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PAPER
Thermal conductivity of a thick 3D textile
composite using an RVE model with specialized
thermal periodic boundary conditions
To cite this article: Hye-gyu Kim and Wooseok Ji 2021 Funct. Compos. Struct. 3 015002
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Funct. Compos. Struct. 3 (2021) 015002
https://doi.org/10.1088/2631-6331/abd7cd
Functional Composites and Structures
PAPER
RECEIVED
24 November 2020
REVISED
16 December 2020
ACCEPTED FOR PUBLICATION
Thermal conductivity of a thick 3D textile composite using an RVE
model with specialized thermal periodic boundary conditions
Hye-gyu Kim and Wooseok Ji
30 December 2020
Department of Mechanical Engineering, Ulsan National Institute of Science and Technology (UNIST), Ulsan, Republic of Korea
PUBLISHED
E-mail: wsji@unist.ac.kr
19 January 2021
Keywords: 3D textile composite, thermal conductivity, homogenization, periodic boundary condition, representative volume element
Abstract
Finite element analysis is performed to virtually measure homogenized thermal conductivity of a
thick 3D woven textile composite (T3DWC). Temperature-dependent thermal and mechanical
properties of constituents are considered for the measurements over a wide range of temperature.
A two-step homogenization approach is adopted here to simplify the analysis at the microscopic
level without losing heterogeneity of the material at the macroscopic scale. First-step
homogenization is carried out at a tow level using an analytical homogenization scheme. Fiber
tows are homogenized and assigned with effective elastic and thermal properties. The solid tows are
then implemented into a representative volume element considering the unique in-plane periodic
fiber architecture of the thick composite material. Due to the unique in-plane periodicity,
conventional periodic boundary conditions for thermal and mechanical loading conditions are
reformulated. Anisotropic thermal conductivity of T3DWC is obtained from the second-step
homogenization based on virtual thermal tests performed at ambient to elevated temperatures.
1. Introduction
Three-dimensional (3D) textile composites have been considered a prominent material for structural
components in many fields including aerospace and automobile industries [1]. Unlike conventional
laminated composites, 3D textile composites (3DTCs) have unique out-of-plane reinforcements to enhance
through-thickness strength as well as interlaminar shear strength [2]. Consequently, they exhibit much more
improved impact resistance than the laminated composite materials, of which interfaces are the weakest link
and vulnerable to delamination failure. The improved mechanical performance has widened the applicability
of 3DTCs to various types of load-bearing structures. High-temperature applications of 3DTCs are also
possible when high-temperature resistant materials such as ceramic and carbon are used for fibers and a
matrix material.
When combined mechanical and thermal loadings are expected during a service life of a composite
material, an accurate evaluation of its thermo-mechanical properties is typically required in the preliminary
design phase. Many researchers have studied both analytically and numerically to predict thermal properties
of various types of textile composites. Vorel and Šejnoha [3] utilized the Mori–Tanaka averaging scheme
combined with a multi-scale homogenization method to calculate the effective thermal conductivity of plain
weave composites. This scheme even considered defects including a void or delamination. While
homogenized effective properties are only available from such an analytical model, finite element (FE)
models incorporating a representative volume element (RVE) and periodic boundary conditions (PBCs) can
provide more detailed information such as interior temperature fields and heat flux distributions mapped
onto complicated textile architecture. Gowayed et al [2] developed an FE model to predict thermal
conductivity of woven textile composites using a unit cell approach and micro-scale level homogenization
technique. It was the first FE model for thermal analysis to take account of woven fabric geometry other than
laminated composites reinforced with unidirectionally continuous fibers. Dasgupta et al [4] utilized an FE
model of a unit cell with a multi-scale homogenization approach to predict thermal and mechanical
properties of a plain weave composite. They also established a 3D thermal resistance network model to
© 2021 The Korean Society for Composite Materials and IOP Publishing Limited
Funct. Compos. Struct. 3 (2021) 015002
H-G Kim and W Ji
Folding
(a) Plain woven fabric
(b) Folded fabric
(c) Straight warp
(d) 3D textile composite
Figure 1. Construction of the thick 3D woven textile composite.
analytically predict thermal conductivity of the composite. It was shown that results from the analytical and
FE models were in good agreement with experimental measurements. Siddiqui and Sun [5] computed
effective thermal conductivity of a plain weave composite at various temperatures and different fiber volume
fractions. They utilized temperature-dependent thermal conductivity values of the fiber. Jiang et al [6]
predicted thermal conductivity of 3DTC with helical fiber tow geometry. The RVE was discretized into small
subcells and the subcells were subjected to the FE equation of a steady-state heat transfer problem. Effective
thermal conductivity dependent on various braiding angles and different fiber volume fractions was
obtained through a volume average method. Gou et al [7] studied thermal behaviors of a textile composite
having 3D four-directional braided geometry with different tow cross-sections. The effect of cross-section
shapes, braiding angles, and fiber volume fractions on the composite thermal conductivity was investigated.
