# Elastic and Fracture Behavior of Three-Dimensional Ply-to-Ply Angle
Interlock Woven Composites: Through-Thickness, Size Effect, and Multiaxial
Tests

Weixin Li  Yao Qiao  Joel Fenner  Kyle Warren  Marco Salviato
<EMAIL_ADDRESS>Zdeněk P. Bažant  Gianluca Cusatis  Department of
Civil and Environmental Engineering, Northwestern University, Evanston, IL
60208, USA  Department of Aeronautics and Astronautics, University of
Washington, Seattle, WA 98195, USA  Albany Engineered Composites, Rochester,
NH 03867, USA

###### Abstract

This work presents a comprehensive investigation of the elastic and fracture
behavior of ply-to-ply angle interlock three-dimensional woven composites. The
research investigated novel splitting and wedge-driven out-of-plane fracture
tests to shed light on the tensile fracture behavior in the thickness
direction and to provide estimates of the out-of-plane tensile strength and
fracture energy. In addition, size effect tests on geometrically-scaled Single
Edge Notch Tension (SENT) specimens were performed to fully characterize the
intra-laminar fracture energy of the material and to study the scaling of
structural strength in this type of three-dimensional composites. The results
confirmed that size effect in the structural strength of these materials is
significant. In fact, even if the range of sizes investigated was broader than
in any previous size effect study on traditional laminated composites and two-
dimensional textile composites, all the experimental data fell in the
transition zone between quasi-ductile and brittle behavior. This implies
strong damage tolerance of the investigated three-dimensional composites. The
analysis of the data via Bažant’s Type II Size Effect Law (SEL) enabled the
objective characterization of the intra-laminar fracture energy of three-
dimensional composites for the first time. Finally, Arcan rig tests combined
with X-ray micro-computed tomography allowed unprecedented insights on the
different damage mechanisms under multi-axial nominal loading conditions,
particularly tension-dominated and shear-dominated conditions.

††journal: Composites Part C: Open Access

##  1 Introduction

Thanks to their outstanding mechanical performance, textile composites were
developed for various industrial applications including land, marine and air
transportation, wind and tidal energy production, and blast protection of
civil infrastructures and vehicles [1, 2]. Among them, three-dimensional (3D)
woven composites provide extra reinforcement in the out-of-plane direction and
enhanced fracture toughness and damage resistance compared to two-dimensional
(2D) textile composites or conventional laminated composite systems [3, 4, 5,
6]. The mechanical characterization of composite materials is essential not
only to provide a database of their mechanical properties necessary for the
design of structural components but also to advance the understanding of their
deformation and failure mechanisms for the design of next generation material
systems. However, comprehensive laboratory characterization of three
dimensional (3D) woven composites is not straightforward and requires (1)
measuring both in-plane and out-of-plane macroscopic elastic and strength
properties considering material inhomogeneity associated with large periodic
unit cells and complex microstructure; (2) recognizing and quantifying various
damage mechanisms and failure events under different loading conditions; (3)
characterizing the fracture properties and capturing the scaling of their
mechanical response.

The in-plane properties have been usually characterized by preforming uniaxial
tension and compression tests, which can easily provide measures of Young’s
modulus, Poisson’s ratio, and strength in both the longitudinal (warp) and
transverse (weft) directions [7, 8, 9, 4, 10, 11, 12, 13, 14]. Although simple
and straightforward, the means of strain measurement needs to be considered
carefully to assure an adequate representation of the material with large unit
cell size. The in-plane elastic and strength properties of 3D woven composites
have been widely studied and were found to be strongly related to fabric
architecture [15, 16, 17, 18] and geometrical defects such as fiber crimp or
tow waviness [9, 15, 19, 18, 12]. A variety of damage mechanisms and failure
modes have been documented by means of uniaxial tension and compression tests.
For on-axis tension loading, transverse crack in fiber bundle, matrix
cracking, tow/matrix debonding, and tow straightening were usually observed
prior to failure [20, 15, 21, 22, 23, 12] by means of techniques such as
acoustic emission, optical microscopy, and computed tomography. These damage
events contribute to the nonlinear stress-strain response frequently recorded
in tension tests. Tow rupture and pullout were identified as the primary
tensile failure modes. For on-axis compression loading, micro-buckling of
fibers and growth of kink bands were found to be the key initiating event that
ultimately led to failure [7, 8, 24, 25, 19, 26, 12]. Both fiber kinking at
the microscopic level and kinking of fiber bundles at the miniscopic level
were identified, and are strongly related to fiber misalignment and yarn
imperfection (waviness) [25, 19, 26]. Off-axis tension/compression test with a
bias loading direction ($\pm$$$) with respect to the material principal
direction provides information on in-plane (intra-laminar) shear properties,
and the dominant failure mode was recognized as matrix cracking [27, 28, 29].
In-plane shear behaviors were also frequently characterized by various direct
shear methods such as Iosipescu [30] and rail shear [12] tests. However, the
applicability of the direct shear methods was questioned [31] because of the
large periodic unit cells.

Characterization of the out-of-plane properties of 3D textile composites
remains a challenging problem, although the main purpose of 3D reinforcement
is to increase the mechanical performance in the out-of-plane direction. Very
limited literature data for out-of-plane tension is available. Conventional
uniaxial tension test on dog bone or waisted specimens was adopted for thick
2D composites [32, 33]. This method is not suitable for 3D composites because
the contribution of 3D reinforcement usually vanishes at the surfaces of 3D
composite plates, which poses a challenge for load transmission usually
enabled by an adhesive bond [27]. Various designs of ring or curved beam
specimens have been proposed to address the difficulties arising from the
inter-laminar tensile strength measurement of woven laminates [34, 35]. The
L-shaped beam specimen design has been used to determine the out-of-plane
tensile properties of textile composites [36, 37]. Unfortunately, the curved
specimen design requires special treatment at the manufacturing stage and
cannot apply to the characterization of conventional planar composite panels.
A novel test with a cross specimen loaded in compression using U-shaped steel
rings was proposed to generate failure of the out-of-plane reinforcement in
tension, and achieved promising results [27]. However, the non-uniformity of
the tensile stress field generated by tests based the curved specimen and
cross specimen designs needs to be examined carefully. Compression experiment
in the out-of-plane direction is conceptually simple, and has been widely used
[38, 39, 27]. One needs to pay attention to the design of specimen geometry to
assure a sufficient aspect ratio and the specimen alignment during test setup
to avoid invalid test results and failure patterns. For out-of-plane shear
properties, short-beam (3-point bending) was commonly used. It was suggested
that the short-beam test cannot provide a valid measure of the out-of-plane
(or inter-laminar) shear strength of 3D textile composites because the test
would not result in shear failure [40, 8, 41], although it was shown to be a
reliable method to characterize the out-of-plane shear modulus with the
assistance of a full-field measurement [42]. Punch test based on a cutting
mechanism with sharp corners [43, 19], notched shear test based on
asymmetrical loading [27], Iosipescu shear test [44, 37], and V-notched rail
shear test [45] were also adopted to determine the shear properties in the
out-of-plane direction. Their applicability needs further examination in
various aspects, especially in the representativeness of the generated stress
states.

Fracture tests have been widely accepted as a means to assess the
fracture/damage resistance, damage tolerance, and ductility of composite
materials. Both out-of-plane (inter-laminar) [46, 47, 48, 49, 50, 51, 3] and
in-plane (intra-laminar) fracture properties [52, 53, 54, 55] have been
characterized in laboratory for various woven composite systems, and are
useful for the design of structural components. It has been long recognized
that the delamination fracture behavior of laminated and woven composites is
nonlinear or of resistance type and can be described by equivalent linear
fracture mechanics or quasi-brittle mechanics. Double cantilever beam (DCB)
and wedge-driven delamination tests were frequently used to measure their
fracture properties. On the other hand, composite in-plane fracture behaviors
were frequently described by Linear Elastic Fracture Mechanics (LEFM), which,
however, neglects the presence of a non-linear Fracture Process Zone (FPZ)
with finite size. This is not accurate because the size of the FPZ occurring
in the presence of a traction-free crack or notch is large compared to the
typical size of laboratory specimens for carbon-polymer composites due to the
complex mesostructure [56, 57]. Various damage mechanisms such as crack
bridging, fiber-matrix debonding, and fiber pull-out occur and dissipate a
large amount of energy within the FPZ, resulting in a nonuniform stress field
which decreases with increasing crack opening. The size of the FPZ is even
larger for textile composites than other quasibrittle materials (such as
concrete, rock, ceramics, just to name a few) considering the coarse internal
structure, the considerable unit cell size, and the complicated fiber-fiber
and fiber-matrix interactions. Failing to capture the effects of a finite FPZ,
one may obtain apparent measures of material fracture properties which are
size dependent [58]. Although the complex fracturing process and the
significant process zone size have been highlighted and investigated since the
early 90s [59, 8], the application of nonlinear or quasi-brittle fracture
mechanics in experimental characterization of carbon-polymer composites has
fallen behind, presumably due to the difficulties of attaining a stable
fracture test without exhibiting snap-back instability [60]. A possible remedy
is size effect testing based on a energetic size effect law, which not only
has been applied to identify the fracture properties of various quasi-brittle
materials [61, 62, 63, 64] including carbon-polymer composites [65, 66, 67,
68], but also provide a new insight into the scaling characteristics of their
mechanical properties.

Another important, yet overlooked, aspect of three-dimensional woven
composites is their behavior under multi-axial loading conditions. Multi-axial
properties of composites were typically characterized by using tabular or
cruciform specimens in the literature [69, 70, 71, 72, 73, 74, 75, 76]. The
former one can provide various combinations of local multi-axial stresses
(e.g. tension/shear, compression/shear, etc.) depending on the global loading
condition whereas the latter one is generally used to investigate the
tension/tension behavior of the materials. However, the drawbacks for the
tabular specimens can be the difficulties in manufacturing specimens with this
geometry and the possible buckling failure in the tests. The fabrication of
the cruciform specimens is relatively easier compared to the aforementioned
specimens but the experiments require an advanced testing system and the range
of the multiaxiality ratio is also not sufficiently wide. Alternatively, to
overcome the foregoing disadvantages, the Arcan rig test with different
modifications was used to provide a variety of multi-axial stress states in
the materials with less efforts [77, 78, 79, 80, 81, 82, 83, 84, 85, 86]. By
leveraging these methods, the complete envelope of failure stresses and
related damage mechanisms for the composites under multi-axial loading
condition can be successfully achieved. On similar grounds, further
experimental characterization on the fatigue multi-axial behavior of
composites (e.g. fatigue lifetime, endurance limit, stiffness degradation,
fatigue damage progression, etc.) can also be obtained [87, 88, 89, 78, 90,
91, 92, 93, 94, 95]. These multi-axial properties are significantly important
for the correct calibration of either continuum or discrete computational
models used for the real and complex composite structures under various
loading conditions.

As a first step towards filling the several knowledge gaps discussed
previously, this work presents a comprehensive investigation of the elastic
and fracture behavior of ply-to-ply angle interlock three-dimensional woven
composites. A summary of all the tests performed is given in Table 1 while a
detailed description of all the tests and their results is provided next.

Table 1: Summary of the in-plane and out-of-plane tests performed in this
work.

