Notch failure in laminated composites under opening mode: the Virtual
Isotropic Material Concept
A.R. Torabi1, E. Pirhadi
Fracture Research Laboratory, Faculty of New Science and Technologies, University of Tehran, Tehran, Iran
Abstract
Experimental and theoretical fracture investigations are performed on laminated
glass/epoxy composite specimens with different numbers of ply and lay-up configurations
weakened by central U-shaped notches of various tip radii and loaded under pure opening
mode. Since the main purpose of this research is to evaluate the load-carrying capacity (LCC)
of the U-notched glass/epoxy composite specimens, it is experimentally recorded by the
testing machine and the failure behavior of the specimens is carefully observed. To predict
the experimentally obtained LCCs or the last-ply-failure (LPF) loads theoretically, a novel
concept, called Virtual Isotropic Material Concept (VIMC), is proposed and linked to two
well-known stress-based brittle fracture models in the context of the linear-elastic notch
fracture mechanics (LENFM), namely the maximum tangential stress (MTS) and the mean
stress (MS) criteria. It is shown that both VIMC-MTS and VIMC-MS combined criteria can
predict the experimental results well without the need for ply-by-ply failure analysis, the firstply-failure (FPF) prediction, and the progressive damage modeling (PDM) etc. Thanks to
both VIMC and LENFM, the LPF prediction is direct, rapid, and convenient, because only
two distinct fracture parameters, namely the ultimate tensile strength and the trans-laminar
fracture toughness of laminate are necessary for LPF prediction.
Keywords: Virtual Isotropic Material Concept (VIMC); Laminated composite; Notch;
Fracture; Load-carrying capacity; Last-ply-failure (LPF)
1.
1
Introduction
Corresponding author: A.R. Torabi, Tel: +98 21 61115775, E-mail address: a_torabi@ut.ac.ir
1
In the past five decades, many experimental and numerical research works have been
performed to evaluate failure in laminated composite specimens under different loading
conditions. However, most of them have dealt with cracked samples and not with notched
ones. In practical applications of composite laminates, drilling holes for mechanical
fastenings or cut-outs for access are usually inevitable. These holes and cut-outs with various
shapes, such as circular holes, rectangular cut-outs with round corners and bean-shaped holes
with two U-ends etc., concentrate stresses at their neighborhood and increase the risk of
initiation of damage from their borders. The damage initiated may grow and lead to final
fracture depending on the loading and service conditions. Increasing of the application of
laminated composites in aerospace structures, particularly in the primary structures, makes it
necessary to have reliable criteria for predicting failure in laminated composite components
weakened by notches and ensuring the structural integrity.
A careful literature survey on fracture prediction of laminated composites indicates that there
are generally three different approaches for this purpose; i) the models based on fracture
mechanics [1-5], ii) the stress-fracture models [6-17], and iii) the progressive damage models
[18-25]. One of the well-known failure models based on the linear elastic fracture mechanics
(LEFM) for composite materials in macroscopic level is the WEK-fracture model, known as
the inherent flaw model (IFM). This model was presented by Waddoups et al. [2], which is
known by two parameters, namely the un-notched strength and the characteristic dimension
(the inherent flaw length). Whitney and Nuismer [6] suggested two stress-based fracture
criteria known as the point stress (PS) criterion and the average stress (AS) criterion.
Additionally, Pipes et al. [16] and Kim et al. [17] modified the PS criterion and proposed a
three-parameter model with an exponential relationship between the characteristic dimension
and the notch size. Another criterion, called Mar-Lin criterion, was presented by Mar and Lin
2
[13], in which failure in multi-layer composites occurs through the propagation of a crack
lying on matrix material.
Some other researchers have focused on damage zone, which is less attractive from
the practical viewpoint than other closed-form criteria. Backlund and Aronsson [3, 4] have
proposed the damage zone model (DZM) to evaluate the residual strength of composite
laminates. Eriksson and Aronsson [26] have developed a new criterion, called the damage
zone criterion (DZC), based on the assumption that a damage zone is present in the maximum
stress region of the laminate; the maximum strength region represents the point in which the
tensile stress of the notched laminate reaches the tensile strength of the un-notched laminate.
Khatibi et al. [27-29] have also evaluated the residual strength of composite laminates
weakened by a sharp notch using an effective crack growth model (ECGM).
In completion of damage mechanism and complying with damage zone model, the
progressive damage models have been employed to take into account the damage and
resulting changes in the stress distributions that occur throughout the loading process [23–
29]. Another criterion based on damage mechanisms, is the continuum damage mechanics
(CDM) proposed originally by Ladeveze [30] for characterizing and analyzing the produced
damage by material stiffness loss. Mohammadi et al. [31-33] have frequently analyzed the
laminated composites in wide range of failure modes, loadings, and layups by using the CDM
to estimate the residual stiffness. Moreover, Mohammadi and co-workers [33] have
conducted a parametric study on the influence of diameter of hole and characterized the
initiation and growth of damage parameters in the Ladeveze model.
As conclusions about important damage models it should be added that the DZM
requires some properties such as the un-notched strength, apparent fracture energy, and
stiffness parameters needing more experimental works. The ECGM criterion is based on the
global equilibrium condition with the aid of an iterative technique on the results of large
3
variation of apparent fracture energy (𝐺𝑐 ) values to evaluate the residual strength of
composite laminates. So, taking average 𝐺𝑐 values as the basic input parameter in ECGM
analyses will be erroneous. At last, in the CDM criterion two complicacies exist; first, the
dependency of it on damage and inelastic strains to estimate residual stiffness, second, this
model requires different user-defined subroutines for various lay-up configurations, stacking
sequences, failure modes, and loading conditions.
To the best of authors’ knowledge, no paper or technical report is available in the
literature dealing with fracture analysis of U-notched composite laminates. Most of the
fracture investigations on notched composite laminates or laminas have been performed on
O-notched (open hole) and cracked (slit) specimens. In order to simplify the procedure of the
last-ply-failure (LPF) prediction in U-notched laminated composites, it is tried in the present
study to use directly the linear elastic notch fracture mechanics (LENFM). For this purpose,
the composite laminate is equated with a virtual isotropic plate of the same thickness and
some brittle fracture criteria in the context of LENFM are utilized for failure prediction.
Hence, in next paragraph, some papers already published on brittle fracture of U-notched
components made of isotropic materials are reviewed.
Generally, several main failure theories, namely the cohesive zone model (CZM), the
strain energy density (SED), the J-integral, the maximum tangential stress (MTS), and the
mean stress (MS) criteria, based on stress or energy equations have been suggested and
utilized for evaluating fracture in notched brittle specimens. The cohesive zone model (CZM)
has been successfully employed by proposing suitable softening curves for some engineering
materials, such as steel, aluminum, polymethyl-methacrylate (PMMA), and PVC [34-36].