The numerical results showed a good agreement with experimental results. Dong et al [1] also considered 3D
four-directional braided composites. They created a meso-scale RVE model of a unit cell as well as a full-scale
model having the same dimensions with an actual specimen used in their test. The full-scale model showed
better agreement with the experimental results than the meso-scale model because the full-scale model
included the exterior surface geometry of the specimen that the meso-scale cannot implement.
In the presentation, an FE model is developed to obtain effective thermal conductivity of a thick 3D
woven composite (T3DWC) using an RVE model. The RVE model implements not only complicated 3D
geometry of fiber tows but also temperature-dependent thermal and mechanical properties. T3DWC
considered in the presented analysis does not have a periodicity in the thickness direction. The fiber
architecture of the RVE model along the thickness direction is thus identical to that of the original composite
material. The RVE model is created based on the planar periodicity of the material only. Due to the unique
in-plane periodicity, conventional PBCs for thermal and mechanical loading conditions are reformulated.
Virtual thermal tests are performed over a wide range of temperature to obtain homogenized thermal
conductivity of T3DWC, which is both directionally and temperature-dependent.
This paper is organized as follows. Section 2 introduces the textile geometry of T3DWC and materials
properties of constituents. Section 3 describes the finite element analysis (FEA) approach in detail to measure
the homogenized thermal conductivity of T3DWC. Section 4 discusses the results from FEA, and finally
section 5 presents our conclusions.
2. Thick 3D woven textile composite
2.1. Textile geometry
Figure 1 shows the fabric geometry of T3DWC considered in the present study. As illustrated in figure 1, the
T3DWC preform consists of a continuously folded stack (b) of a two-dimensional (2D) plain weave fabric (a)
and straight longitudinal tows (c). The definition of an RVE model for T3DWC with detailed geometrical
dimensions is given in figure 2. Three kinds of tows in the RVE are classified as curved warp, curved weft, and
straight warp according to their shapes and directions as depicted in figure 2(b). The curved warp and weft
compose the 2D fabric in figure 1(a). The tows are separated by the matrix material as shown in figure 2(b).
The separation is important for constructing a reliable mesh because, without the matrix material, the tows
will meet at a point, which may significantly degrade the quality of mesh elements and cause numerical
issues [8]. Moreover, the matrix layer mainly supports shear stresses and transfer loads between the tows [9].
2
Funct. Compos. Struct. 3 (2021) 015002
H-G Kim and W Ji
59(
6WUDLJKWZDUS
&XUYHGZHIW
&XUYHGZDUS
Top view
0DWUL[OD\HU
Front view
(a)
Side view
(b)
3
=
4
5Ő
=
6WUDLJKW
(c)
Figure 2. (a) Definition of the RVE model, (b) classification of three kinds of tows in the RVE and (c) cross-sections and
undulations of the three tows. All dimension units are millimeters.
The curved tows are assumed to have a sinusoidal undulation and an elliptic cross-section as shown in
figure 2(c). The sectional area of the curved weft is set to be 75% of that of the curved warp. The
cross-section of the straight warp is modeled as a rounded rectangle, of which the area is the same as the
curved warp. The RVE model has total 41.48% of the tow volume fraction; 16.30% for the curved warp,
17.93% of the curved weft and 7.25% of the straight warp. For simplification purposes, residual tensile and
compressive stresses around the folding curves are ignored.
