A New Method for Evaluation of Mechanical Properties of Glass/Epoxy Composites at Low Temperatures
Для расчета прочности однонаправленного композита при различных видах нагружения в условиях комнатной температуры и -60°C использованы аналитические модели, учитывающие микромеханические характеристики композитов. В отличие от стандартных методов, базирующихся на результатах испытаний образцов из од...
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Інститут проблем міцності ім. Г.С. Писаренко НАН України
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irk-123456789-957252016-03-03T03:01:57Z A New Method for Evaluation of Mechanical Properties of Glass/Epoxy Composites at Low Temperatures Shokrieh, M.M. Torabizadeh, M.A. Fereidoon, A. Научно-технический раздел Для расчета прочности однонаправленного композита при различных видах нагружения в условиях комнатной температуры и -60°C использованы аналитические модели, учитывающие микромеханические характеристики композитов. В отличие от стандартных методов, базирующихся на результатах испытаний образцов из однонаправленного композита, в данной работе измеряются микромеханические характеристики стекловолокон и эпоксидной матрицы. С помощью разных аналитических моделей выполнен расчет четырех различных модулей упругости и характеристик прочности при комнатной температуре и -60°C.С целью верификации результатов расчета проведены экспериментальные исследования. Показано, что наилучший расчет модулей упругости однонаправленного композита при комнатной и низкой температурах обеспечивает применение упругой модели. Получено хорошее соответствие между расчетными и экспериментальными данными по механическим характеристикам этого материала при исследованных температурах. Розрахунок міцності однонаправленого композита при різних типах навантаження в умовах кімнатної температури і -60°С проводиться за допомогою аналітичних моделей з урахуванням мікромеханічних характеристик композита. На відміну від стандартних методів, що базуються на результатах випробувань зразків з однонаправленого композита, в даній роботі проводиться вимірювання мікромеханічних характеристик скловолокон і епоксидної матриці. За допомогою різних аналітичних моделей виконано розрахунок чотирьох різних модулей пружності і характеристик міцності за кімнатної температури і-60°С. Із метою верифікації результатів розрахунків проведено експериментальні дослідження. Показано, що найкращий розрахунок модулей пружності однонаправленого композита за кімнатної і низької температур забезпечує використання пружної моделі. Отримано хорошу відповідність між розрахунковими й експериментальними даними щодо механічних характеристик цього матеріалу при досліджуваних температурах. 2012 Article A New Method for Evaluation of Mechanical Properties of Glass/Epoxy Composites at Low Temperatures / M.M. Shokrieh, M.A. Torabizadeh, A. Fereidoon // Проблемы прочности. — 2012. — № 1. — С. 119-136. — Бібліогр.: 16 назв. — англ. 0556-171X http://dspace.nbuv.gov.ua/handle/123456789/95725 539.4 en Проблемы прочности Інститут проблем міцності ім. Г.С. Писаренко НАН України |
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Научно-технический раздел Научно-технический раздел Shokrieh, M.M. Torabizadeh, M.A. Fereidoon, A. A New Method for Evaluation of Mechanical Properties of Glass/Epoxy Composites at Low Temperatures Проблемы прочности |
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Для расчета прочности однонаправленного композита при различных видах нагружения в условиях комнатной температуры и -60°C использованы аналитические модели, учитывающие микромеханические характеристики композитов. В отличие от стандартных методов, базирующихся на результатах испытаний образцов из однонаправленного композита, в данной работе измеряются микромеханические характеристики стекловолокон и эпоксидной матрицы. С помощью разных аналитических моделей выполнен расчет четырех различных модулей упругости и характеристик прочности при комнатной температуре и -60°C.С целью верификации результатов расчета проведены экспериментальные исследования. Показано, что наилучший расчет модулей упругости однонаправленного композита при комнатной и низкой температурах обеспечивает применение упругой модели. Получено хорошее соответствие между расчетными и экспериментальными данными по механическим характеристикам этого материала при исследованных температурах. |
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Article |
| author |
Shokrieh, M.M. Torabizadeh, M.A. Fereidoon, A. |
| author_facet |
Shokrieh, M.M. Torabizadeh, M.A. Fereidoon, A. |
| author_sort |
Shokrieh, M.M. |
| title |
A New Method for Evaluation of Mechanical Properties of Glass/Epoxy Composites at Low Temperatures |
| title_short |
A New Method for Evaluation of Mechanical Properties of Glass/Epoxy Composites at Low Temperatures |
| title_full |
A New Method for Evaluation of Mechanical Properties of Glass/Epoxy Composites at Low Temperatures |
| title_fullStr |
