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Micromechanical analysis of stress relaxation response of fiber-reinforced polymers
Composites Science and Technology 69 (2009) 1286–1292
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Micromechanical analysis of stress relaxation response of fiber-reinforced polymers Mohammad Tahaye Abadi * Aerospace Research Institute, Ministry of Science, Research and Technology, P.O. Box 14665-834, Tehran, Iran
a r t i c l e
i n f o
Article history: Received 6 August 2008 Received in revised form 28 January 2009 Accepted 28 February 2009 Available online 9 March 2009 Keywords: A. Polymer–matrix composites C. Stress relaxation Micromechanics
a b s t r a c t The paper describes a micromechanical method to determine the stress relaxation response of polymer composites consisting of linearly viscoelastic matrices and transversely isotropic elastic fibers. A representative unit cell is subjected to some prescribed axial and shear loadings to study and quantify the time-dependent behavior of composite materials. Closed-form analytical expressions are derived describing the anisotropic viscoelastic response of composite materials as functions of matrix and fiber properties. The present analytical expressions are employed to determine the stress relaxation behavior of a graphite/epoxy composite and the results are compared with the finite element analysis of the micromechanical model. Very good correlation between analytical expressions and numerical results is illustrated for the linearly anisotropic viscoelastic response of composite materials. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction Many polymer composites exhibit viscoelastic behavior, in which the magnitude of the stress components is a function of the deformation history and it depends on strain, strain rate, temperature and time. The viscoelastic materials have memory and this memory typically fades with time. The long molecular chains of polymer matrix are the source of viscoelasticity phenomenon in polymer composites. It has been an active research area for many years to predict the viscoelastic properties of the composites based on their constituents. Many analytical as well as numerical modeling algorithms have been forwarded and the experimental studies have been conducted to examine the overall viscoelastic properties of composites. The micromechanical methods provide efficient tools to characterize the behavior of the composite materials. A representative volume element of the compound material is analyzed due to the known properties of their constituents and loading cases. In the micromechanical approach, the heterogeneous structure of the composite material is replaced by a homogeneous medium with anisotropic properties [1]. Hashin [2] was the first to discuss the micromechanical method to model the viscoelastic response of unidirectional composites. Based on an energy balance approach with the aid of elasticity theory, Whitney and Riley [3] obtained closedform analytical expressions describing the elastic modulus of composites. Since it has been difficult to derive analytical expressions determining the viscoelastic behavior of composite materials, the micromechanical models were numerically evaluated. Aboudi [4,5] developed a unified micromechanical theory based on the * Tel.: +98 21 88366030; fax: +98 21 88362011. E-mail address: [email protected] 0266-3538/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.compscitech.2009.02.036
generalized method of cells (GMC) to predict the overall behavior of composite materials with elastic or inelastic constituents. In his work, each cell was divided into a convenient number of subcells and the compatibility of displacements and equilibrium of tractions were implemented at the various interfaces. Yancey and Pindera [6] applied a Laplace transform inversion scheme to the micromechanical model proposed by Aboudi to discuss the creep compliances of a unidirectional graphite/epoxy composite, and the predictions were compared with experimental data at different temperatures. Barbero and Luciano [7] developed explicit analytical expressions for the relaxation modulus in the Laplace domain and a numerical method was used to obtain the Laplace inversion in time domain. Brinson and Lin [8] and Fisher and Brinson [9] used the finite element method to analyze the two-phase and three-phase viscoelastic composites, respectively, and the results were compared with the Mori– Tanaka model [10,11]. Naik et al. [12] used the finite element method to predict the viscoelastic properties of composites considering the periodic boundary conditions. In the present research work, closed-form analytical expressions are derived to determine the stress relaxation response of polymer composites consisting of linearly viscoelastic matrices and transversely isotropic elastic fibers. A micromechanical approach is used to derive the analytical expressions in time domain. A representative unit cell is subjected to some prescribed axial and shear loadings in order to study and quantify the timedependent behavior of composite materials. The predictions of the present algebraic expressions are employed to determine the stress relaxation behavior of a graphite/epoxy composite and the results are compared with the finite element analysis of micromechanical model. Very good correlation between analytical expressions and numerical results is illustrated for the linearly anisotropic viscoelastic response of composite materials.
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2. Mechanical models of composite constituents Most polymers used as the matrix of composite materials exhibit viscoelastic behavior, in which the stress tensor depends on the history of deformation, but the history dependent behavior fades with time. To determine the stress tensor, a function is needed to describe the effects of history dependent behavior in the polymer materials. Experimental studies [13,14] have shown that the Poisson’s ratio of polymers is a time-dependent variable and the constitutive equations based on constant Poisson’s ratio have poorly confirmed by experimental data. The stress components in a linear viscoelastic material at a specific time t can be written as [15]
rmijðtÞ ¼
Z t Em Em m m m 0 0 e d þ e R e d s ðtsÞ ijðsÞ kkðtÞ ij ijðtÞ 3ð1 2mm 1 þ mm 0 0Þ 0
ð1Þ
Table 1 The parameters of the Prony series describing the stress relaxation function in the epoxy materials. k
gk a
ska
kk
vk ðs1 Þ
1 2 3 4 5 6 7 8 9 10 11 12
0.012623 0.02552 0.062518 0.076863 0.128583 0.147175 0.181631 0.171479 0.083365 0.034882 0.012037 0.009644
1.0145E09 1.0145E08 1.0145E07 1.0145E06 1.0145E05 1.0145E04 1.0145E03 1.0145E02 1.0145E01 1.0145E+00 1.0145E+01 1.0145E+02
0.001179 0.002388 0.005884 0.0073 0.012382 0.014445 0.018246 0.01769 0.008762 0.003693 0.001278 0.001025
1.0154E09 1.0162E08 1.0188E07 1.0198E06 1.0234E05 1.0248E04 1.0273E03 1.0268E02 1.0205E01 1.0170E+00 1.0154E+01 1.0152E+02
a
rmijðtÞ and emijðtÞ are, respectively, the components of stress and m m strain tensors, em ijðtÞ the component of deviatoric strain (eij ¼ eij m m m ekk dij =3Þ, RðtsÞ the relaxation kernel, E0 and m0 the initial elastic
According to Ref. [12,16].