2.2. Material properties
2.2.1. Effective mechanical and thermal properties of tows
The tows are composed of numerous unidirectional micrometer-sized fibers as illustrated in figure 3. Since it
is practically impossible to consider every individual fiber filament in the FE-based RVE model, the tows are
modeled as a monolithic solid by homogenizing the fibers and the matrix material. When both the fiber and
matrix materials are isotropic linear elastic, the homogenized tow becomes a transversely isotropic elastic
material, of which the principal axis conforms to the fiber direction. The team of Waas implemented a
3
Funct. Compos. Struct. 3 (2021) 015002
H-G Kim and W Ji
Figure 3. Microscopic view of the tow with the material coordinate system. 1-axis is parallel to the fiber direction.
concentric cylinder model to homogenize the tows in two-dimensional triaxial braided composites and
successfully predicted elastic properties of the textile composites with bias tows at different angles [10, 11]. In
this paper, the Halpin–Tsai homogenization method [12] is used to obtain effective elastic and thermal
properties of the solid tow. The elastic stiffnesses are computed from the following equations:
E11 = Vf Ef + Vm Em
(1a)
E22 = E33 = Em
1 + ξηVf
,
1 − ηVf
η=
Ef /Em − 1
,
Ef /Em + ζ
ζ =2
(1b)
G12 = G13 = Gm
1 + ξηVf
,
1 − ηVf
η=
Gf /Gm − 1
,
Gf /Gm + ζ
ζ =1
(1c)
ν12 = Vf νf + Vm νm
(1d)
where E is a Young’s modulus, G is a shear modulus, ν is a Poisson’s ratio, and V is a volume fraction.
Subscripts m and f denote the matrix and fiber, respectively while 1, 2 and 3 are the material orientations as
defined in figure 3. Effective thermal properties are obtained from
k22 = k33 = km
C = Vf Cf + Vm Cm
(2a)
k11 = Vf kf + Vmm
(2b)
1 + ξηVf
,
1 − ηVf
α11 =
η=
kf /km − 1
,
kf /km + ξ
ξ=1
Vf αf Ef + Vm αm Em
Vf Ef + Vm Em
]
[
]
Em
Ef
α22 = α33 = Vf αf −
Vm νf (αm − αf ) + Vm αm +
Vf νm (αm − αf ) .
E11
E11
(2c)
(2d)
[
(2e)
Here, C is specific heat capacity, k is thermal conductivity, α is the linear coefficient of thermal expansion.
4
Funct. Compos. Struct. 3 (2021) 015002
H-G Kim and W Ji
Table 1. Mechanical and thermal properties of the constituents at selected temperatures.
Density (kg m−3 )
Elastic modulus, E (GPa)
Shear modulus, G (GPa)
Poisson’s ratio, ν
Linear coefficient of thermal expansion, α (10−6 ◦ C−1 )
Thermal conductivity, k (W m−1 ◦ C−1 )
Specific heat capacity, C (J kg−1 ◦ C−1 )
SiC matrix [9]
SiC fiber [10]
3160a
414.4 (25 ◦ C)
403.5 (500 ◦ C)
378.2 (1600 ◦ C)
178.6 (25 ◦ C)
174.1 (500 ◦ C)
163.6 (1600 ◦ C)
0.16a
1.164 (25 ◦ C)
4.385 (500 ◦ C)
5.552 (1600 ◦ C)
112.5 (25 ◦ C)
55.15 (500 ◦ C)
25.03 (1600 ◦ C)
719.5 (25 ◦ C)
1090 (500 ◦ C)
1347 (1600 ◦ C)
2740a
270a
116.38a
0.16a,b
3.279 (25 ◦ C)c
4.570 (500 ◦ C)c
5.350 (1600 ◦ C)c
7.77 (25 ◦ C)
10.1 (500 ◦ C)
15.5 (1600 ◦ C)
670.0 (25 ◦ C)
1170 (500 ◦ C)
1360 (1600 ◦ C)
a
These values are assumed to be independent on temperature changes.
Poisson’s ratio of the SiC fiber is assumed to be the same as that of the matrix material.
c
Kier et al [11].
b
2.2.2. Temperature-dependent properties
The properties of each constituent material can vary over temperature. Temperature-dependent tow
properties can be estimated using the same homogenization scheme and parameters in equations (1a) and
(2a) assuming that they are invariant with respect to temperature changes. For example, when Ef and Em can
be expressed as a function of temperature, i.e. Ef = Ef (T) and Em = Em (T), E11 at temperature T can be
computed from
E11 (T) = Vf Ef (T) + Vm Em (T) .
(3)
Here, the volume fractions are again assumed to be independent of temperature. The textile composite
considered in the presentation is a SiC/SiC composite. Table 1 lists material properties of each constituent
measured at reference temperature and some high-temperature points. These data are obtained from
technical data sheets and literature [13–15]. The fiber volume fraction in all the three tow types in figure 2(b)
is set to 70%. Figure 4 displays the effective mechanical and thermal properties of the homogenized tow that
are varying over temperature. Fiber and matrix properties are also plotted in figure 4 for comparison
purposes.