Also the strain energy density (SED) criterion has been proposed by Yosibash et al. [37]
showing very good correlation to a large amount of experimental data regarding brittle failure
of Alumina-7% Zirconia and PMMA specimens. More recent developments in this context
4
have also been published (see for instance [38-40]). A criterion based on the J-integral has
been proposed by Matvienko and Morozov [41] for a body with a U-notch considering linear
elastic and elastic–plastic behavior of material. Gomez et al. [42] have published a paper
dealing with various criteria that exist for investigating fracture initiation at U-notches under
mode I loading. Moreover, the SED model has been frequently used by several investigators
to assess brittle fracture of U-notched components made of different engineering materials
under mode I (i.e. opening mode) and mixed mode I/II (i.e. combined tension-shear mode)
loading conditions [43-46].
Beside SED failure criterion, two other well-known brittle fracture models are the
maximum tangential stress (MTS) and the mean stress (MS) criteria. The MTS criterion,
proposed originally by Erdogan and Sih [47] for predicting mixed mode brittle fracture in
sharp cracked bodies, has been extended by Ayatollahi and Torabi [48] to U-notched domains
and frequently used for predicting the onset of mixed mode brittle fracture in different
notched brittle members. The MS criterion in its mixed mode format has been previously
formulated by Seweryn and Lukaszewicz [49] for V-notched components and presented as a
closed-form expression. Recently, Torabi [50] has successfully made use of the point stress
(PS) and the mean stress (MS) fracture criteria for predicting mode I failure of U-notched
graphite plates tested and reported by Berto et al. [51]. Additionally, Torabi [52] has utilized
the MTS fracture curves to predict theoretically the fracture loads of several U-notched
graphite specimens reported in [51] under mixed mode I/II loading conditions and found very
good agreement between the theoretical and experimental results. Another research work has
been published by Torabi et al. [53] in which the load-carrying capacity (LCC) of U-notched
Brazilian disc (UNBD) specimens made of graphite has been successfully predicted by using
the mean stress (MS) and point stress (PS) criteria under mode I loading conditions.
5
Considering the absence of the LPF prediction of the U-notched laminated composite
specimens in the literature and also the fact that such a prediction by means of any mentioned
failure models is very complex and time-consuming (because they need several material
properties, ply-by-ply analysis, and the first-ply-failure (FPF) prediction etc.), a novel
concept, called the Virtual Isotropic Material Concept (VIMC), is proposed by which a
laminated composite specimen is equated with a virtual isotropic specimen of exactly the
same dimensions. The VIMC allows to utilize brittle fracture criteria, which are valid for
isotropic materials, to predict LPF of notched laminated composite specimens. To verify the
validity of the proposed concept, numerous experiments are carried out on laminated
glass/epoxy composite specimens with different numbers of ply and various lay-up
configurations weakened by U-notches under pure mode I loading (i.e. the opening mode). It
is revealed that VIMC combined with MTS and MS brittle fracture models can predict the
experimentally obtained LCCs well, easily, and rapidly.
At last, it should be noted that the proposed novel concept permits to engineers
predicting failure of laminated composite components weakened by conventional notches
(i.e. V and U-notches) without regarding lay-up configurations, fibers, and epoxy-based
resins used in fabrication of laminated composite specimens. Also, the engineers always
accept warmly the simple ways in which any complicated subroutines have not been used.
2.
Experiments
2.1.
Material characterization
First, for manufacturing the composite plates, an epoxy resin with its hardener,
namely Epon 828 and Siclo-Aliphatic-Amin, respectively, and a type of unidirectional Eglass cloth fiber are chosen. The composite plates are fabricated by using the vacuum bagautoclave molding technique including bleeders located on top and bottom surfaces in order
6
to ensure that the processing technique is similar to fabrication of glass fiber reinforced
polymer (GFRP) composites, which are widely used in many industries, such as aerospace
industries. In this study, three lay-up configurations, namely unidirectional ([0]8 ), cross-ply
([0⁄90 /0⁄90]𝑠 ), and quasi-isotropic ([0 ∕ 90 ∕ ±45]𝑠 ) are considered. The fiber volume
fraction of the composite plates is determined to be between 56% and 59% for the three layup configurations. The thickness of each lamina is approximately equal to 0.28 mm and the
total thickness of each laminate configuration is about 2.9-3.1 mm for 8 layers. In order to
examine if the novel concept proposed in the present contribution is independent of the
number of layers, the composite plates of the three different layups with 16 layers and total
thickness of 5.6-5.8 mm are also considered in the experiments.
Two major parameters, namely the ultimate tensile strength (𝜎𝑢 ) and the trans-laminar
fracture toughness (𝐾𝑇𝐿 ), which are necessary for failure prediction by the novel concept, are
determined using the standard procedures described in ASTM D3039 [54] and ASTM E1922
[55], respectively. Three standard tests are carried out according to ASTM D3039 [54] to
determine the material properties for each lay-up configuration, such as the ultimate tensile
strength in the test direction, the ultimate tensile strain, the modulus of elasticity in the test
direction, and the Poisson’s ratio. The dimensions of the coupons for various lay-up
configurations can be easily realized by referring to ASTM D3039 [54]. The tensile tests are
performed using a universal tension-compression testing machine under displacement-control
condition at the loading rate of 2 mm/min, as recommended in ASTM D3039 [54] for
establishing the ultimate tensile strength (𝜎𝑢 ) of the laminate in the desired loading direction.
It should be noted that the novel combined criteria proposed in this research are
independent of 𝐸𝑥 , 𝐸𝑦 , 𝜈𝑥 and 𝜈𝑦 , which are the elastic moduli and the Poisson’s ratios in X
and Y directions, but solely dependent on 𝜎𝑢 and 𝐾𝑇𝐿 . Hence, there is no need to perform two
7
distinct tensile tests along X and Y directions which are done normally in most of the
experimental composite researches.
Three fracture tests are performed for each lay-up configuration with the aim to
measure the trans-laminar fracture toughness (𝐾𝑇𝐿 ) explained in ASTM E1922 [55]. This
code has permitted to have various configurations for test specimens, such as the planar size,
the thickness, the single-edge-notch length, and the stacking sequence with eccentric loading
conditions, providing that the notch-mouth displacement (NMD) values at maximum load
satisfy the criterion
∆𝑉𝑛
𝑉𝑛−0
≤ 0.3, where 𝑉𝑛−0 = 𝑉𝑛 is the NMD of the initial linear portion of
the plot at the maximum load 𝑃 = 𝑃𝑚𝑎𝑥 (see Fig. 1) and ∆𝑉𝑛 is the additional NMD up to the
maximum load 𝑃𝑚𝑎𝑥 . The single-edge notches (i.e. the pre-cracks) in the modified compacttension (CT) specimens suggested by ASTM E1922 [55] are introduced by using a diamond
wheel cutter with a thickness of 0.3 mm. Note that the maximum allowable size of the precrack mouth based on ASTM E1922 [55] is equal to 0.015W, where W is the specimen width.
