Interfacial Fracture Toughness of Multilayer Composite Structures
The interfaces in multilayer composite structures are susceptible to delamination due to the combination of active tensile and shear loads under operating conditions. A four-layer center crack composite beam in four-point bending is simulated to determine the interfacial fracture energy of the multi...
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| Cite this: | Interfacial Fracture Toughness of Multilayer Composite Structures / S.-C. Her, W.-B. Su // Проблемы прочности. — 2015. — № 1. — С. 212-217. — Бібліогр.: 10 назв. — англ. |
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nasplib_isofts_kiev_ua-123456789-1732832025-02-09T23:02:31Z Interfacial Fracture Toughness of Multilayer Composite Structures Сопротивление многослойных композитных структур распространению межфазных трещин Her, S.-C. Su, W.-B. Научно-технический раздел The interfaces in multilayer composite structures are susceptible to delamination due to the combination of active tensile and shear loads under operating conditions. A four-layer center crack composite beam in four-point bending is simulated to determine the interfacial fracture energy of the multilayer structure. The crack is propagating along the interface between the second and third layers. Based on the Euler–Bernoulli theory, the strain energy of the four-layer composite beam is derived. Strain energies before and after the propagation of the interfacial crack are calculated, which results in determining strain energy release rates. Analytical results for those rates are validated with the numerical data obtained by the finite element method. The effect of layer thickness of the composite beam on the interfacial fracture toughness is investigated through a parametric study. В производственных условиях поверхности раздела в многослойных комбинированных структурах проявляют склонность к расслоению в результате комбинации действующих растягивающего усилия и усилия сдвига. Моделируется балка из четырехслойного комбинированного материала с центральной трещиной при четырехточечном изгибе с целью определения энергии разрушения на поверхности раздела многослойной структуры. Трещина развивается вдоль поверхности контакта между вторым и третьим слоями. Величину энергии деформации балки, полученную на основании теории Эйлера Бернулли, рассчитывали до и после распространения трещины по поверхности раздела, что позволило определять скорости выделения энергии деформации. Результаты анализа этих скоростей оценивали с помощью численных данных, полученных методом конечных элементов. Влияние толщины слоя комбинированного материала балки на вязкость разрушения по поверхности контакта изучали при параметрическом исследовании. 2015 Article Interfacial Fracture Toughness of Multilayer Composite Structures / S.-C. Her, W.-B. Su // Проблемы прочности. — 2015. — № 1. — С. 212-217. — Бібліогр.: 10 назв. — англ. 0556-171X https://nasplib.isofts.kiev.ua/handle/123456789/173283 539.4 en Проблемы прочности application/pdf Інститут проблем міцності ім. Г.С. Писаренко НАН України |
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Научно-технический раздел Научно-технический раздел Her, S.-C. Su, W.-B. Interfacial Fracture Toughness of Multilayer Composite Structures Проблемы прочности |
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The interfaces in multilayer composite structures are susceptible to delamination due to the combination of active tensile and shear loads under operating conditions. A four-layer center crack composite beam in four-point bending is simulated to determine the interfacial fracture energy of the multilayer structure. The crack is propagating along the interface between the second and third layers. Based on the Euler–Bernoulli theory, the strain energy of the four-layer composite beam is derived. Strain energies before and after the propagation of the interfacial crack are calculated, which results in determining strain energy release rates. Analytical results for those rates are validated with the numerical data obtained by the finite element method. The effect of layer thickness of the composite beam on the interfacial fracture toughness is investigated through a parametric study. |
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Her, S.-C. Su, W.-B. |
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Her, S.-C. Su, W.-B. |
| author_sort |
Her, S.-C. |
| title |
Interfacial Fracture Toughness of Multilayer Composite Structures |
| title_short |
Interfacial Fracture Toughness of Multilayer Composite Structures |
| title_full |
Interfacial Fracture Toughness of Multilayer Composite Structures |
| title_fullStr |
Interfacial Fracture Toughness of Multilayer Composite Structures |
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Interfacial Fracture Toughness of Multilayer Composite Structures |
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interfacial fracture toughness of multilayer composite structures |
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Інститут проблем міцності ім. Г.С. Писаренко НАН України |
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2015 |
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Научно-технический раздел |
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Interfacial Fracture Toughness of Multilayer Composite Structures / S.-C. Her, W.-B. Su // Проблемы прочности. — 2015. — № 1. — С. 212-217. — Бібліогр.: 10 назв. — англ. |
| series |
Проблемы прочности |
| work_keys_str_mv |
AT hersc interfacialfracturetoughnessofmultilayercompositestructures AT suwb interfacialfracturetoughnessofmultilayercompositestructures AT hersc soprotivleniemnogosloinyhkompozitnyhstrukturrasprostraneniûmežfaznyhtreŝin AT suwb soprotivleniemnogosloinyhkompozitnyhstrukturrasprostraneniûmežfaznyhtreŝin |
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2025-12-01T14:59:18Z |
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1850318429314088960 |
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UDC 539.4
Interfacial Fracture Toughness of Multilayer Composite Structures
S.-C. Her
1
and W.-B. Su
Department of Mechanical Engineering, Yuan Ze University, Chung-Li, Taiwan
1 mesch@saturn.yzu.edu.tw
The interfaces in multilayer composite structures are susceptible to delamination due to the
combination of active tensile and shear loads under operating conditions. A four-layer center crack
composite beam in four-point bending is simulated to determine the interfacial fracture energy of the
multilayer structure. The crack is propagating along the interface between the second and third layers.
