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carbon composites
NEW CARBON MATERIALS Volume 26, Issue 4, Aug 2011 Online English edition of the Chinese language journal Cite this article as: New Carbon Materials, 2011, 26(4): 287–292.
RESEARCH PAPER
Effect of graphitization parameters on the residual stress in 4D carbon fiber/carbon composites SHI Hong-bin1*, TANG Min1, GAO Bo1, SU Jun-ming2 1
The National Key laboratory of Combustion, Thermostructure and Flow of SRM, The 41 st Institute of the Fourth Academy of CASC, Xi’an
710025; 2
The 43 st of the 4th Academy of CASC, Xi’an 710025 China
Abstract:
A representative volume element model was proposed for C/C composites with the fibers oriented in four different
directions. Three of the fiber directions were coplanar, at 120° to each other, and the fourth was perpendicular.
The element truly
reflects the orientations of the carbon fibers and matches the actual configuration of the composites. Based on the model and computational mesoscopic mechanics, the residual thermal stress distribution of the material at different graphitization temperatures, cooling rates, and interface stiffnesses was obtained using a finite element method. The residual thermal stresses of the composite from a low-graphitization temperature or with low interface stiffness are lower than those from a high-graphitization temperature or with high interface stiffness. The higher the cooling rate, the higher the residual thermal stress. Key Words: Carbon/carbon composite; Graphitization temperature; Cooling grads
1
Introduction
4D carbon fiber/carbon composites are manufactured by carbon rod to make the axial enforcement net, and by soft fiber bundles to braid the preform. The preform is densified by repeated coal pitch impregnation and hot pressed in a mold at 600 °C for carbonization, followed by a heat treatment at 2 000 °C for graphitization in an inert environment[1]. It has good property of low ablation and high shock resistance that makes it suitable for an important application of throat-insert in solid rocket motor. Although, carbon/carbon composites are only under the load of temperature in the heat-treatment process, the residual thermal stress in fiber bundle/matrix interface caused by the different thermal expansion properties of fibers and matrix can lead to the debonding of fibers from the interface. Aubard[2] thought that the failure of carbon/carbon composite was caused by the debonding of interface between the fiber bundles and matrix. Hence, the interface has a strong effect on the properties of carbon/carbon composites[3], and it is necessary to study the distribution of residual thermal stress. For three-dimensional carbon/carbon composites, the researchers concentrated mostly on the properties of strength and stiffness and only limited information for analyzing the thermal process is available. The study of varying rule of residual thermal stress is insufficient. The basic properties of a four-directional composite were simulated by using finite
element methods and the thermal-stress distribution in the cooling period are obtained[4]. The maximum residual stresses are very high because of the perfect interface hypothesis. Rao[5] analyzed the interface response of four-directional in-plain composites by using cohesive element and modeled the heat-treatment process from high temperature to room temperature. The result can reflect the real distribution of thermal stress. A parallel-consecutive reaction model of chemistry and kinetics was proposed to simulate homogeneous gas-phase reactions of propylene pyrolysis in CVI processes, and the competition between the heterogeneous reactions of pyrolytic carbon deposition and the homogeneous reactions was also analyzed by a numerical simulation method[6]. In all heat-treatment processes, graphite has a strong effect on the properties of carbon/carbon composites because of higher temperature and longer period when compared with the other processes. So, it is useful to analyze the graphite process to extract helpful information to improve the manufacturing technology. The propose of the present work is to study the residual thermal stress distribution and interface response of a 4D carbon fiber/carbon composite at different graphitization temperatures, cooling grads, and interface stiffness. Their effects on thermal stress are attained and some useful conclusions are obtained, which can provide helpful information for the design and manufacture of carbon/carbon composites.
Received date: 24 March 2010; Revised date: 1 April 2011 *Corresponding author. E-mail: [email protected] Copyright©2011, Institute of Coal Chemistry, Chinese Academy of Sciences. Published by Elsevier Limited. All rights reserved. DOI: 10.1016/S1872-5805(11)60082-6
SHI Hong-bin et al. / New Carbon Materials, 2011, 26(4): 287–292
2
Analysis and calculation model
2.1
P is temperature load matrix.