Considering that W is equal to 50 mm in the present specimens, the limiting value is obtained
equal to 0.75 mm and hence, the introduced pre-crack mouth size (i.e. 0.3 mm) is valid.
Fig. 1. Typical load versus notch-mouth-displacement plot in ASTM E1922 [55].
The trans-laminar fracture toughness 𝐾𝑇𝐿 is calculated from the maximum load
recorded by the tensile testing machine by using the equations established on the basis of the
8
linear elastic stress analysis of the modified single-edge notched specimens. To verify the
validity of the trans-laminar tests, the important criterion mentioned in ASTM E1922 [55] is
checked and it is found that the trans-laminar tests for all of the lay-up configurations satisfy
that criterion. For more clarity, with consideration of the values of the two parameters 𝑉𝑛−0
and ∆𝑉𝑛 equal to 1.8 mm and 0.3 mm, respectively for the quasi-isotropic laminate [0/90/
±45]𝑠 , it is evident that the ratio ∆𝑉𝑛 / 𝑉𝑛−0 is calculated to be 0.16, which is adequately
lower than the limiting value 0.3 (see Fig. 2b).
Two samples of the stress–strain and load–notch-mouth-displacement curves obtained
from the standard tests according to ASTM D3039 [54] and ASTM E1922 [55], respectively
on the laminated composite specimens are represented in Fig. 2. The bulk mechanical
properties of each lay-up configuration are also presented in Table 1.
1000
(a)
stress (MPa)
800
600
400
200
0
0.000
0.005
0.010
0.015
strain
9
0.020
0.025
∆𝑉𝑛 =0.3 mm
𝑉𝑛−0 = 1.8 mm
10
𝑃𝑚𝑎𝑥
(b)
Load (kN)
8
6
4
2
0
0.0
0.5
1.0
1.5
2.0
2.5
Notch-mouth-displacement (mm)
Fig. 2. Stress˗strain and Load˗notch-mouth-displacement curves according to (a): ASTM D-3039 and (b): ASTM E-1922
corresponding to unidirectional ([0]16 ) and quasi-isotropic ([0/90/±45]𝑠 ) laminates.
Table 1. Mechanical properties of various lay-up configurations tested.
Material property
Unidirectional
Cross-ply
Quasi-isotropic
8-ply
16-ply
8-ply
16-ply
8-ply
16ply
Ultimate tensile strength (𝝈𝒖 ) [MPa]
858
876
489
498
425
442
Trans-laminar fracture toughness (𝑲𝑻𝑳 ) [MPa√𝒎]
51.2
47.8
39.2
36.5
42.6
40.2
To check the repeatability of the experiments, three tests are conducted for each layup configuration. Also, to ensure the accuracy and reliability of the experimental results and
also, to check the likely dependence of the Virtual Isotropic Material Concept (VIMC) on the
laminate thickness, all of the ultimate tensile strength and the trans-laminar fracture
toughness tests are repeated for 16-ply laminated specimens. Totally, 36 tests of laminate
characterization are performed; 18 tests for 8-ply and 18 for 16-ply specimens. The average
values of 𝜎𝑢 and 𝐾𝑇𝐿 are summarized in Table 1 for various lay-up configurations and
different ply numbers. As can be seen in Table 1, the values for 8-ply laminates are different
from those for 16-ply laminates. The reason for this behavior is discussed in Section 6.
The obvious fully linear curve up to the peak point in Fig. 2a and the negligible
nonlinear portion in Fig. 2b until the peak point prove that the laminates exhibit brittle
10
manner in the presence and absence of pre-existing defects. Hence, the measured 𝜎𝑢 and 𝐾𝑇𝐿
values are valid and they can be used in various brittle fracture criteria for predicting failure
in glass/epoxy laminated composites.
In order to compute the value of the trans-laminar fracture toughness 𝐾𝑇𝐿 for each
lay-up configuration, the maximum load recorded in the load−notch-mouth-displacement
curve (see Fig. 2b) can be substituted into the closed-form expression of the mode I stress
intensity factor (SIF) 𝐾𝐼 given by ASTM E1922 [55] and the corresponding critical value can
be computed. If the laboratory-scaled pre-cracked specimen geometry does not meet ASTM
E1922 exactly, the value of 𝐾𝑇𝐿 can be calculated by directly applying the maximum load to
the finite element (FE) model of pre-cracked specimen and computing the mode I SIF under
the linear-elastic conditions.
2.2.
Fracture testing of glass/epoxy laminates weakened by U-notches
The specimen used in this study to perform the fracture tests on U-notched composite
laminates is a rectangular plate with a central bean-shaped slit with two U-ends as shown in
Fig. 3. The specimen length (L) and width (W) are constant and equal to 160 mm and 50 mm,
respectively. The total slit length (i.e. the tip-to-tip distance) 2a is also kept constant and
equal to 25 mm. The specimen thickness depends trivially on the number of layers in the
laminated composite tested. With the aim to investigate the effects of the notch tip radius 𝜌
on the load-carrying capacity (LCC) of the U-notched specimens, various notch tip radii 𝜌 =
1, 2, 4 𝑚𝑚 are considered in the experiments. Fig. 3 represents the U-notched test specimen
with its geometrical parameters schematically. A high-precision water-jet cutting machine is
utilized to fabricate the U-notched specimens from the composite plates. To polish the
notches as the cut surfaces, brass rods are utilized. These rods are used in a drill press with 25
𝜇𝑚 lapping compound followed by 9 𝜇𝑚 diamond suspension compound that provide
smooth and defect-free notches. The parameters 𝜌, 2a, L, W, and P in Fig. 3 are the notch tip
11
radius, twice the notch length (i.e. the total bean-shaped slit length), the specimen length, the
specimen width, and the remotely applied tensile load, respectively. Some of the U-notched
composite specimens are shown in Fig. 4. As shown in Fig. 4, the bean-shaped slits are cut
from the center of the specimens such that the bisector line of the U-ends is perpendicular to
the applied load direction.
Fig. 3. Schematic of the U-notched rectangular specimen with its geometrical parameters.
Fig. 4. Some U-notched rectangular laminated composite specimens with various notch tip radii.
