Numerical evaluation of the coefficients of thermal ... - Springer Link

Journal of Mechanical Science and Technology 29 (3) (2015) 1187~1197 www.springerlink.com/content/1738-494x

DOI 10.1007/s12206-015-0231-x

Numerical evaluation of the coefficients of thermal expansion of fibers in composite materials using a lamina-scale cost function with quasi-analytical gradients† Jae Hyuk Lim1, Jean-Baptiste Charpentier2 and Dongwoo Sohn3,* 1

Satellite Mechanical Department, Korea Aerospace Research Institute, 169-84 Gwahak-ro, Yuseong-gu, Daejeon, 305-806, Korea 2 Mechanical Engineering Department, École Nationale Supérieur des Mines de Saint-Étienne, 158, cours Fauriel F-42023 Saint-Étienne cedex 2, France 3 Division of Mechanical Engineering, College of Engineering, Korea Maritime and Ocean University, 727 Taejong-ro, Yeongdo-gu, Busan, 606-791, Korea (Manuscript Received August 13, 2014; Revised November 6, 2014; Accepted December 4, 2014)

----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

Abstract In this work, the coefficients of thermal expansion (CTEs) of fibers in composite materials that contain microstructures are numerically evaluated using a lamina-scale cost function with quasi-analytical gradients. To consider the effects of fiber arrangements and local defects, such as interface debonding and voids, a variety of representative volume elements are modeled with a number of finite element meshes. Then, the CTEs of fibers are evaluated by minimizing a lamina-scale cost function that represents the difference between the measured CTEs and the computed CTEs by means of a computational homogenization scheme for the composite lamina. The descent direction of the cost function is obtained using quasi-analytical gradients that take partial derivatives from prediction models, such as the Schapery model and Hashin model defined in an explicit manner, which accelerates the minimization procedure. To verify the performance of the proposed scheme in terms of accuracy and efficiency, the CTEs of constituents calculated using the proposed scheme in a unidirectional composite lamina are compared with experimental values reported in the literature. Furthermore, the convergence behavior of the proposed scheme with quasi-analytical gradients is also investigated and compared with other minimization methods. Keywords: Representative volume element (RVE); Inverse analysis; Coefficient of thermal expansion (CTE); Quasi-analytical gradient; Unidirectional (UD) composite ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

1. Introduction Composite materials have been widely used in the automotive, offshore, sporting, and aerospace industries because of their high stiffness and strength-to-weight ratio. Their good mechanical characteristics, such as low coefficient of thermal expansion (CTE) [1] and coefficient of moisture expansion (CME) as well as long-term fatigue life, make structural engineers choose them for versatile structures with state-of-the-art technology, such as aircraft and spacecraft. During the development period, numerous tests with a proper choice of materials for fiber, matrix, and fiber-volume ratio could be a promising solution by investigating the properties of composite materials to check the compliance of the material with requirements listed in the material specification. However, conducting tests with various specimens is costly and time consuming; therefore, a limited number of tests and specimen samples can be practically considered. To reduce *

Corresponding author. Tel.: +82 51 410 4291, Fax.: +82 51 405 4790 E-mail address: [email protected] † Recommended by Associate Editor Kyeongsik Woo © KSME & Springer 2015

such efforts, several numerical prediction methods for the material properties of composite lamina (or ply) have been suggested and employed in the development of engineering applications. In particular, the prediction methods for CTEs can be broadly classified into five types: the phenomenological model, analytical formula, semi-empirical formula, homogenization model, and computational homogenization with the aid of finite element methods. The phenomenological model is the rule of mixture (ROM) using volume fractions of fiber and matrix [2]. In an analytical formula, such as the Schapery model, an analytic solution is derived by solving equations of motion based on elasticity, but it is accurate only while the isotropic symmetry is maintained [3]. A semi-empirical formula, such as the Chamberlain model [4], was proposed. This model introduces parameters based on the test and experience similar to the Halpin–Tsai model [5] for evaluating stiffness. In addition, the Mori–Tanaka model and the self-consistent model, which are classified as homogenization models, are iterative schemes using the Eshelby tensor and the stiffness matrix, and they are commonly used to predict equivalent

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J. H. Lim et al. / Journal of Mechanical Science and Technology 29 (3) (2015) 1187~1197