Mathematical model
In the graphite process, the product after CVI is placed in a vapor camp stove, which is filled with N2 or Nr, to experience thermal process. To construct the math model, consider the following hypothesis: (1) At the beginning of temperature-hold stage of graphite process, the interface of fiber bundles and matrix are perfect, and the material is in no-stress status. (2) The thermal exchange condition and process cannot be measured, but the surface temperature changing with time can be measured precisely in the graphite process. Thus, temperature loads are given on the unit cell surface for studying the thermal-stress problem. (3) Nomenclature: Eij, Young’ module, GPa; Gij, Shear module, GPa; µij, Poson’ratio, W/mK; Ki, Conductivity, e-6/°C; αi, Thermal expansion coefficient; ρ, Density, kg/m3; C, Specific, J/kg·K；Q， Internal heat source intensity，W/kg。 In the graphite process of carbon/carbon composite, the transient fields Φ ( x, y, z , t ) must satisfy the following equations: ∂Φ ∂ ∂Φ ∂ ∂Φ ρc − (k x )− (k y ) ∂t ∂x ∂x ∂y ∂y ∂ ∂Φ − (k z ) − ρQ = 0 ∂z ∂z
(1)
Γ1 , is the given temperature on Γ1 .
where Φ
on
(2)
The effective integration form of (1) and (2) is as follows: ∂ ∂Φ ∂Φ ∂ ∂Φ ∫Ω ω ( ρc ∂t − ∂x (k x ∂x ) − ∂y (k y ∂y ) , (3) ∂ ∂Φ − (k z ) − ρQ)dΩ + ∫ ω1 (Φ − Φ)dΓ Γ1 ∂z ∂z =0 where ω、ω1 are arbitrary functions. If Γ1 has attained the boundary of Φ = Φ and ω1 = 0 , the equation (3) can be written as follows by order ω = δΦ . ∂Φ ∂δΦ ∂Φ ∂δΦ ∂Φ )− (k x )− (k y ) ∂t ∂x ∂x ∂y ∂y ∂δΦ ∂Φ (k z ) − δΦρQ]dΩ − ∂z ∂z =0
[δΦ ( ρc
(4)
By using finite element to discrete the domain of Ω , the temperature Φ in an element can be obtained by the node temperature of Φ i . ne
Φ = Φ = ∑ N i ( x, y , z )Φ i (t )
(5)
i =1
N i is the space function, bringing equation (5) into (3), and
the node temperature
Φi
•
C Φ + KΦ = P ,
ε 0 = α (Φ − Φ 0 )[11 1 0 0 0]T ,
(6)
where C is specific matrix; K is conductivity matrix; and
(7)
where Φ 0 is the initial temperature field and Φ is the transient temperature field. The constitutive law can be described as equation (8) under the initial strain as follows:
σ = D(ε − ε 0 )
(8)
According to virtual displacement principle 1 ∏ p (u ) = ∫ ( ε T Dε − ε T Dε 0 − u T f )dΩ Ω 2
(9)
− ∫ u T T dΓ Γσ
Discrete the domain of Ω , and the finite element equation is
Ka = P
(10)
In the graphite process, the load P can be described as P = Pε 0 ,
According to the second hypothesis, the boundary condition is given as follows:
Φ=Φ
Thermal expansion only produces normal strain, and the shear strain is zero. It can be considered as initial strain ε 0 , the described equation is
(11)
where Pε is the load produced by temperature strain. 0 The constitutive law of the interface between the fiber bundles and matrix is described as surface-based cohesive behavior[7-10]. It describes the relation of traction and separation. The available traction-separation model assumes initially linear elastic behavior followed by the initiation and evolution of damage. The elastic behavior can be written as follows:
⎧ tn⎫ ⎡Knn Kns Knt ⎤⎧δn⎫ ⎪ ⎪ ⎢ ⎥⎪ ⎪ ⎨ts ⎬=⎢Kns Kss Kst ⎥ ⎨δs ⎬=Kδ ⎪t ⎪ ⎢K K K ⎥ ⎪δ ⎪ ⎩ t ⎭ ⎣ nt st tt ⎦⎩ t ⎭
(12)
The nominal traction stress vector, t , consists of three components: t n , t s , and t t , which represent the normal and the two shear tractions, respectively. The corresponding separations are denoted by δ n , δ s , and δ t . The matrix K represents the stiffness of the interface. According to the work of Zhandarov and Chandra[7-8] on fiber bundles pulled from matrix , the maximum principle is chosen as the criterion of interface debonding. Damage is assumed to initiate when the maximum contact stress ratio (as defined in the expression below) reaches a value of one. This criterion can be represented as follows:
⎧t t t ⎫ max ⎨ n0 , s0 , 0t ⎬ = 1 , ⎩ t n t s tt ⎭
(13)
SHI Hong-bin et al. / New Carbon Materials, 2011, 26(4): 287–292
