiment. The upper half of of the CTOD from the simulation
is mirrored over the $x$ axis and plotted in red solid curve. An exemplary
experimental data is plotted in blue solid curve. The crack tip of both CTODs
are moved the origin. The parabolic fitting of the CTODs are shown in black
and yellow dashed curves respectively. The right part is magnified from the
area in the rectangular box from the left. (c) 2D finite element simulation
setup for gel sample with a weakening region around the crack tip. The
semicircle highlighted in yellow at the crack tip has the radius of
$30\text{\,}\mathrm{\SIUnitSymbolMicro m}$. The shear modulus in this region
is $\mu_{\mathrm{soft}}=$15\text{\,}\mathrm{k}\mathrm{P}\mathrm{a}$$, while
the rest of the material remains
$\mu=$35\text{\,}\mathrm{k}\mathrm{P}\mathrm{a}$$. The dimensions and the
boundaries conditions are the same as in (a). (d) CTOD from simulation (red
solid curve) in (c) and the parabolic fitting (black dashed curve).

## References

  * [1] T. L. Anderson, Fracture mechanics: fundamentals and applications, 4th Edition, CRC press, 2017.
  * [2] J. R. Rice, A Path Independent Integral and the Approximate Analysis of Strain Concentration by Notches and Cracks, Journal of Applied Mechanics 35 (2) (1968) 379–386.  doi:10.1115/1.3601206.
  * [3] M. Williams, Stress singularities resulting from various boundary conditions in angular corners of plates in extension (1952).
  * [4] T. Goldman Boué, R. Harpaz, J. Fineberg, E. Bouchbinder, Failing softly: a fracture theory of highly-deformable materials, Soft Matter 11 (19) (2015) 3812–3821.  doi:10.1039/C5SM00496A.
  * [5] I. Kolvin, J. M. Kolinski, J. P. Gong, J. Fineberg, How Supertough Gels Break, Physical Review Letters 121 (13) (Sep. 2018).  doi:10.1103/PhysRevLett.121.135501.
  * [6] E. Bouchbinder, A. Livne, J. Fineberg, The 1/r singularity in weakly nonlinear fracture mechanics, Journal of the Mechanics and Physics of Solids 57 (9) (2009) 1568–1577.  doi:10.1016/j.jmps.2009.05.006.
  * [7] M. R. Begley, C. Creton, R. M. McMeeking, The elastostatic plane strain mode I crack tip stress and displacement fields in a generalized linear neo-Hookean elastomer, Journal of the Mechanics and Physics of Solids 84 (2015) 21–38.  doi:10.1016/j.jmps.2015.07.005.
  * [8] Y. Tanaka, K. Fukao, Y. Miyamoto, Fracture energy of gels, The European Physical Journal E 3 (4) (2000) 395–401.  doi:10.1007/s101890070010.
  * [9] A. Livne, G. Cohen, J. Fineberg, Universality and Hysteretic Dynamics in Rapid Fracture, Physical Review Letters 94 (22) (Jun. 2005).  doi:10.1103/PhysRevLett.94.224301.
  * [10] T. Baumberger, C. Caroli, D. Martina, Fracture of a biopolymer gel as a viscoplastic disentanglement process, The European Physical Journal E 21 (1) (2006) 81–89.
  * [11] A. Livne, E. Bouchbinder, J. Fineberg, Breakdown of Linear Elastic Fracture Mechanics near the Tip of a Rapid Crack, Physical Review Letters 101 (26) (Dec. 2008).  doi:10.1103/PhysRevLett.101.264301.
URL https://link.aps.org/doi/10.1103/PhysRevLett.101.264301