CTEs [6, 7]. In computational homogenization, several schemes [8-13] with periodic boundary conditions have been proposed with representative volume elements (RVEs) that consist of a number of finite element meshes. Computational homogenization schemes have remarkable advantages over other prediction methods: they can make models that contain microstructures in detail, such as the shape of fiber [14-16], void distribution and geometry [17, 18], interfacial debonding and particle cracking [19, 20], as well as fiber arrangement [21], which have been routinely observed and affect the prediction results; they also provide the best correlation with test data [8]. From a realistic viewpoint, the properties of constituents must be known a priori to perform a trade-off study of composite materials that meet the needs of customers. In the case of the matrix, the elastic properties and CTEs of many resin materials, which typically show isotropic behaviors, are easily obtained from standard tensile tests with prismatic or dog bone-shaped bulk specimens according to the American Society for Testing and Materials (ASTM) D638-02 and D882-02, and thermo-mechanical analyzer (TMA) measurement, respectively. However, in the case of fibers with diameters that are typically a few micrometers, the aforementioned bulk specimen tests cannot be adopted because of the small transverse dimension of the fibers. In addition, most of the fibers used in unidirectional (UD) laminae, such as carbon fibers, show anisotropic behaviors, which makes identifying some material properties difficult. Among the properties of fibers, the longitudinal Young’s modulus can be directly measured by performing a tensile test with a single filament according to the ASTM D3379-89 specification, and the longitudinal CTE is measured by TMA, but not for the material properties in the transverse direction [2]. To determine the elastic properties and CTEs of carbon fibers in a theoretical manner in the transverse direction, Rupnowski et al. [22] conducted an inverse analysis with test data measured in a lamina-scale through the Mori–Tanaka method. They successfully predicted the elastic properties and CTEs of a T650-35 fiber, but other predictions of the elastic properties of an M40J fiber were less accurate than those of the T650-35 fiber because the Mori–Tanaka method cannot take into account the effects of microstructures precisely. Meanwhile, to accurately measure the transverse CTE of fibers with respect to temperature in an experimental manner, Kulkarni and Ochoa [23] set up specific test equipment by transmission electron microscopy (TEM) with image processing and investigated the impact of CTE values on interfacial residual stresses in composites. Pradere et al. [24] and Pradere and Sauder [25] suggested a new method for measuring the transverse CTE of fibers with an experimental device and proper data processing based on inverse analysis, and they finally predicted the CTE values of fibers with high accuracy. In addition, various methods to investigate material or interface properties at a microstructure level have been reported in the Refs. [26, 27].

The objective of this work is to propose a new inverse analysis scheme to evaluate the CTEs of fibers in composite materials in a theoretical manner through minimizing the lamina-scale cost function with approximated gradients, which are termed quasi-analytical gradients. The basic assumption of this work is that the material properties of laminae and resins are already known. In practice, this method is acceptable because they can be measured through a series of tests with bulk specimens, e.g., thickness-tapered and dog bone-shaped specimens for laminae and resins, respectively, in a straightforward manner except the case in which suppliers of materials are reluctant to provide raw data due to a security problem. To evaluate the CTEs of fibers, the lamina-scale cost function is defined as the difference between the measured CTEs and computed CTEs of the lamina. As gradients for searching the descent direction of the cost function, quasi-analytical gradients combined with the prediction formula, such as the Schapery and Hashin models, are proposed to accelerate the speed of computation much faster than conventional numerical gradients and the zero-order methods [28]. Although the quasi-analytical gradients in the present study are not the same as analytical gradients, they are sufficiently accurate for the purpose of finding a descent direction of the cost function and robust enough to meet the descent condition ensuring convergence. Therefore, the minimization of the cost function is successful, and it leads to a converged solution. This process is verified through numerical examples considering the effects of microstructures, such as the fiber arrangement, interface debonding, and voids. The remainder of this paper is organized as follows: The procedure of the inverse analysis with quasi-analytical gradients is presented in Sec. 2. Then, with several examples in Sec. 3, the performance of the proposed scheme is verified in terms of accuracy and convergence, as well as robustness, which is an ability to provide stable results against the presence of local defects. Finally, Sec. 4 contains the concluding remarks.

2. Determination of CTEs of fibers using inverse analysis with quasi-analytical gradients To determine the CTEs of fibers based on the known elastic properties and CTEs of the lamina and matrix in the framework of inverse analyses, a lamina-scale cost function f is defined as f = (a1 (P f ) - a1test ) 2 + (a 2 (P f ) - a 2test ) 2 + (a 3 (P f ) - a 3test ) 2 ,

(1)

where P f = {P1 f , P2f , P3f }T = {a1f ,a 2f ,a 3f }T is an unknown vector composed of the three CTEs of fibers showing orthotropic symmetry as shown in Fig. 1. The subscript 1 indicates the longitudinal direction, and the subscripts 2 and 3 indicate the transverse directions of fibers, respectively. In addition, a1test , a 2test , and a 3test are the measured CTEs of the

J. H. Lim et al. / Journal of Mechanical Science and Technology 29 (3) (2015) 1187~1197 Square RVE (1 ´ 1 ´ 1)

Hexagonal RVE (1 ´ 0.7597 ´ 1.316)

¶f ¶f ¶a1 ¶f ¶a 2 ¶f ¶a 3 , = + + ¶a1f ¶a1 ¶a1f ¶a 2 ¶a1f ¶a 3 ¶a1f

Diamond RVE (1 ´ 1 ´ 1)

¶f ¶f ¶a1 ¶f ¶a 2 ¶f ¶a 3 = + + , ¶a 2f ¶a1 ¶a 2f ¶a 2 ¶a 2f ¶a 3 ¶a 2f

X3

¶f ¶f ¶a1 ¶f ¶a 2 ¶f ¶a 3 , = + + ¶a 3f ¶a1 ¶a 3f ¶a 2 ¶a 3f ¶a 3 ¶a 3f

X1

X2

¶f ¶a ¶a = 2(a1 - a1test )( 1f ) + 2(a 2 - a 2test )( 2f ) f ¶P ¶P ¶P ¶a 3 test + 2(a 3 - a 3 )( f ) . ¶P

Fig. 1. Configuration of an RVE and three microstructure models. X3

X2 X1 l3 + Dl3

l3

l1

l2

l1 + Dl1

Undeformed Configuration

l2 + Dl2

Deformed Configuration (von Mises stress plot)

Fig. 2. Evaluation of the equivalent CTEs using a computational homogenization.