Fig.1 Braid mode and unit cell of 4D carbon fiber/carbon composites: (a) braid pattern ; (b) unit cell Table 1 Properties of fiber bundle
t/°C
E11
E22
µ 21
G12
G23
α1
α2
K1
K2
CT
ρ
25
226.9
19.0
0.2
23.4
6.8
-0.9
10.9
29.3
24.4
633
1 998
800
204.7
15.5
0.24
21.5
5.5
1.0
12.9
23.9
20.3
974
1 968
2 750
148.8
12.9
0.26
20.9
4.5
5.0
16.9
18.3
16.4
1832
1 960
Table 2 Properties of matrix
t/°C
E
µ 21
G
α
K
CT
ρ
25
18
0.35
6.7
6.0
70
650
1 981
800
18
0.35
6.7
6.6
58
1700
1 952
2 750
18
0.35
6.7
6.7
52
1900
1 943
where t 0n 、, t s0 、, and
t 0t represent the maximum traction
values in the normal and shear direction. For linear softening, Abaqus uses an evolution of the damage variable, D, that reduces to the following expression D=
δ mf (δ mmax − δ m0 ) , δ mmax (δ mf − δ m0 )
(14)
where δ mf is effective separation at complete failure and δ m0 is relative to the effective separation at initiation damage. In the preceding expression and in all later references, δ mmax refers to the maximum value of the effective separation attained during the loading history. For carbon/carbon composites, only limited information exists for fracture energy and interfacial strength. Based on fiber-bundle pull-out tests on 4D carbon fiber/carbon composite, we take the normal and shear fracture energy of the interface as 143 J/m2 and the normal and the shear strength of the interface as 12 MPa and 8.4 MPa, respectively. The normal and the shear stiffness of the interface is 10 GPa. 2.2
Finite element model
Because of the high level of heterogeneity within the carbon/carbon composite, these quantities varied very rapidly within a very small size at a given point. To solve the thermal
stress problem with whole body by using finite element methods would be almost impossible because discretization of the body becomes enormous in order to represent detailed structure of the microscopic material constitution. 4D carbon fiber/carbon composites are formed by the spatial repetition of a base cell made of different materials. Thus, it is reasonable to take a unit cell with periodic boundary[11–13] to study the thermal process. According to the braided pattern of 4D carbon fiber/carbon composites ( Fig.1(a)), a unit cell was built, and it was 3.4mm × 4.0mm × 5.88mm in the x, y, and z directions, as shown in Fig.1(b). The diameter of fiber bundles was 1.2 mm. In the finite element analysis method, the coupled temperature-displacement analyses step was used to solve the solution. The unit cell was meshed by C3D8T, the total number is 24636. Fiber bundles seemed as transverse isotropic materials and the matrix was assumed as isotropic materials. Their properties are shown in Tables 1 and 2. In Table 1, 1 represents the fiber direction and 2 the transverse direction. It was difficult to test the mechanical properties at high temperature because of the brittleness of carbon matrix, and only thermal properties were tested at different temperatures. Its mechanical properties were tested at room temperature. Because the modulus was low, it had little effect on the calculation. For studying the effect of graphitization temperature on the residual thermal stress, the graphitization temperatures were given as 2 000 °C, 2 500 °C, and 2 750 °C，respectively. For studying the effect of cooling rate on the residual stress, two kinds of cooling history (grads 1 and grads 2）were used to calculate the stress, and the cooling rate for the latter was
SHI Hong-bin et al. / New Carbon Materials, 2011, 26(4): 287–292
Fig.2 Temperature loading history at different initial temperatures
Fig.4 Residual thermal stress distribution at the temperature loads of 2 000 °C
Fig.3 Temperature loading history at different cool grads
two times larger than that of the former. Temperature loads history were illustrated in Figs.2 and 3. At 2 500 °C, two values of interface stiffness were used to study the effect of interface stiffness on the thermal stress, and the values were 10 GPa and 15 GPa.