A New Method for Evaluation of Mechanical Properties of Glass/Epoxy Composites at Low Temperatures |
| title_full_unstemmed |
A New Method for Evaluation of Mechanical Properties of Glass/Epoxy Composites at Low Temperatures |
| title_sort |
new method for evaluation of mechanical properties of glass/epoxy composites at low temperatures |
| publisher |
Інститут проблем міцності ім. Г.С. Писаренко НАН України |
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2012 |
| topic_facet |
Научно-технический раздел |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/95725 |
| citation_txt |
A New Method for Evaluation of Mechanical Properties of Glass/Epoxy Composites at Low Temperatures / M.M. Shokrieh, M.A. Torabizadeh, A. Fereidoon // Проблемы прочности. — 2012. — № 1. — С. 119-136. — Бібліогр.: 16 назв. — англ. |
| series |
Проблемы прочности |
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2025-07-07T02:43:34Z |
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2025-07-07T02:43:34Z |
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| fulltext |
UDC 539.4
A New Method for Evaluation of Mechanical Properties of Glass/Epoxy
Composites at Low Temperatures
M. M. Shokrieh,
a
M. A. Torabizadeh,
b,1
and A. Fereidoon
c
a Iran University of Science and Technology, Tehran, Iran
b University of Applied Science and Technology, Branch of Khorasan Razavi, Mashhad,
Iran
c University of Semnan, Semnan, Iran
1 Torabizadeh@yahoo.com
ÓÄÊ 539.4
Íîâûé ìåòîä îöåíêè ìåõàíè÷åñêèõ ñâîéñòâ ñòåêëîýïîêñèäíûõ
êîìïîçèòîâ ïðè íèçêèõ òåìïåðàòóðàõ
Ì. Ì. Øîêðè
à
, Ì. À. Òîðàáèçàäå
á,1
, À. Ôåðåéäóí
â
à Èðàíñêèé óíèâåðñèòåò íàóêè è òåõíîëîãèè, Òåãåðàí, Èðàí
á Óíèâåðñèòåò ïðèêëàäíîé íàóêè è òåõíîëîãèè, Ïðîâèíöèÿ Õîðàñàí Ðàçàâè, Ìåøõåä,
Èðàí
â Óíèâåðñèòåò ã. Ñåìíàíà, Èðàí
Äëÿ ðàñ÷åòà ïðî÷íîñòè îäíîíàïðàâëåííîãî êîìïîçèòà ïðè ðàçëè÷íûõ âèäàõ íàãðóæåíèÿ â
óñëîâèÿõ êîìíàòíîé òåìïåðàòóðû è � �60 C èñïîëüçîâàíû àíàëèòè÷åñêèå ìîäåëè, ó÷èòû-
âàþùèå ìèêðîìåõàíè÷åñêèå õàðàêòåðèñòèêè êîìïîçèòîâ.  îòëè÷èå îò ñòàíäàðòíûõ ìåòî-
äîâ, áàçèðóþùèõñÿ íà ðåçóëüòàòàõ èñïûòàíèé îáðàçöîâ èç îäíîíàïðàâëåííîãî êîìïîçèòà, â
äàííîé ðàáîòå èçìåðÿþòñÿ ìèêðîìåõàíè÷åñêèå õàðàêòåðèñòèêè ñòåêëîâîëîêîí è ýïîêñèäíîé
ìàòðèöû. Ñ ïîìîùüþ ðàçíûõ àíàëèòè÷åñêèõ ìîäåëåé âûïîëíåí ðàñ÷åò ÷åòûðåõ ðàçëè÷íûõ
ìîäóëåé óïðóãîñòè è õàðàêòåðèñòèê ïðî÷íîñòè ïðè êîìíàòíîé òåìïåðàòóðå è � �60 C. Ñ
öåëüþ âåðèôèêàöèè ðåçóëüòàòîâ ðàñ÷åòà ïðîâåäåíû ýêñïåðèìåíòàëüíûå èññëåäîâàíèÿ. Ïîêà-
çàíî, ÷òî íàèëó÷øèé ðàñ÷åò ìîäóëåé óïðóãîñòè îäíîíàïðàâëåííîãî êîìïîçèòà ïðè êîìíàò-
íîé è íèçêîé òåìïåðàòóðàõ îáåñïå÷èâàåò ïðèìåíåíèå óïðóãîé ìîäåëè. Ïîëó÷åíî õîðîøåå
ñîîòâåòñòâèå ìåæäó ðàñ÷åòíûìè è ýêñïåðèìåíòàëüíûìè äàííûìè ïî ìåõàíè÷åñêèì õàðàê-
òåðèñòèêàì ýòîãî ìàòåðèàëà ïðè èññëåäîâàííûõ òåìïåðàòóðàõ.
Êëþ÷åâûå ñëîâà: ìåõàíè÷åñêèå õàðàêòåðèñòèêè, ìèêðîìåõàíè÷åñêèå ìîäåëè,
íèçêàÿ òåìïåðàòóðà, ïðîãíîçèðîâàíèå, îäíîíàïðàâëåííûé êîìïîçèò.
N o t a t i o n
E E1 2, – longitudinal and transverse Young moduli of unidirectional composites,
respectively
G12 – shear modulus of unidirectional composites
E Ef m, – fiber longitudinal and matrix Young moduli, respectively
© M. M. SHOKRIEH, M. A. TORABIZADEH, A. FEREIDOON, 2012
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2012, ¹ 1 119
G Gf m, – fiber and matrix shear moduli, respectively
V Vf m, – fiber and matrix volume fractions, respectively
�12 – major Poisson’s ratio of unidirectional composites
� �f m, – Poisson’s ratios of fiber and matrix, respectively
� �f u m u, ,, – fiber and matrix tensile strengths, respectively
� m u
c
, – matrix compressive strength
� �f Y m Y, ,, – shear yield stresses of fiber and matrix, respectively
X Xt c, – longitudinal tensile and compressive strengths, respectively
Y Yt c, – transverse tensile and compressive strengths, respectively
S – shear strength of unidirectional composites
� �f m, – fiber and matrix tensile strains, respectively
� �f m, – coefficients of thermal expansion of fiber and matrix, respectively
Introduction. One of the persistent difficulties in the design and analysis of
composites is prediction of laminate fracture under uniaxial and/or combined
loading by using either unidirectional (UD) composite data or micromechanics
with pristine constituent material properties. However, in contrast to isotropic
materials, experimental evaluation of these data is quite costly and time consuming
because they are functions of several variables: the individual constituents of the
composite material, fiber volume fraction, stacking sequence of plies, processing,
etc. Thus, the need and motivation for developing analytical models to find these
parameters are very important. The difficulty of prediction of laminate’s strength
has been compounded multiply by the availability of many and diverse failure
criteria. It became apparent, therefore, that some kind of formalized comparisons
between various failure theories with measured data would be instructive and very
useful. Micromechanical models to predict the mechanical behavior of UD
composites have been studied by several authors [1–7].