UD lamina; and P = {P1 , P2 , P3}T = {a1 ,a 2 ,a 3}T is a vector that comprises the calculated CTEs of the UD lamina expressed as a function of Pf. Using a computational homogenization scheme proposed by Karadeniz and Kumlutas [8], the equivalent CTEs of lamina P can be calculated in a simple and rapid manner. The only task involved in implementing the homogenization scheme is to apply periodic boundary conditions at each pair of RVE surfaces. Subsequently, a thermomechanical analysis is conducted with the temperature excursion of the entire nodes. Finally, as shown in Fig. 2, the equivalent CTEs of lamina of the RVE are calculated with analysis results using the following equation: a ij =

Dl j li DT

1189

, (i, j =1, 2,3) ,

(2)

where li, ∆li, and ∆T are the length of RVE in the i-direction, change in length of RVE in the j-direction, and temperature excursion, respectively. The subscripts i and j indicate either the longitudinal or transverse direction, the same as in Eq. (1). In this case, the CTE that corresponds to the i- and j-directions is denoted by αij. Here, αij is also rewritten into a vector notation a i Î {a1 ,a 2 ,a 3 ,a 4 ,a 5 ,a 6 } from a matrix notation a ij Î {a11 ,a 22 ,a 33 ,a 23 ,a13 ,a12 } . For simplicity, vector notation is used in the following descriptions. To minimize the lamina-scale cost function in Eq. (1), gradients of the cost function with respect to the CTEs of constituents can be defined using the chain rule [29] as follows: ìï ¶f ¶f ¶f ¶f =í f , , f f f ¶P ïî ¶a1 ¶a 2 ¶a 3

T

üï ý , ïþ

(3)

(4)

However, the partial derivatives associated with the CTEs of fibers (∂α1 / ∂Pf, ∂α2 / ∂Pf, and ∂α3 / ∂Pf) are generally unknown. Typically, to circumvent this problem, numerical gradients or the zero-order methods have been widely adopted in many engineering problems [28]. In the case of numerical gradients, a first-order forward-difference of the cost function is adopted in a straightforward manner ¶f æ ¶f ö »ç ÷ ¶P f è ¶P f ø

numerical

=

f (P f + DP f ) - f (P f ) . DP f

(5)

In general, the computational cost with the numerical gradients is directly proportional to the number of independent design variables. It typically requires (n + 1) times evaluation of the cost function with n independent design variables; for an orthotropic material with three CTEs of fibers, a total of four evaluations of the cost function are required. The computational cost for evaluating the CTEs with the numerical gradients is rather expensive. Furthermore, if a complex RVE is discretized with many degrees of freedoms to represent many fibers and local defects, the corresponding computational cost increases significantly. Moreover, the numerical gradient is easily exposed to problems that involve a truncation error. The zero-order method, as another alternative for minimization, typically requires more iterations than the gradient-based methods because it finds the minimum point without gradients. Thus, it requires a considerable computation time. In this work, to reduce the aforementioned heavy computational cost, quasi-analytical gradients are proposed for the UD composites, and they provide the descent direction of the cost function explicitly. Therefore, they accelerate the speed of the minimization procedure. These gradients will be discussed in the following section. 2.1 Quasi-analytical gradients for fibers in UD composites The unknown partial derivatives ∂P / ∂Pf in the aforementioned analytical gradients in Eq. (3) may be obtained from the Schapery [3] and Hashin [30] models in an approximate manner as follows. To predict the CTEs of the UD composite lamina showing transversely isotropic symmetry by the Schapery model, a

1190


modification for transverse isotropic fiber by Strife and Prewo [31] can be considered

a L = a1 =

E1f a1f V f + E ma mV m , E1f V f + E mV m

aT = a 2 = a 3 =

(1 + n 12f )a 2f V f

(6) m

m

m

+ (1 + n )a V - a1n 12 ,

¶a 2 ¶a ¶a ¶a = { 2f , 2f , 2f } ¶P f ¶a1 ¶a 2 ¶a 3 = {-

n 12 E1f V f E1

n 12 E1f V f E1

(8)

where c is the gradient vector of the cost function, d is the search direction, and k is the iteration count. In the case of the steepest descent method with analytical gradients, a relation d(k) = −c(k) holds, and the descent condition is strongly satisfied as 2

( -c ( k ) × c ( k ) ) = - c ( k ) < 0 .

2.2 Inverse analysis procedure

(10)

The detailed minimization process for the inverse analysis is described in Fig. 3. To minimize the cost function, the following treatments are conducted:

,0,(1 + v12f )V f }T .

Hashin [30] suggested a modification of α2 to provide physical meaning because Eq. (7) suggested by Strife and Prewo [31] was simply derived from the Schapery model in a heuristic manner without any theoretical consideration. aT = a 2 = a 2f V f çç1 + n 12f è

(15)

In the case of the proposed inverse analysis with quasianalytical gradients, it takes unknown partial derivatives in Eq. (4) from the prediction models, i.e., d(k) = −cquasi(k) ≈ −c(k). Confirming whether the descent condition is fully satisfied is difficult because it depends on the degree of equivalence between the prediction model and analytical solution. Several examples are presented in Sec. 3 to verify this fact numerically.