3
Result and analysis
The distribution of residual stress of the unit cell under the temperature loading history at 2 000 °C in Fig.3 is illustrated in Fig.4. It can be noticed that the opposite surface of the unit cell has the same stress distribution, and the displacement continuity and stress continuity are both satisfied with the applied periodic boundary. Because of the different expand property, there are some level of stress distribution in the fibers and the matrix. The maximum stress occurred at the interface of matrix and fibers because in the high temperature after insulation, the material is in no-stress mode, and when it turns into low temperature, the fibers will suffer compress loads and the matrix will suffer tensile loads. That is in accordance with the result ofAly-Hassan et al.[14]. At the joint action of the two forces, stress concentration is formed in the interface. In addition, maximum stress occurred in the dominoes, which has the nearest distance of fiber bundles, and this embodied the interaction of each fiber bundles. Corresponding to the three kinds of temperature loads
Fig.5 Residual thermal stress-time curves with different temperature
(2 000 °C, 2 500 °C，2 750 °C illustrated in Fig.2), the maximum residual stress are 8.95 MPa, 9.86 MPa, and 10.89 MPa, respectively, and the stress-time curve of the maximum stress point is illustrated in Fig.5. In the same cooling rate, the maximum stress increases with the graphitization temperature, and the maximum stress generated by the graphitization temperature of 2 500 °C is 10.1% higher than that of 2 000 °C. The two different slopes either embodied the change of cooling rate or the nonlinear change of heat generation. In the case of guaranteeing the graphitization temperature, controlling the cooling grad is useful to protect interface from debonding. According to the two kinds of cooling rates (grads 1 and grads 2), the stress-time curves of the maximum stress points, after graphite, is illustrated in Fig.6. The maximum Mises stress are 9.86 MPa and 10.72 MPa, respectively, and the latter is 8.7% higher than the former. It is verified that the maximum stress increases with the increase in cooling grad. Corresponding to the two values of interface stiffness (10 GPa, 15 GPa), the stress-time curve of the maximum stress points at the temperature loads of 2 500 °C is illustrated in Fig.7, and it can be easily found that the maximum stress increases with the interface stiffness. The maximum stress is 9.86 MPa and 14.62 MPa, respectively. The maximum stress is 48.3% higher when the interface stiffness increased by 50%.
SHI Hong-bin et al. / New Carbon Materials, 2011, 26(4): 287–292
corresponding to the four temperature loads is shown in Fig.8. It can be seen that the separation increased with the graphitization temperature and cooling grads. This is in accordance with the result of stress analysis because the separation and the thermal stress have the same changing tendency.
4
Conclusions
In graphitizing process, the following effects are observed:
Fig.6 Residual thermal stress-time curves with different cooling grads
(1) The change of residual thermal stress is proportional to graphitization temperature, cooling grad, and interface stiffness. (2) Graphitization temperature is the main factor that affects the residual thermal stress among the factors investigated. (3) Interface with weak stiffness can either enhance the strength or decrease the residual thermal stress for the carbon/carbon composite. (4) The maximum stress occurred in the domain, which has the great interaction of fiber bundles. The distribution of residual thermal stress can be adjusted by changing the distance of fiber bundles. Generally, the thermal-stress information of carbon/carbon composite in graphitization process can give a good guidance for improving the mechanical properties during manufacture.
Fig.7 Residual thermal stress-time curves with different interface
References
stiffness [1] Su Jun-ming, Cui Hong, SU Zhe-an, et al. Research of 4D C/C composite material throat insert by mix weaved with carbon fiber and axial carbon rod[J]. Carbon(Chinese), 2004(1): 12-16. [2] Aubard X, Cluzel C, Guitard L, et al. Damage modeling at two scales for 4D carbon/carbon composites[J]. Computers & Structeres, 2000, 78: 83-91. [3] Liu Chang, Wei Yong-liang. Observation of the interfacial microstructure
of
C/C
composites[J].Carbon
Techniques,
1998(5): 11-14. [4] WAN Qiong， LI Fu-guo， LIANG Hong，et al.Finite element simulation of key properties of three-dimensional four-step
braided
composite[J].
and
Journal of Aeronautical
Materials, 2005, 25(1): 30-35. [5] Rao M V, Mahajan P, Mittal R K. Effect of Architecture on Fig.8 Normal separation along the path A-B with different graphitization temperatures and cooling grades
In the normal range of cooling grad and graphitization temperature, the main factor to affect the residual stress is graphitization temperature. In the process of graphitization, graphite crystals turn into ordered structure from disorder structure, the higher the temperature the bigger the total energy. The significant change of total energy made the obvious change in residual stress. The normal separation of path A to B in interface
mechanical
properties
of
carbon/carbon
composites[J].