Carbajal and Mujika [8] have determined the compressive strength of
unidirectional composites by three-point bending tests. Unidirectional carbon/
epoxy composites T6T/F593 from Hexcel Composites are tested by three-point
bending with different thicknesses and spans. Liang et al. [1] studied the ultimate
strength of continuous composite beams in combined bending and shear. They
developed a three-dimensional finite element model which was in good agreement
with experimental results. Huang [2] illustrated a micromechanical prediction of
ultimate strength of transversely isotropic fibrous composite. The biggest difference
of his theory, relative to those existing micromechanics models, was that the
constituent nonlinear deformation had been taken into account reasonably. They
found good correlation between the predicted strengths and available experimental
data. Naik and Kumar [3] evaluated and compared some of the existing models
proposed for predicting the compressive strength of UD composites along the fiber
direction. They found that Lo–Chim [9] model gave good results as it was
correlated with the same set of experimental results for obtaining experimental
parameters. Camanho et al. [10] proposed a new analytical closed-form model to
predict the in situ shear strength of composite laminates as a function of ply
thickness and location. Hatta et al. [11] reviewed various strengths of carbon-
M. M. Shokrieh, M. A. Torabizadeh, and A. Fereidoon
120 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2012, ¹ 1
carbon composites. The topics reviewed included tensile, shear, compressive and
fatigue strengths, as well as fiber/matrix interfacial strength of C/Cs. Sun and Jun
[4] considered the effect of fiber misalignment and non-linear behavior of the
matrix on fiber microbuckling and the compressive strength of a unidirectional
fiber composite. Their results showed reasonable correlation with available
experimental data. Wilczynski [5] described a model to predict the mechanical
properties of unidirectional fibrous composites in longitudinal compression. He
illustrated an analytical solution for both straight and initially curved reinforcing
fibers. He found that his results showed 30% higher values than experimental ones.
Huang [6] simulated the overall thermal-mechanical properties of a fibrous
composite beyond of an elastic deformation range using a recently developed
micromechanics model (the bridging model). King et al. [7] presented a method to
estimate the influence of the matrix and the interfacial bond strength itself on the
composite shear strength for a given fiber/matrix composite. Analytical prediction
of these effects has been achieved using a finite element micromechanics model.
Numerical results generated using this model have shown a close fit with
experimental data.
The aim of the present paper is to use micromechanical properties of
composite at room temperature and � �60 C to predict mechanical properties of
UD composites. This includes of four elastic moduli and strengths of laminated
composites. By a set of experimental tests, mechanical properties of glass fiber and
epoxy matrix at room temperature and � �60 C are determined. Then by using
different proposed micromechanics models, full characterization of UD composite
is performed.
Micromechanical Properties. Before modeling the nine parameters of
unidirectional composites (including four elastic moduli and five strengths), micro-
mechanical properties of glass fiber and epoxy matrix should be determined. An
experimental program is conducted to characterize the fiber and resin. The
specimens are tested to failure at room and low temperatures. Figures 1 and 2 show
typical test specimens of fiber and matrix at room temperature before and after
tensile test, respectively. All tests were conducted under displacement control
condition using an Instron 5582 machine adaptable for cryogenic service by an
environmental chamber. The displacement rate was 2 mm/min. Environmental
chamber has the ability to cool down the temperature to � �196 C by evaporating
liquid cryogenic nitrogen. During the tests, a pressurizing device is used to control
the cooling time from room temperature to � �60 C and maintain an evaporating
pressure of 152 kPa. Figure 3 shows experimental equipment for mechanical
testing at both room and low temperatures.
The measured properties are summarized in Table 1.
As shown in Table 1, micromechanical properties of composite (both fiber
and matrix) change with decreasing temperature. However, the difference is that
changes in mechanical properties of epoxy matrix are much higher than those of
glass fiber. For example, with temperature reduction longitudinal stiffness of fiber
increases 34% whereas that for matrix is about 80%. The respective increases in
the tensile strength of fiber and matrix at � �60 C are 14 and 40%, respectively.
Therefore, the role of matrix in changing the mechanical properties of composite is
stronger than that of glass fiber.
A New Method for Evaluation of Mechanical Properties ...
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2012, ¹ 1 121
122 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2012, ¹ 1
T a b l e 1
Experimental Material Properties of Glass Fiber and Epoxy Matrix
at Room Temperature and � �60 C
Mechanical
properties
Glass fiber Epoxy matrix
23�C � �60 C 23�C � �60 C
E, GPa 45.94 65.40 2.30 4.20
G , GPa 18.40 23.17 0.88 1.40
�, % 3.0 2.0 6.5 5.0
�u , MPa 1100.0 1260.0 150.2 210.5
�u , MPa 35.00 40.07 34.00 48.10
� 0.2 – 0.3 –
� �10 6 , ( )� �C 1 5 – 62 –
V 0.55 0.55 0.45 0.45
a
b
Fig. 1. Typical test specimens for glass fibers (a) and epoxy resin ML506 (b).
a
b
Fig. 2. Typical tested specimens for glass fibers (a) and epoxy resin ML506 (b) at room temperature.