Type | Specimen geometry | Dimensions [mm] | Direction | Measured properties
---|---|---|---|---
In-plane on-axis tension | Coupon | 250$\times$18.5$\times$7.2 |  Warp |  $E_{1}$, $\nu_{12}$, $F_{1t}$
| 250$\times$35.6$\times$7.2 |  Weft |  $E_{2}$, $\nu_{12}$, $F_{2t}$
In-plane on-axis compression | Coupon | 200$\times$18.5$\times$7.2 |  Warp |  $E_{1}$, $\nu_{12}$, $F_{1c}$
| 200$\times$35.6$\times$7.2 |  Weft |  $E_{2}$, $\nu_{12}$, $F_{2c}$
In-plane off-axis tension/compression | Coupon | 220$\times$30.5$\times$7.2 |  $+$$$ | $G_{12}$, $F_{12}$
| 220$\times$30.5$\times$7.2 |  $-$$$
Out-of-plane compression | Prism | 28.8$\times$17.5$\times$17.5 |  Out-of-plane |  $E_{3}$, $\nu_{31}$, $\nu_{32}$, $F_{3c}$
Tensile splitting test | Cube | 28.8$\times$28.8$\times$28.8 |  Out-of-plane | $F_{3t}$
In-plane fracture and size effect test | SENT specimen | 300$\times$55$\times$7.2 |  Warp | $G_{f}$, $c_{f}$
| 246$\times$40.2$\times$7.2 |
| 200$\times$27.5$\times$7.2 |
Wedge-driven out-of-plane fracture test | V-notch prism | 100$\times$28.8$\times$24 |  Out-of-plane | $G_{F3}$
In-plane multi-axial fracture tests | Center-notched specimen | 250$\times$20$\times$7.2 |  Warp | $\sigma^{multi}_{Nc}$, $\tau^{multi}_{Nc}$
| 250$\times$40$\times$7.2 |  Weft

##  2 Materials

The material investigated in this work is a three-dimensional woven composite
with a ply to ply angle interlock fiber architecture. The preforms were woven
with 48K Hexcel IM7 carbon fiber, and Cytec PR520 toughened epoxy resin was
injected using a resin transfer molding (RTM) process. A geometric
representation of the three-dimensional woven architecture is illustrated in
Fig. 1. The particular architecture has 6$\times$6 columns of warp and weft
tows in each unit cell. Column spacing was adjusted to achieve a target warp
content of 60%. Warp tows also work as binders to provide through-thickness
reinforcement. The overall fiber volume fraction is 0.58. Dimensional
measurement were performed on the material surface and the repeated unit cell
is approximately 23 mm in the warp direction and 14.5 mm in the weft
direction, as illustrated in Fig. 2.

Figure 1: Geometric representation of a ply to ply angle interlock
architecture of a 3D woven composite.

Planar panels with various thicknesses were prepared by Albany Engineered
Composites (AEC). The thinnest composite panels have 6 warp layers and the
nominal thickness is 7.2 mm, whereas the thickest panels of 28.8 mm in
thickness have 24 warp layers. The core warp layers are repeated through the
thickness, and the in-plane unit cell dimensions remain the same throughout
the configurations. The pictures of a sample lateral surface by sectioning the
thin panels showed wavy warp and weft tows, as shown in Fig. 2. It is
difficult to quantify the tow waviness or the amount of crimp from the cross
section because they are distinctive at different section planes. However,
since waving over the thickness, warp tows span 23 mm horizontally whereas
weft tows 14.5 mm, the overall waviness of weft tows is expected to be greater
than that of warp tows.

Figure 2: Images of a 3D woven composite sample and representation of unit
cell.

Specimens were cut from the panels into designed geometries and nominal
dimensions (Table 1) by using a sliding table tile saw with a water-cooled
diamond blade. Surface layers damaged by cutting were removed by grinding and
were further polished by fine grit sandpapers to ensure the high specimen
quality. The actual dimensions of the prepared specimens were measured before
each test was carried on.

Table 2: Summary of measured elastic and strength properties Description | Symbol [units] | Measured value | Sections
---|---|---|---
In-plane Young’s modulus in direciton 1 |  $E_{1}$ [GPa] | 71.4 | 3.1
In-plane Young’s modulus in direction 2 |  $E_{2}$ [GPa] | 37.8 | 3.1
In-plane Poisson’s ratio |  $\nu_{12}$ [-] | 0.218 | 3.1
In-plane shear modulus |  $G_{12}$ [GPa] | 5.6 | 3.2
Out-of-plane Young’s modulus |  $E_{3}$ [GPa] | 8.4 | 3.3
Out-of-plane Poisson’s ratio in plane 3-1 |  $\nu_{31}$ [-] | 0.056 | 3.3
Out-of-plane Poisson’s ratio in plane 3-2 |  $\nu_{32}$ [-] | 0.396 | 3.3
In-plane tensile strength in direction 1 |  $F_{1t}$ [MPa] | 893 | 3.1
In-plane tensile strength in direction 2 |  $F_{2t}$ [MPa] | 271 | 3.1
In-plane compressive strength in direction 1 |  $F_{1c}$ [MPa] | 327 | 3.1
In-plane compressive strength in direction 2 |  $F_{2c}$ [MPa] | 196 | 3.1
In-plane shear strength |  $F_{12}$ [MPa] | 80 | 3.2
Out-of-plane tensile strength |  $F_{3t}$ [MPa] | $>12$ | 3.4
Out-of-plane compressive strength |  $F_{3c}$ [MPa] | 648 | 3.3

##  3 Characterization of elasticity and strength

This section discusses the tests for characterizing the elastic behaviors,
strengths, and pre-peak responses of the investigated 3D woven composites. The
on-axis and off-axis uniaxial tests were used to measure the composite in-
plane behaviors, and the out-of-plane compression tests the elastic properties
and compressive strength in the third direction through the panel thickness.
As a first try, we proposed the tensile splitting test to estimate the out-of-
plane tensile strength of the material.

For all the tests, the geometry of the specimens have to be determined in a
way that the specimens are large enough compared to the unit cell size such
that the measurements are representative while not exceeding the capacity of
the loading systems and load cells. The specimen geometry specification for
the conducted tests was reported in Table 1. The large unit cell size also
poses a challenge for the measurement of strain since commercially available
strain gauges were normally designed for materials with relatively small unit
cells such as laminate composites and hence fail to provide a sufficient gauge
length. To avoid this difficulty, a Digital Image Correlation (DIC) system
(Correlated Solutions [96]) was adopted to provide an in-situ full-field
strain measurement on the specimen surfaces. This also enables one to
correlate the non-linearity observed in the loading curves with the abnormal
strain distribution detectable on the specimen surfaces.

The in-plane tension and compression tests were performed on a closed-loop
servohydraulic Instron machine with a 222.4 kN (50 kip) load cell and
hydraulic wedge grips, while the out-of-plane tests on a MTS machine with
978.6 kN (220 kip) loading capacity and proper fixtures as discussed latter.
All of the tests were operated in a stroke-control mode, and the loading rate
was adjusted such that the corresponding strain rate is within the quasi-
static range.

###  3.1 On-axis uniaxal test

Rectangular coupons cut along the warp direction were labeled $None$, while
the ones along the weft direction $None$. The loading direction was aligned
with long axis, and hence $None$ specimens were warp loaded and $None$
specimens weft loaded. Typical mechanical responses of the warp loaded
($None$) and weft loaded ($None$) specimens in the on-axis uniaxial tension
tests are shown in Fig. 3. Figure 3a plotted axial stress versus axial strain
curves measured in the experiment, whereas Fig. 3b plotted the typical
relation between the transverse and the axial strains in the elastic region.
The axial stress was approximated by a nominal stress calculated as the load
applied on the specimens divided by the initial area of the specimen cross
section. The axial and the transverse strains, on the other hand, were
obtained by averaging the corresponding strain fields over the selected area
of interest (AOI) centered at the middle of the specimen images. The AOI was
selected properly such that it spans more than one repeated unit cell of the
material, and the results with the AOI centered at various positions of
different sizes were compared to ensure that the calculated average strains
did not depend on the selection of AOI.

The red solid lines and the blue dashed lines shown in Fig. 3a represent the
obtained stress and strain relations of the specimens loaded in the warp and
weft directions, respectively. Each of the curve can be described by a bi-
linear response, which is especially evident for the weft loaded specimens.
The first linear segment is related to the elastic behavior, and the axial
modulus was calculated as the slope of the segment. The results of the axial
modulus with their mean values and Coefficient of Variance (COV) are listed in
Table 3. The axial modulus for the specimens loaded in the warp direction is
larger than the one in the weft direction, which reflects the difference of
fiber content evaluated in these two directions. The Poisson’s ratio can be
also estimated from Fig. 3b as the slope of transverse strain versus axial
strain curve. The scattered data were fitted by a linear function through the
least square method to reduce the measurement error and uncertainty arising
from the heterogeneous deformation in the investigated material. Initiation of
the nonlinear response, or knee point in the stress-strain curves, was
believed to be related to transverse crack in weft fiber bundle [21] or matrix
cracking [97]. Similar to the previous work by Warren et al.[97], one observed
high surface strain concentrations after the knee point, as revealed by Fig.
4. Figure 4a, b, and c show the typical surface strain fields of the warp
loaded specimen in the direction of loading prior to the knee point, after the
knee point, and prior to failure, respectively. Similarly, Fig. 4d, e, and f
show the results of the weft loaded specimens. The strain concentrations were
prevalent after the knee points for both specimens, and it can be seen from
Fig. 4c and f that the strain concentrations coincide with the locations where
surface cracks occurred. As discussed in the following section, the surface
cracks in the tension tests resulted from matrix cracking. Therefore, one may
conclude that the presence of the knee points is strongly related to matrix
cracking. It is also worth noting that the strain concentrations are notably
evident for the weft loaded specimens, suggesting significant modulus
reduction due to matrix cracking as shown in the corresponding stress-strain
curves. The nonlinear segment of the stress-strain curves was believed to be
related to tow straightening, which imparts strain on the surrounding matrix.
As a consequence, since the higher waviness of weft tows allows more tow
straightening, it may contribute to the greater strain to failure for the weft
loaded specimens as one may note from Fig. 3a. However, the tensile strength
when evaluated in the weft direction is significantly lower compared to the
strength in the warp direction, as shown in Fig. 3a and Table 3. This could be
the result of lower fiber content in the weft direction than the one in the
warp direction.

Figure 3: Mechanical responses of on-axis loaded specimens in tension tests: (a) axial stress versus axial strain curves; (b) typical relation between transverse and axial strains. Figure 4: Typical surface strain fields in the direction of tension loading on (a)-(c) warp loaded specimen and (d)-(f) weft loaded specimen in which (a) and (d) represent a state in the elastic region, (b) and (c) represents a state right beyond the elastic region, and (c) and (f) represent a state prior to failure.  Table 3: Results of tension and compression tests on on-axis loaded specimens |  |  Axial modulus (COV) [GPa] |  Poisson’s ratio (COV) [-] |  Strength (COV) [MPa]
---|---|---|---|---
Tension | warp |  76.8 (2.06%) |  0.226 (7.45%) |  893 (3.02%)
weft |  35.2 (1.40%) |  0.105 (2.04%) |  271 (3.03%)
Compression | warp |  71.4 (3.55%) |  0.210 (3.92%) |  327 (2.43%)
weft |  40.3 (0.62%) |  0.106 (3.96%) |  196 (2.72%)

The typical failure mechanisms in the on-axis tension tests can be recognized
from Fig. 5, and coincide with a previous study by Warren et al.[12]. Tow
rupture and withdrawal from surrounding tow were the main failure mechanisms.
Matrix cracking adjacent to tows can be also identified on the surface of the
specimens, and is responsible for surface cracks one observed in Fig. 4c and
f. The failure planes were mostly located between the tow/matrix interfaces.
When evaluated in the weft direction, as shown in Fig. 5b, the failure path
was found to be inclined with respect to the weft tows, and the pattern is
repeatable among all investigated specimens. On the other hand, evaluation of
the specimens loaded in the warp direction shows both inclined failure path as
demonstrated in Fig. 5a, and the path normal to the loading direction similar
to the one shown in Ref. [97]. In addition, extensive debonding between the
matrix and the surrounding tows was also observed on the lateral surfaces of
the specimens loaded in the both warp and weft directions. The brooming type
failure in the lateral view is the result of reinforcement failure in the
through-thickness direction following tow rupture.

Figure 5: Typical failed specimens in on-axis tension test loaded in the (a)
warp direction (b) weft direction.

The experimental results of the compression tests are shown in Fig. 6a for the
axial stress versus axial strain curves and Fig. 6b for the typical relation
between the transverse and the axial strains in the elastic region. Similar to
the results of the tension tests, the axial moduli in compression and the
Poisson’s ratios of both warp and weft loaded specimens can be estimated from
the initially linear portion of the stress-strain curves shown in Fig. 6a and
fitted curves in Fig. 6b, respectively. They are also reported in Table 3. The
stress-strain curves were almost linear until failure for the specimens
compressed in the warp direction, whereas a small portion of pre-peak
nonlinearity can be observed for the weft loaded specimens. Similar to the
tension results, the compressive strength evaluated in the warp direction is
higher than the strength in the weft direction as reported in Table 3, which,
once again, could be attributed to the difference of fiber content and tow
waviness in these two directions.

Figure 6: Mechanical responses of on-axis loaded specimens in compression
tests: (a) axial stress versus axial strain curves; (b) typical relation
between transverse and axial strains.