C=-

æ

(14)

(9)

,(1 + v12f )V f ,0}T ,

¶a 3 ¶a ¶a ¶a = { 3f , 3f , 3f } f ¶P ¶a1 ¶a 2 ¶a 3 = {-

(c( k ) × d ( k ) ) < 0 ,

(7)

where α1 and α2 (= α3) are the longitudinal and transverse CTEs of the composite lamina, respectively. The superscripts f and m indicate a fiber that shows transversely isotropic symmetry and a matrix that shows isotropic symmetry, respectively. Major Poisson’s ratio and Young’s modulus of the composite lamina are obtained by the ROM as n 12 = n 12f V f + n mV m and E1 = E1f V f + E mV m , where the volume fractions of fiber and matrix are denoted by V f and V m. From the Schapery model, the gradients ∂P / ∂Pf can be derived as ¶a1 ¶a ¶a ¶a E fV f = { 1f , 1f , 1f } = { 1 ,0,0}T , f E1 ¶P ¶a1 ¶a 2 ¶a 3

with quasi-analytical gradients, the descent condition to find the minimum values should be satisfied, which is a necessary condition that ensures convergence, as [28]

a1f a 2f

ö m m m ÷÷ + a V (1 + n ) - n 12a1 . (11) ø

From the Hashin model, ∂α2 / ∂Pf are obtained as ¶a 2 V f E1f = {V f n 12f - n 12 ,V f ,0}T , f E1 ¶P

(12)

¶a 3 V f E1f = {V f n 12f - n 12 ,0,V f }T . f E1 ¶P

(13)

Inserting Eqs. (8)-(10), (12) and (13) into Eq. (4) can obtain a set of approximated gradients, termed quasi-analytical gradients ∂f quasi / ∂Pf, in a straightforward manner. These quasianalytical gradients may be not the same as the analytical gradients ∂f / ∂Pf, but they could be an excellent substitute to finding the descent direction of the cost function. To ensure the convergence of the proposed inverse analysis

æ ¶f ¶f ¶f / » -ç f f ç ¶P f ¶P ¶P è

ö ÷÷ ø

quasi

P f (k + 1) = P f (k ) + DP f o C ,

æ ¶f /ç ç ¶P f è

ö ÷÷ ø

quasi

,

(16) (17)

where ∆Pf is the step size, and C is the normalized descent direction with respect to the CTEs of the fiber Pf. In Eq. (17), an operator ◦ indicates the element-wise product of two vectors, i.e., (DP f o C)i = ΔPi f Ci , i = 1, 2, 3. This numerical scheme is implemented using Matlab [32] combined with finite element software Abaqus [33]. Matlab creates an input file with the updated CTE of constituents and evaluates the cost function at the current step based on the Abaqus results using a batch file (Run.bat) operation. The total numbers of nodes are 10,290, 19,089, and 9,933 for square, hexagonal, and diamond arrays, respectively, as seen in Fig. 1. The element type used is an eight-node brick element (C3D8I in Abaqus) that has incompatible modes. The initial point of Pf is simply set as {0 ppm/°C, 0 ppm/°C, 0 ppm/°C}. The step sizes ∆Pf initially are set as {1 ppm/°C, 1 ppm/°C, 1 ppm/°C}, and then reduced by the factor 0.618, the golden section ratio, whenever the oscillation of solutions is detected in the minimization process. As the stopping criterion, the magnitudes of the lamina-scale cost function (tolo) are used with a tolerance of 10−8. This approach is equivalent


Start

Initial guess for P f (k ) at k = 0 P f = {a1f , a 2f , a 3f }T P f (k + 1) = P f (k ) + DP f o C

Run unit-cell problems using Abaqus (Run.bat)

Update fiber CTEs Update Abaqus input files k = k +1

Read analysis results & Evaluate a cost Function

Calculate the descent direction using quasi-analytical gradients

No f (P f (k )) < tolo

C=-

Yes

æ ¶f ¶f ¶f / » -ç f ç ¶P ¶P f ¶P f è

ö ÷ ÷ ø

quasi

æ ¶f /ç f ç ¶P è

ö ÷ ÷ ø

quasi

Print

P f = P f (k )

Abaqus Matlab

End

Fig. 3. Flowchart of inverse analysis.

to a conventional steepest descent method used to find the minimum point [28].

3. Numerical results and discussion To verify the performance of the proposed scheme with quasi-analytical gradients, three benchmark problems are solved. The first example deals with P75/Epoxy930, P100/Aluminum2024, and T300/Epoxy5028 UD composite laminae, the material properties of the constituents of which are fully known as verified by Karadeniz and Kumlutas [8]. The accuracy of the proposed scheme is checked by comparing computed CTEs through inverse analysis with measured CTE values of fibers. To demonstrate the accuracy and efficiency of the proposed scheme, i.e., the steepest descent method with two quasi-analytical gradients, its convergence behavior is compared with that of the steepest descent method with the numerical gradient and a zero-order method, namely, the Nelder–Mead simplex method [34] supported by Matlab. To check the accuracy of the proposed scheme, its solutions are also compared with results by inverse analysis with the Mori–Tanaka scheme, which is known as an analytical solution while the transversely isotropic assumption is valid. Furthermore, the fiber arrangement effect, which is one of the classical microstructural characteristics, is investigated with three regular microstructure models (square, hexagonal, and diamond arrays) because their realistic fiber arrays are not known a priori and are difficult to directly extract from the microstructures. However, applications of the proposed approach are not limited to a class of problems with regular RVEs.