Based on the Euler–Bernoulli theory, the strain energy of the four-layer composite beam is derived.
Strain energies before and after the propagation of the interfacial crack are calculated, which results in
determining strain energy release rates. Analytical results for those rates are validated with the
numerical data obtained by the finite element method. The effect of layer thickness of the composite
beam on the interfacial fracture toughness is investigated through a parametric study.
Keywords: interfacial fracture toughness, multi-layered structure, strain energy release rate.
Introduction. Multi-layered composite structures have received substantial attention
in a variety of engineering applications, such as wear resistant materials, microelectronic
devices and composite laminates used in aircraft structures. While multi-layered composites
are widely used in structural components due to their favorable mechanical properties, such
as high stiffness-to-weight and strength-to-weight ratios, the main drawback of such
materials is their low interlaminar fracture toughness, which usually causes delamination
when subjected to impact. Delamination or interlaminar separation is the most predominant
and life-limiting failure mechanism in composite structures. Delaminations may develop
during manufacturing; it may also result from impact damage, or from three-dimensional
interlaminar stresses at stress-free edges and other discontinuities. In general, delamination
corresponds to a crack-like discontinuity between the plies and it may typically extend
during application of mechanical or thermal loads, or both during service life of composites
[1, 2]. Fracture toughness is usually used as a generic term for measures of material
resistance to extension of a crack. Fracture toughness testing has been recognized as the
key to provide accurate toughness values needed in the linear elastic fracture mechanics
[3]. A number of experimental and analytical techniques have been proposed to estimate the
fracture toughness. Double cantilever beam (DCB) test is nowadays standardized for the
measurement of the mode I interlaminar critical strain energy release rate [4]. For mode II
there are several tests proposed in the literature [5–7]: end notched flexure (ENF), end
loaded split (ELS), and four-point end notched flexure test (4ENF). Mode III interlaminar
fracture of carbon/epoxy laminates was evaluated using the edge crack torsion (ECT) test
and four-point bending plate test (de Morais and Pereira) [8, 9].
The aim of this paper is to investigate the interfacial fracture toughness of a four-layer
composite structure with an interfacial crack. A four-point bending test is used to evaluate
the interfacial fracture energy of a four-layer beam. An analytical expression for the strain
energy release rate is presented as a function of the material properties and thickness of the
four-layer structure. The effect of thickness ratio between the adjacent layers of the
interfacial crack is studied by performing a parametric study.
1. Strain Energy Release Rate of a Four-Layer Composite Beam. In this study, the
procedure for evaluation of the interfacial fracture energy of a four-layer composite beam is
preceded in two steps. Firstly, the strain energy release rate of a four-layer composite beam
© S.-C. HER, W.-B. SU, 2015
212 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2015, ¹ 1
is derived on the basis of the Euler–Bernoulli beam theory. Then, the analytical expression
for the strain energy release rate is validated using the finite element method.