where
modulus and Poisson’s ratio, respectively. The summation is taken over the repeating indices and the superscript m denotes to the variables defined in the matrix of composite material. Viscoelastic properties of materials are usually obtained via stress relaxation tests. As an example, consider a viscoelastic material subjected to only shear strain in plane 2–3 as follows:
em23ðtÞ ¼
1 2
0
t<0
ð2Þ
c0 t P 0
where c0 is a constant shear strain. Substituting Eq. (2) in (1) results in
Z t Em 0 c0 R d s 1 ðt s Þ 2ð1 þ mm 0 0Þ Z t ¼ rm RðtsÞ ds 23ð0Þ 1
rm23ðtÞ ¼
ð3Þ
0
Experimental studies of polymers have revealed that the relaxation function can be given by
RðtÞ ¼
K X gk k¼1
sk
e
t=sk
ð4Þ
where gk and sk are the material constants. Using Eqs. (3) and (4), the drop of shear stress can be described by Prony series as follows: K X rm23ðtÞ Z t 1 m ¼ RðtsÞ ds ¼ gk ð1 et=sk Þ r23ð0Þ 0 k¼1
ð5Þ
The relaxation behavior of polymers can be evaluate in tension test with the following program: m 11ðtÞ
e
¼
0
t<0
ð6Þ
e0 t P 0
where e0 is a constant value. According to Eq. (1), the axial stresses along and normal to the extension direction become Z t m m Em e 1 2mðtÞ 2ð1 þ mðtÞ Þ 2 m ð7aÞ rm11ðtÞ ¼ 0 0 þ R ð1 þ m Þd s ðt s Þ ðsÞ 3 1 2mm 1 þ mm 1 þ mm 0 0 0 0 Z m m t E e 1 2mðtÞ 1 þ mðtÞ 1 ð7bÞ rm22ðtÞ ¼ 0 0 þ RðtsÞ ð1 þ mm ðsÞ Þds m m 3 1 2mm 1 þ m m 1 þ 0 0 0 0 m m m where mm ðtÞ is the Poisson’s ratio (mðtÞ ¼ e22ðtÞ =e11ðtÞ Þ which is a time-
dependent property of viscoelastic materials unlike the linear elastic materials. Since the axial stress normal to the extension direction is zero, Eq. (7b) yields the integral equation describing the Poisson’s ratio of isotropic linear viscoelastic materials as follows:
mmðtÞ ¼ m0 þ
1 2mm 0 3
Z
t 0
RðtsÞ ð1 þ mm ðsÞ Þds
ð8Þ
Substituting Eq. (8) in (7a) gives the axial stress required to produce and hold prescribed strain
rm11ðtÞ ¼ Em0 e0
1 2mm ðtÞ
ð9Þ
1 2mm 0
As an example, the initial engineer constants of the epoxy materials are given by [12,16]
Em 0 ¼ 8:3 GPa;
mm0 ¼ 0:4
ð10Þ
To describe the stress relaxation function of the epoxy materials, twelve parameter sets have been used for gk and sk , as listed in Table 1. The fibers usually behave as elastic materials in the working temperature of polymer–matrix composites. In the present work, a transversely isotropic elastic model is used to model the fiber materials. The elastic properties of the carbon fiber are given by
Gf12 ¼ Gf13 ¼ 8:96 GPa
Ef1 ¼ 233 GPa;
Ef2 ¼ 23:1 GPa;
Gf23
mf12 ¼ mf13 ¼ 0:200; mf23 ¼ 0:400
¼ 8:27 GPa;
ð11Þ in which the fiber axis is along the one direction. 3. Stress relaxation response of fiber-reinforced polymers Based on the Boltzmann principle, the stress tensor of a viscoelastic material at the current instant is a functional on the entire history of strain tensor in the interval [1; t]. Assuming this functional to be linear and applying Rizes theorem, the stress components of a nonaging viscoelastic material can be written as [17]
rijðtÞ ¼
Z
t
1
vijklðtsÞ deklðsÞ ¼ vijklð0Þ eklðtÞ
Z
t
1
@ vijklðtsÞ @s
eklðsÞ ds
ð12Þ
where limt!1 eklðtÞ ¼ 0. The relaxation test is performed by applying constant strain components, ekl , as the following program:
eklðtÞ ¼
0 ekl
t<0
ð13Þ
tP0
Combining Eqs. (12) and (13) yields the stress tensor as follows:
rijðtÞ ¼ vijklð0Þ
Z
t
1
@ vijklðtsÞ @s
ds ekl ¼ C ijklðtÞ ekl
ð14Þ
In the present research work, the time-dependent functions of C ijklðtÞ are determine for transversely isotropic viscoelastic materials, in which the general response can be described by five time-dependent functions. It should be noted that these functions are material parameters and independent of strain tensor and the status of loading. The previous research works [15,18–20] have shown that the creep and the stress relaxation responses can be related to each
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other. The numerical solution of a Volterra integral equation yields the creep behavior based on the stress relaxation results [15]. 4. Micromechanical models The micromechanical model is commonly used to determine the properties of composite materials. A repeating unit cell based on a pre-determined fiber packing is assumed to represent the microstructure of the composite. The response of the unit cell under specified loading conditions are analyzed in order to study and quantify the viscoelastic property of the unit cell. In the micromechanical models, it is assumed that the fibers are regularly spaced and perfectly bonded to the matrix material. In the present research work, a micromechanical model based on the mechanics of materials approach is used to determine the viscoelastic properties of polymer composites as a function of their constituent properties. In the present micromechanical model, a simplified representative unit cell is subjected to the specific deformations and the variation of deforming force is investigated throughout the time domain. The unit cell is a section cut from a single layer of fiber-reinforced material. The section consists of side-by-side alternating region of fiber and matrix and the fibers arranged in the parallel arrays, as shown in Fig. 1. In the following sections, the response of the unit cell in different deformation fields is analyzed in order to determine the algebraic expressions predicting the viscoelastic properties of the composite materials as functions of fiber and matrix properties. In the present research work, the finite element analysis of micromechanical model is used to verify the derived closed-form analytical expressions predicting the relaxation behavior of composite materials. A unit cell with square fiber packing is used in the finite element analysis to examine the viscoelastic properties of composite materials with different fiber volume fractions. The dimensions of unit cell are chosen by 2b1 2b2 2b3 and different diameters are considered for the circular cross section of fiber depending on the fiber volume fraction. Since the deformation along fiber direction is uniform, the minimum number of elements is considered based on a convenient aspect ratio for generated mesh. Due to the material and geometric symmetries, one-eighth of unit cell is meshed, as shown in Fig. 2. To represent the periodic patterns of the microstructures, boundary conditions of unit cell follow from the symmetries including translations, reflections and rotations [21]. In the finite element analysis of micromechanical model, a prescribed deformation is applied to the unit cell and it is kept constant throughout
Fig. 1. The unit cell considered in the mechanics of materials approach.
Fig. 2. The mesh geometry used in one-eighth of the unit cell with square fiber packing.
the time domain in order to characterize the stress relaxation behavior. To apply the boundary conditions to one-eighth of unit cell, it is required that the displacement of some planes of unit cell is set to a specific value [21]. A virtual point is defined in the finite element model in order to apply a prescribed displacement component to the nodes located on a specific plane. The displacement of these nodes and the virtual point are set the same. Therefore, the force component required to deform the virtual point is equivalent to the integral of corresponding forces applied to the nodes located on the specific plane. The stress on the plane is calculated based on the force applied to the virtual point and the area of plane. As shown in Fig. 3, the force required to keep a constant deformation
Fig. 3. The applied displacement and variation of force in the finite element analysis of relaxation test.
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is reduced as time elapses and it is determined in the finite element analysis as well as the closed-form analytical expressions in the following section. In the finite element analysis, the time considered to apply the deformation is set to a very small value (t 1 ¼ 1011 s) compared to the total relaxation time (t1 tf Þ. 5. Loading of unit cell 5.1. Longitudinal axial loading To study the behavior of composite materials along the fiber direction, the unit cell shown in Fig. 1 is subjected to a fixed axial strain e0 in the one direction. Since the bond between of the fiber and matrix is assumed to be perfect, both of them are stretched by the same amount in the one direction. Treating the two constituents as if they are in a one-dimensional state of stress, the stress in the constituents of composite material can be determined using Eq. (9) and the axial modulus of the elastic fiber. Since the sum of forces in the fiber and matrix is equal to the total force, the stress in the composite material becomes:
r11ðtÞ ¼ Ef1 V f þ Em0 V m
1 2mm ðtÞ
1 2m
m 0
e0
ð15Þ
where V f and V m are the volume fractions of fiber and matrix, respectively (V f þ V m ¼ 1). According to the above equation, the axial stress depends on the Poisson’s ratio of matrix materials that it can be determined by solving the integral Eq. (8). The Laplace transformation of integral equation gives
m^mðsÞ ¼
mm0
1 2mm 0 ^ ^m RðSÞ ð1 þ m þ ðsÞ Þ s 3
ð16Þ
^ ^m where m ðsÞ and RðSÞ are the Laplace transformations of the functions mmðtÞ and RðtÞ , respectively. Using Eqs. (4) and (16), Laplace transformation of Poisson’s ratio becomes
^m ðsÞ
m ¼
K X 1 2mm gk =sk 0 þ s 3 1 þ s=sk k¼1
mm0
!