The experimentally obtained last-ply-failure (LPF) loads of the U-notched laminated
specimens for the three lay-up configurations are summarized in Tables 2-4. Considering the
three notch tip radii, three lay-up configurations, repeating each test three times, and two
12
numbers of ply, 54 fracture tests are totally carried out on the U-notched composite
specimens. As is evident in the three Tables, the averaged LPF load increases as the notch tip
radius increases from 𝜌=1 mm to 𝜌=4 mm, because of the decrease of the stress concentration
in the notch tip vicinity. Moreover, it can be seen in Tables 2-4 that for a specific notch tip
radius 𝜌, as the number of plies changes from 8 to 16 (i.e. the laminate thickness becomes
larger by approximately a factor of 2), the LPF load does not grow by a factor of 2. The
reason for this phenomenon is discussed in Section 6.
Fig. 5 shows three U-notched laminated specimens with 𝜌=1 mm during loading (a)
and 𝜌=2 mm and 𝜌=4 mm after failure (b, c). Two sample load-displacement curves obtained
from testing the U-notched glass/epoxy laminated specimens under pure mode I loading
conditions are also depicted in Fig. 6. As can be seen in Fig. 6, the load-displacement curves
are linear up to the LPF and fracture occurs abruptly.
Table 2. The experimentally obtained LCCs of the U-notched laminated specimens
for the unidirectional laminates [0]8 and [0]16 .
Number of layers
𝝆 (𝒎𝒎)
𝑷𝟏 (𝑵)
𝑷𝟐 (𝑵)
𝑷𝟑 (𝑵)
𝑷𝒂𝒗𝒈 (𝑵)
1
8-ply
21800
22900
26400
23700
2
8-ply
25800
28200
24900
26300
4
8-ply
26500
27100
28600
27400
1
16-ply
41100
38900
45100
41700
2
16-ply
42500
44300
44900
43900
4
16-ply
45100
44800
42700
44200
Table 3. The experimentally obtained LCCs of the U-notched laminated specimens
for the quasi-isotropic laminates [0/90/+45/−45/]s and [0/90/+45/−45/]2s .
Number of layers
𝝆 (𝒎𝒎)
𝑷𝟏 (𝑵)
𝑷𝟐 (𝑵)
𝑷𝟑 (𝑵)
𝑷𝒂𝒗𝒈 (𝑵)
1
8-ply
12600
13800
16500
14300
2
8-ply
15700
17300
17700
16900
4
8-ply
16800
17900
17300
17200
1
16-ply
25900
27300
27200
26800
2
16-ply
28800
28400
30100
29100
4
16-ply
29100
29500
30950
29850
13
Table 4. The experimentally obtained LCCs of the U-notched laminated specimens
for the cross-ply laminates [0/90/0/90]s and [0/90/0/90]2s .
Number of layers
𝝆 (𝒎𝒎)
𝑷𝟏 (𝑵)
𝑷𝟐 (𝑵)
𝑷𝟑 (𝑵)
𝑷𝒂𝒗𝒈 (𝑵)
1
8-ply
15800
16100
17600
16500
2
8-ply
18900
19300
16700
18300
4
8-ply
19400
19000
18300
18900
1
16-ply
25800
26900
26800
26500
2
16-ply
29700
31200
31800
29800
4
16-ply
30400
31200
29900
30900
(a) 𝜌 = 1 𝑚𝑚
(b) 𝜌 = 2 𝑚𝑚
(c) 𝜌 = 4 𝑚𝑚
Fig. 5. The U-notched laminated specimens; (a) during loading and (b, c) after failure.
50
Load (kN)
40
30
20
10
0
0.0
0.5
1.0
1.5
Displacement (mm)
(a)
14
2.0
2.5
50
Load (kN)
40
30
20
10
0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Displacement (mm)
(b)
Fig. 6. The load-displacement curves for 16-ply unidirectional laminated composite specimens;
(a) 𝜌 = 1 𝑚𝑚 and (b) 𝜌 = 4 𝑚𝑚.
In Section 6, a full discussion is provided regarding the experimental observations on
failure of U-notched composite specimens (see for instance Fig. 5 (b, c)). Moreover, some
pictures captured from the damage zone near the notch tip by means of the scanning electron
microscope (SEM) are presented and discussed.
3.
The Virtual Isotropic Material Concept
As widely reported in literature and with paying a careful attention to the overall
behavior of the tested U-notched composite specimens, the linearity of the load-displacement
curves is confirmed. This important manner is a basis to develop a novel concept, called the
Virtual Isotropic Material Concept (VIMC). In this new concept, an equivalent brittle
material, which is a virtual isotropic material, is defined and employed for simulating the
bulk behavior of the tested laminated composites. This means that the real laminated
composite material having generally orthotropic behavior is equated with a virtual brittle
material having isotropic behavior for predicting the last-ply-failure (LPF) load of the Unotched laminated composite specimens by means of using directly the linear elastic notch
fracture mechanics (LENFM) principles. It is noteworthy that the VIMC does not require
15
performing experiments for measuring the longitudinal (𝐸𝑥 ), lateral (𝐸𝑦 ) and shear (𝐺𝑥𝑦 )
moduli, consuming time and cost. Only, two important characteristics of the composite
laminate, namely the ultimate tensile strength (𝜎𝑢 ) and the trans-laminar fracture toughness
(𝐾𝑇𝐿 ), are essential in VIMC. As mentioned before, 𝜎𝑢 and 𝐾𝑇𝐿 can be obtained from two
distinct tests according to ASTM D3039 [54] and ASTM E1922 [55], respectively.
Fig. 7 represents the VIMC schematically. As seen in Fig. 7, a laminated composite
material with orthotropic behavior having several material parameters, e.g. 𝐸𝑥 , 𝐸𝑦 , 𝐺𝑥𝑦 , 𝜈𝑥𝑦 ,
𝐾𝑇𝐿 , 𝜎𝑓 (fracture stress) etc., is equated with a virtual material with isotropic behavior having
the elastic modulus E equal to 𝐸𝑥 or 𝐸𝑦 , depending on the direction of loading on the
composite laminate, the fracture toughness 𝐾𝐼𝐶 or 𝐾𝐶 , and the ultimate tensile strength 𝜎𝑢 . It
has been well established in literature that in all of brittle fracture criteria in the context of
LENFM, 𝜎𝑢 and 𝐾𝐼𝐶 are the two significant material properties for failure prediction of
notched components.
≅
(a) Orthotropic laminated composite with
(b)
𝐸𝑥 , 𝐸𝑦 , 𝐺𝑥𝑦 , 𝜈𝑥𝑦 ,𝐾𝑇𝐿 , 𝜎𝑓 etc.
Linear elastic isotropic brittle material with
E, 𝜈, 𝐾𝐼𝐶 (𝑜𝑟 𝐾𝐶 ) , 𝜎𝑢
Fig. 7. The Virtual Isotropic Material Concept (VIMC) schematically.