1191

The second example is used to determine the CTEs of an IM7 fiber embedded in Epoxy977 matrix. To evaluate them, three different fiber elastic properties in the literature [23, 35, 36] that exhibit differences are adopted for the simulation. In addition, the CTE values of an IM7 fiber with various fiber elastic properties are compared with the test result [23]. Thus, we discuss the effect of fiber elastic properties and fiber volume fractions on the CTE prediction of fibers using the proposed scheme. As the third example, RVEs that contain interface debonding and local voids are used to identify CTEs because these kinds of defects are frequently reported in UD composites and lead to some deviation in the prediction of the CTEs of fibers. Comparing these CTE values with the results of RVEs without defects verifies the robustness of the proposed scheme in the presence of local defects. Several assumptions in material properties are adopted through numerical simulations: The fraction of voids is negligible; moreover, fibers and matrices are linearly elastic. In addition, the interfaces between inclusions and matrices are fully bonded; therefore, interface failures, such as delamination, fiber/matrix debonding, and interfacial slip, are not considered unless otherwise specifically stated similar to another work [22]. All experimental values used in the work are also assumed to be sufficiently accurate. Notably, the CTEs of fibers can be predicted with high accuracy when material properties of matrices and composite laminae are obtained accurately. 3.1 Investigation of accuracy and efficiency of the proposed scheme In the first example, the CTEs of fibers in P75/Epoxy930, P100/Aluminum2024, and T300/Epoxy5028 UD laminae were investigated with the proposed scheme. The elastic properties and CTEs of the resins and laminae used in this section are summarized in Tables 1 and 2. To check the computation time, a personal computer with Intel® Core(TM) I7-3930K with 3.2 GHz CPU and 32 GB RAM is used. The cost function in Eq. (1) is minimized to determine the CTEs of fibers using the proposed scheme. To check its performance in terms of accuracy and convergence behavior, the lamina-scale cost function of the P75/Epoxy930 lamina with a square RVE is minimized by using the steepest descent method with the numerical gradient, and a zero-order method is also carried out. Their convergence behaviors are depicted in Figs. 4 and 5, and the analysis results are summarized in Table 3. The figures show that the iteration counts with the gradients methods is approximately 40, which is much less than 150 in the zero-order method. The computation times with the proposed quasi-analytical gradients combined with the Schapery model and the Hashin model are 2,308 and 2,499 seconds, respectively. The proposed scheme with the quasi-analytical gradients has the advantage of low computational time, which is only one-fifth of that using the numerical

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J. H. Lim et al. / Journal of Mechanical Science and Technology 29 (3) (2015) 1187~1197 Comparison of Convergence Behavior

6

Table 1. Material properties of fibers and matrices [8].

10

Quasi-Analytical Gradient(Schapery) Quasi-Analytical Gradient(Hashin) Numerical Gradient Zero-Order(Nelder-Mead)

4

Fiber

P75

P100

T300

10

E1f (GPa)

550.40

796.63

233.13

10

E2f (GPa)

9.52

7.24

23.11

G12f (GPa)

6.9

6.9

8.97

n 12f

0.2

0.2

0.2

n 23f

0.4

0.4

0.4

Matrix

Epoxy930

Aluminum2024

Epoxy5028

E m (GPa)

4.35

73.11

4.35

Cost Function Values

2

0

10

-2

10

-4

10

-6

10

-8

10

11279 sec

20 2308 sec

νm

0.37

0.33

0.37

α m (ppm/°C)

43.92

23.22

43.92

9836 sec

-10

10

40 2499 sec

60

80

100

120

140

160

Iteration Count

Fig. 4. Comparison of convergence behaviors in terms of cost function values.

Table 2. Coefficients of thermal expansion and fiber volume fraction in UD composites [8].

Composite

a b

P75/ Epoxy930 (s ± m)

P100/ Aluminum2024 (s ± m)

T300/ Epoxy5028 (s ± m)

1.52 ± 0.076a

−0.08 ± 0.05b

a1 (ppm/°C) a 2 (ppm/°C)

−1.04 ± 0.05a 34.5 ± 1.73

26.2 ± 1.31

25.60 ± 1.28a

Vf

0.48

0.40

0.68

a

a

5% COV is assumed. Minimum measurement error is assumed.

Table 3. Comparison of convergence behaviors and computation time with respect to minimization schemes with a square RVE for P75/Epoxy930 UD lamina. Steep descent method Minimization scheme

Quasi-analytical gradients Numerical gradients

(a)

Zero-order method (Nelder– Mead simplex)

Schapery model

Hashin model

Converged CTEs a Lf , aTf (ppm/°C)

−1.47, 7.74

−1.47, 7.74

−1.47, 7.74

−1.47, 7.74

Number of iteration

35

38

43

150

Elapsed time (sec)

2308

2499

11279

9836

gradient (11,279 seconds) and one-fourth of that by the zeroorder method (9,836 seconds). The converged values are the same. To check the microstructure effect due to the fiber arrangement, the CTE values of fibers are extracted with three RVEs for each of the three UD laminae (P75/Epoxy930, P100/Aluminum2024, and T300/Epoxy5028) using inverse analysis with the quasi-analytical gradient from the Schapery model. According to the analysis results in Table 4, uncertainty in the CTEs of fibers due to different fiber arrangements in RVEs is almost negligible for all UD lamina. This conclusion is consistent with the results reported in another work

(b) Fig. 5. Comparison of convergence behaviors in terms of CTE values of fibers: (a) the gradients methods; (b) the zero-order method.