M. M. Shokrieh, M. A. Torabizadeh, and A. Fereidoon
Four Elastic Moduli. In this section, different available micromechanics
models for prediction of elastic moduli of UD composites are illustrated. To
prevent repeating, the explication of derivation of relations is not discussed here
[12].
Strength of Material Approach.* Longitudinal Young Modulus. This method
has some assumptions to model the behavior of UD composites, such as: perfect
bond between fibers and matrix; elastic moduli, diameter and space between fibers;
continuous and parallel fibers; fibers and matrix are linearly elastic and the
composite is free of voids. By these assumptions, this method illustrates the
following relation:
E E V E Vf f m m1 � � . (1)
Transverse Young Modulus. In the same manner, the following equation is
illustrated by this method for this elastic modulus:
1
2E
V
E
V
E
f
f
m
m
� � . (2)
In-Plane Shear Modulus. According to shear deformation and stresses in fiber
and matrix, in-plane shear modulus of UD composites can be predicted as
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2012, ¹ 1 123
Fig. 3. Experimental set up for mechanical testing at room and low temperatures.
* Rule of mixture.
A New Method for Evaluation of Mechanical Properties ...
1
12G
V
G
V
G
f
f
m
m
� . (3)
Major Poisson’s Ratio. The major Poisson’s ratio is defined as the negative
of the ratio of the normal strain in the transverse direction to the normal strain in
the longitudinal direction, when a normal load is applied in the longitudinal
direction. Thus it can be expressed as
� � �12
�f f m mV V . (4)
Semi-Empirical Models. Due to some difficulties in previous method, semi-
empirical models have been developed for design purposes. The most useful of
these models include those of Halpin and Tsai [12] because they can be used over a
wide range of elastic properties and fiber volume fractions.
Longitudinal Young Modulus. The Halpin–Tsai equation for the longitudinal
Young modulus is the same as that obtained through the strength of materials
approach.
Transverse Young Modulus. The transverse Young modulus in this model is
given by
E
E
V
Vm
f
f
2
1
1
�
�
�
, (5)
where
�
�
�
( )
( )
.
E E
E E
f m
f m
1
(6)
The term � is called the reinforcing factor and depends on fiber geometry, packing
geometry and loading conditions. Halpin and Tsai obtained the value of the
reinforcing factor by elastic solutions for a square array equal to 2.
In-Plane Shear Modulus. The Halpin–Tsai equation for in-plane shear modulus
is
G
G
V
Vm
f
f
12
1
1
�
�
�
, (7)
where
�
�
�
( )
( )
.
G G
G G
f m
f m
1
(8)
The value of �
1 for circular fibers in a square array gives reasonable results
only for fiber volume fractions of up to 50%. Hewitt and Malherbe [12] suggested
choosing a function:
�
�1 40 10V f . (9)
124 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2012, ¹ 1
M. M. Shokrieh, M. A. Torabizadeh, and A. Fereidoon
Major Poisson’s Ratio. The Halpin–Tsai equation for the major Poisson’s ratio
is the same as that obtained using the strength of materials approach.
Elasticity Approach. In addition to the two previous models, expressions for
the elastic moduli based on elasticity are also available. Elasticity accounts for
equilibrium of forces, compatibility and Hooke’s law relationship in three
dimensions, whereas the two previous methods were not satisfy these
simultaneously.
Longitudinal Young Modulus. This model evaluates the longitudinal Young
modulus as follows:
E E V E V E E V Vf f m f m f f f m f1
21 2 1� � � � � �( ) ( ) ( )� �
� � � � �[ ( )E V V Vf m f m f m f2 12� � �
� � � � � � � �E V V Vm f f f f f f f( )] .1 2 22 2 1� � � � (10)
Although the preceding expression can be written in a compact form by using
definitions of shear and bulk modulus of the material, this is not discussed here.
Note that the first two terms of Eq. (10) represent the mechanics of materials
approach result given by Eq. (1).
Transverse Young Modulus. This model yields an exact solution for the
transverse shear modulus. However, the transverse Young modulus can be found as
follows:
E G2 23 232 1� �( ) ,� (11)
where G23 and � 23 are transverse shear and Poisson’s ratio that is discussed in
detail in Appendix.
In-Plane Shear Modulus. The shear modulus by elastic solution can be found
as following:
G G
G V G V
G V G Vm
f f m f
f f m f
12
1 1
1 1
�
� � �
� � �
( ) ( )
( ) ( )
. (12)
Major Poisson’s Ratio. This elastic modulus of UD composite is calculated in
this model by the following equation:
� � � � �12 1� � � � �f f m f f m f mV V V V( ) ( )
� � � � � �( )2 22 2E E E E E Ef m m f f m m f m f� � � �
� � � � �[ ( )E V V Vf m f m f m f2 12� � �
� � � � � � � �E V V Vm f f f f f f f( )] .1 2 22 2 1� � � � (13)
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2012, ¹ 1 125
A New Method for Evaluation of Mechanical Properties ...
Strengths of UD Composites. In this section, method of evaluation of
strength from the individual properties of the fiber and matrix by using different
models is shown. The strength parameters for a UD lamina are much harder to
predict than the stiffnesses because the strengths are more sensitive to the material
and geometric non-homogeneities, fiber–matrix interface, fabrication process and
environmental conditions. Several theoretical and empirical models are available
for some of the strength parameters.