Composite Structure, 2008, 83: 131-142. [6] Li Hejun, Li Aijun, Bai Ruicheng, et al. Numerical simulation of chemical vapor infiltration of propylene into composites with reduced
multi-step
kinetic
models[J].Carbon,
2005,
43:
2937-2950. [7] Zhandarov S, Mäder E. Characterization of fiber/matrix interface strength: applicability of different tests, approaches and parameters [J].Composites Science and Technology，2005, 65: 149-160. [8] Chandra N，Ananth C R. Analysis of interfacial behavior in
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MMCs and IMCs using thin-slice push-out tests[J].Composites
model and application of unified periodic boundary condictions
Science and Technology，1995, 54: 87-100.
in micro-mechanical analysis of composites[J].Solid and
[9] Hatta H, Goto K, Ikegaki S, et al. Tensile strength and fiber/matrix interfacial properties of 2D- and 3D-carbon/carbon composites[J].J Eur Ceram Soc ,2005, 25: 535-542. [10] Sakai M, Matsuyama R, Miyajimal T. The pull-out and failure of a fibre bundle in a carbon fibers[J]. Carbon，2000, 39: 635-645.
Structures, 2006, 43: 266-278. [13] Xi Kun, Xu Xiwu. Prediction of elastic constants and simulation of stress field of 3D braided composites based on the finite element method[J]. Acta material composite finical, 2007, 24(3): 178-185.
[11] Zihui Xia, Yunfa Zhang. A unified periodical boundary
[14] Aly-Hassan M S, Hatta H, Wakayama S, et al. Comparison of
condiction for representation volume elements of composites
2D and 3D carbon/carbon composites with respect to damage
and applications[J]. Solid and Structures, 2003(40): 1907-1921.
and fracture resistance[J]. Carbon, 2003, 41: 1069-1078.
[12] Zihui Xia, Chuwei Zhou. On selection of repeated unit cell
RESEARCH PAPER
Effect of graphitization parameters on the residual stress in 4D carbon fiber/carbon composites SHI Hong-bin1*, TANG Min1, GAO Bo1, SU Jun-ming2 1
The National Key laboratory of Combustion, Thermostructure and Flow of SRM, The 41 st Institute of the Fourth Academy of CASC, Xi’an
710025; 2
The 43 st of the 4th Academy of CASC, Xi’an 710025 China
Abstract:
A representative volume element model was proposed for C/C composites with the fibers oriented in four different
directions. Three of the fiber directions were coplanar, at 120° to each other, and the fourth was perpendicular.
The element truly
reflects the orientations of the carbon fibers and matches the actual configuration of the composites. Based on the model and computational mesoscopic mechanics, the residual thermal stress distribution of the material at different graphitization temperatures, cooling rates, and interface stiffnesses was obtained using a finite element method. The residual thermal stresses of the composite from a low-graphitization temperature or with low interface stiffness are lower than those from a high-graphitization temperature or with high interface stiffness. The higher the cooling rate, the higher the residual thermal stress. Key Words: Carbon/carbon composite; Graphitization temperature; Cooling grads
1
Introduction
4D carbon fiber/carbon composites are manufactured by carbon rod to make the axial enforcement net, and by soft fiber bundles to braid the preform. The preform is densified by repeated coal pitch impregnation and hot pressed in a mold at 600 °C for carbonization, followed by a heat treatment at 2 000 °C for graphitization in an inert environment[1]. It has good property of low ablation and high shock resistance that makes it suitable for an important application of throat-insert in solid rocket motor. Although, carbon/carbon composites are only under the load of temperature in the heat-treatment process, the residual thermal stress in fiber bundle/matrix interface caused by the different thermal expansion properties of fibers and matrix can lead to the debonding of fibers from the interface. Aubard[2] thought that the failure of carbon/carbon composite was caused by the debonding of interface between the fiber bundles and matrix. Hence, the interface has a strong effect on the properties of carbon/carbon composites[3], and it is necessary to study the distribution of residual thermal stress. For three-dimensional carbon/carbon composites, the researchers concentrated mostly on the properties of strength and stiffness and only limited information for analyzing the thermal process is available. The study of varying rule of residual thermal stress is insufficient. The basic properties of a four-directional composite were simulated by using finite
element methods and the thermal-stress distribution in the cooling period are obtained[4]. The maximum residual stresses are very high because of the perfect interface hypothesis. Rao[5] analyzed the interface response of four-directional in-plain composites by using cohesive element and modeled the heat-treatment process from high temperature to room temperature. The result can reflect the real distribution of thermal stress. A parallel-consecutive reaction model of chemistry and kinetics was proposed to simulate homogeneous gas-phase reactions of propylene pyrolysis in CVI processes, and the competition between the heterogeneous reactions of pyrolytic carbon deposition and the homogeneous reactions was also analyzed by a numerical simulation method[6]. In all heat-treatment processes, graphite has a strong effect on the properties of carbon/carbon composites because of higher temperature and longer period when compared with the other processes. So, it is useful to analyze the graphite process to extract helpful information to improve the manufacturing technology. The propose of the present work is to study the residual thermal stress distribution and interface response of a 4D carbon fiber/carbon composite at different graphitization temperatures, cooling grads, and interface stiffness. Their effects on thermal stress are attained and some useful conclusions are obtained, which can provide helpful information for the design and manufacture of carbon/carbon composites.