  * [12] A. Livne, E. Bouchbinder, I. Svetlizky, J. Fineberg, The Near-Tip Fields of Fast Cracks, Science 327 (5971) (2010) 1359–1363.  doi:10.1126/science.1180476.
  * [13] I. Kolvin, G. Cohen, J. Fineberg, Topological defects govern crack front motion and facet formation on broken surfaces, Nature Materials 17 (2) (2017) 140–144.  doi:10.1038/nmat5008.
  * [14] X. Zhao, X. Chen, H. Yuk, S. Lin, X. Liu, G. Parada, Soft materials by design: Unconventional polymer networks give extreme properties, Chemical Reviews 121 (8) (2021) 4309–4372.  doi:10.1021/acs.chemrev.0c01088.
  * [15] A. Taureg, J. M. Kolinski, Dilute concentrations of submicron particles do not alter the brittle fracture of polyacrylamide hydrogels, arXiv preprint arXiv:2004.04137 (2020).
  * [16] R. Long, C.-Y. Hui, Crack tip fields in soft elastic solids subjected to large quasi-static deformation—a review, Extreme Mechanics Letters 4 (2015) 131–155.
  * [17] J. K. Knowles, E. Sternberg, An asymptotic finite-deformation analysis of the elastostatic field near the tip of a crack, Journal of Elasticity 3 (2) (1973) 67–107.
  * [18] Y. Liu, B. Moran, Asymptotic path-independent integrals for the evaluation of crack-tip parameters in a neo-hookean material, International Journal of Fracture 224 (1) (2020) 133–150.
  * [19] S. A. Chester, L. Anand, A thermo-mechanically coupled theory for fluid permeation in elastomeric materials: application to thermally responsive gels, Journal of the Mechanics and Physics of Solids 59 (10) (2011) 1978–2006.
  * [20] N. Bouklas, R. Huang, Swelling kinetics of polymer gels: comparison of linear and nonlinear theories, Soft Matter 8 (31) (2012) 8194–8203.
  * [21] T. Baumberger, O. Ronsin, Environmental control of crack propagation in polymer hydrogels, Mechanics of Soft Materials 2 (1) (2020) 1–38.
  * [22] C. Yang, T. Yin, Z. Suo, Polyacrylamide hydrogels. i. network imperfection, Journal of the Mechanics and Physics of Solids 131 (2019) 43–55.
  * [23] M. Wang, M. Adda-Bedia, J. M. Kolinski, J. Fineberg, How hidden 3d structure within crack fronts reveals energy balance, Journal of the Mechanics and Physics of Solids 161 (2022) 104795.
  * [24] S. Van der Walt, J. L. Schönberger, J. Nunez-Iglesias, F. Boulogne, J. D. Warner, N. Yager, E. Gouillart, T. Yu, scikit-image: image processing in python, PeerJ 2 (2014) e453.
  * [25] Y. Qi, Z. Zou, J. Xiao, R. Long, Mapping the nonlinear crack tip deformation field in soft elastomer with a particle tracking method, Journal of the Mechanics and Physics of Solids 125 (2019) 326–346.
  * [26] T. Goldman, A. Livne, J. Fineberg, Acquisition of Inertia by a Moving Crack, Physical Review Letters 104 (11) (Mar. 2010).  doi:10.1103/PhysRevLett.104.114301.
  * [27] L. B. Freund, Dynamic fracture mechanics, Cambridge university press, 1998.
  * [28] R. W. Style, T. Sai, N. Fanelli, M. Ijavi, K. Smith-Mannschott, Q. Xu, L. A. Wilen, E. R. Dufresne, Liquid-liquid phase separation in an elastic network, Phys. Rev. X 8 (2018) 011028.  doi:10.1103/PhysRevX.8.011028.
  * [29] D. B. Allan, T. Caswell, N. C. Keim, C. M. van der Wel, R. W. Verweij, soft-matter/trackpy: Trackpy v0.5.0 (Apr. 2021).  doi:10.5281/zenodo.4682814.
URL https://doi.org/10.5281/zenodo.4682814

  * [30] J. C. Crocker, D. G. Grier, Methods of digital video microscopy for colloidal studies, Journal of colloid and interface science 179 (1) (1996) 298–310.
  * [31] T. Benkley, C. Li, J. Kolinski, Particle tracking based estimation of the deformation gradient tensor, Unpublished.
  * [32] P. H. Geubelle, W. G. Knauss, Finite strains at the tip of a crack in a sheet of hyperelastic material: I. homogeneous case, Journal of Elasticity 35 (1-3) (1994) 61–98.
  * [33] J. Slootman, V. Waltz, C. J. Yeh, C. Baumann, R. Göstl, J. Comtet, C. Creton, Quantifying rate-and temperature-dependent molecular damage in elastomer fracture, Physical Review X 10 (4) (2020) 041045.
  * [34] R. Long, C.-Y. Hui, J. P. Gong, E. Bouchbinder, The fracture of highly deformable soft materials: A tale of two length scales, Annual Review of Condensed Matter Physics 12 (2021) 71–94.
  * [35] Z. Xiong, P. J. Jacques, A. Perlade, T. Pardoen, Characterization and control of the compromise between tensile properties and fracture toughness in a quenched and partitioned steel, Metallurgical and Materials Transactions A 50 (8) (2019) 3502–3513.
  * [36] Y. Cohen, O. Ramon, I. Kopelman, S. Mizrahi, Characterization of inhomogeneous polyacrylamide hydrogels, Journal of Polymer Science Part B: Polymer Physics 30 (9) (1992) 1055–1067.
  * [37] J. Liu, C. Yang, T. Yin, Z. Wang, S. Qu, Z. Suo, Polyacrylamide hydrogels. ii. elastic dissipater, Journal of the Mechanics and Physics of Solids 133 (2019) 103737\.
  * [38] Y. Wang, T. Yin, Z. Suo, Polyacrylamide hydrogels. iii. lap shear and peel, Journal of the Mechanics and Physics of Solids 150 (2021) 104348.
  * [39] S. Hassan, J. Kim, et al., Polyacrylamide hydrogels. iv. near-perfect elasticity and rate-dependent toughness, Journal of the Mechanics and Physics of Solids 158 (2022) 104675.
  * [40] J. Kim, T. Yin, Z. Suo, Polyacrylamide hydrogels. v. some strands in a polymer network bear loads, but all strands contribute to swelling, Journal of the Mechanics and Physics of Solids 168 (2022) 105017.
  * [41] Y. Wang, G. Nian, J. Kim, Z. Suo, Polyacrylamide hydrogels. vi. synthesis-property relation, Journal of the Mechanics and Physics of Solids (2022) 105099.
  * [42] J. P. Gong, Y. Katsuyama, T. Kurokawa, Y. Osada, Double-network hydrogels with extremely high mechanical strength, Advanced materials 15 (14) (2003) 1155–1158.
  * [43] J. Hutchinson, Plastic stress and strain fields at a crack tip, Journal of the Mechanics and Physics of Solids 16 (5) (1968) 337–342.
  * [44] J. R. Rice, G. Rosengren, Plane strain deformation near a crack tip in a power-law hardening material, Journal of the Mechanics and Physics of Solids 16 (1) (1968) 1–12.
  * [45] M. Ambati, T. Gerasimov, L. De Lorenzis, A review on phase-field models of brittle fracture and a new fast hybrid formulation, Computational Mechanics 55 (2) (2015) 383–405.
  * [46] J. Bleyer, J.-F. Molinari, Microbranching instability in phase-field modelling of dynamic brittle fracture, Applied Physics Letters 110 (15) (2017) 151903.
  * [47] Z. I. Kalcioglu, R. Mahmoodian, Y. Hu, Z. Suo, K. J. Van Vliet, From macro-to microscale poroelastic characterization of polymeric hydrogels via indentation, Soft Matter 8 (12) (2012) 3393–3398.