As one can see from Table 3, the elastic properties measured from the
compression test are consistent with the measurements from the tension tests.
Accordingly, the axial moduli evaluated in the tension and compression tests
were averaged, and reported in Table 2 for $E_{1}$ representing in-plane
modulus in the warp direction whereas $E_{2}$ in the weft direction.
Similarly, the in-plane Poisson’s ratio, $\nu_{12}$ was calculated from the
results of the warp-loaded specimens, and reported in Table 2. One may also
calculate $\nu_{21}$ from the weft-loaded specimens, however, it is related to
$\nu_{12}$ through the relation $\nu_{12}/E_{1}=\nu_{21}/E_{2}$, which can be
examined by using the results reported in Table 3.

The dominant failure modes in the on-axis compression tests are tow
microbuckling, formation of kink band, and matrix/tow debonding, as shown in
Fig. 7. Matrix cracking can be also observed in the front view of the failed
specimens. No brooming type failure was observed for the investigated
material, which was commonly observed in 3D woven with orthogonal fiber
architecture, 2D textile, and unidirectional composites. This could be
attributed to the high tow waviness of the preform and the absence of straight
tows (stuffer) for the investigated material. As a consequence, microbuckling
rather than macroscopic compressive failure of tows was observed, and a low
angled failure path with respect to the loading direction can be found, which
also leads to the relatively low compressive strength and the high ratio of
tensile to compressive strengths of the material characterized in this work.

Figure 7: Typical failed specimens in on-axis compression test loaded in the
(a) warp direction (b) weft direction.

###  3.2 Off-axis uniaxial test

Off-axis Specimens were cut and loaded in directions $+$$$ and $-$$$ with
respect to the principal material axis (warp and weft direction) in order to
measure in-plane shear properties. Axial stress versus axial strain curves of
the $\pm$$$ off-axis specimens under tension and compression are shown in Fig.
8a and b, respectively. The axial stress and strain were calculated in the
same way as the ones for the case of on-axis loading. The stress-strain
responses are consistent for all loading configurations, and are generally
nonlinear. Three regions can be identified from the stress-strain curves,
corresponding to three different stages of loading. Figures 9a-c show the
strain fields in the loading direction at the three loading stages in a
typical off-axis tension test. Fig. 9a represents the strain field in the
elastic loading stage, and one can observe the inhomogeneous deformation as
revealed by the presence of strain concentrations of which the location was
randomly distributed. The location of the strain concentrations started
following the surface weave pattern of the material when the inelastic region
was attained, as shown in Fig. 9b. Strain localization with a band of
localized strain inclined to the loading direction was observed in Fig. 9c as
the post-peak region of the stress-strain curve was reached.

Figure 8: Axial stress versus axial strain responses for the specimens loaded
in the $\pm$$$ off-axis direction under (a) tension and (b) compression.
Figure 9:  Typical strain fields in the direction of tension loading on a
$+$$$ off-axis loaded specimen, which represents a state (a) in the elastic
region, (b) right beyond the elastic region, and (c) prior to failure. Table
4: Results of tension and compression tests on $\pm$$$ specimens

| Tension | Compression
---|---|---
| $+$$$ | $-$$$ | $+$$$ | $-$$$
Axial modulus (COV) [GPa] | 19.1 (3.08%) | 18.5 (3.80%) | 18.1 (1.82%) | 17.8 (0.95%)
Poisson’s ratio (COV) [-] | 0.645 (5.18%) | 0.642 (0.95%) | 0.671 (0.47%) | 0.662 (3.52%)
Strength (COV) [MPa] | 160 (0.70%) | 159 (0.21%) | 161 (0.65%) | 161 (1.50%)

The axial modulus and the Poisson’s ratio can be determined from the initial
linear segments of the measured responses for each case, and are reported in
Table 4. The obtained elastic properties are generally consistent with only
small deviation. Great consistency of the pre-peak nonlinearity and the peak
load can be also found for the investigated $+$$$ and $-$$$ off-axis specimens
under both tension and compression, suggesting the same damage mechanism and
deformation mode. The mechanical responses obtained from the tests with off-
axis loading are related to the in-plane shear behaviors of the material.

Given the axial modulus at an angle of $\theta=+$$$ or $-$$$ to the warp (or
weft) direction of the material, denoted by $(E_{y})_{\pm$$}$ in which the
subscript $y$ represents the vertical loading direction in the tests, and the
corresponding Poisson’s ratio, denoted by $(\nu_{xy})_{\pm$$}$, one can
estimate the in-plane shear modulus, $G_{12}$ of the material according to the
theory of elasticity for orthogonal materials [98], which read

$G_{12}=\frac{(E_{y})_{\pm$$}}{2[1+(\nu_{xy})_{\pm$$}]}$ (1)

Substituting the values of the axial modulus $(E_{y})_{\pm$$}$ and the
Poisson’s ratios $(\nu_{xy})_{\pm$$}$ reported in Table 4, one can obtain the
measure of $G_{12}$ with a mean of 5.6 GPa and a COV of 4.35 %. The mean value
of $G_{12}$ is also reported in Table 2.

The in-plane shear strength of the material, $F_{12}$ could be also related to
the off-axis tensile strengths, denoted by $(F_{yt})_{\pm$$}$, and the
compressive strength, denoted by $(F_{yc})_{\pm$$}$, as listed in Table 4. If
one adopts the maximum stress theory as the failure criterion and realizes
that the material failed in shear in the actual tests, the shear stress along
the material axes could be considered to attain its maximum value (i.e.
$F_{12}$) when the axial stress reaches the peak (i.e. $(F_{yt})_{\pm$$}$ or
$(F_{yc})_{\pm$$}$). Note that the shear stress along the material axes
$\tau_{12}$ is related to the axial stress $\sigma_{y}$ through the relation
$\tau_{12}=-\sigma_{y}\sin\theta\cos\theta$. As a consequence, $F_{12}$ can be
approximated as $F_{12}\approx(F_{yt})_{\pm$$}/2$ or
$F_{12}\approx(F_{yc})_{\pm$$}/2$ given $\theta=$$$. Aware of the consistency
of the off-axis tensile and compressive strengths, one can average
$(F_{yt})_{\pm$$}$ and $(F_{yc})_{\pm$$}$, and thus obtain a measure of
$F_{12}$ as 80 MPa. It is also possible to adopt more complex failure criteria
considering the effect of stress interaction and the biaxial state of stress,
such as Tsai-Hill and Tsai-Wu criterion, in the estimation of $F_{12}$. The
accuracy of the estimation, however, is an open question in the absence of
mechanical characterization under biaxial or multiaxial state of stress. It is
worth noting that the measure of $F_{12}$ based on the maximum stress theory
and the off-axis strengths is in the same order of magnitude as the in-plane
shear strength data reported in Ref. [97] for 3D woven composites with similar
architectures and constituents. However, the shear properties were evaluated
by using V-notched rail shear method in Ref. [97], and may not be comparable
with the data presented in this work. In addition, as mentioned in Section 1,
the accuracy of direct shear methods for 3D woven composites with large unit
cells may be also questionable.

Figure 10: Typical $\pm$$$ off-axis specimens failed in (a) tension (b)
compression.

The failure modes of the typical $\pm$$$ off-axis specimens under tension and
compression loads are shown in Fig. 10a and b, respectively. Extensive matrix
cracking can be identified from the front view of the failed specimens,
whereas matrix/tow debonding can be seen from the lateral view in both tension
and compression tests. The failure modes observed in Fig. 7 were captured in
the post-peak region in which matrix cracking already reached a state of crack
saturation, and the behaviors were governed by fiber re-orienting, referred to
as trellising [99]. Because no biased reinforcement was presented for the
investigated material, the trellising behavior can be recognized from the
observation that the angle between the originally orthogonal warp and weft
tows changed from $None$ (less than $None$ in the tension tests and greater
than $None$ in the compression test). The trellising accompanying frictional
behavior between tows consumes a larger amount of energy, and leads to larger
strain measures as shown in Fig. 8.

###  3.3 Out-of-plane compression

The uniaxial compression test was used to characterize the out-of-plane
Young’s modulus $E_{3}$, Poisson’s ratios $\nu_{31}$ and $\nu_{32}$, and
compressive strength $F_{3c}$. Prismatic specimens were cut in the third
direction through the panel thickness of 28.8 mm. The nominal width and depth
of the specimens are 17.5 mm to ensure a aspect ratio (height:width) of about
1.6. The prepared specimens were placed between two steel loading platens with
Teflon sheets inserted to reduce the friction of the contact surfaces. Loading
was along the out-of-plane direction, and two sets of tests were conducted
with the images of the specimens captured on plane 1-3 and plane 2-3
respectively to enable DIC analysis on these two planar surfaces for strain
measurements.

Fig. 11a shows the stress-strain responses of the specimens under out-of-plane
compression, and Fig. 11b plots the typical transverse strain response versus
axial strain measurements in the elastic region. The stress and strain
measurements were calculated by using the same method adopted in the in-plane
tests as described above. The stress-strain curves are mostly linear until
failure, and $E_{3}$ and $F_{3c}$ were calculated as the slope and peak of the
curves, respectively. Similarly, Poisson’s ratios can be calculated from Fig.
11b. Note that the strain measurements conducted on plane 3-1 and plane 3-2
with two sets of test enable one to independently obtain two constants,
$\nu_{31}$ and $\nu_{32}$. The obtained properties are also summarized in
Table 2. It is also worth noting that the obtained out-of-plane compressive
strength is about three time larger than the in-plane compressive strengths
measured in the warp or weft directions.

Figure 11: Mechanical responses of specimens under out-of-plane uniaxial
compression: (a) axial stress versus axial strain curves; (b) typical relation
between transverse and axial strains.

Fig. 12a shows the typical vertical displacement field of the specimen loaded
in the out-of-plane direction prior to failure. The surface deformation
revealed by the DIC system is generally uniform up to failure. This
observation is consistent with the nearly linear stress-strain response
plotted in Fig. 11a. All specimen failed in a catastrophic way right after the
load reached its peak. The explosive ejection of fragments perpendicular to
the loading direction was observed (see Fig. 12b). A similar failure mode has
been also reported in Ref. [27]. As shown in Fig. 12c, some fragments
exhibited a failure plane across plane 2-3 with a angle of about $None$ to the
loading direction, which coincides with the plane of maximum shear stress
under uniaxial compression. No extensive fiber microbuckling and kinking were
noticed.

Figure 12: (a) Typical vertical displacement field obtain from DIC analysis in
out-of-plane compression test. (b)-(c) Photograph of failed specimens under
out-of-plane compression.

###  3.4 Tensile splitting test

Tensile splitting test provides an indirect method for measurement of tensile
strength of materials and has been widely used in laboratory for mechanical
characterization of quasi-brittle and brittle materials such as rock,
concrete, and ceramic. Given the difficulties of performing direct uniaxial
tension tests on 3D woven composites in the out-of-plane direction associated
with the high inter-laminar tensile strength featured by 3D reinforcement, the
possibility of applying tensile splitting test for 3D woven composites are
explored.

Cuboid specimens with an edge length of 28.8 mm, which is equal to the
thickness of the thick composite panels, were prepared for the tensile
splitting tests. Two steel pins as well as the attached bearing blocks were
used to transmit the load provided by the machine and to apply a line load on
the specimens placed between them. To reduce high stress concentration at the
line of contact, two strips of plywood cushion were placed between the loading
pins and the specimens. The specimens were positioned carefully to ensure that
the line of vertical loading is perpendicular to the out-of-plane direction.
Either warp or weft directions of the specimens could be positioned vertically
(parallel to the loading direction). A continuously increasing compressive
load was provided by the MTS machine until failure occurred.