K X 1 2mm gk =sk 0 1 3 1 þ s=sk k¼1
!1
ð17Þ The inverse Laplace transformation of Eq. (17) becomes
tmðtÞ ¼ tm0 þ
K X
kk ð1 et=vk Þ
Fig. 4. The drop of longitudinal stress as a function of time domain and fiber volume fraction for a graphite/epoxy composite.
DW DW f DW m ¼ þ ¼ V f ef22 þ V m em 22ðtÞ W W W f f m m ¼ ðV v 12 þ V tðtÞ Þe0
e22ðtÞ ¼
ð20Þ
where W is the width of unit cell, DW f and DW m are the width changes of fiber and matrix, respectively. Substituting Eq. (18) in (20) yields the Poisson’s ratio of composite material, i.e.
"
K X m12 ¼ 22ðtÞ ¼ V f v f12 þ V m tm0 þ kk ð1 et=v Þ e0 k¼1 k
# ð21Þ
Fig. 5 shows the variations of Poisson’s ratio t12 with time domain and fiber volume fraction for a graphite/epoxy composite determined by Eq. (21) and finite element analysis of micromechanical model. The accuracy of algebraic expression for t12 is quite obvious and it is similar to the accuracy of the results for longitudinal stress in the composite materials.
ð18Þ
k¼1
where kk and vk are the constants depending on gk , sk and mm 0 . Table 1 gives the calculated constants expressing the time-dependent Poisson’s ratio of the epoxy materials. Substituting Eq. (18) in (15) determines the axial stress in the composite material, i.e.
(
r11ðtÞ ¼
Ef1 V f
" þ
m Em 0V
K X 2 1 kk ð1 et=vk Þ 1 2mm 0 k¼1
#)
e0
ð19Þ
Fig. 4 illustrates the predictions of the algebraic expression for the drop in longitudinal stress (r 1 ¼ 1 r11ðtÞ =r11ð0Þ Þ as a function of time domain and fiber volume fraction in the graphite/epoxy composite. The finite element results of micromechanical model with square fiber packing are used for comparisons. The derived expression is quite accurate for the prediction of the relaxation longitudinal stress in the fiber-reinforced material. The axial stretching strain contracts the composite material in the two direction due to Poisson’s effects. Because the fiber and matrix have different Poisson’s ratios, they will not contract the same amount in the widthwise and the combined contraction of the fiber and matrix results in the overall contraction of the composite in the two direction. Hence, the transverse axial strain of composite material can be written as
Fig. 5. The Poisson’s ratio t12 as a function of time domain and fiber volume fraction for a graphite/epoxy composite.
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5.2. Transverse axial loading To characterize the relaxation response of composite materials in the transverse direction, the unit cell is subjected to a fixed strain in the two direction. The overall change in transverse dimension of the unit cell is the sum of dimension change of the two constituents. Due to unit cell geometry, the transverse strain can be written as
e0 ¼
DW DW f DW m ¼ þ W W W
ð22Þ
Since there is no direct path to transmit the transverse stress to the fiber, a portion of the transverse stress is transmitted through the fiber, and a portion is transmitted around the fiber, through the matrix material. Therefore, a stress partitioning factor is introduced to account for the division of the stress in each of the two constituents. The volume fraction of matrix that is subjected to a constant stress level r22 , is less than its real existing volume fraction, namely, qV m , where q is the stress partitioning factor (0 < q < 1). Due to viscoelastic behavior of matrix material, the stress partitioning factor is a time-dependent variable and it depends on the fiber volume fraction in the general form. The volume fraction of fiber that is subjected to stress level r22 is assumed to be V f . As a result of the stress partitioning factor, the strain in the direction two defined by Eq. (22) becomes
e0 ¼ ðV f ef22 þ qðV f ;tÞ V m em22ðtÞ ÞðV f þ qðV f ;t ÞV m Þ1
r22ðtÞ
Vf
V f þ qðV f ;tÞ V m
Ef2
þ
qðt;V f Þ V m 1 2mm 0 1 2mm Em ðtÞ 0
! ð24Þ
It is assumed that the stress partitioning factor can be expressed by a Prony series, i.e.
qðV f ;tÞ ¼ q0 þ fðV f Þ
K X
5.3. Longitudinal shear loading
ð23Þ
Considering a one-dimensional stress state in the two constituents, the transverse strain in the fiber and matrix is related to the stress level using Eq. (9) and the transverse modulus of elastic fiber. Then, the strain in the direction two can be written as
e0 ¼
Fig. 6. The drop of transverse stress as a function of time domain and fiber volume fraction for a graphite/epoxy composite.
gk ð1 et=sk Þ
ð25Þ
To evaluate the viscoelastic shear behavior of fiber-reinforced composite, the unit cell is subjected to a constant shear strain c0 in the 1–2 plane and the required shear stress r12 is determined. Since there is no direct path to transmit the shear stress to the fiber materials, similar to transverse loading; a stress partitioning factor is introduced to account for the division of the stress in each of the two constituents. The volume fraction of matrix that is subjected to stress level r12 is denoted by jV m , where j is the stress partitioning factor depending on fiber volume fraction and loading time. Similarly, the shear strain of composite material can be expressed by
c0 ¼ ðV f cf12 þ jðV f ;tÞ V m cm12ðtÞ ÞðV f þ jðV f ;tÞ V m Þ1
ð28Þ
k¼1
where q0 is a constant and f is a function depends on the fiber volume fraction. Substituting Eqs. (18) and (25) in (24) results in the transverse stress, namely, f
r22ðtÞ ¼
h
Vf f E2
þ
t=sk
k¼1 gk ð1 e PK gk ð1et=sk Þ ðV f Þ PK k¼1 t=v
V þ q0 þ fðV f Þ q0 þf
PK
2
1
k ð1e k¼1 k 12mm 0
kÞ
i
Þ V
m
e0
ð26Þ
r12ðtÞ ¼
Vf þ
h
j0 þ g ðV f Þ h
f
V f G12
Vm Em 0
Fig. 6 shows the drop of transverse stress (r 2 ¼ 1 r22ðtÞ =r22ð0Þ Þ predicted by Eq. (26) and the finite element analysis considered in different time domain and fiber volume fraction. It is a common procedure to determine the stress partition factors by numerical analysis of micromechanical model describing the microstructure of anisotropic elastic materials. In the present research work, a numerical procedure is used to determine the stress partition factors. For the graphite/epoxy composite, the parameters defining the stress partitioning factor are given by
q0 ¼ 0:385; f ðV f Þ ¼ ½9:0ðV f Þ3 9:7ðV f Þ2 þ 3:9 V f 0:385
Considering the pure shear deformation of the viscoelastic matrix according to Eq. (5) and longitudinal shear modulus of elastic fiber, Eq. (28) can be written as
K .X
gk
þ
PK
j0 þgðV f Þ
g
k¼1 k ð1
PK
g ð1e k¼1 k
PK
Gm 0 1
i et=sk Þ V m i c0 t=sk
Þ Vm
t=sk
Þ
g ð1e k¼1 k
In which the stress partitioning factor for longitudinal shear stress is approximated by
jðt;V f Þ ¼ j0 þ g ðV f Þ
K X
gk ð1 et=sk Þ
Since the transverse load is transmitted to the fiber through the matrix material, the viscoelastic properties of the matrix material have considerably effects on the drop of transverse stress and it is more than the drop of longitudinal stress shown in Fig. 4.