# SRNI-CAR: A Comprehensive Dataset for Analyzing the Chinese Automotive
Market

Ruixin Ding
Adam Smith Business School
University of Glasgow
<EMAIL_ADDRESS>Bowei Chen
Adam Smith Business School
University of Glasgow
<EMAIL_ADDRESS>James M. Wilson
Zhi Yan
Adam Smith Business School
University of Glasgow
<EMAIL_ADDRESS>CIAD UMR 7533
UTBM
<EMAIL_ADDRESS>Yufei Huang
Trinity Business School
Trinity College Dublin
<EMAIL_ADDRESS>
###### Abstract

The automotive industry plays a critical role in the global economy, and
particularly important is the expanding Chinese automobile market due to its
immense scale and influence. However, existing automotive sector datasets are
limited in their coverage, failing to adequately consider the growing demand
for more and diverse variables. This paper aims to bridge this data gap by
introducing a comprehensive dataset spanning the years from 2016 to 2022,
encompassing sales data, online reviews, and a wealth of information related
to the Chinese automotive industry. This dataset serves as a valuable
resource, significantly expanding the available data. Its impact extends to
various dimensions, including improving forecasting accuracy, expanding the
scope of business applications, informing policy development and regulation,
and advancing academic research within the automotive sector. To illustrate
the dataset’s potential applications in both business and academic contexts,
we present two application examples. Our developed dataset enhances our
understanding of the Chinese automotive market and offers a valuable tool for
researchers, policymakers, and industry stakeholders worldwide.

###### Index Terms:

Automotive industry, big data, sales forecasting, text mining, machine
learning

##  I Introduction

The automotive industry has been one of the important economic sectors in many
countries and regions. It also has an important role in global energy
consumption, energy sustainability and environmental impact. At present, China
has become the world’s largest market for new energy vehicles (NEVs) and one
of the largest automobile markets. Studying the Chinese automobile market
reveals its profound impact on the global automotive ecosystem in terms of
market growth opportunities, innovation cooperation, market diversity,
environmental sustainability, and more. For a long time, the automotive
industry has been facing many challenges in the marketing process, such as
fierce market competition, the diversity of consumer demand, and the pressure
of environmental protection policies. With the growth of social media and
digital advertising, consumer engagement on the Web has greatly increased, and
rational and effective digital marketing has become another serious challenge
in the automotive industry [1].

The imperative for precise and efficient forecasting, rooted in historical
data, becomes evident to address the challenges mentioned above. A substantial
cohort of automotive industry stakeholders including executives, marketers,
and academics are actively in pursuit of more efficacious methods for market
analysis. Traditional forecasting approaches face formidable hurdles when
contending with vast datasets, often falling short in ensuring the fidelity of
sales predictions [2]. While the advent of cutting-edge data-driven methods
such as data mining and machine learning has led to significant advancements
in knowledge discovery and predictive capabilities, their efficacy hinges
crucially on the availability of comprehensive and accurate datasets. However,
currently available datasets are often fragmented, making it difficult for
researchers to integrate industry information, automakers’ behavior, consumer
demand, market feedback, and sales forecasts. It is increasingly important to
create a dataset that can meet the multiple business analytics needs of the
automotive industry. Therefore, we collected car model sales data, consumer
online review data, and automotive industry news and information data from
different sources, and integrated them into a comprehensive dataset, called
_SRNI-CAR_.

TABLE I: Summary of variables in the datasets for automotive market or
consumer analytics.