Longitudinal Tensile Strength. A simple mechanics of material approach
model is applied with some basic assumptions such that fiber and matrix are
isotropic, homogeneous and linearly elastic and the failure strain for the matrix is
higher than that of fiber (in the case of polymeric matrix composites). By these
assumptions, the ultimate strength of the composite in longitudinal direction is
predicted as
X V E Vt f u f f u m f� � �� �, , ( ).1 (14)
Because the fibers carry most of the load in polymeric matrix composites, it is
assumed that, when the fibers fail at the strain of � f , the whole composite fails.
Longitudinal Compressive Strength. The model used for calculating the
longitudinal tensile strength for a unidirectional lamina cannot also be used for its
longitudinal compressive strength because the failure modes are different. Three
typical failure modes may occur: Fracture of matrix and/or fiber–matrix bond due
to tensile strains in the matrix and/or bond; microbuckling of fibers in shear;
extensional mode and shear failure of fibers. Based on these three modes,
compressive strength of UD composites can be as follows:
Ultimate Tensile Strains in Matrix Failure Mode. Using the maximum strain
failure theory, if the transverse strain exceeds the ultimate transverse tensile strain,
the lamina is considered to have failed in the transverse direction. Thus
X
E
c
t
u� 1 2
12
( )
.
�
�
(15)
However, for the value of ( )�2
t
u one can use the empirical formula,
( ) ( ),/� �2
1 31t
u m fV� � (16)
or the mechanics of materials formula,
( ) .� �2 1 1t
u m
m
f
d
s
E
E
� �
�
�
�
�
�
�
�
�
�
�
�
�
(17)
In the above relation, d is diameter of the fibers and s is center-to-center spacing
between the fibers. For circular fibers in a square array,
d
s
V f
�
�
�
�
4
1 2
�
/
. (18)
126 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2012, ¹ 1
M. M. Shokrieh, M. A. Torabizadeh, and A. Fereidoon
Shear/Extensional Fiber Microbuckling Failure Mode. Local buckling models
for calculating longitudinal compressive strengths have been developed. Since
these results are based on advanced topics, only the final expressions are given:
X S Sc
min[ , ],1 2 (19)
where
S V V
E
E
V E E
Vf f
m
f
f m f
f
1 2 1
3 1
� �
�
�
�
�
�
�
�
( )
( )
, (20)
S
G
V
m
f
2 1
�
. (21)
This model has been introduced by Rosen [13].
Shear Stress Failure of Fibers Mode. A unidirectional composite may fail due
to direct shear failure of fibers. In this case, based on the rule of mixtures,
compressive strength of the unidirectional composite is as follow:
X V Vc f Y f m Y m
�2[ ]., ,� � (22)
Lo–Chim Model. Lo and Chim [9] proposed a model that is based on the fact
that compression failure takes place by local microbuckling of fibers as opposed to
the buckling of the entire length of the fiber. Buckling load of a simply supported
Timoshenko beam was obtained using the energy balance principle. The beam
properties were replaced by the composite properties. For glass/epoxy composites
the relation is
X
G
G
E
c
�
�
�
�
�
�
�
12
2 12
1
15 12 6. [ ]
.
�
(23)
Budiansky Model. Budiansky [14] unified Rosen and Argon formula for
elastic-ideally plastic composites. The expression given by Budiansky is
X
G
c
m
�
12
1 � �
. (24)
Here G12 is defined as the effective longitudinal shear modulus of the composite,
given by right-hand term of Rosen Eq. (21). Budiansky and Fleck [15] plotted the
results of various researchers as X c vs. G12 graph for different � �m values.
The data fell well below the elastic kinking line, and were consistent with their
equation for a range of values of � �m near 4.
Wilczynski Model. This model predicts the compressive strength of UD
composites based on failure mode of laminate as [5],
X S S Sc
min[ , , ],1 2 3 (25)
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A New Method for Evaluation of Mechanical Properties ...
where
S V Vm u f f1 1� � �� �, ( ), (26)
S V V
f u
f f2 1� � �
�
�
�
,
( ), (27)
S
R
V V
c
f f3 1� � �
�
�( ). (28)
In the above equations, �� E Ef m and Rc , critical microbuckling load, can be
calculated as
R E E
V
Vc m f
m
m
�
�
�
8
1
1
. (29)
Transverse Tensile Strength. An expression of mechanics of material approach
model for finding the transverse tensile strength of a unidirectional lamina is given
by [12]:
Y Et
t
u� 2 2( ) ,� (30)
where
( ) .� �2 1t
u
m
f
m
d
s
E
E
d
s
� � �
�
�
�
�
�
�
�
�
� (31)
The preceding expression assumes that the fiber is perfectly bonded to the matrix.
If the adhesion between the fiber and matrix is poor, the composite transverse
strength will be further reduced.
Transverse Compressive Strength. Equation (30), which was developed for
evaluating transverse tensile strength, can be used to find the transverse compressive
strengths of a lamina. The actual compressive strength is again lower due to
imperfect fiber/matrix interfacial bond and longitudinal fiber splitting. Using
compressive parameters in Eq. (30),
Y Ec
c
u� 2 2( ) ,� (32)
where
( ) ,� �2 1c
u
m
f
m
cd
s
E
E
d
s
� � �
�
�
�
�
�
�
�
�
� (33)
�m
c is the ultimate compressive failure strain of matrix, which is calculated from
Hooke’s law.