Received date: 24 March 2010; Revised date: 1 April 2011 *Corresponding author. E-mail: [email protected] Copyright©2011, Institute of Coal Chemistry, Chinese Academy of Sciences. Published by Elsevier Limited. All rights reserved. DOI: 10.1016/S1872-5805(11)60082-6
SHI Hong-bin et al. / New Carbon Materials, 2011, 26(4): 287–292
2
Analysis and calculation model
2.1
P is temperature load matrix.
Mathematical model
In the graphite process, the product after CVI is placed in a vapor camp stove, which is filled with N2 or Nr, to experience thermal process. To construct the math model, consider the following hypothesis: (1) At the beginning of temperature-hold stage of graphite process, the interface of fiber bundles and matrix are perfect, and the material is in no-stress status. (2) The thermal exchange condition and process cannot be measured, but the surface temperature changing with time can be measured precisely in the graphite process. Thus, temperature loads are given on the unit cell surface for studying the thermal-stress problem. (3) Nomenclature: Eij, Young’ module, GPa; Gij, Shear module, GPa; µij, Poson’ratio, W/mK; Ki, Conductivity, e-6/°C; αi, Thermal expansion coefficient; ρ, Density, kg/m3; C, Specific, J/kg·K；Q， Internal heat source intensity，W/kg。 In the graphite process of carbon/carbon composite, the transient fields Φ ( x, y, z , t ) must satisfy the following equations: ∂Φ ∂ ∂Φ ∂ ∂Φ ρc − (k x )− (k y ) ∂t ∂x ∂x ∂y ∂y ∂ ∂Φ − (k z ) − ρQ = 0 ∂z ∂z
(1)
Γ1 , is the given temperature on Γ1 .
where Φ
on
(2)
The effective integration form of (1) and (2) is as follows: ∂ ∂Φ ∂Φ ∂ ∂Φ ∫Ω ω ( ρc ∂t − ∂x (k x ∂x ) − ∂y (k y ∂y ) , (3) ∂ ∂Φ − (k z ) − ρQ)dΩ + ∫ ω1 (Φ − Φ)dΓ Γ1 ∂z ∂z =0 where ω、ω1 are arbitrary functions. If Γ1 has attained the boundary of Φ = Φ and ω1 = 0 , the equation (3) can be written as follows by order ω = δΦ . ∂Φ ∂δΦ ∂Φ ∂δΦ ∂Φ )− (k x )− (k y ) ∂t ∂x ∂x ∂y ∂y ∂δΦ ∂Φ (k z ) − δΦρQ]dΩ − ∂z ∂z =0
[δΦ ( ρc
(4)
By using finite element to discrete the domain of Ω , the temperature Φ in an element can be obtained by the node temperature of Φ i . ne
Φ = Φ = ∑ N i ( x, y , z )Φ i (t )
(5)
i =1
N i is the space function, bringing equation (5) into (3), and
the node temperature
Φi
•
C Φ + KΦ = P ,
ε 0 = α (Φ − Φ 0 )[11 1 0 0 0]T ,
(6)
where C is specific matrix; K is conductivity matrix; and
(7)
where Φ 0 is the initial temperature field and Φ is the transient temperature field. The constitutive law can be described as equation (8) under the initial strain as follows:
σ = D(ε − ε 0 )
(8)
According to virtual displacement principle 1 ∏ p (u ) = ∫ ( ε T Dε − ε T Dε 0 − u T f )dΩ Ω 2
(9)
− ∫ u T T dΓ Γσ
Discrete the domain of Ω , and the finite element equation is
Ka = P
(10)
In the graphite process, the load P can be described as P = Pε 0 ,
According to the second hypothesis, the boundary condition is given as follows:
Φ=Φ
Thermal expansion only produces normal strain, and the shear strain is zero. It can be considered as initial strain ε 0 , the described equation is
(11)
where Pε is the load produced by temperature strain. 0 The constitutive law of the interface between the fiber bundles and matrix is described as surface-based cohesive behavior[7-10]. It describes the relation of traction and separation. The available traction-separation model assumes initially linear elastic behavior followed by the initiation and evolution of damage. The elastic behavior can be written as follows:
⎧ tn⎫ ⎡Knn Kns Knt ⎤⎧δn⎫ ⎪ ⎪ ⎢ ⎥⎪ ⎪ ⎨ts ⎬=⎢Kns Kss Kst ⎥ ⎨δs ⎬=Kδ ⎪t ⎪ ⎢K K K ⎥ ⎪δ ⎪ ⎩ t ⎭ ⎣ nt st tt ⎦⎩ t ⎭
(12)
The nominal traction stress vector, t , consists of three components: t n , t s , and t t , which represent the normal and the two shear tractions, respectively. The corresponding separations are denoted by δ n , δ s , and δ t . The matrix K represents the stiffness of the interface. According to the work of Zhandarov and Chandra[7-8] on fiber bundles pulled from matrix , the maximum principle is chosen as the criterion of interface debonding. Damage is assumed to initiate when the maximum contact stress ratio (as defined in the expression below) reaches a value of one. This criterion can be represented as follows:
⎧t t t ⎫ max ⎨ n0 , s0 , 0t ⎬ = 1 , ⎩ t n t s tt ⎭
(13)
SHI Hong-bin et al. / New Carbon Materials, 2011, 26(4): 287–292
Fig.1 Braid mode and unit cell of 4D carbon fiber/carbon composites: (a) braid pattern ; (b) unit cell Table 1 Properties of fiber bundle
t/°C
E11
E22
µ 21
G12
G23
α1
α2
K1
K2
CT
ρ
25
226.9
19.0
0.2
23.4
6.8
-0.9
10.9
29.3
24.4
633
1 998
800
204.7
15.5
0.24
21.5
5.5
1.0
12.9
23.9
20.3
974
1 968
2 750
148.8
12.9
0.26
20.9
4.5
5.0
16.9
18.3
16.4
1832
1 960
Table 2 Properties of matrix
t/°C
E
µ 21
G
α
K
CT
ρ
25
18
0.35
6.7
6.0
70
650
1 981
800
18
0.35
6.7
6.6
58
1700
1 952
2 750
18
0.35
6.7
6.7
52
1900
1 943
where t 0n 、, t s0 、, and
t 0t represent the maximum traction
values in the normal and shear direction. For linear softening, Abaqus uses an evolution of the damage variable, D, that reduces to the following expression D=
δ mf (δ mmax − δ m0 ) , δ mmax (δ mf − δ m0 )
(14)
where δ mf is effective separation at complete failure and δ m0 is relative to the effective separation at initiation damage. In the preceding expression and in all later references, δ mmax refers to the maximum value of the effective separation attained during the loading history. For carbon/carbon composites, only limited information exists for fracture energy and interfacial strength. Based on fiber-bundle pull-out tests on 4D carbon fiber/carbon composite, we take the normal and shear fracture energy of the interface as 143 J/m2 and the normal and the shear strength of the interface as 12 MPa and 8.4 MPa, respectively. The normal and the shear stiffness of the interface is 10 GPa. 2.2
Finite element model
Because of the high level of heterogeneity within the carbon/carbon composite, these quantities varied very rapidly within a very small size at a given point. To solve the thermal
stress problem with whole body by using finite element methods would be almost impossible because discretization of the body becomes enormous in order to represent detailed structure of the microscopic material constitution. 4D carbon fiber/carbon composites are formed by the spatial repetition of a base cell made of different materials. Thus, it is reasonable to take a unit cell with periodic boundary[11–13] to study the thermal process. According to the braided pattern of 4D carbon fiber/carbon composites ( Fig.1(a)), a unit cell was built, and it was 3.4mm × 4.0mm × 5.88mm in the x, y, and z directions, as shown in Fig.1(b). The diameter of fiber bundles was 1.2 mm. In the finite element analysis method, the coupled temperature-displacement analyses step was used to solve the solution. The unit cell was meshed by C3D8T, the total number is 24636. Fiber bundles seemed as transverse isotropic materials and the matrix was assumed as isotropic materials. Their properties are shown in Tables 1 and 2. In Table 1, 1 represents the fiber direction and 2 the transverse direction. It was difficult to test the mechanical properties at high temperature because of the brittleness of carbon matrix, and only thermal properties were tested at different temperatures. Its mechanical properties were tested at room temperature. Because the modulus was low, it had little effect on the calculation. For studying the effect of graphitization temperature on the residual thermal stress, the graphitization temperatures were given as 2 000 °C, 2 500 °C, and 2 750 °C，respectively. For studying the effect of cooling rate on the residual stress, two kinds of cooling history (grads 1 and grads 2）were used to calculate the stress, and the cooling rate for the latter was
SHI Hong-bin et al. / New Carbon Materials, 2011, 26(4): 287–292
Fig.2 Temperature loading history at different initial temperatures
Fig.4 Residual thermal stress distribution at the temperature loads of 2 000 °C
Fig.3 Temperature loading history at different cool grads
two times larger than that of the former. Temperature loads history were illustrated in Figs.2 and 3. At 2 500 °C, two values of interface stiffness were used to study the effect of interface stiffness on the thermal stress, and the values were 10 GPa and 15 GPa.