3. Virtual tests to measure thermal conductivity
3.1. Finite element RVE model
Virtual tests using the FE-based RVE model are performed to measure directional thermal conductivity of
T3DWC. The RVE model defined in figure 2 is meshed with four-node linear tetrahedral elements with a
global size of 0.12 mm, resulting in total 105 648 nodes and 590 793 elements. Mechanical and thermal
properties of the fiber tows and matrix material displayed in figure 4 are implemented. The virtual tests are
conducted through steady-state coupled temperature–displacement analysis using a commercial FE software
package, ABAQUS. Linear constitutive equations are used for both mechanical deformation and thermal
conduction with the assumption of small deformation. Thermal and geometric boundary conditions for the
numerical tests are described in the next sections.
3.2. Specialized periodic boundary conditions
3.2.1. Equations of the periodic boundary conditions
When a material has a geometrically periodic microstructure, an RVE modeling technique takes advantage of
the periodicity and models the material as an infinite repetition of a smallest periodic unit cell [16]. T3DWC,
considered in the presentation, has a periodicity in the x- and y-directions only as illustrated in figure 5.
Therefore, conventional PBCs should be reformulated to take account of the unique in-plane periodicity for
both displacement and temperature fields. As shown in figure 6, PBCs are imposed between X0 and X1 faces
5
Funct. Compos. Struct. 3 (2021) 015002
H-G Kim and W Ji
(a) Elastic and shear moduli
(b) Thermal conductivity
(c) Coefficient of linear thermal expansion
(d) Specific heat capacity
Figure 4. Mechanical and thermal properties of the fiber, matrix, and homogenized tow.
Textile composite
Composite structure
Periodic RVE
Figure 5. Periodic RVE modeling approach for analysis of a large thick composite structure.
and between Y0 and Y1 faces while Z0 and Z1 faces are independent of each other. For the displacement field,
PBC equations in [17] are modified here to consider the in-plane periodicity. PBC equations for faces are
ui (X1) − ui (X0) = aε0ix
6
(4a)
Funct. Compos. Struct. 3 (2021) 015002
H-G Kim and W Ji
E
12
9
11
F
8
5
10
G
7
A
4
1
B
H
6
D
3
2
C
Figure 6. X0 face is coupled with X1 face while Y0 and Y1 faces are the matching pair for periodic boundary conditions.
ui (Y1) − ui (Y0) = bε0iy .
(4b)
ui (2) − ui (4) = aε0ix
(5a)
ui (10) − ui (12) = aε0ix
(5b)
ui (3) − ui (1) = bε0iy
(5c)
ui (11) − ui (9) = bε0iy
(5d)
ui (5) − ui (7) = aε0ix − bε0iy
(5e)
ui (6) − ui (8) = aε0ix + bε0iy .
(5f )
ui (C) − ui (A) = aε0ix + bε0iy
(6a)
ui (B) − ui (D) = aε0ix − bε0iy
(6b)
ui (G) − ui (E) = aε0ix + bε0iy
(6c)
For edges;
For vertices;
7
Funct. Compos. Struct. 3 (2021) 015002
H-G Kim and W Ji
ui (F) − ui (H) = aε0ix − bε0iy .
(6d)
In equations (4)–(6), a and b are the lengths of the RVE in the x- and y-directions, respectively. The
subscript, i, denotes a direction between x, y, and z. ε0ix and ε0iy are macroscopic strains to prescribe specific
boundary conditions, i.e. aε0ix and bε0iy can be a change in length or shear displacement of the RVE model
defined in a global manner.
PBCs for the temperature are constructed in a similar fashion. Equations (4) through (6) are rewritten by
replacing ui with a temperature change ∆T. Temperature PBCs for faces are thus expressed as
∆T (X1) − ∆T (X0) = (∆T)x
0
(7a)
0
(7b)
∆T (Y1) − ∆T (Y0) = (∆T)y .
For edges;
0
∆T (2) − ∆T (4) = (∆T)x
(8a)
0
∆T (10) − ∆T (12) = (∆T)x
(8b)
∆T (3) − ∆T (1) = (∆T)y
0
(8c)
0
(8d)
∆T (11) − ∆T (9) = (∆T)y
0
0
(8e)
0
0
(8f )
∆T (5) − ∆T (7) = (∆T)x − (∆T)y
∆T (6) − ∆T (8) = (∆T)x + (∆T)y .