In-Plane Shear Strength. Strength of Material Approach. The procedure for
finding the ultimate shear strength for a unidirectional lamina is using a mechanics
of material approach as follows:
128 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2012, ¹ 1
M. M. Shokrieh, M. A. Torabizadeh, and A. Fereidoon
S G
d
s
G
G
d
s
m
f
m u� � �
�
�
�
�
�
�
�
�
�12 121 ( ) ,,� (34)
where ( ) ,�12 m u is the ultimate shearing strain of the matrix, which is calculated
from Hooke’s law.
Tsai Model. Based on the Tsai model [16], the shear strength can be studied
using the following equation,
S
K
f s
m s
m�
� �1 1 1� �
�
( )
,
,
(35)
where
� s
m
f
G
G
� �
�
�
�
�
�
1
2
1 , (36)
and K m s, is matrix stress concentration factor in shear. An exact determination of
K m s, is difficult because the behavior of matrix is nonlinear near failure. However,
if the matrix is ductile, stress concentration can be assumed to be unity.
King Model. King et al. [7] predicted the in-plane shear strength of UD
composites by the following expression,
S
E
E
m
m
m u
n n
�
�
�
�
�
�
�
�
�
�
�
�
�
2
2
1
1
�
�
� ,
/
. (37)
With some experimental evaluation, they found that parameter n can be assumed
equal to 2 for glass-epoxy composites.
Results and Discussion. Four Elastic Moduli. A comparison of the four
elastic moduli using the mechanics of materials approach, Halpin–Tsai equations,
and elastic models with experimental data points performed with fiber volume
fraction of 55%, for glass fiber/epoxy matrix composites, which their micro-
mechanical properties are listed in Table 1 is showed in Figs. 4–7. All data in the
following figures are plotted for room temperature.
As shown in above figures, the mechanics of material approach predicts the
lower bound of all four elastic moduli at room temperature, while the Halpin–Tsai
model estimates the upper bound in each elastic modulus. Among all applied
models, elastic solution has the best fit with available experimental data for fiber
volume fraction 55% at room temperature and � �60 C. Tables 2 and 3 compare the
predicted values for elastic moduli and those obtained from experimental tests at
room temperature and � �60 C, respectively.
Strengths of UD Composites. By employing described methods for predicting
the five strengths of a unidirectional composite with micromechanical properties of
glass fiber/epoxy matrix at room temperature and � �60 C, strengths of UD
composites can be estimated. Tables 4 and 5 illustrate the estimated values for five
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2012, ¹ 1 129
A New Method for Evaluation of Mechanical Properties ...
strengths of composite in comparison with experimental evaluated values at room
temperature and � �60 C, respectively.
As is evident from the above Tables, some analytical models for predicting the
compressive strengths of UD lamina fail to match the experimental results. Several
factors may contribute to this discrepancy, including: irregular spacing of fibers
causing premature failure in matrix-rich area; less than perfect bonding between
the fibers and the matrix and poor alignment of fibers. In addition, there is
controversy concerning the techniques used in measuring compressive strengths.
Moreover, predicting transverse tensile strength of UD lamina is quite complicated.
Under a transverse tensile load, factors other than the individual properties of the
130 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2012, ¹ 1
T a b l e 2
Comparison of Predicted Elastic Moduli with Experimental Data at Room Temperature
Method E1 , GPa E2 , GPa G12 , GPa �12
Rule of mixture [12] 21.43 4.74 1.84 0.275
Halpin–Tsai [12] 21.43 8.13 2.71 0.275
Elastic model [12] 21.44 6.54 2.63 0.266
Experimental 19.94 5.83 2.11 0.244
T a b l e 3
Comparison of Predicted Elastic Moduli with Experimental Data at � �60 C
Method E1 , GPa E2 , GPa G12 , GPa �12
Rule of mixture [12] 30.03 8.47 2.89 0.275
Halpin–Tsai [12] 30.03 13.78 4.16 0.275
Elastic model [12] 30.04 10.44 4.06 0.266
Experimental 28.65 11.03 4.21 0.244
Fig. 4. Theoretical values of longitudinal Young modulus as a function of fiber volume fraction
compared with experimental values for unidirectional glass/epoxy lamina at room temperature.
M. M. Shokrieh, M. A. Torabizadeh, and A. Fereidoon
fiber and matrix are important. These include the bond strength between the fiber
and the matrix, the presence of voids and the presence of residual stresses due to
thermal expansion mismatch between the fibers and matrix. The prediction of the
ultimate shear strength is complex. Similar parameters such as weak interface, the
presence of voids and Poisson’s ratio mismatch, make modeling quite complex.
The above results show that by imposing some errors, micromechanical models
based on mechanical properties of fiber and matrix can be used instead of
experimental evaluation of mechanical properties of UD composites at room
temperature and � �60 C. By this method, time consuming and costly tests on UD
composites are eliminated, and simple characterization tests on fiber and matrix are
only needed.
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2012, ¹ 1 131
Fig. 5. Theoretical values of transverse Young modulus as a function of fiber volume fraction
compared with experimental values for unidirectional glass/epoxy lamina at room temperature.
Fig. 6. Theoretical values of in-plane shear modulus as a function of fiber volume fraction compared
with experimental values for unidirectional glass/epoxy lamina at room temperature.
A New Method for Evaluation of Mechanical Properties ...