Figure 13 shows the load-displacement responses of the investigated specimens
in the tensile splitting tests. Four tests were recorded and can be
categorized into two groups: test 1, 2, and 3 were performed on the specimens
loaded in the warp direction, whereas test 4 in the weft direction. All curves
can be described by an initial stage with a gentle slope followed by a segment
with linear growth of load and a small portion of pre-peak nonlinearity. The
gentle slope at the initial stage can be attributed to adjusting contact
between the specimens and the loading pins. The peak load with a mean of 44.4
kN and a COV of 3.84% was reported for the warp loaded specimens, whereas 22.9
kN for the weft loaded one. Peak load in tensile splitting test was believed
to be related to the tensile strength of material, and in this work, out-of-
plane tensile strength $F_{3t}$ of the investigated 3D woven composite through
the formula: $\sigma_{t}=kP_{max}/{Dt}$ in which the parameter $k$ depends on
specimen geometry and loading condition. For anisotropic materials, the
parameter $k$ also depends on the elastic constants. In this work, finite
element analysis was performed to interpret the experimental results as shown
in Appendix A. The parameter $k$ was estimated as 0.213 and 0.356 for
specimens loaded in warp and weft directions, respectively. Accordingly, one
obtains $F_{3t}$ of mean 12.5 MPa and COV 5.14% from test 1, 2, and 3, whereas
$F_{3t}\approx 11.0$ MPa from test 4. The values of $F_{3t}$ obtained from
these two groups of tests are in a good agreement.

Figure 13: Experimental results of tensile splitting tests: (a) load-
displacement responses, (b)-(c) photograph of a failed specimen. Figure 14:
Strain fields obtained from DIC analysis: (a) $\varepsilon_{xx}$, (b)
$\varepsilon_{yy}$, and (c) $\varepsilon_{xy}$ prior to failure, and (d)-(f)
$\varepsilon_{xx}$ strain field prior to, upon, and after peak load.

The photographs of a typical failed specimen is shown in Fig. 13b and c. A
splitting failure mode is usually expected for tensile splitting tests.
However, in addition to the splitting failure developed near the center of the
specimen, one may also notice from Fig. 13 that a wedge was formed from the
upper half of the specimen. The formation of wedges was commonly observed in
tensile splitting test on concrete or rocks, and was believed to result from
compression-shear failure in the vicinity of the load application zone due to
high stress concentrations [100]. Some researchers reported that the formation
of wedges occurred after the development of main cracks driven by the tensile
stress in the center of a specimen [101, 102], whereas some reported that
failure initiation was driven by compression-shear rather than tensile
stresses [103, 104]. Thanks to the DIC technique, one can examine the two
possibilities by keeping track of the occurrence of failure associated the
evolution of the surface strain fields during a test.

Figure 14a, b, and c show the horizontal normal strain ($\varepsilon_{xx}$),
the vertical normal strain ($\varepsilon_{yy}$), and the shear strain
($\varepsilon_{xy}$) obtained from the DIC analysis for test 3 prior to
failure, respectively. It can be seen that significant strain concentrations
for both $\varepsilon_{xx}$, $\varepsilon_{yy}$, and $\varepsilon_{xy}$ were
developed in the vicinity of the top and bottom load application zones. Fig.
14d, e, and f show the snapshots of the $\varepsilon_{xx}$ strain field load
obtained prior to, upon, and after the load reached its peak. As one can see,
surface cracks initiated from the bottom (Fig. 14d) and top (Fig. 14e) load
application zones, and propagated to develop a failure path through the
specimen (Fig. 14f) along the loading direction. This observation is also
echoed by the numerical analysis presented in Appendix A.1 which showed large
stress concentration in the load application zones. For this reason, the
tensile strength calculated based on the measured peak load should be regarded
as a lower bound of $F_{3t}$ of the material. Possible ways to improve the
test setup for achieving a more reliable strength measure is outside the scope
of this work but were offered in Appendix A.2 for the interest of readers.

##  4 Characterization of in-plane fracture properties through size effect
testing

Size effect testing was employed in this work for measuring the in-plane
fracture properties of the investigated 3D woven composite. Recognizing the
occurrence of the non-negligible FPZ in the presence of a large stress-free
crack for quasi-brittle materials, the size effect method was developed based
on the energetic size effect law associated with a given structural geometry
[105]. It is easy to implement given that only peak load measurements are
needed and provides an alternative method of characterizing the nonlinear
fracture properties of quasi-brittle materials which cannot be easily
estimated by measuring the specimen post-peak responses in a fracture test.
The size effect method has been successfully applied to obtain the intra-
laminar fracture properties of laminated and two-dimensional textile
composites [65, 66, 68] and to characterize the inter-laminar fracturing
behaviors [106]. In this work, the in-plane fracture energy and a measure of
the FPZ length of the investigated material were obtained through size effect
testing.

###  4.1 Test description

The size effect tests were performed on Single Edge Notched Tension (SENT)
specimens. The SENT specimen design was adopted because it offers the simplest
specimen geometry compared to other typical fracture tests such as Double Edge
Notched Tension (DENT) and Compact Tension. In addition, single edge notch is
preferred to double edge notch because the scenario that only one of the
cracks can propagate was frequently observed [107] and the double edge notch
was also associated with asymmetric response. As illustrated in Fig. 15,
geometrically similar SENT specimens of four increasing sizes, with a size
ratio of 1:1.5:2:3.5, were tested to investigate the size dependence of the
measured fracture properties. The geometric specifications of the specimens
are reported in Table 5. Note that two-dimensional geometric similarity with
scaled planar dimensions and a constant thickness (7.2 mm) was considered in
this work. The notches were made by means of a hand saw which provided a width
of roughly 1 mm. Although only blunt notches were prepared, trial tests
confirmed that there is no significant difference of results obtained with
blunt and sharp (notch tip radius $\approx 20$\mathrm{\SIUnitSymbolMicro m}$$)
notches, which could be attributed to the fact that the fracturing behavior of
the material can be similar if the notch radius is sufficiently small compared
to the Irwin’s characteristic length and the size of the Repetitive Unit Cell
(RUC) [64, 108, 109] and this length is definitely large for the investigated
three-dimensional woven composites.

Figure 15: Geometry and dimensions of the SENT specimens. Units: mm Table 5: Geometric specifications of the SENT specimens. Size |  Width, $D$ [mm] |  Notch length, $a_{0}$ [mm] |  Gauge length, $L$ [mm] |  Thickness, $t$ [mm]
---|---|---|---|---
Size 1 | 27.5 | 14.6 | 100 | 7.2
Size 2 | 40.2 | 10.7 | 146 | 7.2
Size 3 | 55 | 7.3 | 200 | 7.2
Size 4 | 96.25 | 51.1 | 350 | 7.2

The tests were conducted on an Instron machine, and hydraulic wedge grips were
used for gripping specimens. Note that although the grip length was not
scaled, the size effect data should not be affected because gripping has no
appreciable effect on the stored energy and fracture always occurred away from
the grips [68]. Similar to the in-plane tension and compression tests, the
fracture tests were carried out under a stroke control mode in a quasi-static
setting.

###  4.2 Experimental results

The load-displacement responses of the SENT specimens during the fracture
tests are shown in Fig. 16a. For the large and medium specimens, the response
can be described by a bilinear curve, whereas for the small specimens, the
load-displacement curves are mostly linear with a small portion of pre-peak
nonlinearity. The bilinear or pre-peak nonlinear curves may be an indication
of hardening inelastic behaviors or ductility. Nevertheless, specimens
exhibited snap-back instability for all investigated sizes. Although a few
small and medium sized specimens regained partial load bearing capacity after
reaching the peak load, a sudden drop of load was always observed for all of
the specimens. None of these tests yield a complete load-displacement curve
with a gradual post-peak response, and all of the specimens failed in a
catastrophic (dynamic) way shortly after the peak load. Figure 16b-e shows the
typical failure patterns of the investigated specimens of various sizes.
Extensive tow rupture and pull-out can be noticed. The test results are
summarized in Table 6. The failure paths, although mostly found to be along
the extension of the notches, are zigzag and have the tendency to following
the weaving pattern of the material.

Table 6: Experimental results of fracture tests on SENT specimens.  Size |  Nominal strength, $\sigma_{Nu}$ [MPa] |  Apparent fracture toughness, $K_{IC}$ [MPa$\sqrt{\text{m}}$] |  Apparent fracture energy, $G_{f}$ [N/mm]
---|---|---|---
Size 1 |  435.6 | 72.5 | 76.1 | 140.9 | 157.4
480.5 | 80.6 | 177.5
445.4 | 75.0 | 153.8
Size 2 |  474.5 | 94.0 | 91.7 | 229.9 | 223.7
465.7 | 93.8 | 237.0
434.4 | 87.3 | 204.3
Size 3 |  424.5 | 98.4 | 98.5 | 252.0 | 253.1
397.5 | 91.9 | 218.1
451.1 | 105.1 | 289.3
Size 4 |  325.9 | 101.5 | 107.6 | 270.9 | 305.5
368.5 | 115.2 | 349.2
340.3 | 106.2 | 296.6

The specimen nominal strength, $\sigma_{Nu}$, defined as the average stress at
failure distributed on the unnotched cross section, $\sigma_{Nu}=P_{max}/Dt$,
is reported in Table 6. One may notice that $\sigma_{Nu}$ varies with the
specimens size. Indeed, according to strength-based criteria, the nominal
strength is assumed to be size independent. On the other hand, classic Linear
Elastic Fracture Mechanics (LEFM) predicts a decrease of $\sigma_{N}$
proportional to $D^{-1/2}$, which is also not the case as one can note from
Table 6. As a consequence, one can conclude that neither strength-based
criteria nor LEFM can capture the variation of the nominal strength with the
structural size.

Figure 16: (a) Load-displacement curves of the SENT specimens of various
sizes. Typical failure patterns of SENT specimens of width (b) $D=27.5$ mm (c)
$D=40.2$ mm (d) $D=55$ mm (e) $D=96.25$ mm.

###  4.3 Analysis of in-plane fracture tests by size effect law

For analyzing the fracture test data, LEFM is widely used by assuming an
infinitesimal FPZ compared to the characteristic size of the investigated
specimens. However, as one may notice from Table 6, the apparent fracture
toughness and fracture energy calculated from the measured peak load according
to LEFM are size-dependent. In addition, LEFM predicts that the nominal
strength of a structure should be inversely proportional to the square root of
the structural size, which is not the case as indicated in Section 4.2. These
two observations imply that LEFM is not applicable for the investigated
material. Indeed, researchers have found that typical carbon-epoxy composites
have a remarkable FPZ size [56, 57, 65, 68] indicating that quasi-brittleness
has to be considered.

To gain an insight into the quasi-brittleness of the composite system under
study, the size effect data of the fracture tests were analyzed by means of
type II Size Effect Law (SEL) , which can be derived from an equivalent linear
elastic fracture mechanics approach. The type II size effect occurs when a
large notch or traction-free crack exists at maximum load, and the resulting
SEL bridges the region between strength-based criteria and classic LEFM
theory. The derivation of SEL is summarized in Appendix B. The dimensionless
form can be written as

$\frac{\sigma_{Nu}}{\sigma_{0}}=\frac{1}{\sqrt{1+D/D_{0}}}$ (2)

where $\sigma_{0}=\sqrt{E^{*}G_{f}/(c_{f}g^{\prime}(\alpha_{0}))}$ and
$D_{0}=c_{f}g^{\prime}(\alpha_{0})/g(\alpha_{0})$; $E^{*}$ is the effective
elastic modulus for orthotropic materials; $G_{f}$ and $c_{f}$) represent the
initial fracture energy and effective FPZ length of the material;
$g(\alpha_{0})$ and $g^{\prime}(\alpha_{0})$ denote the dimensionless energy
release rate and its derivative, and $\alpha_{0}=a/D$ the dimensionless
initial crack length. Note that $\sigma_{0}$ has the dimension of stress, and
$D_{0}$ has the dimension of length. It is key that the SEL is endowed with
the characteristic length $D_{0}$ to describe the transition from ductile to
brittle behavior with increasing structure size.

The parameters $\sigma_{0}$ and $D_{0}$ are constants given that the
investigated specimens are geometrically similar, and can be determined by
either linear or nonlinear regression analysis of the experimental data. For
simplicity, a linear regression analysis is adopted in this work, which is
enabled by reformulating Eq. 2 as

$Y=AX+C$ (3)

in which $X=D$, $Y=\sigma_{Nu}^{-2}$, $A={C}/{D_{0}}$, and
$C={\sigma_{0}^{-2}}$. The fitting of the experimental data through the linear
regression analysis based on Eq. 3 was conducted. The results are shown in
Fig. 17a, and provided the estimates of the following parameters: $A=0.0554$
GPa-2mm-1 and $C=2.89$ GPa-2. Correspondingly, $D_{0}$ and $\sigma_{0}$ can be
estimated from the coefficients, and they are $D_{0}=52.14$ mm and
$\sigma_{0}=589$ MPa.