For vertices;
0
0
(9a)
0
0
(9b)
0
0
(9c)
0
0
(9d)
∆T (C) − ∆T (A) = (∆T)x + (∆T)y
∆T (B) − ∆T (D) = (∆T)x − (∆T)y
∆T (G) − ∆T (E) = (∆T)x + (∆T)y
∆T (F) − ∆T (H) = (∆T)x − (∆T)y
0
0
where (∆T)x and (∆T)y are prescribed temperature differences between X0 and X1 faces and between Y0
and Y1 faces, respectively.
On the right-hand sides of equations (4a)–(9a), some terms appear repeatedly. For convenience purposes,
two fictitious nodes denoted as F1 and F2 are additionally introduced to the model in order to replace the
repeated terms with the degrees of freedom (DOFs) of the two nodes. The fictitious nodes have four DOFs:
ux , uy , uz and ∆T. They can be defined in terms of the prescribed strains and temperature as follows:
F1 : ux = aε0xx ,
uy = aε0yx ,
uz = aε0zx ,
∆T = (∆T)x
0
(10a)
F2 : ux = bε0xy ,
uy = bε0yy ,
uz = bε0zy ,
∆T = (∆T)y .
0
(10b)
In this manner, simply by defining DOFs of the nodes F1 and F2 , all the PBCs in equations (4a)–(9a) are fully
determined. Again, these fictitious nodes are not a physical part of the RVE model.
8
Funct. Compos. Struct. 3 (2021) 015002
H-G Kim and W Ji
(a) Nodes exactly matching
(b) Nodes not exactly matching
(c) Barycentric coordinates
Figure 7. PBCs between exact matching nodes and non-matching nodes.
3.2.2. Node matching between the faces
The PBCs in equations (4a)–(9a) can be implemented into an FE model only if the projected position of two
matching nodes are identical as illustrated in figure 7(a). However, due to complicated microstructural
geometry, a node may not have its pair on the opposite face as shown in figure 7(b). In this case, DOF values
should be interpolated at the projected location for PBCs. When a tetrahedron element with first-order
shape functions is used, the barycentric coordinate system [18] can be used to obtain a linear interpolation of
an arbitrary function ϕ at the point P inside the triangle in figure 7(b). The interpolated ϕ at P is expressed as
ϕ (P) = w1 ϕ (P1 ) + w2 ϕ (P2 ) + w3 ϕ (P3 )
(11)
where the point P is the location of an exact matching point corresponding to the node Q, and the points, P1 ,
P2 , and P3 are the nodes of the triangular face that contains the point P. w1 , w2 , and w3 are the weight
functions defined by the ratio of areas, which are expressed as
w1 =
A1
,
A1 + A2 + A3
w2 =
A2
,
A1 + A2 + A3
w3 =
A3
A1 + A2 + A3
(12)
where A1 , A2 , and A3 are the area of the triangles indicated in figure 7(b). Equation (11) is applied to
mismatching nodes on faces only. Node on the edges are uniformly created using a meshing tool to
fundamentally avoid node mismatches.
3.3. Boundary conditions for virtual thermal tests
3.3.1. Effective thermal conductivity
The effective thermal conductivity of the RVE model is defined from the proportional coefficient between
the average heat flux and the average thermal gradient. The linear relation is expressed as
⟨q⟩ = k⟨∇T⟩
(13)
where k is the thermal conductivity tensor, q is the heat flux vector, ∇T is the thermal gradient vector. The
operator ⟨ · ⟩ is the volume average of a variable. For the thermal conductivity along the x-direction, for
9
Funct. Compos. Struct. 3 (2021) 015002
H-G Kim and W Ji
example, the RVE model is subjected to a heat flux in the direction only and thermally insulated in the other
two directions. Then equation (13) is simplified and the conductivity in the specific direction can be isolated.
Since the volume averages of the heat flux and the thermal gradient are equal to the heat flux and
temperature change on the surfaces of the RVE model, equation (13) is finally written as
(∆T)x
Qx
= kx
bc
a
(14)
and kx is determined with known Qx and (∆T)x . Qx is a total heat flux across the surface normal to the
measurement direction and (∆T)x is the temperature difference between the two surfaces normal to the
direction. Either Qx or (∆T)x can be given as a thermal boundary condition through DOFs of the fictitious
nodes. When Qx is specified, (∆T)x will be calculated as a response of the RVE model [19]. The input and
output result in the effective thermal conductivity according to equation (14) since thermal conductivity is
basically the ratio of heat flux to temperature difference. In a similar manner, ky and kz can be determined.