Variable |  [3] |  [4] |  [5] |  [6] |  [7] |  [8] | Our dataset
---|---|---|---|---|---|---|---
Vehicle attribute related variables (e.g., model type, model size, energy type, etc.) | ✓ |  | ✓ | ✓ | ✓ |  | ✓
Brand related variables (e.g., brand energy type, brand country-of-origin, etc.) | ✓ | ✓ |  | ✓ |  |  | ✓
Price variables (e.g., official price, transaction price, etc.) | ✓ |  | ✓ | ✓ |  | ✓ | ✓
Date-related variables (e.g., year of purchase, year of review, model launch date, etc.) | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓
Consumer experience variables (e.g., experience duration, mileage, etc.) |  |  |  |  |  | ✓ | ✓
User ratings (e.g., overall rating, exterior rating, interior rating, etc.) |  |  | ✓ |  |  | ✓ | ✓
Online reviews (e.g., advantage, features comments, comfort comments, etc.) |  | ✓ | ✓ |  |  | ✓ | ✓
Industry news and information (e.g., title, text, information label, pageview, etc.) |  |  |  |  |  |  | ✓
Sales | ✓ | ✓ | ✓ | ✓ | ✓ |  | ✓
Website search variables (e.g., search volume of brand, etc.) |  | ✓ |  |  |  |  |
Economic factor (e.g., consumer price index, gas prices, etc.) |  |  |  |  | ✓ |  |
Vehicle design (e.g., vehicle pictures, design fluency, etc.) |  |  |  | ✓ |  |  |

Our work makes two significant contributions. Firstly, we have curated a more
comprehensive and scalable data resource. It not only consolidates industry
news, development insights, automotive marketing data, consumer online
reviews, and sales information but also introduces valuable variables
previously absent, such as model launch dates and brand inception dates.
Therefore, our dataset supports a broader spectrum of research possibilities
compared to existing publicly available datasets in the automotive domain.
Furthermore, it enhances analytical accuracy and interpretability. Secondly,
our dataset possesses substantial business value in the automotive sector.
Sales data aids automakers and marketers in discerning market trends, while
review data facilitates the identification of consumer preferences, evaluation
of marketing effectiveness, and product strengths and weaknesses analysis.
Incorporating industry news, development insights, and automotive publicity
empowers automakers to grasp industry trends and market competition.
Integrated analysis across multiple data provides superior decision support
for product planning, business expansion, and marketing strategy formulation.
Our dataset can be used to improve product quality, and align with consumer
demands. Additionally, our dataset can be used by government policymakers,
enabling the formulation of pertinent automotive industry policies and
regulations. Thus, the development of this dataset holds profound significance
for a myriad of stakeholders within the automotive industry.

##  II Review of Existing Automotive Datasets

The automobile industry has garnered significant attention from researchers
and analysts, particularly for sales forecasting. A review of prior studies
reveals that researchers commonly incorporate a range of variables
encompassing vehicle attributes, brand characteristics, pricing, temporal
factors, user experiences, ratings, reviews, website search data, economic
indicators, and vehicle design to prognosticate car sales. However, as shown
in Table I, none of the datasets employed in these studies encompass all these
pertinent variables. In the realm of nonlinear modeling, the reduction of
variables often proves counterproductive to enhancing predictive model
performance [9]. Consequently, the absence of comprehensive variables
constitutes a notable deficiency within the existing datasets in the
automotive industry.

The limitations of the current automotive datasets are noteworthy. Firstly,
they often omit vital information about brand creation and model launch dates,
which is crucial for understanding market dynamics [10]. Secondly, they fail
to differentiate between the growing number of new energy vehicle brands and
do not clearly identify brand origins, despite their influence on consumer
perceptions [11]. Thirdly, these datasets often lack detailed comments and
ratings for specific vehicle attributes, limiting their usefulness for sales
forecasting and preference analysis [4]. Fourthly, they usually provide only
aggregate pricing data for vehicle series, omitting separate pricing for
individual models and making it challenging to analyze the impact of discounts
[8]. Furthermore, important data related to the automotive industry, such as
model sentiment and review articles, are often missing, and aligning them with
sales data remains challenging [12]. These limitations highlight the need for
more comprehensive datasets encompassing critical temporal, attribute-
specific, and contextual factors for robust automotive market analysis and
forecasting.

##  III Data Collection, Preparation and Description

We collected sales and online review data from PCAuto and Dongchedi, and
automotive industry news and information data from Autohome. They are three
most authoritative and largest automobile media platforms in China.

While the collected raw data encompass a broader range of variables than
previous studies, they still lack essential elements necessary to address the
evolving requirements of business analysis and academic research. To address
these gaps, we introduced additional variables related to brand country-of-
origin classification, vehicle entry order, and brand entry order. We also
synchronized monthly sales data with online reviews to enhance our ability to
derive valuable business insights. Furthermore, we manually incorporated
actual transaction prices and official guide prices for specific car models
within each series to investigate pricing and discount impacts. To ensure data
validity, we diligently identified and addressed missing and outlier values
across three data sources. Sales data required minimal processing, containing
no missing values, outliers, or duplicates. For online reviews, we selectively
removed instances with multiple missing values and those with missing values
in less than 1% of variables other than sales. Notably, data with only a
missing value in sales were retained, comprising 9.68% of the total dataset.
It is important to note that imported car sales data are not included,
resulting in the absence of corresponding sales data in the review dataset.
Finally, we meticulously eliminated duplicates to produce the refined dataset
intended for publication.