[21]. Therefore, for simplicity, a square fiber RVE is chosen for the following analyses. Meanwhile, in Table 4, marginal deviations of the two transverse CTE values ( a 2f and a 3f ) are present even though they are not noticeable. The deviations are caused by a tolerance limit of 10−8 in the minimization process of cost functions. Adopting a much smaller tolerance limit could reduce these deviations, but it leads to a high

1193


Table 4. Values of CTEs obtained by the proposed scheme according to fiber arrangement. CTEs of P75 (ppm/°C)

a Lf = a1f a

f 2

7.74

a Tf = (a 2f + a 3f ) / 2 CTEs of P100 (ppm/°C)

a =a a

−1.47 7.74

a 3f

f L

Square RVE Hexagonal RVE Diamond RVE (V f = 48.00%) (V f = 48.03%) (V f = 48.01%)

f 1

f 2

7.74

−1.47

−1.47

7.76

7.77

7.74

7.77

7.75

Table 6. Elastic properties of IM7 fiber in the literature. Fiber properties

Sample #1 [35]

Sample #2 [36]

Sample #3 [23]

E1f (GPa)

276.0

270.96

276.0

f 2

19.0

17.24

19.5

f 12

27.0

27.58

70.0

G (GPa)

7.0

8.27

5.735

f 12

0.2

0.32

0.28

E (GPa) G (GPa) f 23

n

7.77

Square RVE Hexagonal RVE Diamond RVE (V f = 40.00%) (V f =40.03%) (V f = 40.00%) −1.55

−1.55

−1.55

1.14

0.57

0.70

Table 7. Material properties of Epoxy977 and CTEs of IM7/Epoxy977 lamina. Matrix properties Epoxy977 [23] Lamina properties m

a 3f

1.14

a Tf = (a 2f + a 3f ) / 2 CTEs of T300 (ppm/°C)

1.14

0.61

0.67

0.59

E (GPa) ν

0.69

m

m

Square RVE Hexagonal RVE Diamond RVE (V f = 68.01%) (V f = 68.00%) (V f = 68.00%)

α (ppm/°C)

−0.55

−0.53

−0.55

a 2f

12.49

11.82

12.52

f 3

12.50

11.77

12.51

Fiber CTE (ppm/°C)

12.50

11.80

12.52

a Lf

a Tf = (a 2f + a 3f ) / 2

3.45

αL (ppm/°C)

0.39

αT (ppm/°C)

23.4

44.69

Vf

0.65 ± 0.3

0.18

Table 8. Comparison results of IM7 fiber CTEs with various fiber elastic properties and fiber volume fractions.

a Lf = a1f

a

IM7/977 [23]

a

f T

Sample #1 (s ± m)

Sample #2 (s ± m)

Sample #3 (s ± m)

−0.196±0.045 −0.155±0.042 −0.064±0.051 3.73±1.65

4.52±2.58

6.43±2.44

Test results (s ± m) [23] −0.4±0.8 5.6±0.8

Table 5. Comparison of results of the proposed scheme and the inverse analysis scheme by the Mori–Tanaka method with a square RVE.

CTEs of fibers (ppm/°C)

Inverse analysis with a computational homogenization

Inverse analysis with the Mori-Tanaka method

Test result [8]

a1f (s ± m)

a1f (s ± m)

a1f a 2f (s ) (s )

a 2f (s ± m)

a 2f (s ± m)

P75

−1.47±0.05 7.74±3.53 −1.51±0.06 7.71±3.50 −1.35 6.84

P100

−1.55±0.11 1.09±9.52 −1.55±0.08 0.84±9.45 −1.40 6.84

T300 −0.55±0.053 12.50±1.77 −0.53±0.053 11.72±1.80 −0.54 10.08

computational cost and numerous iterations. To further check the accuracy of the proposed scheme, all computed CTEs results are compared with those obtained by inverse analysis with the Mori–Tanaka scheme and test results [8]. Here, ±5% coefficient of variation (COV) in lamina CTEs is assumed based on the values given in Table 2, which is conventional in CTE measurement results. The fiber CTEs are then evaluated using the lamina CTEs with ±5% COV. The results are summarized in Table 5 in the form of mean (s) and standard deviation (m). All analysis results obtained by the proposed scheme are consistent with those obtained by inverse analysis with the Mori–Tanaka method. Moreover, the measured fiber CTEs from test results are within the range of standard deviation of the computed CTEs in consideration of the 5% COV of the lamina CTEs. Accordingly, the accuracy of the proposed scheme is confirmed.

3.2 Effect of fiber elastic properties and fiber volume fractions in UD composites In the second example, the effects of fiber elastic properties and fiber volume fractions are investigated. According to the Refs. [22, 37, 38], many measurement methods have been proposed to evaluate the transverse elastic properties of fibers. Hence, all elastic properties of fibers are assumed to be already known in this work. However, their values show some discrepancy due to the uncertainty that arises from the small dimension in the transverse direction around a few micrometers in measurement results. The IM7 fiber is one of the widely adopted carbon fibers in industry, and its elastic properties are easily found in three Refs. [23, 35, 36]. However, some deviations exist, as listed in Table 6. They are termed Samples #1 [35], #2 [36], and #3 [23], respectively. Among them, Sample #1 was used in the third worldwide failure exercise [35], and the others were obtained experimentally. The material properties of the Epoxy977 matrix and the IM7/Epoxy977 lamina are listed in Table 7. In addition to the uncertainty in fiber elastic properties of the three samples, the uncertainty of fiber volume fractions V f = 0.65 ± 0.03 is also considered as in the Ref. [23]. The calculated CTEs are listed in Table 8. The results are also expressed in terms of probability density function (PDF) with the assumption of a normal distribution, as shown in Fig. 6. Despite