ð30Þ
k¼1
The variation of longitudinal shear stress is shown in Fig. 7 that it is computed by Eq. (29) and the finite element analysis of micromechanical model in different time domain and fiber volume fractions. For the graphite/epoxy composite, the initial stress partitioning factor and the corresponding function of fiber volume function are numerically calculated as follows:
ð27Þ
k¼1
ð29Þ
j0 ¼ 0:6562; g ðV f Þ ¼ ½10:2ðV f Þ3 12:6ðV f Þ2 þ 4:7 V f 0:6562
K .X
gk
ð31Þ
k¼1
The longitudinal shear stress predicted by algebraic equation correlates well with the numerical results of micromechanical method, as shown in Fig. 7.
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partitioning factor was set to a constant value of u ¼ 0:8889 and it is independent of time and fiber volume fraction, namely, hðV f Þ ¼ 0. Hence, the volume of the matrix materials that is subjected to a constant stress level varies linearly with the fiber volume fraction. Thus, Eq. (32) can be written as
r13ðtÞ ¼
f
V Gf13
V f þ u0 V m m þ m PuK 0 V G0 1
t=sk
g ð1e k¼1 k
Þ
c0
ð33Þ
As shown in Fig. 8, the above equation accurately predicts the relaxation of transverse shear stress of fiber-reinforced materials with different fiber volume fractions. 6. Discussion
Fig. 7. The variations of longitudinal shear stress as a function of time domain and fiber volume fraction for a graphite/epoxy composite.
5.4. Transverse shear loading The analysis of transverse shear stress, r23 , required to keep a prescribed shear strain in 2–3 plane is the same way used for the analysis of longitudinal shear stress in the relaxation test. The result is
r23ðtÞ
h i P V f þ u0 þ hðV f Þ Kk¼1 gk ð1 et=sk Þ V m h i ¼ c0 P f
V f G23
þ
u0 þhðV f Þ
K
g ð1et=sk Þ k¼1 k
PK
Gm 0 1
g ð1et=sk Þ k¼1 k
ð32Þ
Vm
where u0 is a constant value and h is a function of fiber volume fraction. Fig. 8 illustrates the predictions of the algebraic expression for r23 as a function of time domain and fiber volume fraction in the graphite/epoxy composite. The finite element results of micromechanical model with square fiber packing are used for comparisons. In order to obtain a good agreement between algebraic expression and numerical results for transverse shear loading, the stress
Fig. 8. The variations of transverse shear stress as a function of time domain and fiber volume fraction for a graphite/epoxy composite.
The analytical expressions provide a convenient procedure to model the relaxation response of viscoelastic polymers reinforced with elastic fibers. Their predictions correlate well with the numerical results of micromechanical model. The present model can be used to analyze different parameters in the design of composite materials for a specific viscoelastic property. Furthermore, the analytical expressions can be used to determine the creep response of composite materials. Using the matrix and fiber properties and the fiber volume fraction, the analytical expressions determine the stress required to stretch the composite material in the fiber direction and the Poisson’s ratio m12 in the different time domain of relaxation test. There is a considerable drop in the axial stress immediately after applying the initial strain, as shown in Fig. 4. After time more than 101.45 s, the variation of stress highly reduces in the graphite/ epoxy composites. This time duration corresponds to the largest sk considered in the viscoelastic matrix. The time required that the stress state to reach a stable condition reduces in the composite materials having high fiber volume fraction, as shown in Fig. 4. The effect of fiber reinforcing polymer materials on the reduction of Poisson’s ratio m12 fades with time, as shown in Fig. 5. The variation of Poisson’s ratio with loading time is similar for the composite materials with different fiber volume fractions. In a composite materials subjected to the axial transverse strain and longitudinal shear strain, the required stress can be predicted by analytical expressions as functions of constituent properties and the stress partitioning factor which depends on the loading time and fiber volume fraction. The Prony series describing the stress relaxation function in the polymer matrix provide an accurate approximation of the stress partitioning factor in the time domain. To determine the relationship between the stress partitioning factor and the fiber volume fraction in the transverse axial and longitudinal shear loadings, the corresponding stress is required at the start and sufficiently long time after loading in the stress relaxation test. The stress values can be determined by analyzing the finite element analysis of micromechanical model or measuring in the stress relaxation test. To characterize the stress relaxation response of composite material subjected to the transverse shear strain, a constant stress partitioning factor is considered in the derivation of analytical expression. Hence, the shear stress at the start of relaxation test is sufficient in order to determine the constant stress partitioning factor in the composite materials. As illustrated in Fig. 8, the shear stress highly depends on the fiber volume fraction at the start of relaxation test. However, the long time shear stress has lower dependency to the fiber volume fraction, because the shear stress reduces in the viscoelastic matrix and lower shear stress is transmitted to the fibers. Hence, the initial shear strain of the fiber decreases as the relaxation test proceeds. Therefore, the fibers have negligible reinforcing effects on the viscoelastic polymer materials for a sufficiently long time after loading.
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7. Conclusions A procedure is developed to determine the viscoelastic behavior of composite materials consisting of linearly viscoelastic matrices and transversely isotropic elastic fibers. The analysis of a simplified unit cell subjected to the prescribed axial and shear deformations yields the closed-form analytical expressions that accurately predict the relaxation response of composite materials. Defining a stress partitioning factor provides a convenient way to account for the division of the transverse axial stress and shear stress in each of the constituents of viscoelastic composite. The stress partitioning factor depends on the time domain and fiber volume fraction in the transverse axial loading and longitudinal shear loading, while it is a constant value in the transverse shear loading. The present model can be conveniently used for parametric studies and the finite element analysis of viscoelastic composite structures. The procedure can be extended to the characterization of other nonlinear and time-depended behaviors of composite materials. References [1] Nemat-Nasser S, Hori M. Micromechanics: overall properties of heterogeneous materials. Amsterdam: Elsevier Science Publishers; 1993. [2] Hashin Z. Analysis of composite materials – a survey. J Appl Mech 1983;105:481–504. [3] Whitney JM, Riley MB. Elastic properties of fiber reinforced composite materials. J AIAA 1966;4:1537–42. [4] Aboudi J. Mechanics of composite materials a unified micromechanical approach. Amsterdam: Elsevier Science Publishers; 1991. [5] Aboudi J. Micromechanical modeling of finite viscoelastic multiphase composites. Math Phys 2000;54:114–34.