3.3.2. In-plane thermal conductivity
In order to obtain the effective thermal conductivity along the x-direction, the following thermal boundary
conditions are imposed on the RVE model:
F1 : Qx = specified,
∆T = free
(15a)
F2 : ∆T = 0
(15b)
X0 face : T = specified.
(15c)
Again, F1 and F2 are the fictitious nodes controlling the PBC equations. When Qx is specified on the node F1 ,
because DOFs of the whole nodes on the face are linked to the corresponding DOFs of the control point, the
concentrated heat flux at F1 is uniformly loaded onto the face. The heat flow induces a temperature change
across the RVE in the measurement direction. There is no free temperature change because the face
temperature is already constrained. Therefore, a very small amount of heat difference is exerted for the
temperature distribution in the RVE to be almost constant at the value specified on the X0 face. In this
manner, it can be said that thermal conductivity is determined at the specific temperature. In the presented
numerical analysis, the heat flux of Q = 1 W is applied in the virtual tests. According to ASTM E1225 [20]
documenting the standard test method for measuring thermal conductivity, the magnitude of ∆T should be
smaller than 30 K when a test sample reaches steady state. Additionally, to correctly implement the thermal
expansion, the initial temperature of the RVE is set to the room temperature (T0 = 25 ◦ C) before the heat
flux is supplied.
In addition to the thermal conditions in equation (15), zero average total strains should be enforced in
the x-direction because the RVE represents the entire composite structure having infinite in-plane
dimensions as illustrated in figure 5. The boundary conditions for the zero strains are
F1 : ux = 0,
uy = 0
(16a)
F2 : ux = 0,
uy = 0
(16b)
where ux and uy are displacement DOFs in the x- and y-directions, respectively. There is no such constraint
in the z-direction and thermal expansion is allowed in the direction. Since the PBCs using fictitious points
merely control relative differences of DOFs between two matching nodes on the opposing faces, it is
necessary to constraint a motion of one node to avoid a global rigid-body motion. The following constraints
are enforced on a corner node of the RVE model:
ux = 0,
uy = 0,
uz = 0.
(17)
In a similar manner, the thermal boundary conditions for the y-direction thermal conduction test are given
as
F1 : ∆T = 0
10
(18a)
Funct. Compos. Struct. 3 (2021) 015002
H-G Kim and W Ji
F2 : Qy = 1 W,
∆T = free
Y0 face : T = specified.
(18b)
(18c)
3.3.3. Thermal conductivity along the thickness direction
Since there is no PBC in the z-direction, heat flux can be directly given to the surface. The thermal and
displacement boundary conditions for this case are
F1 : ∆T = 0,
ux = 0,
uy = 0
(19a)
F2 : ∆T = 0,
uy = 0,
uy = 0
(19b)
Z1 face : Qz = 0.02 W
(19c)
Z0 face : T = specified.
(19d)
The smaller heat flux Q = 0.02 W is used for the z-direction tests because the height of the RVE model, c, is
larger than the in-plane widths, a and b. The specific heat flux value leads to a reasonable temperature
difference. The same initial temperature T0 = 25 ◦ C is set on the RVE model before the heat flux is exerted.
Again, the RVE is assumed to experience zero average total strains and no temperature difference in the xand y-directions. All the displacement DOFs of a corner node is fully constrained to avoid a global
rigid-body motion.
3.3.4. Coefficient of linear thermal expansion (CTE)
The effective coefficient of thermal expansion (CTE) of T3DWC in the z-direction can be simultaneously
obtained from the aforementioned thermal conduction tests because the RVE model can freely expand in
that direction only. The effective CTE can be calculated from
αz =
1 H1 − H0
H0 T1 − T0
(20)
where H0 is the reference height of the RVE model at reference temperature T0 , and H1 is the extended (or
shortened) height due to temperature T1 . In the present numerical tests, H0 is set to c with the reference
temperature of 25 ◦ C. H1 is obtained by averaging the distance between the Z0 and Z1 faces after T1 is
applied to the RVE. The effective CTE values are computed from the same simulations to compute thermal
conductivity. In the numerical analysis, heat flux is prescribed for each case as well as from new experimental
setup in which uniform temperature is given instead. Additionally, for comparison purposes, the effective
CTE values without the in-plane constraints in equation (16) are also calculated. In this case, the PBCs are
still active but have no strain constraints while the temperature inside the RVE is set to be constant.