1.1. Modeling of the Strain Energy Release Rate. A four-layer composite beam
consists of four different materials subjected to four point bending adopted in this work to
investigate the interfacial fracture toughness. A central notch is cut through the thickness of
the top two layers, and a symmetric crack is situated along the interface between the second
and third layers as shown in Fig. 1. The four-layer beam width is b. The thicknesses of four
layers are h1, h2, h3, and h4, respectively. The Young moduli of the four layers are E1,
E2 , E3, and E4, respectively. The crack length is 2a. The specimen is subjected to
four-point bending. The interfacial crack between the two supports is under constant
moment conditions. The strain energy release rate should exhibit steady state characteristics,
at least when the crack length significantly exceeds the thickness of the cut layer. Figure 2
shows the free-body diagram of the right half of the specimen subjected to a moment M
that produces pure bending.
In the right-hand side of Fig. 2 the bending stresses in four layers can be written as
�1
1�
ME
EI
y, � 2
2�
ME
EI
y, � 3
3�
ME
EI
y, � 4
4�
ME
EI
y, (1)
EI E bh h h h h y bh� � � � � �
�
�
�
�
�
�
�
�
��1 1
3
1 2 3 4
2
1
1
12
1
2
� � � � �
�
�
�
�
�
�
�
�
��E bh h h h y bh2 2
3
2 3 4
2
2
1
12
1
2
� � � �
�
�
�
�
�
�
�
�
�� �E bh h h y bh E bh h3 3
3
3 4
2
3 4 4
31
12
1
2
1
12
1
2
4
2
4�
�
�
�
�
�
�
�
�
�y bh ,
Fig. 1. Four-point bending test.
Fig. 2. Free-body diagram for the right half of four-point bending test specimen.
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2015, ¹ 1 213
Interfacial Fracture Toughness of Multilayer Composite Structures
y
E h h h h h E h h h h E h h h
�
� � � � � � � �1 1 1 2 3 4 2 2 2 3 4 3 3 32 2 2 2 2 2( ) ( ) ( 4 4 4
2
1 1 2 2 3 3 4 42
)
( )
,
�
� � �
E h
E h E h E h E h
where �1 , � 2 , � 3 , and � 4 are the bending stresses in the first, second, third, and fourth
layers, respectively, EI and y are the flexural stiffness and neutral axis of the four-layer
composite beam, and y is the distance measured from the neutral axis of the four-layer
composite beam.
The strain energy stored of the right-hand side of Fig. 2 is as follows:
W dV
M
EI
dxij ij� �� �1
2 2
2
� � . (2)
In the left-hand side of Fig. 2 the bending stresses in the top two layers are assumed to
be negligible, and bending stresses in the bottom two layers can be expressed as
� 3
3� �
ME
EI
y, � 4
4� �
ME
EI
y, (3)
EI
b
E h E h E E h h
h h
E h E h
�
� � �
�
�
�
�
�
�12
23 3
3
4 4
3
3 4 3 4
3 4
2
3 3 4 4
( )
�, ��
� �
�
y
E h h h E h
E h E h
3 3 3 4 4 4
2
3 3 4 4
2
2
( )
( )
,
where � 3 and � 4 are the bending stresses in the third and fourth layers, respectively, EI
�
and �y are the flexural stiffness and neutral axis of the bottom two-layer composite beam,
and y is the distance measured from the neutral axis of the bottom two-layer composite
beam.
Since there is negligible strain energy in the top two layers above the crack, the strain
energy of the left-hand side of Fig. 2 is deduced from the bottom two layers as follows:
W dV
M
EI
dxij ij� � �� �1
2 2
2
� � . (4)
As the crack length is extended from a to a a�� , the difference of the strain energy
in the specimen before and after crack propagation is recognized to be the difference of the
strain energy stored in the left-hand side of Eq. (4) and right-hand side of Eq. (2) of Fig. 2,
� �
��
W
M
EI
dx
M
EI
dx M a
EI EI
aa
� � � � � �
�
�
�
�
���
2 2
0
2
02 2
1 1
. (5)
The strain energy release rate is defined as
G
W
AA
�
�
lim ,
�
�
�0
� �A b a� , (6)
where �W represents the variation of the strain energy in the specimen and �A is the
change of the crack area.
214 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2015, ¹ 1
S.-C. Her and W.-B. Su
Substituting Eq. (5) into Eq. (6) leads to obtaining the strain energy release rate as
follows:
G
W
A
M
b EI EIA
� � � �
�
�
�
�
�
�
lim .