Now, the U-notched laminated composite specimens tested in Section 2 are virtually
considered as the isotropic homogeneous specimens with exactly the same geometries,
16
making it possible to avoid the first-ply-failure (FPF) analysis and the progressive damage
modeling (PDM) for the LPF prediction and instead, to predict the LPF directly and simply
by linking VIMC to LENFM criteria. It should be necessary to note that approximately all of
the presented failure or damage models for predicting the strength of the laminated composite
specimens have utilized layer-by-layer analyses so far. But this new concept does not need to
perform conventional procedures for prediction of strength of laminated composite specimens
especially containing stress raisers.
The first significant parameter, i.e. the ultimate tensile strength, is normally defined
for both isotropic and orthotropic materials as a material property or an inherent property
with the difference that this parameter for orthotropic materials is dependent on the applied
load direction. So, for correct prediction of LPF of U-notched laminated specimens, the
direction along which the tensile strength of un-notched laminate is measured should be
perpendicular to the notch bisector line in the U-notched laminate. Moreover, by introducing
the pre-crack in the composite laminate along a direction the same as the direction of the Unotch bisector line, the trans-laminar fracture toughness (𝐾𝑇𝐿 ) obtained from ASTM E1922
[55] can be utilized instead of the fracture toughness (𝐾𝐼𝐶 ) in LENFM brittle fracture criteria.
It should be noted that the fracture mechanics related failure modes in laminated
composites are delamination, intra-laminar fracture, and trans-laminar fracture. Delamination
and intra-laminar fracture have been under extensive investigations for several years [56-63].
In spite of the fact that the importance of trans-laminar fracture toughness measurement was
recognized many years ago, it has received relatively little attention from the scientific
community until now. Laffan [64] has stated that this little attention is at least partially due to
a lack of confidence in composites, resulting in them not being used in primary structures.
The latest failure mode corresponding to the ultimate tensile strength in laminated composites
is the fiber breakage; thus, conducting the trans-laminar fracture toughness test for measuring
17
the capacity of the laminated composite weakened by a crack to withstand against breakage is
necessary.
In next section, two well-known brittle fracture criteria in the context of LENFM,
namely the maximum tangential stress (MTS) and mean stress (MS) criteria, are briefly
described and linked to VIMC to form two novel combined criteria to predict the
experimentally obtained LCCs of the U-notched laminated composite specimens reported in
Tables 2-4.
4.
Brittle fracture criteria
4.1.
Maximum tangential stress criterion
The maximum tangential stress (MTS) criterion is a well-known brittle fracture
criterion originally developed by Erdogan and Sih [47] for estimating mixed mode I/II failure
of cracked members. This criterion has been extended to U-notched domains by Ayatollahi
and Torabi in 2009 [48] and its effectiveness has been well demonstrated. According to Unotch MTS (U-MTS) criterion, brittle fracture occurs in a U-notched component under pure
mode I loading, when the value of tangential stress at a specified critical distance 𝑟𝑐 ahead of
the notch tip attains the material critical stress 𝜎𝑐 . The first expression for 𝑟𝑐 has been
proposed by Ritchie et al. [65] as follows
𝑟𝑐 =
1 𝐾𝐼𝑐 2
( )
2𝜋 𝜎𝑐
(1)
where 𝐾𝐼𝑐 is the plain-strain fracture toughness of material. The critical stress 𝜎𝑐 has also
been known in literature as the inherent strength 𝜎0 , proposed by Taylor [66]. It has been
stated in [66] that for most of brittle materials, 𝜎0 can be considered equal to the ultimate
tensile strength 𝜎𝑢 and for some materials like polymers, it can be larger than 𝜎𝑢 , even
several times.
18
4.2.
Mean stress criterion
The U-notch mean stress (UMS) criterion suggests that brittle fracture occurs under
mode I loading, when the average value of tangential stress over a particular critical distance
𝑑𝑐 ahead of the notch tip reaches 𝜎𝑐 [40]. The critical distance has been originally suggested
by Seweryn [67] which has been frequently used by many researchers to predict brittle failure
in notched members. The 𝑑𝑐 expression is [67]
𝑑𝑐 =
2 𝐾𝐼𝑐 2
( )
𝜋 𝜎𝑐
(2)
Comparing Eqs. 1 and 2, it is evident that 𝑑𝑐 is four times 𝑟𝑐 .
In order to employ UMTS and UMS criteria for predicting LCC of the U-notched
laminated composite specimens (i.e. the LPF prediction), they should be combined with the
VIMC. Hence, the VIMC-MTS and VIMC-MS combined criteria are utilized in forthcoming
sections for predicting the experimentally obtained LCCs. The most important point is that
the critical distances 𝑟𝑐 and 𝑑𝑐 are equal to
𝑟𝑐 =
1
2𝜋
( 𝜎𝑇𝐿 )2
𝐾
𝑑𝑐 =
2
𝜋
( 𝜎𝑇𝐿 )2
𝑢
𝐾
𝑢
for VIMC-MTS
(3)
for VIMC-MS
(4)
where 𝐾𝑇𝐿 and 𝜎𝑢 are the trans-laminar fracture toughness and the ultimate tensile strength of
the laminated composite material. The values of 𝑟𝑐 and 𝑑𝑐 for different lay-up configurations
tested are presented in Table 5. These values are simply calculated by using the data provided
in Table 1.
19
Table 5. The values of the critical distances 𝑟𝑐 and 𝑑𝑐 for various lay-up configurations.
Unidirectional
Critical distances
Cross-ply
Quasi-isotropic
8-ply
16-ply
8-ply
16-ply
8-ply
16-ply
𝒓𝒄 (𝒎𝒎)
0.5
0.4
1
0.8
1.5
1.3
𝒅𝒄 (𝒎𝒎)
2
1.6
4
3.2
6
5.2
At the first glance, the presented new equations have been utilized in the past for
similar brittle criterions whereas the new developed equations for critical distances have been
modified by appropriate parameters which have more influence in strength prediction
correctly.
In next section, the finite element (FE) analysis is performed and the failure concepts
of the VIMC-MTS and VIMC-MS criteria are directly applied to the stress distributions
resulted from the FE analysis for predicting the LPF of the U-notched composite specimens.
5.
Finite element analysis
In order to implement the VIMC-MS and VIMC-MTS criteria for LPF prediction of
U-notched composite specimens, two-dimensional (2D) finite element (FE) models of the Unotched specimens are created and analyzed under pure mode I loading conditions. It is
necessary to note that one of the advantages of VIMC is the independency of the LPF
prediction from the layer-by-layer modeling and failure analysis of the notched laminated
composite. Hence, the FE models are considered to be virtually made of the isotropic
material. The analyses are performed under the plane-stress assumption due to small
thickness of the specimens compared to the length and width of them. The specimens are
simulated and meshed in the well-known commercial FE software ABAQUS 6.14. Moreover,
the models are meshed with the eight-node plane-stress quadratic elements. In Fig. 8a, a
structured mesh pattern produced with careful partitioning for the whole model is shown.