The fracture properties of the material can be estimated from the fitting of
the size effect data. The initial fracture energy $G_{f}$ and the effective
FPZ length $c_{f}$ can be determined from $D_{0}$ and $\sigma_{0}$ supposing
that $g(\alpha_{0})$ and $g^{\prime}(\alpha_{0})$ are given (both can be
calculated numerically as demonstrated in Ref. [68]). They were estimated as
$G_{f}=466$ N/mm and $c_{f}=13.6$ mm.

Figure 17b plots the normalized strength $\sigma_{Nu}/\sigma_{0}$ as a
function of the normalized characteristic size $D/D_{0}$ in double logarithmic
scale, as a result of the fitting of the experimental data by SEL. The red
circles represent the experimental data, and the black solid line represents
SEL as shown in Eq. 2. It can be seen that SEL depicts a smooth transition
from the strength criterion characterized by a horizontal asymptote (plastic
limit) when $D\to 0$, in which no size effect on structural strength is
expected, to LEFM by an inclined asymptote of slope $-1/2$ when $D\to\infty$,
which represents the strongest size effect possible. The intersection of the
two asymptotes corresponds to $D=D_{0}$, referred to as the transitional size.
The experimental data fall within the transition zone of the SEL curve. As a
consequence, one may conclude that the failure of the investigated 3D woven
composites containing traction-free cracks exhibit a significant size effect.

Figure 17:  Analysis of size effect data: (a) linear regression fitting of
experimental data; (b) plot of size effect data for 3D woven composite.

It is of vital importance to realize the size dependence of the measured
mechanical properties in various situations such as material design, component
manufacturing, and safety evaluation, especially, in the presence of a
traction-free crack. In particular, these conclusions ought to be incorporated
into a design process to enable extrapolation from small scale laboratory
tests to real size structures.

###  4.4 Comparison of size effect experimental data for 3D and 2D composites

The size effect data of the 3D woven composite investigated in this work are
also compared to the data of a two-dimensional textile composite. The latter
ones were reported in Ref. [68], and the material consisted of DGEBA based
epoxy resin matrix reinforced by a twill 2 $\times$ 2 fabric made of carbon
fibers. The fiber volume fraction was reported to be $0.54$, which is close to
the one of the 3D woven material investigated in this work. However, their
elastic and strength properties are significantly different, most likely due
to the contrasting fabric architectures. Note that the unit cell size of the
twill 2 $\times$ 2 composite is about 7 mm, which is more than two times
smaller than that of the 3D woven composite. The same as in the current work,
the size effect data were obtained from the fracture tests on geometrically
similar SENT specimens with increasing size, which were made of $[$$]_{8}$
twill 2 $\times$ 2 laminates. The investigated specimen width $D$ ranged from
20 to 80 mm.

The comparison of the size effect data is shown in Fig. 18. Fig. 18a depicts
the normalized strength $\sigma_{Nu}/\sigma_{0}$ as a function of the
normalized characteristic size $D/D_{0}$ in double logarithmic scale with
$\sigma_{0}$ and $D_{0}$ obtained by the SEL fitting of each set of data.
Comparing with the SEL curve, one can observe that the 3D woven data are close
to the plastic limit while the twill 2 $\times$ 2 are near the LEFM asymptote.
Considering that the investigated specimen size ranges are in the same order
of magnitude, it is fair to conclude that the twill 2 $\times$ 2 is more
brittle than the investigated 3D woven composite. In other words, the 3D woven
composite exhibit higher ductility.

Figure 18:  Comparison of size effect data of 3D woven and twill 2$\times$2
composites: (a) plot of normalized nominal strength versus normalized specimen
characteristic size; (b) relation between normalized strength and notch
length.

Thanks to the high ductility, the 3D woven composite is presumably endowed
with higher damage tolerance compared to the 2D textile composite. As
illustrated in Fig. 18b, the nominal strength of the damaged specimen
$\sigma_{Nu}$ was normalized by the strength of the undamaged one, i.e.
tensile strength in the warp direction $F_{1t}$, and was plotted against the
degree of damage characterized by the notch length $a_{0}$. It can be seen
that within a comparable range of $a_{0}$, a more severe reduction of the
normalized strength was observed for the twill 2 $\times$ 2 composite than for
the 3D woven composite. Consequently, one may conclude that the 3D woven
composite is less vulnerable to the presence of a large traction-free crack.
This conclusion agrees with early findings that certain 3D polymer composites
offer reduced notch sensitivity and superior damage tolerance far beyond those
of 2D laminates [110, 8, 41], which provides various benefits including
extended allowable damage region, reduced maintenance cost, and potentially
longer service life.

##  5 Wedge-driven out-of-plane fracture test

The study of delamination is of particular importance for 3D woven composites
which feature enhanced resistance to out-of-plane fracture, and is presented
in this section. Strictly speaking, the term delamination is somewhat
confusing for 3D textile composites since there is not a clear laminated
structure. Nevertheless, the word out-of-plane fracture is used
interchangeably with delamination in this work for the sake of convenience.

Conventional delamination testing using the double cantilever beam specimens
[111] was originally designed to determine the inter-laminar fracture
properties of laminated composites and may be also applicable to 2D fabric
composites. However, the conventional method may not be suitable to determine
the out-of-plane fracture properties of 3D woven composites for the following
reasons. (1) The DCB specimens frequently suffer from bending failure prior to
delamination given the large delamination resistance and the low in-plane
compressive strength of 3D woven composites. To avoid the undesired bending
failure, tabbed DCB specimens were frequently used [112, 113]. Although it
seems effective, the effect of tabs has to be analyzed, and methods of
preventing delamination between tabs and specimens are needed. (2) Special
gripping systems [114, 113] should be used to avoid the failure of adhesive
bonding specimens and loading devices when a conventional piano hinge or end
block loading system is used. In particular, the gripping system has to be
designed in such a way that it can withstand strong gripping force given that
scaled-up specimens have to be used considering the materials’ large RUC. (3)
The applicability of LEFM to describing the mode I inter-laminar fracture
behavior of 3D woven composites has not been examined yet. This would require
a thorough fracture scaling analysis such as the size effect study completed
by [106] for laminated composites. A possible way to overcome the
aforementioned difficulties is a wedge testing method. It was adopted in this
work to investigate the out-of-plane fracture properties of the investigated
3D woven composite.

###  5.1 Test description

The experimental setup of the wedge tests performed in this work consists of a
prismatic specimen with a V-shaped notch, a steel wedge, and a servo-
controlled testing machine. The steel wedge was driven against the stationary
specimen and to drive the propagation of crack initiated from the tip of the
notch. As depicted in Fig. 19a, the out-of-plane direction of the material was
placed perpendicular to the potential crack path emanated from the notch tip,
and the weft direction was parallel to it. The V-shaped notch was cut through
the warp direction, and the notch half-angle is the same as the wedge half-
angle $\alpha$ which measures $7^{\circ}$ in this work. The specimen dimension
is also depicted in Fig. 19a.

Figure 19: (a) Schematic diagram of wedge test setup and geometry of specimen.
(b) Free-body diagram demonstrating forces applied on the wedge.

Efforts were made to minimize the effect of friction resulting from sliding
between the wedge and the notch surface. First of all, the wedge surface was
polished to ensure a smooth contact. In addition, a stack of plastic sheets
with smooth surfaces (made of grade G-10 glass fabric/epoxy) was bonded to the
notch surfaces to reduce the contact area. Friction was further reduced by
applying a dry film graphite lubricant to the contact surfaces. Ultimately,
the coefficient of kinetic friction between the wedge surfaces and the plastic
sheets, $\mu$, measures 0.15.

The tests were performed under the stroke control mode with a constant
crosshead speed of about 1.5 mm/min. The load and displacement data were
recorded during the tests.

###  5.2 Experimental results

The load-displacement curves of three independent tests are plotted in Fig.
20a, which feature a rapid increase of load in the pre-peak region and a slow
reduction of load in the post-peak region. All of the specimens exhibited saw-
tooth post-peak response which is characterized by a stick-slip behavior
corresponding to the observation that the crack was repeatedly arrested until
sufficient energy was accumulated to enable further crack propagation. Such a
stick-slip behavior has been commonly observed in woven composites [115, 47,
51, 114] and is related to the complex microstructure with a large RUC.

Figure 20: (a) Load-displacement curves of the wedge tests. (b) Typical failed
specimen.

A typical failed specimen after the test is shown in Fig. 20b. One can find
that the crack path is wavy, partially following the weaving pattern of the
weft tows. In particular, a strip of damage zone instead of a single
throughgoing crack characterizes the main failure pattern of the specimens. A
large amount of broken fibers bridging the failure surfaces is noticeable.
Furthermore, pullout of warp tows can be also observed. Consequently, one may
highlight that the fracturing process during the tests involved breaking the
interweaving fiber bundle structure in all three dimensions and thus created a
large fracture process zone. This particular failure pattern resembles the one
observed in the in-plane fracture tests discussed in the above section, and
thus differentiates the out-of-plane fracture behavior of 3D woven composites
from the typical delamination failure of laminated composites which is
featured by a straight line crack separating the originally bonded interfaces
of laminates. Hence, one has a clue that the out-of-plane fracture behavior of
3D woven composites should be also of the quasibrittle type, resembling the
in-plane fracture behavior, although it has to be confirmed by further
investigation such as size effect tests.

Thanks to the use of wedge setup, the tests were mechanically stable, and one
can estimate the fracture energy using the work-of-fracture method given a
complete load-displacement curve. Provided that the energy dissipation due to
inelastic deformation other than fracturing is negligible, the fracture energy
$G_{F}$ can be calculated as $G_{F}=W/Lb$ where $W=$ the area under the
complete stable load-displacement curve representing the total amount of the
energy dissipation during the entire test, $L=$ length of the broken ligament,
and $b=$ specimen thickness. Note that although a residual load was observed
at the end of the tests, it is relatively small compared to the peak, and it
is expected that the associated remaining energy has negligible influence on
the calculation of the fracture energy. The out-of-plane fracture energy
calculated in this way has a mean of 42.8 N/mm and a COV of 7%.

###  5.3 Correction of friction effect

The merit of the wedge-type test is that it provides a solution to the
fracture test in the face of strict requirements for post-peak stability
[116], and is free from the complex setup and the difficulties of avoiding the
unwanted failure experienced by a DCB test [117]. Accompanying the wedge test,
the work-of-fracture method provides a straightforward solution to the
estimation of the fracture property using the energy approach without assuming
the applicability of LEFM theory and the need for seeking closed-form
expression of the energy release rate or stress intensity factor.
Nevertheless, the question of friction contributions can become a concern for
a wedge-type test, which is probably why the wedge-type test method has never
been widely used despite its obvious advantages [118, 119, 117]. It is
necessary to quantify the effect of friction.

Given the measured friction coefficient $\mu$ and the wedge half-angle
$\alpha$, one can subtract the contribution of energy dissipation due to
frictional sliding. According to the static equilibrium condition for the
wedge (see Fig. 19b for the free-body diagram), the driving force $F$ is
balanced by the normal force $F_{N}$ applied on the wedge surfaces and the
frictional force $F_{f}$, i.e., $F=2(F_{N}\sin\alpha+F_{f}\cos\alpha)$. By
assuming that the kinetic friction condition was met throughout the tests, one
obtains $F_{f}=\mu F_{N}$. Accordingly, the frictional force can be estimated
as $F_{f}=\mu F/2(\sin\alpha+\mu\cos\alpha)$. In addition, neglecting the
deformation of the steel wedge and the machine, one can also approximate the
amount of sliding as $u_{s}=u\cos\alpha$, where $u$ is the recorded crosshead
displacement. Accordingly, the energy dissipated by friction can be written as
$W_{f}=F_{f}u_{s}=k_{f}W$, where
$k_{f}=\mu\cos\alpha/2(\sin\alpha+\mu\cos\alpha)$. Given this simple
calculation and neglecting the other possible energy dissipation mechanisms,
one can correct the calculation of fracture energy discussed above by a factor
of $1-k_{f}$. The corrected fracture energy measures 31 N/mm. It can be seen
that the frictional sliding accounts for $27\%$ of total dissipated energy.
Attempts to further reduce the effect of friction will be made in the future
work.