4. Results and discussion
4.1. Effective thermal conductivities
For selected temperatures, the effective thermal conductivities in the three directions, kx , ky , and kz are
calculated from the corresponding virtual conduction tests and the results are plotted in figure 8 together
with the thermal conductivities of constituent materials. The thermal conductivities of T3DWC from the
three different directions are very similar to each other. They exhibit decreasing behavior as the temperature
increases. Especially, ky and kz are hard to distinguish because the initial plain-woven 2D fabric is laid in the
yz-plane and the volumes of the curved warp and the curved weft tows in the RVE model are comparable
while their thermal properties are the same. kx follows the straight warp fibers penetrating the 2D fabric
through its thickness direction and thus the values are slightly different from the others as seen in figure 8.
However, the difference eventually diminishes as the temperature elevates. It is interesting to note that, since
T3DWC is a mixture of the fiber tows and matrix material, as can be seen in figure 8, the composite thermal
conductivities are situated between those of the two constituent materials.
In the present study, virtual tests to obtain thermal conductivity are performed on the RVE model by
prescribing a heat flux at a control point and specifying temperature on a surface normal to a measurement
direction. Temperature differences between two opposite faces induced by the heat flow are summarized in
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Funct. Compos. Struct. 3 (2021) 015002
H-G Kim and W Ji
Figure 8. Thermal conductivity of T3DWC in the x- and y-directions. Constituent properties are also displayed for reference.
Table 2. Resultant temperature differences in the virtual conduction tests.
x-direction
Temperature
T (◦ C)
25
50
100
300
500
700
900
1200
1500
Difference
∆T (◦ C)
0.6047
0.6340
0.6916
0.9085
1.1063
1.2880
1.4565
1.6899
1.9050
y-direction
∆T/T (%)
Difference
∆T (◦ C)
2.42
1.27
0.69
0.30
0.22
0.18
0.16
0.14
0.13
0.9685
1.0173
1.1135
1.4823
1.8259
2.1458
2.4444
2.8569
3.2324
z-direction
∆T/T (%)
Difference
∆T (◦ C)
∆T/T (%)
3.87
2.03
1.11
0.49
0.37
0.31
0.27
0.24
0.22
0.465
0.486
0.534
0.711
0.876
1.035
1.179
1.380
1.570
1.86
0.97
0.53
0.24
0.18
0.15
0.13
0.12
0.10
table 2. The ratios of the temperature difference to the temperature at which thermal conductivity is
computed are decreasing as the temperature increases because thermal conductivity is larger at lower
temperatures. The greatest ratio can be found from the y-direction case at 25 ◦ C. For comparison purposes,
at the same temperature (25 ◦ C) and with the same temperature difference (0.9685 ◦ C), the conductivities of
the tows and matrix material are examined. km changes 0.21% and the tow conductivities, k11 and k22 ,
change 0.17% and 0.15%, respectively, according to the results in figure 4(b). It can be expected that the
deviations of computed composite thermal conductivity do not exceed these values. The resulting range is
sufficiently small enough to ensure the measurement of linear thermal conductivity.
Figure 9 shows an example for the distributions of temperature and heat flux mapped onto the RVE
model and fiber tows only. The example is corresponding to the virtual conduction test performed at 500 ◦ C
for measuring thermal conductivity along the x-direction. Figure 10 is corresponding to the y-direction test
at the same temperature. Figure 11 shows results from the z-direction conduction test at the temperature of
500 ◦ C. Note that, for z-direction tests, total heat flow of 0.02 W is applied while 1 W is used for the in-plane
thermal tests. For all the cases from figure 9 to figure 11, the temperature uniformly changes along the
heating direction while the heat flux distributions are highly dependent on the heterogeneity of T3DWC. The
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Funct. Compos. Struct. 3 (2021) 015002
H-G Kim and W Ji
Temperature (ƒC)
Heat flux (103 W/m2)
500ƒC
1W
=17.8Ő103 W/m2
(a) Temperature
(b) Heat flux magnitude
Figure 9. Typical distributions of temperature and heat flux magnitude in RVE (500 ◦ C, x-direction and flux = 1 W).
Temperature (ƒC)
Heat flux (103 W/m2)
500ƒC
1W
=23.2Ő103 W/m2
(a) Temperature
(b) Heat flux magnitude
Figure 10. Typical distributions of temperature and heat flux magnitude in RVE (500 ◦ C, y-direction and flux = 1 W).
non-uniform distributions are mainly caused by different thermal conductivity values between the
constituent materials. The matrix material transfers heat more effectively than the fiber tows according to the
results in figures 9(b), 10(b) and 11(b) due to its higher thermal conductivity as shown in figure 4(b).