[6] Yancey RN, Pindera MJ. Micromechanical analysis of the creep response of unidirectional composites. J Eng Mater Tech 1990;112:157–63. [7] Barbero EJ, Luciano R. Micromechanical formulas for the relaxation tensor of linear viscoelastic composites with transversely isotropic fibers. Int J Solids Struct 1995;32(13):1859–72. [8] Brinson LC, Lin WS. Comparison of micromechanics methods for effective properties of multiphase viscoelastic composites. Compos Struct 2003;41:353–67. [9] Fisher FT, Brinson LC. Viscoelastic interphase in polymer–matrix composites: theoretical models and finite element analysis. Compos Sci Technol 2003;61:731–48. [10] Mori T, Tanaka K. Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Mettall 1973;21:571–4. [11] Benveniste Y. A new approach to the application of Mori–Tanaka’s theory in composite materials. Mech Mater 1987;6:147–57. [12] Naik A, Abolfathi N, Karami G, Ziejewski M. Micromechanical viscoelastic characterization of fibrous composites. J Compos Mater 2008;42:1179–204. [13] Bertilsson H, Delin M, Kubat J, Rychwalski WR, Kubat MJ. Strain rates and volume changes during short-term creep of PC and PMMA. Rheol Acta 1993;32:361–9. [14] Lu H, Zhang X, Knauss WG. Uniaxial, shear, and Poisson relaxation and their conversion to bulk relaxation: studies on poly (methyl methacrylate). Polym Compos 2004;18(2):211–22. [15] Drozdov AD. Mechanics of viscoelastic solids. John Wiley & Sons; 1998. [16] Wang J, Qian Z, Liu S. Processing mechanics for flip chip assemblies. In: Proceedings of the SMT conference and exhibition, micro materials, Berlin; 1997. p. 287–93. [17] Betten J. Creep mechanics. Springer; 2008. [18] Grzywinski GG, Woodford DA. Creep analysis of thermoplastics using stress relaxation data. Polym Eng Sci 1995;35:1931–7. [19] Touati D, Cederbaum G. On the prediction of stress relaxation from known creep of nonlinear materials. J Eng Mater Technol 1997;119:121–5. [20] Test Ladanyi B, Melouki M. Determination of creep properties of frozen soils by means of the borehole stress relaxation. Can Geo J (CGJOAH) 1993;30:170–86. [21] Li S. Boundary conditions for unit cells from periodic microstructures and their implications. Compos Sci Technol 2008;68:1962–74.
Contents lists available at ScienceDirect
Composites Science and Technology journal homepage: www.elsevier.com/locate/compscitech
Micromechanical analysis of stress relaxation response of fiber-reinforced polymers Mohammad Tahaye Abadi * Aerospace Research Institute, Ministry of Science, Research and Technology, P.O. Box 14665-834, Tehran, Iran
a r t i c l e
i n f o
Article history: Received 6 August 2008 Received in revised form 28 January 2009 Accepted 28 February 2009 Available online 9 March 2009 Keywords: A. Polymer–matrix composites C. Stress relaxation Micromechanics
a b s t r a c t The paper describes a micromechanical method to determine the stress relaxation response of polymer composites consisting of linearly viscoelastic matrices and transversely isotropic elastic fibers. A representative unit cell is subjected to some prescribed axial and shear loadings to study and quantify the time-dependent behavior of composite materials. Closed-form analytical expressions are derived describing the anisotropic viscoelastic response of composite materials as functions of matrix and fiber properties. The present analytical expressions are employed to determine the stress relaxation behavior of a graphite/epoxy composite and the results are compared with the finite element analysis of the micromechanical model. Very good correlation between analytical expressions and numerical results is illustrated for the linearly anisotropic viscoelastic response of composite materials. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction Many polymer composites exhibit viscoelastic behavior, in which the magnitude of the stress components is a function of the deformation history and it depends on strain, strain rate, temperature and time. The viscoelastic materials have memory and this memory typically fades with time. The long molecular chains of polymer matrix are the source of viscoelasticity phenomenon in polymer composites. It has been an active research area for many years to predict the viscoelastic properties of the composites based on their constituents. Many analytical as well as numerical modeling algorithms have been forwarded and the experimental studies have been conducted to examine the overall viscoelastic properties of composites. The micromechanical methods provide efficient tools to characterize the behavior of the composite materials. A representative volume element of the compound material is analyzed due to the known properties of their constituents and loading cases. In the micromechanical approach, the heterogeneous structure of the composite material is replaced by a homogeneous medium with anisotropic properties [1]. Hashin [2] was the first to discuss the micromechanical method to model the viscoelastic response of unidirectional composites. Based on an energy balance approach with the aid of elasticity theory, Whitney and Riley [3] obtained closedform analytical expressions describing the elastic modulus of composites. Since it has been difficult to derive analytical expressions determining the viscoelastic behavior of composite materials, the micromechanical models were numerically evaluated. Aboudi [4,5] developed a unified micromechanical theory based on the * Tel.: +98 21 88366030; fax: +98 21 88362011. E-mail address: [email protected] 0266-3538/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.compscitech.2009.02.036
generalized method of cells (GMC) to predict the overall behavior of composite materials with elastic or inelastic constituents. In his work, each cell was divided into a convenient number of subcells and the compatibility of displacements and equilibrium of tractions were implemented at the various interfaces. Yancey and Pindera [6] applied a Laplace transform inversion scheme to the micromechanical model proposed by Aboudi to discuss the creep compliances of a unidirectional graphite/epoxy composite, and the predictions were compared with experimental data at different temperatures. Barbero and Luciano [7] developed explicit analytical expressions for the relaxation modulus in the Laplace domain and a numerical method was used to obtain the Laplace inversion in time domain. Brinson and Lin [8] and Fisher and Brinson [9] used the finite element method to analyze the two-phase and three-phase viscoelastic composites, respectively, and the results were compared with the Mori– Tanaka model [10,11]. Naik et al. [12] used the finite element method to predict the viscoelastic properties of composites considering the periodic boundary conditions. In the present research work, closed-form analytical expressions are derived to determine the stress relaxation response of polymer composites consisting of linearly viscoelastic matrices and transversely isotropic elastic fibers. A micromechanical approach is used to derive the analytical expressions in time domain. A representative unit cell is subjected to some prescribed axial and shear loadings in order to study and quantify the timedependent behavior of composite materials. The predictions of the present algebraic expressions are employed to determine the stress relaxation behavior of a graphite/epoxy composite and the results are compared with the finite element analysis of micromechanical model. Very good correlation between analytical expressions and numerical results is illustrated for the linearly anisotropic viscoelastic response of composite materials.
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2. Mechanical models of composite constituents Most polymers used as the matrix of composite materials exhibit viscoelastic behavior, in which the stress tensor depends on the history of deformation, but the history dependent behavior fades with time. To determine the stress tensor, a function is needed to describe the effects of history dependent behavior in the polymer materials. Experimental studies [13,14] have shown that the Poisson’s ratio of polymers is a time-dependent variable and the constitutive equations based on constant Poisson’s ratio have poorly confirmed by experimental data. The stress components in a linear viscoelastic material at a specific time t can be written as [15]
rmijðtÞ ¼
Z t Em Em m m m 0 0 e d þ e R e d s ðtsÞ ijðsÞ kkðtÞ ij ijðtÞ 3ð1 2mm 1 þ mm 0 0Þ 0
ð1Þ
Table 1 The parameters of the Prony series describing the stress relaxation function in the epoxy materials. k
gk a
ska
kk
vk ðs1 Þ
1 2 3 4 5 6 7 8 9 10 11 12
0.012623 0.02552 0.062518 0.076863 0.128583 0.147175 0.181631 0.171479 0.083365 0.034882 0.012037 0.009644
1.0145E09 1.0145E08 1.0145E07 1.0145E06 1.0145E05 1.0145E04 1.0145E03 1.0145E02 1.0145E01 1.0145E+00 1.0145E+01 1.0145E+02
0.001179 0.002388 0.005884 0.0073 0.012382 0.014445 0.018246 0.01769 0.008762 0.003693 0.001278 0.001025
1.0154E09 1.0162E08 1.0188E07 1.0198E06 1.0234E05 1.0248E04 1.0273E03 1.0268E02 1.0205E01 1.0170E+00 1.0154E+01 1.0152E+02
a
rmijðtÞ and emijðtÞ are, respectively, the components of stress and m m strain tensors, em ijðtÞ the component of deviatoric strain (eij ¼ eij m m m ekk dij =3Þ, RðtsÞ the relaxation kernel, E0 and m0 the initial elastic
According to Ref. [12,16].