TABLE II: Description of variables in SRNI-CAR. Data |  Variables |  Description
---|---|---
Sales |  Car series |  Name of the car series.
Brand |  Name of the brand.
Year |  Year in which the car series was sold.
Month |  Month in which the car series was sold.
Car model type |  Car model category: Sedan, SUV and MPV.
Brand energy type |  Brand category based on energy type of vehicle produced.
Size |  Vehicle size category: mini, minivan, minibus, small, compact, mid-size, larger than mid-size, full-size.
Brand country of origin |  Country in which the brand was created.
Model launch date |  Year when the car series was launched on the Chinese market.
Brand establishment date |  Year when the brand was created.
Brand entered China date |  Year when the brand officially entered the Chinese market.
Sales |  Total sales of the car series in the month.
Reviews |  Car series |  Name of the car series.
Brand |  Name of the brand.
Size |  Vehicle size category: mini, minivan, small, compact, mid-size, larger than mid-size, full-size.
Car model type |  Car model category: Sedan, SUV and MPV.
User ID |  Name that users use when making online reviews.
Year of review |  Year when the user reviews.
Month of review |  Month when the user reviews.
Specific model purchased |  Specific model of a car series purchased by a user.
Official price |  Official prices for specific models purchased.
Car energy type |  Vehicle energy type: gasoline vehicle, diesel vehicle, hydrogen vehicle, and so on.
Brand energy type |  Brand category based on energy type of vehicle produced.
Brand country of origin |  Country in which the brand was created.
Brand establishment date |  Year when the brand was created.
Brand entered China date |  Year when the brand officially entered the Chinese market.
Model launch date |  Year when the car series was officially launched on the Chinese market.
Year of purchase |  Year when the user purchased the model.
Month of purchase |  Month when the user purchased the model.
Sales |  Total sales of the car series in the month when the user purchased the model.
Experience duration |  Months between purchase date and review posting date.
Province |  Province in which the user purchased the model.
City |  City in which the user purchased the model.
Transaction price |  Real transaction price of the model.
Average energy consumption |  Gasoline, diesel, electricity, or hydrogen consumed for every 100 kilometers traveled.
Mileage |  Kilometers the user has driven the model at the date the review was posted.
Overall rating |  User’s overall rating of the vehicle purchased.
Exterior rating |  User’s rating of the exterior of the vehicle.
Interior rating |  User’s rating of the interior of the vehicle.
Space rating |  User’s rating of the space of the vehicle.
Features rating |  User’s rating of the feature of the vehicle.
Power rating |  User’s rating of the power of the vehicle.
Energy consumption rating |  User’s rating of the energy consumption of the vehicle.
Driving rating |  User’s rating of the driving of the vehicle.
Comfort rating |  User’s rating of the comfort of the vehicle.
Advantage |  The advantages of the model as perceived by the user.
Disadvantage |  The disadvantage of the model as perceived by the user.
Exterior comments |  User’s comments on the exterior of the vehicle.
Interior comments |  User’s comments on the interior of the vehicle.
Space comments |  User’s comments on the space of the vehicle.
Features comments |  User’s comments on the feature of the vehicle.
Power comments |  User’s comments on the power of the vehicle.
Energy consumption comments |  User’s comments on the energy consumption of the vehicle.
Driving comments |  User’s comments on the driving of the vehicle.
Comfort comments |  User’s comments on the comfort of the vehicle.
News information |  Title |  Title of the information.
Pageview |  Number of times the information was viewed.
Number of comments |  Number of comments the information received.
Text |  Text content contained in the information.
Release date |  Date on which this information was published.
Author |  Person who posted the information.
Source |  Source of the information.
Information type |  Whether the information is original, compiled, a press release, or reprinted from another platform.
Information label |  Labels chosen by the author that summarizes the information, based on its content.

As shown in Table II, SRNI-CAR consist of sales, online reviews, and
automotive industry news and information, over the period from 2016 to 2022.
The dataset can be downloaded from the following address:

https://srni-car.github.io

The sales data comprises 1,236 car series, including 518 sedans, 598 SUVs, and
121 MPVs, with 39,496 observations. It encompasses 155 car brands, with 107
traditional and 48 new energy brands, originating primarily from Germany,
France, Korea, the Czech Republic, the USA, Japan, Sweden, Italy, the UK, and
China. This data is valuable for research in sales forecasting, first-mover
advantages, brand country-of-origin, and consumer preferences. It’s stored as
a 3.6 MB CSV file. The online review data is based on 217,292 comments from
car owners in 358 cities, covering 13,039 specific models across 672 car
series, ranging from 26,800 to 14.88 million yuan. It includes 10,977
traditional and 2,062 new energy models under 127 car brands, categorized into
eight aspects by vehicle attributes. It also contains official and real
transaction prices for precise analysis of discounts’ impact, which can
support research in sales forecasting, consumer behavior, product evaluation,
and fake review detection, stored in a 480 MB CSV file. The automotive
industry news and information data contains 83,590 items that covers a wide
range of topics. Each data items is accurately labeled, aiding researchers in
selecting relevant data. This data is stored as a 224.1 MB CSV file,
beneficial for analyzing industry trends, marketing strategies, consumer
perceptions, and automotive technology development.