The DCB test method has been adopted by several researchers to determine the
delamination resistance of 3D woven composites [112, 51, 114, 113, 120]. Most
of the work reported the out-of-plane fracture energy as the energy release
rate for steady state crack propagation, which was calculated using either
Modified Beam Theory (MBT) or Modified Compliance Calibration (MCC) method
according to the ASTM standard [111]. The reported values varied greatly, and
depends strongly on the fiber architecture and the constituent materials. In
addition, the validity of these measurements needs further investigation, and
some of the tests failed to attain the steady-state crack propagation.
Nevertheless, the out-of-plane fracture energy obtained in this work is
comparable with some measurements reported in literature. For instance, [113]
reported that the delamination resistance of a 3D carbon fiber woven material
produced by 3TEX could reach 20-30 N/mm; [120] measured the out-of-plane
energy release rate of a composite material with a 3D angle-interlock woven
fabric and reported that the maximum energy release rate could be larger than
20 N/mm. It is obvious that the out-of-plane fracture energy of 3D woven
composites is significantly higher than the one of laminated composites with
typical value of less than 1 N/mm. This confirms that the 3D architecture does
feature a great enhancement of the delamination resistance.

##  6 In-plane multi-axial fracture test

The multi-axial fracturing behavior of polymers and their fiber-reinforced
laminates was recently investigated in the literature [78, 79, 80, 81, 82, 83,
84, 85, 86]. However, to the best of authors’ knowledge, both notch-free and
notched three-dimensional woven composites under global multi-axial stress
states were rarely reported so far in the open literature. The lack of
experimental data definitely has hindered the understanding of the mechanical
behavior of three-dimensional woven composites under complex loading
scenarios. In this section, the effects of an intra-laminar central notch on
the three-dimensional woven composites under multi-axial quasi-static load
were studied. Two different geometries were prepared for the in-plane multi-
axial tests including the rectangular specimens cut along the warp and weft
directions. The intra-laminar central notch was manufactured by firstly
drilling a hole using a 0.8 mm tungsten carbide drill bit and then completing
the notch with a 0.8 mm wide diamond-coated saw. The dimensions of the
specimens are illustrated in Figure 21.

###  6.1 Testing description

The multi-axial behavior of notched three-dimensional woven composites was
evaluated by leveraging a modified Arcan rig as illustrated in Figure 21 which
comprises four identical 17-4 PH stainless steel plates (two fronts and two
backs) to clamp the specimens by friction through twelve M14 high-strength
bolts. The global multi-axial loads were applied by rotating the modified
Arcan rig with the angle $\theta$ between the loading direction and the
longitudinal direction of the specimen. As illustrated in Figure 21, a mixed
tension and shear can be achieved by an intermediate angle between $0^{\circ}$
and $90^{\circ}$ whereas two extreme angles $\theta=0^{\circ}$ and
$\theta=90^{\circ}$ represent the uni-axial tension and pure shear loading
conditions respectively. To have a better description on the loading
configuration, multiaxiality ratio can be defined as $\lambda=$
arctan$(\tau_{N}/\sigma_{N})$ where $\sigma_{N}=P$cos$\theta/(tD)$ is the
nominal normal stress and $\tau_{N}=P$sin$\theta/(tD)$ is the nominal shear
stress. The information on the multi-axial behavior of notched three-
dimensional woven composites was achieved by investigating six sets of
multiaxiality ratios with $\lambda=0,0.262,0.524,0.785,1.047$ and 1.571 in
this study. The foregoing multi-axial tests were performed on a servo-
hydraulic 8801 Instron machine with closed-loop control and the displacement
rate is 0.02 mm/s for all the specimens.

Figure 21: Test setup and Arcan rig used for the multi-axial tests on notched
three-dimensional woven composites. Both warp and weft directions oriented
towards the longitudinal direction of the specimen were investigated in this
work.

###  6.2 Damage detection and quantification

Speckled specimens were used for the acquisition of the entire strain field on
the material surface by means of an open source Digital Image Correlation
system programmed in MATLAB software and developed at Georgia Tech [121, 122].
The detailed three-dimensional damage morphology at about the peak load was
detected by means of a NSI X5000 X-ray micro-tomography scanning system [123]
with a X-ray tube setting of 110 kV in voltage and 160 $\mu$A in current. It
is worth mentioning here that the sub-critical damage was better visualized by
utilizing a dye penetrant [90, 124] in all the scans and the related solutions
can be found in [125, 126]. After scanning, the damage characteristics were
further analyzed through the reconstructed specimens via the software ParaView
[127].

About 100 reconstructed images were sliced throughout the thickness of the
specimen leading to about 2 $\times$ $10^{9}$ pixels for the gauge volume of
the specimen and the colors of all the pixels were converted into the grey
scale values in order to quantify the sub-critical damage, By setting a range
of the grey scale threshold, the crack volume can be measured by leveraging
the pixels with the values higher than this threshold divided by the total
pixels in the gauge volume of the specimen. This range was selected about
145-155 and carefully taken on the location of the specimen where no
significant damage was observed.

###  6.3 Experimental results and analysis

The representative nominal stress and strain curves for the three-dimensional
woven composites in presence of an intra-laminar central notch under multi-
axial loads were plotted in Figure 22. For both warp and weft scenarios, the
mechanical response of the materials in the longitudinal direction of the
specimen exhibits less non-linearity up to the peak load with the curves
departure from the linear part at least after 35% of the strain at the peak
load. However, the emergence of the material quasi-ductility in the transverse
direction of the specimen was observed with increasing proportions of shear to
tension since departure from linear behavior occurs only after 20% of the
strain at the peak load and it is followed by stable post-peak softening.
Similar phenomenon was also reported in other related studies [78, 79, 80, 81,
82]. The non-linearity of the materials under shear-dominated loads can be
explained mainly due to the formation of the significant matrix and debonding
cracks for the energy dissipation in this case rather than the plasticity of
the material as it will be shown through the morphological study in the next
section. On the other hand, it is interesting to notice from the figure that
the evolution of the nominal stress and strain curves can be dependent on the
global multi-axial stress states. This implies the significant effects of the
local stress states on the plastic deformation and fracture behavior of the
materials which was also reported in other studies for various materials [128,
129, 130, 131].

Figure 22: Nominal stress vs. strain curves in (a) longitudinal direction and
(b) transverse direction obtained from the multi-axial tests on the notched
three-dimensional woven composites with the longitudinal direction oriented in
both warp and weft directions. Note that the nominal strain was measured based
on the difference in the displacements close to the two ends of the gauge area
of the specimen by means of the Digital Imaging Correlation (DIC).

The correlation between the nominal normal and shear strength as a function of
the multiaxiality ratio for the notched three-dimensional woven composites was
further investigated and plotted in Figure 23. As illustrated in Figure 23a
for the warp tows oriented towards the longitudinal direction of the specimen,
the failure envelope features two distinguishable segments which represent
both tension dominated ($\lambda=$ 0 to 0.262) and shear-dominated ($\lambda=$
0.262 to 1.571) loading conditions. This is an indication of two different
damage mechanisms with respect to the multiaxiality ratio as it will be
clarified in the next section. On the other hand, the forgoing failure envelop
is located in the area featuring the boundary with the in-plane tensile
strength ($F_{1t}$) and shear strength ($F_{12}$) measured from the previous
uni-axial tensile tests on the notch-free specimens. This aspect is not
surprising but mainly attributed to the significant reduction on the nominal
normal strength of the specimen since the nominal shear strength does not
exhibit sensitive degradation for the existence of an intra-laminar central
notch in the specimen compared to the one estimated from the uni-axial tensile
tests on the previous off-axis specimens. In addition, the experimental
results obtained from the previous Mode I fracture tests were further used
through the LEFM analysis to predict the nominal normal and shear strength of
the geometry in this case with the warp tows parallel to the longitudinal
direction of the specimen. As can be noted from Figure 23a, the prediction by
means of the LEFM analysis can lead to a remarkable over-estimation for both
nominal normal and shear strength and this mismatch confirms the quasi-
brittleness of the investigated three-dimensional woven composites which
cannot be simply analyzed by leveraging the LEFM approach. It is worth
mentioning here that the LEFM prediction on the nominal shear strength of the
specimen in Figure 23a is based on the assumption that the apparent Mode II
fracture energy $G^{II}_{f}$ of the material with the warp tows oriented
toward the longitudinal direction of the specimen is equivalent to the
apparent Mode I values as summarized in Table 6 due to the lack of data on
this aspect in the open literature. In fact, the larger Mode II fracture
energy of the laminates compared to the Mode I counterparts was typically
reported in the literature [106] and this can lead to even more pronounced
over-prediction on the forgoing nominal shear strength of the specimen.

Figure 23: Failure envelops for the notched three-dimensional woven composites
with the longitudinal direction of the specimen towards the (a) warp tows and
(b) weft tows. Figure (a) compares the failure envelop measured from the
multi-axial tests with the associated results obtained from the uni-axial
tensile tests and predicted through the LEFM analysis. Figure (b) has a
similar comparison but includes the previous results measured from the multi-
axial tests on the notched $[0/90]_{2s}$ GFRP laminates.

Similar analysis was further performed on the notched three-dimensional woven
composites with the weft tows in the longitudinal direction of the specimen.
As it can be noted from Figure 23b, the failure envelope of the foregoing
specimens under multi-axial loads is also composed of two noticeable parts
representing the transition of the damage mechanism from tension to shear.
Moreover, this aforementioned failure envelop was reasonably compared with the
one in the previous study [78] for the notched cross-ply $[0/90]_{2s}$
laminates made of glass fiber and thermoset polymer (GFRP) in the
consideration of similar in-plane dimensions. As illustrated in Figure 23b,
the mechanical behavior of the foregoing cross-ply laminates is similar to the
one of the notched woven composites plotted in this figure under tension-
dominated loading conditions. However, this does not happen in the shear-
dominated scenarios since this notched woven composites exhibit higher nominal
shear strength indicating the better shear resistance of the materials than
the two-dimensional GFRP laminates with similar fiber orientations. On the
other hand, the investigated intra-laminar central notch leads to the
reduction on both nominal tensile and shear strength of the notched woven
composites with the weft tows in the longitudinal direction of the specimen
and this aspect differs from the warp case as plotted in Figure 23a showing
almost no degradation on the nominal shear strength of the material with the
non-negligible intra-laminar central notch. This can be explained by either
the contribution of the warp tows to the better shear performance or the
geometrical effects considering the different specimen sizes for both warp and
weft cases. This requires further computational studies which will be covered
in the future publications.

###  6.4 Morphological Characteristics and Damage Mechanisms

####  6.4.1 Warp tow as longitudinal direction

In the case that the warp tows are oriented towards the longitudinal direction
of the specimens and these notched three-dimensional woven composites are
subjected to multi-axial load, the damage mechanisms can be reasonably
categorized into two different scenarios. The damage characteristics belong to
_Mechanism A_ when the tension-dominated loading conditions
($\lambda=0,0.262$) are applied on the specimens. In this mechanism, the
fracturing features almost at the peak load are characterized by a remarkable
distributed region of high deformation at both sides of the notch as
illustrated in Figure 24a through the two-dimensional DIC analysis. This
deformed region is mainly formed by the significant splitting in the
transverse direction distributed in the gauge area of the specimen and the
noticeable splitting in the longitudinal direction at the notch tip as can be
clearly seen from Figure 24b by leveraging micro-computed tomography analysis.
Thanks to this three-dimensional damage inspection technique, the significant
damage through the entire thickness of the specimen almost at the peak load
was also observed as illustrated in Figure 24b showing the multiple debonding
cracks spreading along the interfaces between tows and matrix as similarly
reported in other related studies [132]. Interestingly, these debonding cracks
are characterized by a distance about two times the thickness of the specimen
and this important feature is similar to the experimental morphology of the
cross-ply laminates under uni-axial tension exhibiting individual transverse
matrix crack at a distance of roughly one to three times the thickness of the
lamina [133, 134, 135, 136]. Based on the foregoing description, the damage
evolution of _Mechanism A_ can be summarized into the following:

  1. 1.

splitting initiates in the transverse direction and has a distributed
development (Figure 24b);

  2. 2.

splitting in the longitudinal direction initiates and develops near the notch
tip (Figure 24b);

  3. 3.

matrix and debonding cracks initiate and develop through the specimen
thickness (Figure 24b);

  4. 4.

fibers break mainly in the longitudinal direction together with the further
splitting propagation mainly in the transverse direction near the notch tip.