20
Also, as seen in Fig. 8 (b, c), refined meshes with the minimum size of about 0.006 mm are
used at the notch neighborhood due to the high level of stress gradient. About 118074
elements are totally utilized in each FE model.
As mentioned in Section 4, the tangential stress distribution on the notch bisector line
is needed for LCC prediction, which can be obtained directly from the FE analysis.
Now, the critical loads of the U-notched laminated composite specimens can be
theoretically predicted by using the VIMC-MTS and VIMC-MS criteria. For this purpose,
first, an arbitrary load equal to 2000 N is applied to the FE model and the tangential stress
distribution on the notch bisector line is obtained. Now, the value of tangential stress at the
critical distance 𝑟𝑐 reported in Table 5 (called 𝜎𝜃𝜃 ) is computed. According to VIMC-MTS
criterion, fracture occurs when the tangential stress at 𝑟𝑐 attains the ultimate tensile strength
of the un-notched laminated composite 𝜎𝑢 reported in Table 1. Therefore, because of the
linearity of the analyses, the fracture load predicted by VIMC-MTS criterion can be easily
given as
𝑃𝑉𝐼𝑀𝐶−𝑀𝑇𝑆 =
𝜎𝑢
∗ 2000 𝑁
𝜎𝜃𝜃
(5)
The procedure of LCC prediction for the U-notched composite specimens in accordance with
VIMC-MS criterion is the same as that described above for VIMC-MTS criterion with the
main difference that, the average of the tangential stresses over the specified critical distance
𝑑𝑐 (called ̅̅̅̅̅)
𝜎𝜃𝜃 should reach the ultimate tensile strength of the laminate (𝜎𝑢 ). Therefore, LPF
load of the U-notched composite samples can be predicted by VIMC-MS criterion as
𝑃𝑉𝐼𝑀𝐶−𝑀𝑆 =
𝜎𝑢
∗ 2000 𝑁
𝜎𝜃𝜃
̅̅̅̅̅
(6)
The failure concepts of VIMC-MTS and VIMC-MS criteria are schematically represented in
Fig. 9. Additionally, the distribution of the tangential stress contours in the vicinity of Unotches in the tested specimens with 3 mm thickness is depicted in Fig. 10 for a constant
21
tensile load equal to 20 kN for two different notch tip radii. The butterfly wing shape in Fig.
10a is quiet symmetric, showing that the U-notched specimen is under pure mode I loading.
(b)
(a)
(c)
Fig. 8. A sample mesh pattern for the tested U-notched specimens; (a) the whole specimen, (b) the vicinity of notch border,
and (c) very close to the notch round border.
𝒓𝒄
A
B
C
(a) VIMC-MTS concept
𝒅𝒄
A
B
D
(b) VIMC-MS concept
Fig. 9. The failure concepts of VIMC-MTS (a) and VIMC-MS (b) criteria. The points A and B denote the center of curvature
of notch and the notch tip, respectively. The point C specifies the point at which the tangential stress is monitored. The line
BD is also the distance over which the tangential stress is monitored.
22
(a)
(b)
Fig. 10. Contours of the tangential stress in the vicinity U-notches in the tested specimens with 3 mm thickness for a load
equal to 20 kN; (a) 𝜌 = 1 𝑚𝑚 and (b) 𝜌 = 2 𝑚𝑚.
6.
Results and discussion
The variations of the last-ply-failure (LPF) load of the U-notched laminated
specimens versus the notch tip radius are presented in Figs. 11-16 for the three lay-up
configurations and two numbers of ply. The discrepancies between the experimental and
theoretical results for 8-ply and 16-ply specimens are also presented in Tables 6 and 7,
respectively. It is obvious from Figs. 11-16 and Tables 6 and 7 that both VIMC-MTS and
VIMC-MS criteria can successfully predict the experimentally obtained LCCs of the Unotched laminated composite specimens under pure mode I loading conditions.
Considering all the notch tip radii, the lay-up configurations, and the numbers of ply,
the average discrepancies of VIMC-MTS and VIMC-MS criteria are obtained equal to
approximately 8.7% and 9.8%, respectively, indicating that both criteria are generally
23
successful in predicting LCCs of the U-notched glass/epoxy composite laminates tested.
Since the predictions by the new concept are good for various lay-up configurations and
different numbers of ply, it seems that the two combined criteria based on the LENFM are
independent of the glass/epoxy laminate properties. Remembering that the LPF prediction in
notched laminated composites by the existing models, like the progressive damage models,
are very complicated, time-consuming, and not user-friendly, the easy, fast, and user-friendly
aspects of VIMC-MTS and VIMC-MS criteria shine.
Table 6. The discrepancies between the theoretical and experimental results for the
U-notched 8-ply laminated composite specimens.
Discrepancy for VIMC-MTS criterion (%)
Discrepancy for VIMC-MS criterion (%)
𝝆 (𝒎𝒎)
Unidirectional
Cross-ply
Quasi-isotropic
Unidirectional
Cross-ply
Quasi-isotropic
1
9.1
11.4
13.5
10.2
12.3
14.7
2
8.9
9.8
12.4
7.6
9.8
13.2
4
7.3
8.4
6.9
8.6
9.2
10.6
Table 7. The discrepancies between the theoretical and experimental results for the
U-notched 16-ply laminated composite specimens.
Discrepancy for VIMC-MTS criterion (%)
Discrepancy for VIMC-MS criterion (%)
𝝆 (𝒎𝒎)
Unidirectional
Cross-ply
Quasi-isotropic
Unidirectional
Cross-ply
Quasi-isotropic
1
8.5
8.8
9.7
9.3
7.2
12.4
2
6.7
4.3
10.3
9.1
6.8
11.3
4
5.1
6.7
7.6
7.4
6.3
8.6
24
Last-ply-failure load (kN)
40
30
20
10
VIMC-MTS criterion
VIMC-MS criterion
Experiments
0
0
1
2
3
4
5
Notch tip radius (mm)
Fig. 11. Variations of the last-ply-failure (LPF) load of the U-notched 8-ply unidirectional laminated specimen versus the
notch tip radius for the two combined criteria and the experimental data.
Last-ply-failure load (kN)
30
25
20
15
10
VIMC-MTS criterion
VIMC-MS criterion
Experiments
5
0
0
1
2
3
4
5
Notch tip radius (mm)
Fig. 12. Variations of the last-ply-failure (LPF) load of the U-notched 8-ply quasi-isotropic laminated specimen versus the
notch tip radius for the two combined criteria and the experimental data.