##  IV Automotive Analytics Examples

Two application examples are presented to showcase the potential of our
dataset in automotive analytics.

###  IV-A Automobile Sales Forecasting

For automotive manufacturers, proficiency in sales forecasting holds critical
significance in shaping both product development and marketing strategies. The
intricate interplay among an array of factors influencing sales, including
price dynamics[13, 14], strategic discounting, consumer sentiment gleaned from
online reviews, and numerical ratings [15], necessitates a comprehensive
analytical framework. Moreover, analysts must diligently consider factors like
first-mover advantages, brand country-of-origin (COO), brand attributes,
product attributes, and the increasingly salient energy characteristics of
vehicles, especially within the context of China’s fervent pursuit of NEVs
[16]. Unlike previous studies constrained by data limitations, our dataset
offers an opportunity for an extensive and all-encompassing examination of
these pivotal determinants of sales.

Figure 1: Top variables with the most significant contributions in two data
instances.

Figure 2: Shap values corresponding to the variables influencing car sales.

Figure 3: Importance of variables in predicting sentiment across eight review
categories. Figure 4: Word clouds created from review comments associated with
the eight vehicle attributes.

To assess the impact of these factors on sales, we harnessed online review
data spanning 2019 to 2021 and opted for the XGBoost model, lauded for its
efficiency and precision in sales forecasting [17]. We optimized model
parameters using grid search cross-validation to bolster predictive
performance. To add robustness, one can explore alternative machine learning
methodologies but that is not the most significance of our example here.
Categorical predictors were judiciously transformed into dummy variables.
Furthermore, we employed SnowNLP for sentiment analysis of textual comments,
yielding sentiment scores on a scale from 0 to 1 as supplementary predictive
variables. Finally, to interpret XGBoost model outputs, we applied the Shapley
additive explanations (in short SHAP), mainly for its efficacy in elucidating
feature contributions without the encumbrance of multi-collinearity concerns
[18, 19].

In our analysis, we were able to obtain localized insights, presented in the
form of Fig. 2, which depict the contribution of each variable for every data
instance—commonly referred to as Shap values. In a locally interpreted force
plot, we can determine the contribution of features in this data instance by
observing the length of the arrows. Additionally, red arrows represent
features that positively impact the prediction, while blue arrows indicate a
negative impact. For example, in the first data instance, we can observe that
model launch date has the largest contribution to the prediction, but is
negatively impacting it. On the other hand, brand entered China date, although
its contribution is relatively lower, still positively impacts the prediction.
Therefore, this data instance reveals that the launch date of the car model is
a significant factor influencing its sales, albeit with a negative effect.
Simultaneously, brand entered China date is also an important factor, despite
its relatively lower contribution, as it positively affects the sales. Within
the SHAP framework, variable importance is gauged as the average of absolute
Shap values across all data instances for each variable. Fig. 2
comprehensively outlines the variables employed in our experiment, showcases
the experimental outcomes, and ranks these variables based on their
significance in predicting sales. Notably, it visually underscores how the
contribution of each variable to sales evolves as the variable’s value
changes.

We found several intriguing and valuable findings. Firstly, the sequence of
model and brand entry wields a significantly higher influence on sales
compared to other variables. Notably, alongside vehicle size, the launch date
of a model, brand entry date into China, and the brand’s founding date emerge
as the three most critical factors. This implies the existence of both model
and brand first-mover advantages in the Chinese automobile market, albeit with
the caveat that late-mover advantages are still relevant based on the brand’s
entry date into China. Secondly, within the realm of price-related variables,
although discounts exert a promotional effect on sales, official prices and
transaction prices exert more substantial impacts, with official prices
retaining the highest importance. However, it is worth noting that the
relationship between price and sales is not linear. Thirdly, sentiment
extracted from review text, categorized by vehicle attributes, exerts a
notably higher impact on sales compared to corresponding rating data. This
suggests that the review text serves as a more representative source of
consumer sentiment regarding a model. Fourthly, within the Chinese automotive
market, compact models and those from German, Japanese, and American brands
maintain their popularity among consumers. Lastly, concerning new energy
vehicles and associated brands, consumers exhibit a preference for pure
electric vehicles (PEVs) and hybrid electric vehicles (HEVs), alongside a high
level of acceptance for NEV brands.

In summary, these findings underscore the imperative for automaker managers
and marketers to meticulously analyze and weigh these factors when devising
enterprise development and marketing strategies. Such insights are also
instrumental for government departments in evaluating the effectiveness of new
energy vehicle policies.