132 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2012, ¹ 1
T a b l e 4
Comparison of Predicted Strengths with Experimental Data at Room Temperature
Strengths of composites Theoretical models (MPa) Experimental results (MPa)
Longitudinal tensile strength (X t ) Eq. (14) 636.05 700.11
Transverse tensile strength (Yt ) Eq. (30) 91.78 69.85
Transverse compressive strength (Yc) Eq. (32) 58.56 122.12
Longitudinal compressive strength
(X c)
Eqs. (15, 19, 22)
Lo–Chim (Eq. 23)
Budiansky (Eq. 24)
Wilczynski (Eq. 25)
75.90
382.86
391.10
636.05
570.37
In-plane shear strength (S) Eq. (34)
Tsai (Eq. 35)
King (Eq. 37)
20.71
50.99
105.89
68.89
T a b l e 5
Comparison of Predicted Strengths with Experimental Data at � �60 C
Strengths of composites Theoretical models (MPa) Experimental results (MPa)
Longitudinal tensile strength (X t ) Eq. (14) 740.25 784.98
Transverse tensile strength (Yt ) Eq. (30) 121.62 75.20
Transverse compressive strength (Yc) Eq. (32) 75.05 186.22
Longitudinal compressive strength
(X c)
Eqs. (15, 19, 22)
Lo–Chim (Eq. 23)
Budiansky (Eq. 24)
Wilczynski (Eq. 25)
96.98
547.51
622.22
740.25
731.94
In-plane shear strength (S) Eq. (34)
Tsai (Eq. 35)
King (Eq. 37)
29.82
71.54
148.49
91.22
Fig. 7. Theoretical values of major Poisson’s ratio as a function of fiber volume fraction compared
with experimental values for unidirectional glass/epoxy lamina at room temperature.
M. M. Shokrieh, M. A. Torabizadeh, and A. Fereidoon
Conclusions. In order to evaluate mechanical properties of UD lamina at
room temperature and � �60 C, instead of direct examination of UD composites,
individual properties of the fiber and matrix at room temperature and � �60 C were
determined experimentally. Then, by using different micromechanical models,
mechanical properties (including four elastic moduli and five strengths) of UD
composites were predicted. The present research was focused on a unidirectional
continuous fiber-reinforced lamina. This is because it forms the basic building
block of a composite structure, which is generally made of several unidirectional
lamina stacked at various angles. Based on the present study and the results
obtained by experiments, the following conclusions can be drawn:
1. Mechanical properties of fiber and matrix change with decreasing
temperature to � �60 C. However, changes in mechanical properties of epoxy
matrix are much higher than those of glass fiber. Therefore, the role of matrix in
changing the mechanical properties of composites under low temperature is more
critical than that of glass fiber.
2. The strength parameters for a UD lamina are much harder to predict than
the stiffnesses because the strengths are more sensitive to the material and
geometric non-homogeneities, fiber–matrix interface, fabrication process and
environmental conditions.
3. For elastic moduli, the mechanics of material approach predicts the lower
bound of all four elastic moduli at room temperature, while the Halpin–Tsai model
estimates the upper bound of each elastic modulus. Among all models, elastic
solution has the best fit with available experimental data for fiber volume fraction
of 55% at room temperature and � �60 C.
4. Several factors may contribute to discrepancy of experimental and predicted
data, including: irregular spacing of fibers causing premature failure in matrix-rich
area; non-perfect bonding between fibers and the matrix and poor alignment of
fibers.
5. Predicting transverse tensile and in-plane shear strengths of UD lamina is
quite complicated. Under transverse tensile and shear loadings, some factors other
than the individual properties of the fiber and matrix are important. These include
the bond strength between the fiber and the matrix, the present of voids and the
present of residual stresses due to thermal expansion mismatch between the fibers
and matrix.
Appendix. The transverse Young modulus in elasticity model can be found as
follows:
E G2 23 232 1� �( ) .� (A1)
The transverse Poisson’s ratio � 23 is given by
� 23
23
23
�
�
�
K mG
K mG
*
*
, (A2)
where
m K
E
� �1 4
12
2
1
* .
�
(A3)
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A New Method for Evaluation of Mechanical Properties ...
The bulk modulus K * of the composite under longitudinal plane strain is
K
K K G V K K G V
K G V K G V
m f m m f m m f
f m m m m f
*
( ) ( )
( ) ( )
.
� � �
� � � (A4)
The bulk modulus K f of the fiber under longitudinal plane strain is
K
E
f
f
f f
� �2 1 1 2( )( )
.
� � (A5)
The bulk modulus K m of the matrix under longitudinal plane strain is
K
E
m
m
m m
� �2 1 1 2( )( )
.