For the x-direction test, the thermal gradient exerts along the 1-direction of the straight tows and along
the 3-direction for the curved warp and weft tows. The transverse conductivities of the fiber tows are larger
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Funct. Compos. Struct. 3 (2021) 015002
H-G Kim and W Ji
Temperature (ƒC)
Heat flux (103 W/m2)
0.02 W
=2.27Ő103 W/m2
500ƒC
(a) Temperature
(b) Heat flux magnitude
Figure 11. Typical distributions of temperature and heat flux magnitude in RVE (500 ◦ C, z-direction and flux = 20 W).
Figure 12. Effective CTEs and constituent CTEs. The curves of X-hear, Y-heat, and Z-heat cases are overlapped by the curve of
constant temperature case.
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Funct. Compos. Struct. 3 (2021) 015002
H-G Kim and W Ji
than the fiber-direction one (see figure 4(b)) and, thus, the amount of heat transferred through the straight
tow is smaller than by the curved tows. On the other hand, for the y-direction test, the thermal gradient
exerts along the 1-direction of the curved tows and along the 3-direction of the straight tows. However, the
curved tows do not transfer heat efficiently compared to the straight tows because the curvatures create an
angle with the direction of the thermal gradient and conductivities projected onto the direction parallel to
the thermal gradient become smaller.
4.2. Effective coefficient of linear thermal expansion
As previously mentioned, the byproduct of the virtual conduction tests is the z-direction CTE, αz , of
T3DWC. Figure 12 displays temperature-dependent CTE values resulted from total five setups. CX, CY, and
CZ are associated with the CTEs obtained from the virtual tests in which the heating is exerted along the x-,
y-, and z-directions, respectively. αz (CT) is the CTE obtained from the setup in which the temperature is
uniform and constant inside the RVE while αz (CF) is calculated without the in-plane constraints in
equation (16). The CTE values of the constituent materials are also displayed in figure 12 for comparison
purposes. α11 and α22 are the CTEs of the fiber tows along the fiber and transverse directions, respectively. αf
and αm are the CTEs of the fiber and matrix material, respectively. The reference temperature of 25 ◦ C is
consistently used for the computation results in figure 12.
αz (CT) is calculated at a specific constant temperature and, thus, used as a benchmark solution to
evaluate CTEs obtained from other setups. Small temperature differences are inevitable as listed in table 2
from the virtual conduction tests. However, the CTE values associated with the CX, CY, and CZ cases are
indistinguishable from αz (CT), implying that the temperature differences in table 2 are negligibly small in
computing CTE values. Note that very small heat fluxes are used for the virtual tests. The maximum error of
CX, CY and CZ measurements against αz (CT) is 0.43% for the CY case at 100 ◦ C. The CTE values of
T3DWC are larger than those of the constituent materials except αz (CF). The difference may result from the
in-plane displacement boundary conditions in equation (16). Due to the fixed boundary conditions,
compressive thermal stresses in the x- and y-directions are developed. The compressive stresses enhances
thermal expansion along the z-direction. In contrast, the CF case has no constraint on in-plane expansion.
Consequently, αz (CF) is closely located with α11 and α22 .
5. Conclusions
Now it is common to utilize the RVE modeling approach to compute homogenized properties of a composite
material. In the present study, the RVE modeling technique has been employed to simulate the
thermo-mechanical response of the 3DTC having complex tow architecture over a wide range of
temperatures. Detailed thermo-mechanical behavior dependent on temperature and the textile configuration
can be revealed through the present numerical analysis. In doing so, we have employed the sequential
two-step homogenization technique to efficiently link the microscopic level with the meso-scale. The
predictive capability of the present numerical approach has been previously proven for various textile
composites [9–11, 16]. In addition, conventional PBCs are customized to the thick composite material
considered here. It is shown that the presented numerical methodology is fundamentally simple yet capable
of dealing with complicated thermal and mechanical response of the complex 3DTC. This method can be
useful at the preliminary design stage to efficiently estimate homogenized thermo-mechanical properties of
composite materials such as ceramic matrix composites and carbon/carbon (C/C) composites, used for
high-temperature applications.
Acknowledgments
The authors are grateful for the financial support from Ulsan National Institute of Science and Technology
(UNIST) through the 2020 Research Fund (Grant No. 1.200031.01).
ORCID iD
Wooseok Ji  https://orcid.org/0000-0002-3136-4259
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