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Table 9. Comparison results of IM7 fiber CTEs due to voids in IM7/Epoxy977 lamina. V v = 1.0% V v = 1.0% V v = 1.0% V v = 1.0%

Fiber CTE (ppm/°C) f L

a =a

f 1

Deviation of a

a

f a L

f 2

a 3f f T

f 2

f 3

a = (a + a ) / 2 Deviation of a (a)

f b T

a =a

f 1

Deviation of a

f T

a

f 2

a

f 3

f 2

f a L

f 3

a = (a + a ) / 2 Deviation of a

3´3 RVE

5´3 RVE

7´3 RVE

−0.152

−0.160

−0.161

−0.161

6.17%

1.23%

0.62%

0.62%

4.639

6.174

6.330

6.330

7.039

6.663

6.641

6.641

5.839

6.419

6.486

6.512

11.10%

2.27%

1.25%

0.85%

V v = 2.2% V v = 2.2% V v = 2.2% V v = 2.2%

Fiber CTE (ppm/°C) f L

Single RVE

f b T

Single RVE

3´3 RVE

5´3 RVE

7´3 RVE

−0.134

−0.159

−0.161

−0.160

17.3%

1.9%

0.6%

1.2%

1.372

5.664

6.013

6.171

7.784

6.755

6.696

6.671

4.578

6.210

6.355

6.421

30.30%

5.45%

3.24%

2.24%

a

(b) Fig. 6. Comparison of IM7 fiber CTEs with test results considering the uncertainty of fiber elastic properties: (a) longitudinal CTEs; (b) transverse CTEs.

the different fiber elastic properties, in the case of the longitudinal direction, all prediction results are within the standard deviation of test results. However, in the case of the transverse direction, only the mean values of PDF with Samples #2 and #3 are located within the range of the standard deviation of test results. The mean value of CTEs with Sample #1 slightly deviates from the range of test values. According to the prediction formulae in Eqs. (7)-(11), this deviation could be attributed to the difference in the major Poisson’s ratios of fiber n 12f , that is, 0.2, 0.32, and 0.28 in the three samples. The results show that the fiber elastic properties, in particular n 12f , have to be carefully measured to increase the accuracy of the evaluation of fiber CTE. 3.3 Effect of local defects In the manufacturing process, the formation of local defects, such as interface debonding and voids, is unavoidable. Despite the localized phenomena, their existence could reduce the accuracy of evaluation of fiber CTEs. As the third example, RVEs that contain interface debonding and voids are chosen in predicting CTEs of fibers. Investigating the effect of local defects on the degradation of accuracy of the CTE prediction verifies the robustness of the proposed scheme with the RVEs

Values compared with a Lf = -0.162 , obtained using the undamaged RVE. b Values compared with a Tf = 6.568 , obtained using the undamaged RVE.

for IM7/Epoxy977 composite. The material properties of IM7/ Epoxy977 lamina and Epoxy977 matrix are the same as in the second example. The elastic properties of the IM7 fiber are chosen as the values of Sample #3, which shows the best correlation with test results in the second example. First, two RVEs that have through-thickness voids along the longitudinal direction of fibers, i.e., the X1-direction, are considered, as shown in Fig. 7. The void volume ratios of the one damaged unit-cell, V v, are 1.0% and 2.22%, respectively. Its location is adjacent to the interface between the fiber and matrix in the X2-direction, which leads to asymmetry of the thermo-mechanical response of RVEs. The void occupies the volume of the matrix. The fiber volume ratio is fixed. To realize the local state, several undamaged unit-cells surround the one damaged unit cell, and finally, 3×3, 5×3, and 7×3 RVEs are prepared. It is unrealistic that the voids penetrate the RVEs along one direction of the RVEs because typical voids are observed in local regions. Nevertheless, these damaged RVEs under the unrealistic and worst-case assumption can be used to investigate the robustness of the proposed scheme. The prediction results of the CTEs are summarized in Table 9, where their deviations from the CTEs of undamaged RVEs are also calculated. As a computational model contains fewer undamaged unit cells, the deviation increase; in the extreme case of a single RVE, the deviation is significantly large. In addition, due to the existence of voids, the transverse CTEs


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Table 10. Comparison results of IM7 fiber CTEs according to interface debonding angle with a 3´3 RVE model. Fiber CTE (ppm/°C)

22.5°

45.0°

67.5°

90.0°

−0.160

−0.160

−0.159

−0.160

1.23%

1.25%

1.88%

1.26%

6.336

6.311

6.368

6.488

6.594

6.688

6.788

6.526

a = (a + a ) / 2

6.465

6.500

6.578

6.507

b

1.57%

1.04%

0.15%

0.93%

112.5°

135.0°

157.5°

180.0°

−0.161

−0.161

−0.161

−0.161

0.63%

0.62%

0.62%

0.62%

6.550

6.543

6.595

6.415

5.806

5.453

5.441

5.550

6.178

6.00

6.018

5.983

(b)

5.94%

8.65%

8.37%

8.91%

Fig. 8. CTEs of IM7 fiber in RVEs containing through-thickness interface debonding with respect to debonding angle: (a) a single RVE model; (b) a 3×3 RVE model.

a Lf = a1f Deviation of a

f a L

a 2f a f T

f 3

f 2

f 3

Deviation of a Tf Fiber CTE (ppm/°C)

a Lf = a1f Deviation of a

f T

a

f 2

a

f 3

f 2

f a L

f 3

a = (a + a ) / 2 Deviation of a

f b T

a

Values compared with a Lf = -0.162 , obtained using the undamaged RVE. b Values compared with a Tf = 6.568 , obtained using the undamaged RVE. Transverse CTE variation due to void existence 6.6 6.4 6.2

o

CTE(ppm/ C)