###  IV-B Consumer Behavior Analytics

Online reviews, generated by consumer post-purchase, significantly reflect and
influence consumer decision-making [20]. Rating and review text are key
components of online reviews, with ratings indirectly affecting sales through
their interaction with the text [21]. Consumers tend to use ratings to
shortlist products and rely on review text for final choices [15],
highlighting the higher business value of review text in expressing customer
satisfaction. Understanding how consumer behavior and preferences impact
review sentiment is crucial for shaping marketing strategies.

Previous studies have shown that the level of user experience and price have
an impact on automotive customer satisfaction. Mileage and experience duration
are two important variables in the automotive industry to measure the level of
experience [8]. In addition, because consumers may have different expectations
for different countries and different types of products, factors such as brand
country-of-origin, brand history, and car type may also have an impact on the
sentiment of the review text. At the same time, previous studies did not take
into account that the factors that affect the sentiment of reviews for
different attributes of vehicles may be different. For example, perhaps the
review for comfort is more likely to be influenced by the level of user
experience than the review of exterior. SRNI-CAR makes it possible to forecast
the sentiment of automobile consumer reviews and to study the above questions.

In this example, we conducted sentiment score forecasts for comments
associated with distinct vehicle attributes, employing the SnowNLP, XGBoost,
and SHAP. The comprehensive experiment was divided into eight segments, each
aligned with specific vehicle attributes. These segments encompassed the
forecasting of sentiment scores for comments pertaining to exterior, interior,
space, features, power, driving, energy consumption, and comfort attributes.

The depicted analysis Fig. 3 underscores the significance of various variables
in predicting sentiment scores for comments linked to diverse vehicle
attributes. The combination of these eight forecasts yields several pivotal
insights. First, it is evident that the user’s experience level, encompassing
driving mileage and experience duration, influences consumer review
sentiments, indirectly affirming its impact on overall customer satisfaction,
with mileage proving more influential than experience duration among the eight
vehicle attributes examined. Second, price-related factors indeed influence
consumer reviews, with transaction and official prices holding greater sway
than discounts, although their influence is less pronounced in reviews
concerning comfort, exterior, and power attributes. Third, the year of vehicle
purchase emerges as a crucial factor in forecasting review sentiment, an
aspect hitherto unexplored, suggesting evolving consumer expectations. Fourth,
vehicle-related attributes, such as size and energy type, demonstrate little
relevance in exterior-related comments, indicating that Chinese consumers lack
a distinct preference for specific model appearances. Fifth, model launch
date, brand establishment date, and entry into the Chinese market date
consistently wield importance across all eight review categories, suggesting
that consumer review sentiment may signify either a first-mover or late-mover
advantage. Sixth, ratings assume the highest significance in reviews
associated with power, with elevated importance in feature, driving, and
space-related reviews, highlighting the divergence between ratings and textual
comments across various review aspects. Last but not least, the country of
origin, size, and energy type exhibit varying impacts on sentiment in comments
across different aspects. Furthermore, as shown in Fig. 4, we conducted a word
frequency analysis of comments pertaining to the eight distinct attributes and
subsequently generated word clouds. The findings elucidate consumers’ distinct
priorities within these attributes. Notably, consumers focus more on the
exterior design when assessing a vehicle’s appearance. Concerning interior
design, particular attention is directed toward the vehicle’s central console.
In evaluating power performance, consumers prioritize acceleration
capabilities, while precision in steering takes precedence in assessments of
handling. Features such as the reverse camera and functionality garner
significant attention. The spaciousness of rear-seat areas is a key
consideration for consumers when assessing vehicle space. In the realm of
comfort, the quality of seats emerges as a pivotal factor. Lastly, regarding
energy consumption, consumers exhibit heightened scrutiny when driving in
urban settings and on highways.

In summary, when exploring consumer behavior within the automobile market, one
must consider the multifaceted nature of review data. Our dataset and
accompanying analysis provide invaluable insights for marketers, facilitating
the evaluation of product competitiveness, the identification of suitable
marketing strategies, and the implementation of precision marketing tactics to
enhance overall marketing effectiveness.

##  V Conclusion

We have diligently created a comprehensive dataset SRNI-CAR for Chinese
automotive market. This paper details the data collection and processing
procedures, expounds upon the intricacies of variables, and highlights the
dataset’s profound relevance in areas such as market demand, consumer
behavior, preferences, and industry development. We believe that SRNI-CAR will
benefit not only the academic research community but also automotive industry
leaders, marketers, and government agencies. We remain committed to
continuously improving this dataset, with plans for updates and expansions,
all aimed at enhancing its practical applications. This paper serves as a
means to make the dataset accessible to a wider audience, facilitating a
deeper understanding of the complexities within the automotive sector.

## Acknowledgment

The first author acknowledges the University of Glasgow Adam Smith Business
School for Student Research Internship funding support. The second author
would like to thank the funding support of Nvidia Accelerated Data Science
Grant and University of Glasgow Adam Smith Business School.