25
Last-ply-failure load (kN)
25
20
15
10
VIMC-MTS criterion
VIMC-MS criterion
Experiments
5
0
0
1
2
3
4
5
Notch tip radius (mm)
Fig. 13. Variations of the last-ply-failure (LPF) load of the U-notched 8-ply cross-ply laminated specimen versus the notch
tip radius for the two combined criteria and the experimental data.
Last-ply-failure load (kN)
60
50
40
30
20
VIMC-MTS criterion
VIMC-MS criterion
Experiments
10
0
0
1
2
3
4
5
Notch tip radius (mm)
Fig. 14. Variations of the last-ply-failure load (LPF) of the U-notched 16-ply unidirectional laminated specimen versus the
notch tip radius for the two combined criteria and the experimental data.
26
Last-ply-failure load (kN)
50
40
30
20
VIMC-MTS criterion
VIMC-MS criterion
Experiments
10
0
0
1
2
3
4
5
Notch tip radius (mm)
Fig. 15. Variations of the last-ply-failure load (LPF) of the U-notched 16-ply cross-ply laminated specimen versus the notch
Last-ply-failure load (kN)
tip radius for the two combined criteria and the experimental data.
40
30
20
10
VIMC-MTS criterion
VIMC-MS criterion
Experiments
0
0
1
2
3
4
5
Notch tip radius (mm)
Fig. 16. Variations of the last-ply-failure load (LPF) of the U-notched 16-ply quasi-isotropic laminated specimen versus the
notch tip radius for the two combined criteria and the experimental data.
It is obvious from Tables 6 and 7 that in both quasi-isotropic laminated composite
members fabricated with 8-ply and 16-ply, the average discrepancies are more than other layup configurations. One of the main important reasons is maybe that the quasi-isotropic
laminated composite specimens have various fiber directions encountering U-notch border.
27
This variety causes that the portion of maximum tangential stress to be distributed along
±45° fiber directions. Since the global approach of novel concept (VIMC) is equality at the
macroscopic level and also the presented lay-up configurations are symmetric, the maximum
tangential stresses taken from FE analyses are synchronized with desired critical distances.
Another important point is that the discrepancies of 8-ply in all of lay-up
configurations are obviously more than 16-ply ones. As the engineers know, many researches
have been done about size and scale effects of experimental specimens on theoretical
predictions. The investigators try to reach a comprehensive approach about independency of
presented criteria from size and scale of tested components either orthotropic or isotropic
materials. The almost all of the comparisons between experimental and theoretical
observations have shown that the important source of errors is related to the various
thicknesses of the test specimens. As the thickness of test specimens increases the
discrepancies between theoretical and experimental criteria would be decreased. This fact has
been proved in this paper correctly. Although, different limiting values exist for various
materials beyond which the increase of thickness does not influence on final results.
Although the prediction of the LPF load of the U-notched laminated composite
specimens is the important aim of this study at the macroscopic level, the microscopic
investigation of the damage zone at the notch neighborhood is essential to better recognize
the failure modes. To this end, several micrographs are taken from the damage zone by
means of the scanning electron microscope (SEM) and interpreted. Visual observations
during the fracture tests are also reported and discussed.
As previously mentioned, the three fracture mechanics related modes of failure in
laminated composites are delamination, intra-laminar fracture, and trans-laminar fracture.
According to [68] in which the techniques for measuring the trans-laminar fracture toughness
28
experimentally associated with trans-laminar (fiber-breaking) failure modes of laminated
composites have been reviewed, the critical damages due to trans-laminar fracture tests are
the fiber-matrix damages and consequently, the fiber pull out or fiber breakage. The fibermatrix damages can occur longitudinally or transversely depending on the position of the
initial crack or cracks. Also, Liu et al. [68] have stated that in the presence of notches in
laminated composites, the conventional damages in un-notched laminated composites such as
the longitudinal cracks (splits) and free-edge delaminations change to the notch-induced
splits (NISs) and the notch-induced delaminations (NIDs). Moreover, they have stated that
the global damages are localized due to the notch presence in laminates and the subcritical
damage states depend on the type of lay-up configuration and the notch size.
With this prelude, according to Fig. 17, the small damages are localized at the
periphery of the U-notch and the clear differences in the shape of bulk damage between the
three lay-up configurations are seen. As can be seen in Fig. 17, two vertical lines form in the
vicinity of the U-notch round border in both sides of the central bean-shaped slit for all of the
three lay-up configurations. However, this phenomenon in the unidirectional and cross-ply
laminates is more visible than that in the quasi-isotropic laminate. In the quasi-isotropic
laminates, the lines are oblique as depicted in Fig. 17b. As shown in Fig. 17, the initial
longitudinal crack is observed in the notch tip vicinity without exception in the lay-up
configuration and consequently, the damage zone propagates across the width of specimens
(i.e. the horizontal axis). Because of the pure mode I loading conditions, as expected, the first
micro-cracks or splits initiate at 𝜃 = 5° − 10° (𝜃 = 0° specifies the notch bisector line) as
represented in Fig.17. The meaning of the initial longitudinal crack is the matrix-cracking
propagation along the fiber due to resin brittleness which occurs suddenly after the first fiber
breakage. If the test continues, considering that the last-ply-failure (LPF) corresponds to
failure of 0° ply aligned with the loading direction, another longitudinal crack propagation
29
smaller than the initial one forms in matrix in the first step after the maximum load record. It
should be explained that during the fracture test, the monotonic load increases until it
suddenly drops from the peak to a significantly lower load, via a loud sound relating to the
fiber breakage. After the fiber breakage, micro-cracks or notch-induced splits (NISs) bond
together and propagate vertically between fiber/matrix bonding. It should be noted that the
NIS and notch-induced delamination (NID) are two kinds of damage previously introduced
by Liu et al. [68] for notched laminated composite specimens, especially for notched crossply laminates. The NID is significantly visible at the notch neighborhood for unidirectional
and cross-ply laminates, while the NIS for quasi-isotropic laminates. These failure evidences
with their locations in the tested U-notched composite specimens are depicted in Fig. 17.
Vertical line
Notch induced splits
Oblique line
Notch
induced
delamination
Fiber
Notch
induced
splits
breakage
Notch
induced
delamination
(a)
(b)
Vertical line
Notch
induced
delamination
(c)
Fig. 17. The close view of damage zone near the notch tip for the three lay-up configurations tested; (a) unidirectional, (b)
quasi-isotropic, and (c) cross-ply laminates.