On the other hand, the specimens exhibit completely different damage
characteristics compared to the previous ones following _Mechanism A_ when the
shear load component is involved ($\lambda=0.524,0.785,1.047$ and $1.571$).
This significant difference was exemplified in Figures 24c-f for the DIC and
micro-computed tomography analyses on the specimens with the multiaxiality
ratio $\lambda=0.7857$ and $1.571$. In these scenarios, the damage
characteristics follow _Mechanism B_ which features localized region of high
deformation at both sides of the notch almost at the peak load as shown in
Figures 24c and 24e. The detailed damage morphology with respect to the
forgoing localized damage region was illustrated in Figures 24d and 24f
showing that the in-plane damage band is the combination of the localized
splitting in both transverse and longitudinal directions whereas the minor
matrix cracks and the pronounced debonding cracks between tows and matrix lead
to the out-of-plane damage band which was not obtained from the two-
dimensional DIC analysis. In addition to this, the splitting in the
longitudinal direction is more pronounced in terms of both length and quantity
as the multiaxiality ratio increases representing the transition from uni-
axial tension to pure shear condition as illustrated in Figure 24. Based on
the foregoing discussion, the damage evolution of _Mechanism B_ consists of
the following phases:

  1. 1.

splitting initiates in the transverse direction and has a localized
development near the notch (Figures 24d,e);

  2. 2.

matrix and debonding cracks initiate and develop through the specimen
thickness (Figures 24d,e);

  3. 3.

splitting in the longitudinal direction initiates and develops in the specimen
with the noticeable longer length for the pure shear loading condition
(Figures 24e);

  4. 4.

fiber breaking leads to the final separation of the specimen.

Figure 24: Analyses of quasi-static damage mechanisms of notched three-
dimensional woven composites with the warp tows oriented toward the
longitudinal direction of the specimens under multi-axial load by Digital
Imaging Correlation (DIC) and micro-computed tomography ($\mu-$CT). These
figures illustrate the detailed damage characteristics almost at the peak load
for the forgoing composites weakened by a 8mm central notch under the
multiaxiality ratio $\lambda$ = 0, 0.78 and 1.571. Note that the symbol
$\varepsilon_{np}$ in the DIC analysis represents the normalized maximum
principle strain. Also note that the warp direction is the longitudinal
direction whereas the weft direction is the transverse direction in this case.
Figure 25: Analyses of quasi-static damage mechanisms of notched three-
dimensional woven composites with the weft tows oriented toward the
longitudinal direction of the specimens under multi-axial load by Digital
Imaging Correlation (DIC) and micro-computed tomography ($\mu-$CT). These
figures illustrate the detailed damage characteristics almost at the peak load
for the forgoing composites weakened by a 8mm central notch under the
multiaxiality ratio $\lambda$ = 0, 0.78 and 1.571. Note that the symbol
$\varepsilon_{np}$ in the DIC analysis represents the normalized maximum
principle strain. Also note that the weft direction is the longitudinal
direction whereas the warp direction is the transverse direction in this case.

####  6.4.2 Weft tow as longitudinal direction

On similar grounds, the damage mechanisms can also be reasonably categorized
into two different situations when the notched three-dimensional woven
composites are subjected to multi-axial load but the weft tows are oriented
towards the longitudinal direction of the specimens. In this case, the
application of tension-dominated loading conditions ($\lambda=0,0.262$ and
$0.524$) on the specimens leads to the damage characteristics following
_Mechanism A_. As illustrated in Figure 25a, a distributed region of high
deformation at both sides of the notch was also observed almost at the peak
load through the DIC analysis and the forgoing region is still the consequence
of the distributed splitting in the transverse direction as shown in Figure
25b through the micro-computed tomography analysis. These aspects are almost
similar to the morphological features as discussed in the previous section for
the warp tows parallel to the longitudinal direction of the specimen but the
difference lies in the disappearance of the noticeable splitting in the
longitudinal direction near the notch tip as clearly shown in Figure 25b. On
the other hand, the matrix and debonding cracks with an approximate distance
of roughly two times the thickness of the specimen were also observed as can
be noted in Figure 25b. In summary, the damage evolution of _Mechanism A_ has
the following steps:

  1. 1.

splitting initiates in the transverse direction and has a distributed
development (Figure 25b);

  2. 2.

matrix and debonding cracks initiate and develop through the specimen
thickness (Figure 25b);

  3. 3.

fibers break mainly in the longitudinal direction together with the further
splitting propagation in the transverse direction near the notch tip.

When the specimens are subjected to shear-dominated loading conditions
($\lambda=0.785,1.047$ and $1.571$), the damage characteristics can be
categorized into _Mechanism B_ which has similar features compared to the
previous ones for the materials with the warp tows parallel to the
longitudinal direction of the specimen under the same global multi-axial
loads. As it can be noted in Figures 25c-f for both DIC and $\mu$-CT analyses
almost at the peak load, the specimens exhibit localized three-dimensional
damage bands at both sides of the notch due to the mixed splitting in both
longitudinal and transverse directions near the notch and the combination of
matrix and debonding cracks through the entire thickness of the specimen.
However, the splitting in the longitudinal direction of the specimen was
significantly reduced in these scenarios which is substantially different from
the previous _Mechanism B_ showing remarkable splitting in the longitudinal
direction of the specimen. Based on these interesting morphological studies,
the damage evolution of this _Mechanism B_ can be summarized in the following:

  1. 1.

splitting initiates in the transverse direction and has a localized
development near the notch (Figures 25d,e);

  2. 2.

matrix and debonding cracks initiate and develop through the specimen
thickness (Figures 25d,e);

  3. 3.

minor splitting initiates and develops in the longitudinal direction (Figures
25d,e);

  4. 4.

fiber breaking leads to the final separation of the specimen.

Figure 26: Crack volume as a function of the multiaxiality ratio almost at the
peak load for the notched three-dimensional woven composites with the
longitudinal direction in both warp and weft directions.

####  6.4.3 Crack volume vs. multiaxiality ratio

The crack volume in the notched three-dimensional woven composites as a
function of the multiaxiality ratio almost at the peak load was further
estimated as illustrated in Figure 26. As can be noted from this figure for
both warp and weft cases, the crack volume increases in a non-linear way as
the multiaxiality ratio increases. This is a confirmation that the material
exhibits more quasi-brittleness when the loading condition transits from
tension to shear. On the other hand, the foregoing phenomenon is also
consistent with the previous nominal stress-strain curves of the notched
three-dimensional woven composites under multi-axial loads since the less
crack volume in the material for the tension-dominated loads leads to the less
non-linearity of the curves whereas the energy dissipation due to the large
crack volume can be the main reason for the significant non-linearity of the
curves under shear-dominated loads.

##  7 Conclusions

A comprehensive experimental study was performed in order to characterize
various mechanical properties of a 3D woven carbon-epoxy composite with ply to
ply angle interlock architecture. The following conclusions can be made based
on the results provided in this work:

1) A database of the elastic, strength, and fracture properties of the 3D
woven composite was provided through a series of quasi-static tests conducted
in the warp, weft, and out-of-plane directions. The elastic properties were
measured based on the theory of elasticity for orthotropic materials.

2) The nonlinear stress-strain responses were observed for the specimens under
on-axis tension loading and $\pm$$$ off-axis tension and compression loading.
The results of DIC analysis revealed that the failure initiation was strongly
related to matrix cracking. In addition, material non-linearity was more
evident for the specimens stretched in the weft direction, which also showed
greater strain to failure although lower modulus and strength were detected.
These observations could be attributed to the higher waviness of weft tows and
lower fiber content in the weft direction.

3) The nonlinear mechanical responses of the specimens under $\pm$$$ tension
and compression loading showed negligible difference, indicating that they
share the same damage mechanism and deformation mode. Shear failure was
recognized for both cases, and shear modulus and strength were determined from
the off-axis specimens.

4) Various failure mechanisms and ultimate failure modes were identified from
the in-plane tension and compression tests on both on-axis and off-axis
specimens.

5) The out-of-plane elastic properties and compressive strengths were
determined from the experimental results of the compression tests conducted in
the out-of-plane direction. The explosive ejection of fragments perpendicular
to the loading direction led to the ultimate failure of the specimens, and
both horizontal and inclined failure planes were observed.

6) Tensile splitting test was proposed to estimate the out-of-plane tensile
strength of the material. The relation between the tensile strength and the
recorded peak load was determined through a finite element simulation. In
addition, the DIC analysis revealed that the failure was initiated in the
vicinity of the load application zones due to significant strain/stress
concentrations, and both vertical crack and wedge formation were observed from
the failed specimens. An improved tensile splitting test was suggested to
reduce the degree of strain/stress concentrations, which was validated by
finite element simulations.

7) The experimental results of the fracture tests on geometrically-scaled
notched specimens showed a remarkable size effect. The strength-based failure
criterion failed to capture the size effect whereas LFEM was also unsuccessful
because the fracture properties determined from the measured peak load were
size dependent. As a consequence, a nonlinear (quasibrittle) fracture
mechanics theory with the capability of accounting for the finiteness of the
FPZ was needed to interpret the fracture test results and to provide correct
scaling analysis of mechanical properties, which is essential for damage
tolerant design of large composite structures.

8) Comparison of the size effect data of the typical 3D and 2D woven
composites suggested that the 3D woven composite provides higher quasi-
ductility and superior damage tolerance which are beneficial in many aspects.
In fact, the Mode I intra-laminar fracture energy, $G_{f}$ was estimated to
the be $466$ N/mm or higher. This value is, on average, about seven times
larger then the fracture energy of typical two-dimensional textile composites
made of similar materials [68].

(9) The nominal stress-strain curves for the investigated center-notched
three-dimensional woven composites exhibit significant non-linearity and
relatively stable post-peak behavior as the loading multiaxiality ratio
increases representing the involvement of shear stresses. This non-linearity
is most likely due to the high volume of damage for energy dissipation and the
plasticity of the material may have less contribution.

(10) The failure envelop constructed by using the critical nominal normal and
shear strength for the investigated center-notched three-dimensional woven
composites cannot be simply predicted through the LEFM method due to the lack
of characterizing the non-linear damage zone ahead of the notch in the
materials. The forgoing multi-axial fracture behavior must be described by
leveraging the quasi-brittle fracture mechanics and related non-linear
computational modeling.

(11) Regarding the damage mechanisms of the investigated center-notched three-
dimensional woven composites under multi-axial scenario, a distributed damage
with the formation of splitting in the transverse direction and the
fiber/matrix debonding cracks throughout the specimen thickness mainly
characterizes the material under tension-dominated loading whereas a localized
damage region features the material under shear-dominated loading. This region
is typically formed by various damage morphologies including matrix/fiber
debonding, matrix cracking, and splitting in both longitudinal and transverse
directions. The foregoing damage characteristics were observed for the
material with either weft or warp tows oriented towards the longitudinal
direction of the specimen.

## Acknowledgements

The work was partially supported under NSF (Funder ID: 10.13039/100000001)
grant No. CMMI-1435923 to Northwestern University. In-kind contribution by
Albany Composites is gratefully acknowledged.

##  Appendix A Analysis of splitting tensile tests by finite element
simulations

A 3D finite element model consisted of a cuboid specimen, two strip cushions,
and two loading pins was created in Abaqus CAE environment [137]. Eight-node
linear brick elements with reduced integration (C3D8R) were adopted for
discretization. The general contact algorithm with hard contact as normal
behavior was applied for the interaction between the different parts of the
model. Mesh refinement with a smallest element size of 0.2 was preformed in
the areas where the specimen is in contact with the cushions to ensure the
accuracy of the numerical results in the region of high strain/stress
gradient. The finite element mesh of the model is shown in Fig. 27a. A linear
elastic orthotropic constitutive model was used for the simulation with the
material properties given in Table 2. The loading pins were defined as rigid
bodies to facilitate the application of the boundary conditions with the
bottom one fixed vertically and the top one assigned a vertical displacement.
Two cases were investigated with case 1 representing specimens loaded in the
warp direction whereas case 2 in the weft direction. The simulations were
performed in Abaqus Implicit 6.13 [137]. Fig. 27b plotted the deformed shape
of the model for case 1 and the horizontal normal stress ($\sigma_{xx}$)
contour on the specimen surfaces under the vertical displacement of -0.5 mm. A
narrow band of positive $\sigma_{xx}$ (tensile stress, highlighted in red) can
be identified.