where
modulus and Poisson’s ratio, respectively. The summation is taken over the repeating indices and the superscript m denotes to the variables defined in the matrix of composite material. Viscoelastic properties of materials are usually obtained via stress relaxation tests. As an example, consider a viscoelastic material subjected to only shear strain in plane 2–3 as follows:
em23ðtÞ ¼
1 2
0
t<0
ð2Þ
c0 t P 0
where c0 is a constant shear strain. Substituting Eq. (2) in (1) results in
Z t Em 0 c0 R d s 1 ðt s Þ 2ð1 þ mm 0 0Þ Z t ¼ rm RðtsÞ ds 23ð0Þ 1
rm23ðtÞ ¼
ð3Þ
0
Experimental studies of polymers have revealed that the relaxation function can be given by
RðtÞ ¼
K X gk k¼1
sk
e
t=sk
ð4Þ
where gk and sk are the material constants. Using Eqs. (3) and (4), the drop of shear stress can be described by Prony series as follows: K X rm23ðtÞ Z t 1 m ¼ RðtsÞ ds ¼ gk ð1 et=sk Þ r23ð0Þ 0 k¼1
ð5Þ
The relaxation behavior of polymers can be evaluate in tension test with the following program: m 11ðtÞ
e
¼
0
t<0
ð6Þ
e0 t P 0
where e0 is a constant value. According to Eq. (1), the axial stresses along and normal to the extension direction become Z t m m Em e 1 2mðtÞ 2ð1 þ mðtÞ Þ 2 m ð7aÞ rm11ðtÞ ¼ 0 0 þ R ð1 þ m Þd s ðt s Þ ðsÞ 3 1 2mm 1 þ mm 1 þ mm 0 0 0 0 Z m m t E e 1 2mðtÞ 1 þ mðtÞ 1 ð7bÞ rm22ðtÞ ¼ 0 0 þ RðtsÞ ð1 þ mm ðsÞ Þds m m 3 1 2mm 1 þ m m 1 þ 0 0 0 0 m m m where mm ðtÞ is the Poisson’s ratio (mðtÞ ¼ e22ðtÞ =e11ðtÞ Þ which is a time-
dependent property of viscoelastic materials unlike the linear elastic materials. Since the axial stress normal to the extension direction is zero, Eq. (7b) yields the integral equation describing the Poisson’s ratio of isotropic linear viscoelastic materials as follows:
mmðtÞ ¼ m0 þ
1 2mm 0 3
Z
t 0
RðtsÞ ð1 þ mm ðsÞ Þds
ð8Þ
Substituting Eq. (8) in (7a) gives the axial stress required to produce and hold prescribed strain
rm11ðtÞ ¼ Em0 e0
1 2mm ðtÞ
ð9Þ
1 2mm 0
As an example, the initial engineer constants of the epoxy materials are given by [12,16]
Em 0 ¼ 8:3 GPa;
mm0 ¼ 0:4
ð10Þ
To describe the stress relaxation function of the epoxy materials, twelve parameter sets have been used for gk and sk , as listed in Table 1. The fibers usually behave as elastic materials in the working temperature of polymer–matrix composites. In the present work, a transversely isotropic elastic model is used to model the fiber materials. The elastic properties of the carbon fiber are given by
Gf12 ¼ Gf13 ¼ 8:96 GPa
Ef1 ¼ 233 GPa;
Ef2 ¼ 23:1 GPa;
Gf23
mf12 ¼ mf13 ¼ 0:200; mf23 ¼ 0:400
¼ 8:27 GPa;
ð11Þ in which the fiber axis is along the one direction. 3. Stress relaxation response of fiber-reinforced polymers Based on the Boltzmann principle, the stress tensor of a viscoelastic material at the current instant is a functional on the entire history of strain tensor in the interval [1; t]. Assuming this functional to be linear and applying Rizes theorem, the stress components of a nonaging viscoelastic material can be written as [17]
rijðtÞ ¼
Z
t
1
vijklðtsÞ deklðsÞ ¼ vijklð0Þ eklðtÞ
Z
t
1
@ vijklðtsÞ @s
eklðsÞ ds
ð12Þ
where limt!1 eklðtÞ ¼ 0. The relaxation test is performed by applying constant strain components, ekl , as the following program:
eklðtÞ ¼
0 ekl
t<0
ð13Þ
tP0
Combining Eqs. (12) and (13) yields the stress tensor as follows:
rijðtÞ ¼ vijklð0Þ
Z
t
1
@ vijklðtsÞ @s
ds ekl ¼ C ijklðtÞ ekl
ð14Þ
In the present research work, the time-dependent functions of C ijklðtÞ are determine for transversely isotropic viscoelastic materials, in which the general response can be described by five time-dependent functions. It should be noted that these functions are material parameters and independent of strain tensor and the status of loading. The previous research works [15,18–20] have shown that the creep and the stress relaxation responses can be related to each
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other. The numerical solution of a Volterra integral equation yields the creep behavior based on the stress relaxation results [15]. 4. Micromechanical models The micromechanical model is commonly used to determine the properties of composite materials. A repeating unit cell based on a pre-determined fiber packing is assumed to represent the microstructure of the composite. The response of the unit cell under specified loading conditions are analyzed in order to study and quantify the viscoelastic property of the unit cell. In the micromechanical models, it is assumed that the fibers are regularly spaced and perfectly bonded to the matrix material. In the present research work, a micromechanical model based on the mechanics of materials approach is used to determine the viscoelastic properties of polymer composites as a function of their constituent properties. In the present micromechanical model, a simplified representative unit cell is subjected to the specific deformations and the variation of deforming force is investigated throughout the time domain. The unit cell is a section cut from a single layer of fiber-reinforced material. The section consists of side-by-side alternating region of fiber and matrix and the fibers arranged in the parallel arrays, as shown in Fig. 1. In the following sections, the response of the unit cell in different deformation fields is analyzed in order to determine the algebraic expressions predicting the viscoelastic properties of the composite materials as functions of fiber and matrix properties. In the present research work, the finite element analysis of micromechanical model is used to verify the derived closed-form analytical expressions predicting the relaxation behavior of composite materials. A unit cell with square fiber packing is used in the finite element analysis to examine the viscoelastic properties of composite materials with different fiber volume fractions. The dimensions of unit cell are chosen by 2b1 2b2 2b3 and different diameters are considered for the circular cross section of fiber depending on the fiber volume fraction. Since the deformation along fiber direction is uniform, the minimum number of elements is considered based on a convenient aspect ratio for generated mesh. Due to the material and geometric symmetries, one-eighth of unit cell is meshed, as shown in Fig. 2. To represent the periodic patterns of the microstructures, boundary conditions of unit cell follow from the symmetries including translations, reflections and rotations [21]. In the finite element analysis of micromechanical model, a prescribed deformation is applied to the unit cell and it is kept constant throughout
Fig. 1. The unit cell considered in the mechanics of materials approach.
Fig. 2. The mesh geometry used in one-eighth of the unit cell with square fiber packing.
the time domain in order to characterize the stress relaxation behavior. To apply the boundary conditions to one-eighth of unit cell, it is required that the displacement of some planes of unit cell is set to a specific value [21]. A virtual point is defined in the finite element model in order to apply a prescribed displacement component to the nodes located on a specific plane. The displacement of these nodes and the virtual point are set the same. Therefore, the force component required to deform the virtual point is equivalent to the integral of corresponding forces applied to the nodes located on the specific plane. The stress on the plane is calculated based on the force applied to the virtual point and the area of plane. As shown in Fig. 3, the force required to keep a constant deformation
Fig. 3. The applied displacement and variation of force in the finite element analysis of relaxation test.