30
Now, for more clarity, two SEM photographs taken in different scales from the
damage zone of the composite specimens tested are depicted in Fig. 18. According to Fig.
18a, the U-notched laminated specimen is broken like a bamboo. The notch domain is
divided into several strips, for which the locations of the fracture surface are different. Under
such a situation, the strips are broken at their weak portions, leading to different locations of
the fracture surface of the strips. The primary damages in the notched laminated composites
are located at the periphery of the notch blunt border and constricted in a small zone. On the
other hand, if the damage zone ahead of the notch tip in laminated composites is not large
with respect to the ligament size, the fracture mechanics criteria can be applied for LPF
prediction. As can be seen in Fig. 18a for the unidirectional laminate, the individual shapes of
bamboo have many fibers bundle that some of which indicate the fiber pull-out mode arising
from the weak fiber/matrix interface and subsequently, interfacial de-bonding would happen.
Generally, in laminated composites with brittle fiber and matrix, three various failure modes,
namely the matrix cracking, the shear crack initiation, and the fiber breakage are seen in most
of experimental investigations. As seen in Fig. 18b for the cross-ply laminate, the
longitudinal matrix cracking is resulted from the coupling of splitting and micro-cracks in 90°
ply failure and subsequently in 0° ply failure. It is also seen in Fig. 18b that the two fibers of
the 90° ply located at the left of the picture are pulled out and then broken.
Fiber pull out
Fiber pull out
Bamboo effects
Bamboo
effects
(a) Unidirectional laminate
31
Fibers pull out
Matrix
cracking
(b) Cross-ply laminate
Fig. 18. Two SEM photographs taken from the damage zone of the U-notched laminated specimens; (a) unidirectional
laminate and (b) cross-ply laminate.
As mentioned in subsection 2.1, the values of the ultimate tensile strength 𝜎𝑢 and the
trans-laminar fracture toughness 𝐾𝑇𝐿 for 8-ply laminates are different from those for 16-ply
laminates. Regarding 𝜎𝑢 , it is seen in Table 1 that for all the three lay-up configurations, the
discrepancy between the values for 8-ply and 16-ply laminates is approximately equal to 1.5
%, which is quite acceptable compared to usual error bands defined for similar experiments.
This discrepancy could be due to the small difference in specimen width and thickness, and
also due to natural errors that happen during the experiments (e.g. human errors, machine
errors etc.). Dealing with 𝐾𝑇𝐿 , however, considerable differences are seen between the values
for 8-ply and 16-ply laminates which may not be attributed to natural testing errors. It has
been reported in ASTM E399 [69] for metallic materials that the thickness-independent
plane-strain fracture toughness of material (𝐾𝐼𝑐 ) can be obtained from the standard test
𝐾
method if the pre-cracked specimen thickness becomes larger than 2.5( 𝜎𝐼𝑐)2 where 𝜎𝑦 is the
𝑦
yield strength of material. Analogous to metallic materials, a critical thickness has also been
proposed in some papers published in literature [70, 71] beyond which the trans-laminar
fracture toughness of material (𝐾𝑇𝐿 ) is independent of thickness. This means that as the
laminate thickness increases, the value of 𝐾𝑇𝐿 obtained from the test decreases until the
thickness reaches the critical thickness. In [70], for graphite/epoxy laminates, it has been
32
reported that 𝐾𝑇𝐿 value does not change when the number of plies is greater than 64. Similar
statements have also been reported in other references, e.g. [72]. Considering 64 as a critical
ply number, in the present glass/epoxy composite laminates, it means a laminate with
approximately 20 mm thickness. Noting that the thicknesses of both 8-ply and 16-ply
laminates tested in the present study are significantly below 20 mm, it is expected that the
values of 𝐾𝑇𝐿 for 8-ply laminates become larger than those for 16-ply laminates, as truly
obtained and presented in Table 1.
It was mentioned in subsection 2.2 that for U-notched laminated composite
specimens, the value of the LPF load does not grow by a factor of 2, as the specimen changes
from 8-ply to 16-ply. It should be noted that the failure behaviors of notched and un-notched
specimens are fundamentally different. In an un-notched rectangular tensile test specimen,
the whole cross-sectional area participates in load-carrying process and failure usually
happens uniformly over the cross-section. For such a specimen, it is truly expected that by
increasing the specimen thickness by a factor of 2, the failure load also increases by the same
factor. However, in a notched rectangular specimen, the stress is concentrated in the notch
vicinity and only a specific part of the cross-sectional area, called the fracture process zone
(FPZ), participates in load-carrying process and hence in the failure of the notched specimen.
As explained in the paragraph above, the fracture phenomenon in pre-cracked laminated
composite specimens is significantly dependent on the laminate thickness. On the other hand,
the fracture behaviors of cracked and notched laminates are very similar and the only
difference is the level of stress concentration and hence, the size of the FPZ. As a
consequence, the notch fracture toughness (NFT) is also dependent on the laminate thickness.
For the U-notched composite specimens tested, by increasing the specimen thickness (i.e.
changing 8-ply to 16-ply) the FPZ size decreases, meaning that lower amount of energy is
absorbed by the notched laminate until LPF. Obviously, this means that the U-notched 16-ply
33
laminate fails at an ultimate tensile load considerably lower than twice the failure load of the
8-ply laminate weakened by the same notch.
7.
Conclusions
The U-notched laminated glass/epoxy composite specimens with three various notch
tip radii, three types of lay-up configuration, and two numbers of ply were fabricated and
tested for failure under pure mode I loading conditions, and the last-ply-failure (LPF) loads
were experimentally recorded. With the aim to avoid using complex, time-consuming, and
not user-friendly models of LPF load prediction, e.g. the progressive damage models that
need ply-by-ply failure analysis, a novel concept, called the Virtual Isotropic Material
Concept (VIMC) was proposed and combined with two brittle fracture criteria, namely the
maximum tangential stress (MTS) and mean stress (MS) criteria for predicting the
experimentally obtained LPF loads. It was demonstrated that both VIMC-MTS and VIMCMS criteria could estimate the experimental results well.
In general, the VIMC-MS criterion could predict the experimental results a bit better
than the VIMC-MTS criterion, but no meaningful difference was seen between the accuracies
of the two criteria. By using the scanning electron microscope (SEM) micrographs and some
close-up pictures taken from the periphery of the damaged U-notches, the major failure
modes were specified. As expected, the matrix cracking mode due to shear and transverse
stresses concentration and subsequently the fiber breakage mode occurred. By using the
VIMC-MTS and the VIMC-MS criteria, one can accurately predict the LPF load of Unotched laminated glass/epoxy composites under pure mode I loading conditions with using
only two distinct laminate parameters, namely the trans-laminar fracture toughness and the
ultimate tensile strength.
34
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