�
�
�0
2
2
1 1
(7)
1.2. Validation of Analytical Model Using Finite Element Method. As it is mentioned
above, the interfacial fracture energy is studied on the basis of the Euler–Bernoulli beam
theory. To verify the analytical model, numerical calculation based on the finite element
method is employed. The finite element analysis has been carried out under plane stress
conditions using ANSYS code. A two-dimensional regular element PLANE82 with 8 nodes
is adopted for the entire structure. To model stresses around the crack tip accurately, a fairly
fine mesh is required. In the finite element analysis the boundary conditions are specified as
shown in Fig. 1. The displacements in the y-direction are zero at the two supports, and
forces in the y-direction are specified at the two loading points. According to the expression
for the strain energy release rate Eq. (6), it is necessary to calculate the difference of strain
energy in the specimen before and after crack propagation. ANSYS is used to calculate
strain energy. In this paper, the numerical value of the crack extension is taken as 1% of the
initial crack length, i.e., �a a� 100. The material properties and thickness of the four-layer
composite beam are listed in Table 1. The width, length, and crack length of the test
specimen are 0.2, 60, and 20 mm, respectively. The bending moment applied to the
specimen is M � 200 N mm� . A typical finite element mesh used in this study is shown in
Fig. 3. The strain energy values in the four-layer beam before and after the crack extension
are calculated by finite element method. Substituting the difference of the strain energy �W
into Eq. (6) results in obtaining the strain energy release rate G� �2 01 105. J/m2. The strain
energy release rate is obtained using the analytical prediction shown in Eq. (7) is
G� �2 02 105. J/m2. Difference between the analytical solution and finite element result is
less than 1%, which demonstrates the accuracy of the present prediction.
T a b l e 1
Material Properties and Thickness of the Four-Layer Beam [10]
Parameter
of beam
Layer
first second third fourth
Material glass copper globtop glass
Young modulus (GPa) 80 130 7 80
Poisson’s ratio 0.3 0.3 0.3 0.3
Thickness (mm) 0.5 0.4 0.53 0.57
Fig. 3. Finite element mesh.
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2015, ¹ 1 215
Interfacial Fracture Toughness of Multilayer Composite Structures
2. Numerical Results and Discussion. The analytical prediction of strain energy
release rate has been proven to be accurate. The aim of the numerical study is to investigate
the influence of layer thickness on the fracture energy. Considering the material properties
listed in Table 1 the effect of layer 2 thickness h2 is examined. The specimen width is
b� 7 mm, load P � 34 N and load span l� 5 mm as it is shown in Fig. 1. The thicknesses
of the first, third, and fourth layers are 1, 0.3, and 1 mm, respectively. The strain energy
release rate varies with the thickness of the second layer as it is plotted in Fig. 4. It shows
that the strain energy release rate increases with an increase of the thickness of the second
layer. The strain energy release rate is converged to 9.7 J/m2 as the second layer thickness
reaches 5 mm. Then the effect of the first layer thickness is studied. Figure 5 illustrates the
variation of the strain energy release rate at different values of thickness ratio. The strain
energy release rate increases with an increase of the thickness ratio h h1 3. Stable strain
energy release rate is found to be achieved in the region of larger h h1 3 and h h h1 2 3( )�
or in the region of smaller h h1 3 and h h h1 2 3( ).�
Fig. 4. Variation of strain energy release rate according to second layer thickness.
Fig. 5. Variation of strain energy release rate according to thickness ratios.
216 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2015, ¹ 1
S.-C. Her and W.-B. Su
Conclusions. In this study, a four-point bending specimen was employed to determine
the interfacial fracture energy of the four-layer composite beam. The strain energy release
rate was studied on the basis of the Euler–Bernoulli beam theory. A simple but accurate
expression relating the strain energy release rate to the thickness and material property of
the four-layer composite beam is presented. A numerical validation was also performed
using finite element method. The difference between the analytical solution and finite
element result is less than 1%. The effect of thickness ratio on the strain energy release rate
is investigated through a parametric study. Numerical results show that the strain energy
release rate increases with an increase of the thickness of the second layer. The analytical
prediction can be used for guidance in the physical design of four-layer structures.
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Received 20. 10. 2014
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2015, ¹ 1 217
Interfacial Fracture Toughness of Multilayer Composite Structures
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