Figure 27: (a) Typical finite element mesh used in the simulation of tensile
splitting tests. (b) Horizontal normal stress $\sigma_{xx}$ contour of a
specimen under vertical displacement of -0.5 mm.

###  A.1 Stress distribution and and estimation of tensile strength

The numerically calculated stress distribution is shown in Fig. 28a and b for
case 1, and Fig. 28c and d for case 2, respectively. Only the horizontal and
vertical normal stress components, $\sigma_{xx}$ and $\sigma_{yy}$, were
plotted, whereas the other components were significantly smaller. Fig. 28a and
c illustrate the stress distribution on the vertical plane of the simulated
specimen along the loading line, whereas Fig. 28b and d show the stress
distribution on the horizontal plane across the center of the specimen. In
order to illustrate the 3D effect on the stress distribution, the stresses on
the middle and front surfaces of the specimens were plotted together. The 3D
effect, however, is not significant for $\sigma_{xx}$ whereas a discrepancy of
$\sigma_{yy}$ sampled on the specimen middle and front surfaces due to the 3D
effect can be observed.

Figure 28: Major stress distribution of a specimen in simulated tensile
splitting test on (a) vertical plane for the case of warp loading, (b)
horizontal plane for the case of warp loading, (c) vertical plane for the case
of weft loading, (d) horizontal plane for the case of weft loading.

As one may notice, the vertical distribution of horizontal tensile stress
$\sigma_{xx}$ is almost uniform on both middle and front surfaces, as shown in
Fig. 28a and c, whereas it attains the maximum value at the center of the
specimen, as shown in Fig. 28b and d. The horizontal tensile stress is usually
associated with the tensile strength of the material when the load reaches its
peak if the ultimate failure mode is tensile splitting driven by the maximum
horizontal tensile stress. In this way, the relation between the maximum
horizontal tensile stress and the applied load can be established, and the
parameter $k$ can be obtained correspondingly. For instance, one obtains the
maximum $\sigma_{xx}$ of 0.87 MPa (in the middle plane) from Fig. 28a and b
for case 1, and the corresponding applied load is 3348 N. The parameter $k$
was estimated as 0.213 accordingly. Similarly, one can obtain the parameter
$k$ for case 2, being 0.356.

Although tensile splitting test provides an accessible method to estimate the
tensile strength of materials, its accuracy is often questioned. Especially,
one may note that along with the uniformly distributed horizontal tensile
stress $\sigma_{xx}$ in the middle of the specimen, a large vertical
compressive stress $\sigma_{yy}$ can be also observed from Fig. 28. As a
consequence, the measured tensile strength is obtained from a multiaxial
stress state, and could be different from the one obtained from a uniaxial
stress state. Another problem which hinders the application of tensile
splitting test is noticeable stress concentrations. Especially, as one can
observe from Fig. 28, the vertical compressive stress $\sigma_{yy}$ in the
vicinity of the load application zones could be 50 times larger than the
maximum horizontal tensile stress, which could be responsible for the
indentation type of failure and the formation of the wedge as one observed in
Fig. 13b and c. Some researchers proposed that although failure in the tensile
splitting test could be sometimes initiated in compression-shear in the
vicinity of loading platens due to stress concentrations, the ultimate failure
is driven by tension [103, 104]. This may explain the typical failure pattern
shown in Fig. 13 b and c, and it is likely that the formed wedge could further
drive crack propagation and eventually lead to the vertical failure path that
is the same as splitting failure mode.

###  A.2 Discussion on tensile splitting test improvement

A remedy to reduce stress concentrations in tensile splitting test is explored
numerically herein. Instead of cuboid specimens, disk-shaped or cylindrical
specimens are used to facilitate the transmission of load from a load frame. A
load of a finite width is applied on the specimen with radius $R$ and
thickness $t$. It was found that the loading width, characterized by an
angular parameter, $2\alpha$ as depicted in Fig. 29a, has a substantial
influence on the stress distribution within the specimen [138]. To explore the
effect of the loading width and the feasibility of the improved tensile
splitting test in determining the out-of-plane tensile strength of the
investigated 3D woven composite, a finite element simulation was performed.
The simulation setup is similar to the one described above, but instead of
applying load through the loading pins, a pressure of 100 MPa is directly
applied (radially) on the top and bottom of the specimen with a finite width,
as illustrated in Fig. 29a. Four different angular widths, $2\alpha=$$$,
$None$, $None$, $None$ are considered. The specimens were loaded in the weft
direction. The typical horizontal normal stress distribution on the surfaces
of the specimen is shown in Fig. 29b. Similar to the one shown in Fig. 27b, a
tensile stress was developed in a region near the center of the specimen.

The numerically calculated stress distribution on the vertical plane across
the center of the specimen is shown in Fig. 29c. All stresses were normalized
by the maximum horizontal tensile stress $\sigma_{xx}^{max}$. It can be found
that the stress distribution in all cases is very similar to the one for the
cuboid specimen as shown in Fig. 28c. However, the stress distribution,
especially the concentrations in the vicinity of the load application zones
vary with the angular width $2\alpha$. The ratio of the maximum compressive
stress, $\max(-\sigma_{ii})$, $i=x$, $y$, to the maximum tensile stress,
$\max(\sigma_{xx})$, can serve as an indicator of the degree of stress
concentration. One may note that the ratio, and thus the stress concentrations
are significantly reduced by increasing $2\alpha$. However, as the angular
width $2\alpha$ increases from $None$ to $None$, the ratio does not decrease
anymore. Therefore, the degree of stress concentration is minimized with
$2\alpha=$$$ or $None$. It is also worth noting that the uniformity of the
horizontal tensile stress in the vertical line is hampered by an increase of
the angular width, which, however, is not desirable. In this case, the angular
width $2\alpha=$$$ is preferred to $2\alpha=$$$.

It is promising that the ratio of the maximum compressive to tensile stress
can be reduced to around $10$ if the angular width $2\alpha=$$$. Therefore, as
long as the ratio of $F_{2c}$ to $F_{3t}$ is larger than 10, the failure
initiation in the improved tensile splitting test could be driven by the
maximum horizontal tensile stress rather than compression-shear in the
vicinity of the specimen load application zones. As a consequence, the tensile
strength of materials can be related to the peak load measured in a tensile
splitting test.

Figure 29: (a) Schematic diagram of the improved tensile splitting test with
cylindrical specimen and distributed load with a angular width $2\alpha$. (b)
Horizontal normal stress $\sigma_{xx}$ contour of a cylindrical specimen under
diametrical compression with $2\alpha=$$$. (c) Distribution of the normalized
horizontal and vertical normal stresses on the vertical plane across the
center of the specimen.

Although conceptually simple, the improved tensile splitting test requires
more efforts to manufacture the disk-shaped or cylindrical specimens and to
setup the loading system to attain the desired load distribution over a
certain width. Thus, the experiments will be left for the future work. In
addition, considering the influence of the material inhomogeneity on the
stress distribution, one may have reasonable doubt about the results obtained
from the finite element simulations with a hidden assumption of material being
homogeneous continuum. This doubt needs to be resolved with the assistance of
a fine scale numerical model capable of capturing the major heterogeneity of
the material, which will be also left for the future work.

##  Appendix B Equivalent elastic crack concept and size effect law

Only Mode I fracture is considered in this study. For an orthotropic structure
containing a crack running in the $x$ direction subjected to an external
nominal stress $\sigma_{N}$ in the $y$ direction, the stress intensity factor
in the sense of LEFM can be written as [139]

$K_{I}=\sigma_{N}\sqrt{D}k(\alpha)=\sigma_{N}\sqrt{\pi
D\alpha}\xi(\alpha,\lambda^{1/4}L/D,\rho)$ (4)

where $\alpha=a/D$ denotes dimensionless crack length,
$k(\alpha)=\sqrt{\pi\alpha}\xi$ is dimensionless stress intensity factor, and
$\xi(\alpha,\lambda^{1/4}L/D,\rho)$ presents a dimensionless function
accounting for material orthotropy, and $\rho$ and $\lambda$ are dimensionless
elastic parameters defined as

$\rho=\frac{\sqrt{E_{x}E_{y}}}{2G_{xy}}-\sqrt{\nu_{xy}\nu_{yx}},\lambda=\frac{E_{y}}{E_{x}}$
(5)

The energy release rate $G(\alpha)$ can be related to the stress intensity
factor $K_{I}$ through the Griffith and Irwin relationship, which reads

$G(\alpha)=\frac{K_{I}^{2}}{E^{*}}=\frac{\sigma_{N}^{2}D}{E^{*}}g(\alpha)$ (6)

where $g(\alpha)=k^{2}(\alpha)$ is dimensionless energy release rate, and
$E^{*}$ is the effective elastic modulus for orthotropic materials, which
reads

$E^{*}=\sqrt{\frac{2E_{x}E_{y}\sqrt{\lambda}}{1+\rho}}$ (7)

According to LEFM, fracture toughness is a critical value of $K_{I}$ when the
peak of $\sigma_{N}$, i.e. nominal strength $\sigma_{Nu}$, is reached.
Identically, fracture energy is a critical value of $G$. These two fracture
parameters calculated in this way are hereinafter referred to as the apparent
fracture toughness, $K_{IcA}$ and the apparent fracture energy, $G_{fA}$. By
substituting the values of $\sigma_{Nu}$ measured in the experiments as
reported in Table 6 and the elastic constants listed in Table 2 ($x$ and $y$
axes coincide with 2 and 1 direction in the material system) into Eq. 4 and
Eq. 6, one can obtain the measures of $K_{IcA}$ and $G_{IcA}$, respectively.
The value of $g(\alpha)$ was calculated numerically in the same way as
discussed in Refs. [68, 140, 58] by using finite element and the quarter
element technique [141]. A uniform remote displacement rather than stress was
assumed as a boundary condition, which is more realistic for the actual test
setup in this work. The results of apparent fracture toughness and apparent
fracture energy are also reported in Table 6. Note that $K_{IcA}$ and $G_{fA}$
are structural characteristics rather than material properties because they
depend on specimen size and geometry.

The equivalent elastic concept provides a simple recipe for the problem of
fracture of a quasibrittle material with a finite, non-negligible FPZ. In this
case, the equivalent crack length $a=a_{0}+c$ at failure must be distinguished
from the actual crack (traction free) length $a_{0}$ where $c=$ elastically
equivalent crack extension giving the same compliance according to LEFM as the
actual crack growth [142]. It has been proven by Bažant et al.[105] that for a
sufficiently large structure, the equivalent crack extension $c=c_{f}$ at
maximum load is a material property, and denotes the elastically equivalent
length of the FPZ. Correspondingly, the energy required for crack growth by
means of the equivalent elastic crack concept can be written as

$G\left(\alpha_{0}+c_{f}/D\right)=\frac{\sigma_{Nu}^{2}D}{E^{*}}g(\alpha_{0}+c_{f}/D)=G_{f}$
(8)

where $G_{f}=$ initial fracture energy of the material, and is assumed to be a
material property. Mathematically, $G_{f}=\lim_{D\to\infty}G_{fA}$ since for
$D\to\infty$, one has $c_{f}/D\to 0$ implying that for an infinitely large
specimen, the FPZ occupies an infinitesimal region of the structure, and the
equivalent crack extension vanishes.

The size effect law can be obtained from Eq. 8 by expressing the nominal
strength $\sigma_{Nu}$ as a function of $D$. The dimensionless function
$g(\alpha_{0}+c_{f}/D)$ can be approximated with its Taylor series expansion
about the value $\alpha_{0}$ and retained only up to the linear term of the
expansion. One obtains:

$\sigma_{Nu}=\sqrt{\frac{E^{*}G_{f}}{Dg(\alpha_{0})+c_{f}g^{\prime}(\alpha_{0})}}$
(9)

where the function $g^{\prime}(\alpha)$ denote the derivative of $g$ with
respect to $\alpha$. This equation has the same form as the classic Bažant SEL
for isotropic case [105], and can be recast in a dimensionless form:

$\frac{\sigma_{Nu}}{\sigma_{0}}=\frac{1}{\sqrt{1+D/D_{0}}}$ (10)

where $\sigma_{0}=\sqrt{E^{*}G_{f}/(c_{f}g^{\prime}(\alpha_{0}))}$ and
$D_{0}=c_{f}g^{\prime}(\alpha_{0})/g(\alpha_{0})$.

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