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is reduced as time elapses and it is determined in the finite element analysis as well as the closed-form analytical expressions in the following section. In the finite element analysis, the time considered to apply the deformation is set to a very small value (t 1 ¼ 1011 s) compared to the total relaxation time (t1 tf Þ. 5. Loading of unit cell 5.1. Longitudinal axial loading To study the behavior of composite materials along the fiber direction, the unit cell shown in Fig. 1 is subjected to a fixed axial strain e0 in the one direction. Since the bond between of the fiber and matrix is assumed to be perfect, both of them are stretched by the same amount in the one direction. Treating the two constituents as if they are in a one-dimensional state of stress, the stress in the constituents of composite material can be determined using Eq. (9) and the axial modulus of the elastic fiber. Since the sum of forces in the fiber and matrix is equal to the total force, the stress in the composite material becomes:
r11ðtÞ ¼ Ef1 V f þ Em0 V m
1 2mm ðtÞ
1 2m
m 0
e0
ð15Þ
where V f and V m are the volume fractions of fiber and matrix, respectively (V f þ V m ¼ 1). According to the above equation, the axial stress depends on the Poisson’s ratio of matrix materials that it can be determined by solving the integral Eq. (8). The Laplace transformation of integral equation gives
m^mðsÞ ¼
mm0
1 2mm 0 ^ ^m RðSÞ ð1 þ m þ ðsÞ Þ s 3
ð16Þ
^ ^m where m ðsÞ and RðSÞ are the Laplace transformations of the functions mmðtÞ and RðtÞ , respectively. Using Eqs. (4) and (16), Laplace transformation of Poisson’s ratio becomes
^m ðsÞ
m ¼
K X 1 2mm gk =sk 0 þ s 3 1 þ s=sk k¼1
mm0
!
K X 1 2mm gk =sk 0 1 3 1 þ s=sk k¼1
!1
ð17Þ The inverse Laplace transformation of Eq. (17) becomes
tmðtÞ ¼ tm0 þ
K X
kk ð1 et=vk Þ
Fig. 4. The drop of longitudinal stress as a function of time domain and fiber volume fraction for a graphite/epoxy composite.
DW DW f DW m ¼ þ ¼ V f ef22 þ V m em 22ðtÞ W W W f f m m ¼ ðV v 12 þ V tðtÞ Þe0
e22ðtÞ ¼
ð20Þ
where W is the width of unit cell, DW f and DW m are the width changes of fiber and matrix, respectively. Substituting Eq. (18) in (20) yields the Poisson’s ratio of composite material, i.e.
"
K X m12 ¼ 22ðtÞ ¼ V f v f12 þ V m tm0 þ kk ð1 et=v Þ e0 k¼1 k
# ð21Þ
Fig. 5 shows the variations of Poisson’s ratio t12 with time domain and fiber volume fraction for a graphite/epoxy composite determined by Eq. (21) and finite element analysis of micromechanical model. The accuracy of algebraic expression for t12 is quite obvious and it is similar to the accuracy of the results for longitudinal stress in the composite materials.
ð18Þ
k¼1
where kk and vk are the constants depending on gk , sk and mm 0 . Table 1 gives the calculated constants expressing the time-dependent Poisson’s ratio of the epoxy materials. Substituting Eq. (18) in (15) determines the axial stress in the composite material, i.e.
(
r11ðtÞ ¼
Ef1 V f
" þ
m Em 0V
K X 2 1 kk ð1 et=vk Þ 1 2mm 0 k¼1
#)
e0
ð19Þ
Fig. 4 illustrates the predictions of the algebraic expression for the drop in longitudinal stress (r 1 ¼ 1 r11ðtÞ =r11ð0Þ Þ as a function of time domain and fiber volume fraction in the graphite/epoxy composite. The finite element results of micromechanical model with square fiber packing are used for comparisons. The derived expression is quite accurate for the prediction of the relaxation longitudinal stress in the fiber-reinforced material. The axial stretching strain contracts the composite material in the two direction due to Poisson’s effects. Because the fiber and matrix have different Poisson’s ratios, they will not contract the same amount in the widthwise and the combined contraction of the fiber and matrix results in the overall contraction of the composite in the two direction. Hence, the transverse axial strain of composite material can be written as
Fig. 5. The Poisson’s ratio t12 as a function of time domain and fiber volume fraction for a graphite/epoxy composite.
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5.2. Transverse axial loading To characterize the relaxation response of composite materials in the transverse direction, the unit cell is subjected to a fixed strain in the two direction. The overall change in transverse dimension of the unit cell is the sum of dimension change of the two constituents. Due to unit cell geometry, the transverse strain can be written as
e0 ¼
DW DW f DW m ¼ þ W W W
ð22Þ
Since there is no direct path to transmit the transverse stress to the fiber, a portion of the transverse stress is transmitted through the fiber, and a portion is transmitted around the fiber, through the matrix material. Therefore, a stress partitioning factor is introduced to account for the division of the stress in each of the two constituents. The volume fraction of matrix that is subjected to a constant stress level r22 , is less than its real existing volume fraction, namely, qV m , where q is the stress partitioning factor (0 < q < 1). Due to viscoelastic behavior of matrix material, the stress partitioning factor is a time-dependent variable and it depends on the fiber volume fraction in the general form. The volume fraction of fiber that is subjected to stress level r22 is assumed to be V f . As a result of the stress partitioning factor, the strain in the direction two defined by Eq. (22) becomes
e0 ¼ ðV f ef22 þ qðV f ;tÞ V m em22ðtÞ ÞðV f þ qðV f ;t ÞV m Þ1
r22ðtÞ
Vf
V f þ qðV f ;tÞ V m
Ef2
þ
qðt;V f Þ V m 1 2mm 0 1 2mm Em ðtÞ 0
! ð24Þ
It is assumed that the stress partitioning factor can be expressed by a Prony series, i.e.
qðV f ;tÞ ¼ q0 þ fðV f Þ
K X
5.3. Longitudinal shear loading
ð23Þ
Considering a one-dimensional stress state in the two constituents, the transverse strain in the fiber and matrix is related to the stress level using Eq. (9) and the transverse modulus of elastic fiber. Then, the strain in the direction two can be written as
e0 ¼
Fig. 6. The drop of transverse stress as a function of time domain and fiber volume fraction for a graphite/epoxy composite.
gk ð1 et=sk Þ
ð25Þ
To evaluate the viscoelastic shear behavior of fiber-reinforced composite, the unit cell is subjected to a constant shear strain c0 in the 1–2 plane and the required shear stress r12 is determined. Since there is no direct path to transmit the shear stress to the fiber materials, similar to transverse loading; a stress partitioning factor is introduced to account for the division of the stress in each of the two constituents. The volume fraction of matrix that is subjected to stress level r12 is denoted by jV m , where j is the stress partitioning factor depending on fiber volume fraction and loading time. Similarly, the shear strain of composite material can be expressed by
c0 ¼ ðV f cf12 þ jðV f ;tÞ V m cm12ðtÞ ÞðV f þ jðV f ;tÞ V m Þ1
ð28Þ
k¼1
where q0 is a constant and f is a function depends on the fiber volume fraction. Substituting Eqs. (18) and (25) in (24) results in the transverse stress, namely, f
r22ðtÞ ¼
h
Vf f E2
þ
t=sk
k¼1 gk ð1 e PK gk ð1et=sk Þ ðV f Þ PK k¼1 t=v
V þ q0 þ fðV f Þ q0 þf
PK
2
1
k ð1e k¼1 k 12mm 0
kÞ
i
Þ V
m
e0
ð26Þ
r12ðtÞ ¼
Vf þ
h
j0 þ g ðV f Þ h
f
V f G12
Vm Em 0
Fig. 6 shows the drop of transverse stress (r 2 ¼ 1 r22ðtÞ =r22ð0Þ Þ predicted by Eq. (26) and the finite element analysis considered in different time domain and fiber volume fraction. It is a common procedure to determine the stress partition factors by numerical analysis of micromechanical model describing the microstructure of anisotropic elastic materials. In the present research work, a numerical procedure is used to determine the stress partition factors. For the graphite/epoxy composite, the parameters defining the stress partitioning factor are given by
q0 ¼ 0:385; f ðV f Þ ¼ ½9:0ðV f Þ3 9:7ðV f Þ2 þ 3:9 V f 0:385
Considering the pure shear deformation of the viscoelastic matrix according to Eq. (5) and longitudinal shear modulus of elastic fiber, Eq. (28) can be written as
K .X
gk
þ
PK
j0 þgðV f Þ
g
k¼1 k ð1
PK
g ð1e k¼1 k
PK
Gm 0 1
i et=sk Þ V m i c0 t=sk
Þ Vm
t=sk
Þ
g ð1e k¼1 k
In which the stress partitioning factor for longitudinal shear stress is approximated by
jðt;V f Þ ¼ j0 þ g ðV f Þ
K X
gk ð1 et=sk Þ
Since the transverse load is transmitted to the fiber through the matrix material, the viscoelastic properties of the matrix material have considerably effects on the drop of transverse stress and it is more than the drop of longitudinal stress shown in Fig. 4.