3
Result and analysis
The distribution of residual stress of the unit cell under the temperature loading history at 2 000 °C in Fig.3 is illustrated in Fig.4. It can be noticed that the opposite surface of the unit cell has the same stress distribution, and the displacement continuity and stress continuity are both satisfied with the applied periodic boundary. Because of the different expand property, there are some level of stress distribution in the fibers and the matrix. The maximum stress occurred at the interface of matrix and fibers because in the high temperature after insulation, the material is in no-stress mode, and when it turns into low temperature, the fibers will suffer compress loads and the matrix will suffer tensile loads. That is in accordance with the result ofAly-Hassan et al.[14]. At the joint action of the two forces, stress concentration is formed in the interface. In addition, maximum stress occurred in the dominoes, which has the nearest distance of fiber bundles, and this embodied the interaction of each fiber bundles. Corresponding to the three kinds of temperature loads
Fig.5 Residual thermal stress-time curves with different temperature
(2 000 °C, 2 500 °C，2 750 °C illustrated in Fig.2), the maximum residual stress are 8.95 MPa, 9.86 MPa, and 10.89 MPa, respectively, and the stress-time curve of the maximum stress point is illustrated in Fig.5. In the same cooling rate, the maximum stress increases with the graphitization temperature, and the maximum stress generated by the graphitization temperature of 2 500 °C is 10.1% higher than that of 2 000 °C. The two different slopes either embodied the change of cooling rate or the nonlinear change of heat generation. In the case of guaranteeing the graphitization temperature, controlling the cooling grad is useful to protect interface from debonding. According to the two kinds of cooling rates (grads 1 and grads 2), the stress-time curves of the maximum stress points, after graphite, is illustrated in Fig.6. The maximum Mises stress are 9.86 MPa and 10.72 MPa, respectively, and the latter is 8.7% higher than the former. It is verified that the maximum stress increases with the increase in cooling grad. Corresponding to the two values of interface stiffness (10 GPa, 15 GPa), the stress-time curve of the maximum stress points at the temperature loads of 2 500 °C is illustrated in Fig.7, and it can be easily found that the maximum stress increases with the interface stiffness. The maximum stress is 9.86 MPa and 14.62 MPa, respectively. The maximum stress is 48.3% higher when the interface stiffness increased by 50%.
SHI Hong-bin et al. / New Carbon Materials, 2011, 26(4): 287–292
corresponding to the four temperature loads is shown in Fig.8. It can be seen that the separation increased with the graphitization temperature and cooling grads. This is in accordance with the result of stress analysis because the separation and the thermal stress have the same changing tendency.
4
Conclusions
In graphitizing process, the following effects are observed:
Fig.6 Residual thermal stress-time curves with different cooling grads
(1) The change of residual thermal stress is proportional to graphitization temperature, cooling grad, and interface stiffness. (2) Graphitization temperature is the main factor that affects the residual thermal stress among the factors investigated. (3) Interface with weak stiffness can either enhance the strength or decrease the residual thermal stress for the carbon/carbon composite. (4) The maximum stress occurred in the domain, which has the great interaction of fiber bundles. The distribution of residual thermal stress can be adjusted by changing the distance of fiber bundles. Generally, the thermal-stress information of carbon/carbon composite in graphitization process can give a good guidance for improving the mechanical properties during manufacture.
Fig.7 Residual thermal stress-time curves with different interface
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