� � (A6)
The transverse shear modulus G23 is given by the acceptable solution of the
quadratic equation:
A
G
G
B
G
G
C
m m
23
2
23
2 0
�
�
��
�
�
�� �
�
�
��
�
�
���
, (A7)
where
A V V
G
G
G
Gf f
f
m
f
m
f
� �
�
�
��
�
�
�� �
�
�
��
�
�
��3 1 12( )
� � � �
�
�
��
�
�
��
�
�
�
�
�
� �
�G
G
G
G
V V
G
G
f
m
m f m
f
m
m f f f m
f
m
3 1
�
��
�
�
��� �
�
�
��
�
�
��
�
�
�
�
�
�
G
G
f
m
m
1 , (A8)
B V V
G
G
G
G
V
f f
f
m
f
m
f
f
m
� � �
�
�
��
�
�
�� �
�
�
��
�
�
��� �3 1 1
2
2( ) (
1
1
)
G
G
f
m �
�
�
��
�
�
��
� � � �
�
�
��
�
�
��
�
�
�
�
�
�� � �
G
G
G
G
V
G
G
G
G
f
m
f
f
m
m f f
f
m
m
f
m
3 1
2
1 1
�
�
��
�
�
�� �
�
�
�
�
�
�V f
� � �
�
�
��
�
�
��� �
�
�
��
�
�
��
�
�
�
�
�
�( )
m
f
m
f
f
m
m f f
G
G
G
G
V1 2 3 , (A9)
C V V
G
G
G
Gf f
f
m
f
m
f
� �
�
�
��
�
�
�� �
�
�
��
�
�
��3 1 12( )
� � � �
�
�
��
�
�
��
�
�
�
�
�
� � �
�
�
G
G
G
G
V
G
G
G
G
f
m
f
f
m
m f f m
f
m
f
m
3 1��
�
�
�� �
�
�
�
�
�
�V f 1 , (A10)
134 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2012, ¹ 1
M. M. Shokrieh, M. A. Torabizadeh, and A. Fereidoon
� �m m� �3 4 , � �f f� �3 4 . (A11)
Then, using Eqs. (A1)–(A6), the transverse Young modulus can be calculated.
Ð å ç þ ì å
Ðîçðàõóíîê ì³öíîñò³ îäíîíàïðàâëåíîãî êîìïîçèòà ïðè ð³çíèõ òèïàõ íàâàíòà-
æåííÿ â óìîâàõ ê³ìíàòíî¿ òåìïåðàòóðè ³ � �60 Ñ ïðîâîäèòüñÿ çà äîïîìîãîþ
àíàë³òè÷íèõ ìîäåëåé ç óðàõóâàííÿì ì³êðîìåõàí³÷íèõ õàðàêòåðèñòèê êîìïî-
çèòà. Íà â³äì³íó â³ä ñòàíäàðòíèõ ìåòîä³â, ùî áàçóþòüñÿ íà ðåçóëüòàòàõ
âèïðîáóâàíü çðàçê³â ç îäíîíàïðàâëåíîãî êîìïîçèòà, â äàí³é ðîáîò³ ïðîâî-
äèòüñÿ âèì³ðþâàííÿ ì³êðîìåõàí³÷íèõ õàðàêòåðèñòèê ñêëîâîëîêîí ³ åïîêñèä-
íî¿ ìàòðèö³. Çà äîïîìîãîþ ð³çíèõ àíàë³òè÷íèõ ìîäåëåé âèêîíàíî ðîçðàõóíîê
÷îòèðüîõ ð³çíèõ ìîäóëåé ïðóæíîñò³ ³ õàðàêòåðèñòèê ì³öíîñò³ çà ê³ìíàòíî¿
òåìïåðàòóðè ³ � �60 Ñ. ²ç ìåòîþ âåðèô³êàö³¿ ðåçóëüòàò³â ðîçðàõóíê³â ïðîâåäåíî
åêñïåðèìåíòàëüí³ äîñë³äæåííÿ. Ïîêàçàíî, ùî íàéêðàùèé ðîçðàõóíîê ìîäó-
ëåé ïðóæíîñò³ îäíîíàïðàâëåíîãî êîìïîçèòà çà ê³ìíàòíî¿ ³ íèçüêî¿ òåìïå-
ðàòóð çàáåçïå÷óº âèêîðèñòàííÿ ïðóæíî¿ ìîäåë³. Îòðèìàíî õîðîøó â³äïîâ³ä-
í³ñòü ì³æ ðîçðàõóíêîâèìè é åêñïåðèìåíòàëüíèìè äàíèìè ùîäî ìåõàí³÷íèõ
õàðàêòåðèñòèê öüîãî ìàòåð³àëó ïðè äîñë³äæóâàíèõ òåìïåðàòóðàõ.
1. Q. Q. Liang, B. Uy, M. A. Bradford, and H. R. Ronagh, “Ultimate strength of
continuous composite beams in combined bending and shear,” J. Constr. Steel
Res., 60, 1109–28 (2004).
2. Z. M. Huang, “Micromechanical prediction of ultimate strength of
transversely isotropic fibrous composites,” Int. J. Solids Struct., 38, 4147–
4172 (2001).
3. N. K. Naik and R. S. Kumar, “Compressive strength of unidirectional
composites: evaluation and comparison of prediction models,” Compos. Struct.,
46, 299–308 (1999).
4. C. T. Sun and A. W. Jun, “Compressive strength of unidirectional fiber
composites with matrix non-linearity,” Compos. Sci. Technol., 52, 577–587
(1994).
5. A. P. Wilczynski, “Longitudinal compressive strength of a unidirectional
fibrous composite,” Ibid, 45, 37–41 (1992).
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(2001).
7. T. R. King, D. M. Blackketter, D. E. Walrath, and D. F. Adams, “Micro-
mechanics prediction of the strength of carbon fiber/epoxy matrix composites:
The influence of the matrix and interface strengths,” J. Compos. Mater., 26,
No. 4, 558–573 (1992).
8. N. Carbajal and F. Mujika, “Determination of compressive strength of
unidirectional composites by three-point bending tests,” Polymer Test., 28,
150–156 (2009).
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2012, ¹ 1 135
A New Method for Evaluation of Mechanical Properties ...
9. K. H. Lo and E. S. M. Chim, “Compressive strength of unidirectional
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Received 14. 09. 2010
136 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2012, ¹ 1
M. M. Shokrieh, M. A. Torabizadeh, and A. Fereidoon
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