ð30Þ
k¼1
The variation of longitudinal shear stress is shown in Fig. 7 that it is computed by Eq. (29) and the finite element analysis of micromechanical model in different time domain and fiber volume fractions. For the graphite/epoxy composite, the initial stress partitioning factor and the corresponding function of fiber volume function are numerically calculated as follows:
ð27Þ
k¼1
ð29Þ
j0 ¼ 0:6562; g ðV f Þ ¼ ½10:2ðV f Þ3 12:6ðV f Þ2 þ 4:7 V f 0:6562
K .X
gk
ð31Þ
k¼1
The longitudinal shear stress predicted by algebraic equation correlates well with the numerical results of micromechanical method, as shown in Fig. 7.
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partitioning factor was set to a constant value of u ¼ 0:8889 and it is independent of time and fiber volume fraction, namely, hðV f Þ ¼ 0. Hence, the volume of the matrix materials that is subjected to a constant stress level varies linearly with the fiber volume fraction. Thus, Eq. (32) can be written as
r13ðtÞ ¼
f
V Gf13
V f þ u0 V m m þ m PuK 0 V G0 1
t=sk
g ð1e k¼1 k
Þ
c0
ð33Þ
As shown in Fig. 8, the above equation accurately predicts the relaxation of transverse shear stress of fiber-reinforced materials with different fiber volume fractions. 6. Discussion
Fig. 7. The variations of longitudinal shear stress as a function of time domain and fiber volume fraction for a graphite/epoxy composite.
5.4. Transverse shear loading The analysis of transverse shear stress, r23 , required to keep a prescribed shear strain in 2–3 plane is the same way used for the analysis of longitudinal shear stress in the relaxation test. The result is
r23ðtÞ
h i P V f þ u0 þ hðV f Þ Kk¼1 gk ð1 et=sk Þ V m h i ¼ c0 P f
V f G23
þ
u0 þhðV f Þ
K
g ð1et=sk Þ k¼1 k
PK
Gm 0 1
g ð1et=sk Þ k¼1 k
ð32Þ
Vm
where u0 is a constant value and h is a function of fiber volume fraction. Fig. 8 illustrates the predictions of the algebraic expression for r23 as a function of time domain and fiber volume fraction in the graphite/epoxy composite. The finite element results of micromechanical model with square fiber packing are used for comparisons. In order to obtain a good agreement between algebraic expression and numerical results for transverse shear loading, the stress
Fig. 8. The variations of transverse shear stress as a function of time domain and fiber volume fraction for a graphite/epoxy composite.
The analytical expressions provide a convenient procedure to model the relaxation response of viscoelastic polymers reinforced with elastic fibers. Their predictions correlate well with the numerical results of micromechanical model. The present model can be used to analyze different parameters in the design of composite materials for a specific viscoelastic property. Furthermore, the analytical expressions can be used to determine the creep response of composite materials. Using the matrix and fiber properties and the fiber volume fraction, the analytical expressions determine the stress required to stretch the composite material in the fiber direction and the Poisson’s ratio m12 in the different time domain of relaxation test. There is a considerable drop in the axial stress immediately after applying the initial strain, as shown in Fig. 4. After time more than 101.45 s, the variation of stress highly reduces in the graphite/ epoxy composites. This time duration corresponds to the largest sk considered in the viscoelastic matrix. The time required that the stress state to reach a stable condition reduces in the composite materials having high fiber volume fraction, as shown in Fig. 4. The effect of fiber reinforcing polymer materials on the reduction of Poisson’s ratio m12 fades with time, as shown in Fig. 5. The variation of Poisson’s ratio with loading time is similar for the composite materials with different fiber volume fractions. In a composite materials subjected to the axial transverse strain and longitudinal shear strain, the required stress can be predicted by analytical expressions as functions of constituent properties and the stress partitioning factor which depends on the loading time and fiber volume fraction. The Prony series describing the stress relaxation function in the polymer matrix provide an accurate approximation of the stress partitioning factor in the time domain. To determine the relationship between the stress partitioning factor and the fiber volume fraction in the transverse axial and longitudinal shear loadings, the corresponding stress is required at the start and sufficiently long time after loading in the stress relaxation test. The stress values can be determined by analyzing the finite element analysis of micromechanical model or measuring in the stress relaxation test. To characterize the stress relaxation response of composite material subjected to the transverse shear strain, a constant stress partitioning factor is considered in the derivation of analytical expression. Hence, the shear stress at the start of relaxation test is sufficient in order to determine the constant stress partitioning factor in the composite materials. As illustrated in Fig. 8, the shear stress highly depends on the fiber volume fraction at the start of relaxation test. However, the long time shear stress has lower dependency to the fiber volume fraction, because the shear stress reduces in the viscoelastic matrix and lower shear stress is transmitted to the fibers. Hence, the initial shear strain of the fiber decreases as the relaxation test proceeds. Therefore, the fibers have negligible reinforcing effects on the viscoelastic polymer materials for a sufficiently long time after loading.
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7. Conclusions A procedure is developed to determine the viscoelastic behavior of composite materials consisting of linearly viscoelastic matrices and transversely isotropic elastic fibers. The analysis of a simplified unit cell subjected to the prescribed axial and shear deformations yields the closed-form analytical expressions that accurately predict the relaxation response of composite materials. Defining a stress partitioning factor provides a convenient way to account for the division of the transverse axial stress and shear stress in each of the constituents of viscoelastic composite. The stress partitioning factor depends on the time domain and fiber volume fraction in the transverse axial loading and longitudinal shear loading, while it is a constant value in the transverse shear loading. The present model can be conveniently used for parametric studies and the finite element analysis of viscoelastic composite structures. The procedure can be extended to the characterization of other nonlinear and time-depended behaviors of composite materials. References [1] Nemat-Nasser S, Hori M. Micromechanics: overall properties of heterogeneous materials. Amsterdam: Elsevier Science Publishers; 1993. [2] Hashin Z. Analysis of composite materials – a survey. J Appl Mech 1983;105:481–504. [3] Whitney JM, Riley MB. Elastic properties of fiber reinforced composite materials. J AIAA 1966;4:1537–42. [4] Aboudi J. Mechanics of composite materials a unified micromechanical approach. Amsterdam: Elsevier Science Publishers; 1991. [5] Aboudi J. Micromechanical modeling of finite viscoelastic multiphase composites. Math Phys 2000;54:114–34.
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