## References

  * [1] C. Homburg and D. M. Wielgos, “The value relevance of digital marketing capabilities to firm performance,” _Journal of the Academy of Marketing Science_ , vol. 50, no. 4, pp. 666–688, 2022.
  * [2] S. Cheriyan, S. Ibrahim, S. Mohanan, and S. Treesa, “Intelligent sales prediction using machine learning techniques,” in _2018 International Conference on Computing, Electronics and Communications Engineering_. IEEE, 2018, pp. 53–58.
  * [3] Z. Xia, S. Xue, L. Wu, J. Sun, Y. Chen, and R. Zhang, “Forexgboost: passenger car sales prediction based on xgboost,” _Distributed and Parallel Databases_ , vol. 38, pp. 713–738, 2020.
  * [4] T. Geva, G. Oestreicher-Singer, N. Efron, and Y. Shimshoni, “Using forum and search data for sales prediction of high-involvement projects,” _Mis Quarterly_ , vol. 41, no. 1, pp. 65–82, 2017.
  * [5] Y. Chen, S. Fay, and Q. Wang, “The role of marketing in social media: How online consumer reviews evolve,” _Journal of interactive marketing_ , vol. 25, no. 2, pp. 85–94, 2011.
  * [6] J. R. Landwehr, A. A. Labroo, and A. Herrmann, “Gut liking for the ordinary: Incorporating design fluency improves automobile sales forecasts,” _Marketing Science_ , vol. 30, no. 3, pp. 416–429, 2011.
  * [7] A. Sa-Ngasoongsong, S. T. Bukkapatnam, J. Kim, P. S. Iyer, and R. Suresh, “Multi-step sales forecasting in automotive industry based on structural relationship identification,” _International Journal of Production Economics_ , vol. 140, no. 2, pp. 875–887, 2012.
  * [8] J.-N. Wang, J. Du, Y.-L. Chiu, and J. Li, “Dynamic effects of customer experience levels on durable product satisfaction: Price and popularity moderation,” _Electronic Commerce Research and Applications_ , vol. 28, pp. 16–29, 2018.
  * [9] M. Hülsmann, D. Borscheid, C. M. Friedrich, and D. Reith, “General sales forecast models for automobile markets and their analysis.” _Trans. Mach. Learn. Data Min._ , vol. 5, no. 2, pp. 65–86, 2012.
  * [10] W. T. Robinson and C. Fornell, “Sources of market pioneer advantages in consumer goods industries,” _Journal of Marketing Research_ , vol. 22, no. 3, pp. 305–317, 1985.
  * [11] G. Häubl, “A cross-national investigation of the effects of country of origin and brand name on the evaluation of a new car,” _International Marketing Review_ , vol. 13, no. 5, pp. 76–97, 1996.
  * [12] G. L. Urban, J. R. Hauser, and J. H. Roberts, “Prelaunch forecasting of new automobiles,” _Management Science_ , vol. 36, no. 4, pp. 401–421, 1990.
  * [13] J. L. Assuncao and R. J. Meyer, “The rational effect of price promotions on sales and consumption,” _Management Science_ , vol. 39, no. 5, pp. 517–535, 1993.
  * [14] M. R. Busse, D. I. Simester, and F. Zettelmeyer, ““the best price you’ll ever get”: The 2005 employee discount pricing promotions in the us automobile industry,” _Marketing science_ , vol. 29, no. 2, pp. 268–290, 2010.
  * [15] N. Hu, N. S. Koh, and S. K. Reddy, “Ratings lead you to the product, reviews help you clinch it? the mediating role of online review sentiments on product sales,” _Decision support systems_ , vol. 57, pp. 42–53, 2014.
  * [16] A. Zahoor, Y. Yu, H. Zhang, B. Nihed, S. Afrane, S. Peng, A. Sápi, C. J. Lin, and G. Mao, “Can the new energy vehicles (nevs) and power battery industry help china to meet the carbon neutrality goal before 2060?” _Journal of Environmental Management_ , vol. 336, p. 117663, 2023.
  * [17] Z. Shilong _et al._ , “Machine learning model for sales forecasting by using xgboost,” in _2021 IEEE International Conference on Consumer Electronics and Computer Engineering_. IEEE, 2021, pp. 480–483.
  * [18] W. E. Marcílio and D. M. Eler, “From explanations to feature selection: assessing shap values as feature selection mechanism,” in _2020 33rd SIBGRAPI conference on Graphics, Patterns and Images_. Ieee, 2020, pp. 340–347.
  * [19] M. L. Baptista, K. Goebel, and E. M. Henriques, “Relation between prognostics predictor evaluation metrics and local interpretability shap values,” _Artificial Intelligence_ , vol. 306, p. 103667, 2022.
  * [20] Y. Chen and J. Xie, “Online consumer review: Word-of-mouth as a new element of marketing communication mix,” _Management science_ , vol. 54, no. 3, pp. 477–491, 2008.
  * [21] X. Li, C. Wu, and F. Mai, “The effect of online reviews on product sales: A joint sentiment-topic analysis,” _Information and Management_ , vol. 56, no. 2, pp. 172–184, 2019.