6

a fT=(a f2+ a f3)/2 (single)

5.8

a fT=(a f2+ a f3)/2 (3x3)

5.6

a Tf=(a 2f+ a 3f)/2 (5x3)

5.4

a Tf=(a 2f+ a 3f)/2 (7x3)

3×3 RVE model

5.2 5 4.8 4.6

5×3 RVE model 0

0.5

1

1.5

2

2.5

3

Void Fraction (%) of one center RVE

Void V f = 65.0% V v = 1.01%

Void V f = 65.0% V v = 2.22%

Single RVE model

7×3 RVE model

Fig. 7. CTEs of IM7 fiber in RVEs containing through-thickness voids.

( a 2f and a 3f ) of the IM7 fiber vary with the void volume ratio. The variation of longitudinal CTE ( a Lf = a1f ) is negligible; the variation of a 2f and a 3f is especially noticeable because of the effect of voids asymmetrically located in the X2-direction. However, by taking the average values of a 2f and a 3f , in the case of V v = 1.0% with the 7´3 RVE model, the maximum deviation of the transverse CTE a Lf = v (a 2f + a 3f ) / 2 is less than 0.85%. In the case of V = 2.2% with the 7×3 RVE model, the maximum deviation of a Lf is less than 2.24%. From the results of the transverse CTEs affected greatly by the void volume fractions in the computa-

(a)

tional models, as shown in Fig. 7, the effect of local defects could be diminished by choosing RVEs that consist of many undamaged unit cells without local defects. Second, through-thickness interface debonding along the X1-direction between the fiber and matrix in a single RVE model and a 3´3 RVE model are considered, as shown in Fig. 8. With the interface debonding lengthened in the circumferential direction from A indicated in Fig. 8, nine RVEs with various debonding angles, 0°(no debonding), 22.5°, 45°, 67.5°, 90°, 112.5°, 135°, 157.5° and 180°, for each model are prepared for the inverse analyses. As shown in Fig. 8(a), in the case of the single RVE model, the transverse CTE varies with the interface debonding angle, whereas the longitudinal CTE remains unchanged; in particular, beyond a 90° angle, the transverse CTE decreases dramatically. The results of the 3´3 RVE model, which contains eight uncracked unit cells, are summarized in Table 10 and Fig. 8(b). In this case, up to a 90° angle, the effect of interface debonding appears with only 0.93% deviation of the CTE values; for the interface bonding from 90° to 180°, the deviation is magnified by 9%. However, the deviation could be reduced much more by adding several uncracked unit cells to the computational model. Real defects in well-manufactured composites are localized, unlike the through-thickness defects with the worst-case assumption. Furthermore, when a computational model that contains several undamaged unit cells is chosen, the local defects in the RVE will not severely contaminate the accuracy of evaluating the CTE of fibers because their effect is easily compensated by the undamaged unit cells. Thus, the proposed

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scheme can robustly provide accurate CTE values of fibers even in the presence of local defects.

4. Conclusion This paper proposed an efficient numerical scheme that can extract the CTEs of constituents in composite materials by means of inverse analysis with approximated gradients, which are termed quasi-analytical gradients, combined with computational homogenization. A cost function composed of the difference between the measured CTEs and the computed CTEs of the UD lamina is minimized with the proposed scheme. The descent direction of the cost function is found with the aid of the proposed quasi-analytical gradients combined with the Schapery model and the Hashin model, which dramatically accelerate the speed of convergence. Benchmark examples demonstrate that the proposed scheme can evaluate the CTEs of fibers as accurately as analytical solutions. Moreover, with typical 5% CTE measurement deviation of lamina, the prediction results show a close correlation with test data. Even when RVE models contain a damaged unit cell, the accuracy of the proposed scheme is maintained by increasing the size of RVEs with many undamaged unit cells. Therefore, the proposed scheme can provide a reliable prediction of CTEs of fibers from the measured CTE of composite materials considering the microstructures, and it could serve as a good engineering tool for investigating material behaviors of constituents that are a few micrometers in diameter. Furthermore, a more realistic prediction would be possible when the proposed scheme is combined with an accurate modeling scheme through image processing and micro-CT to reflect the effects of microstructures in detail.

Acknowledgments This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) and funded by the Ministry of Education (2014R1A1A 2058549). The authors would like to thank the anonymous reviewers for their valuable comments and suggestions, which helped to improve the paper.

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Jae Hyuk Lim received his B.S. degree in Mechanical Engineering from Inha University, Korea in 2000. He received his M.S. and Ph.D. degrees in Mechanical Engineering from the Korea Advanced Institute of Science and Technology (KAIST) in 2002 and 2006, respectively. He is currently a senior research engineer in the Satellite Mechanical Department, Korea Aerospace Research Institute. His research interests include finite element simulations, material modeling and characterization, and design and analysis of satellite structures. Dongwoo Sohn received his B.S. degree from Hanyang University, Korea in 2006. He received his M.S. and Ph.D. degrees from the Korea Advanced Institute of Science and Technology (KAIST) in 2008 and 2011, respectively. He is currently an assistant professor in the Division of Mechanical Engineering, Korea Maritime and Ocean University. His research interests include numerical methods and computational mechanics.