- Email: [email protected]
epoxy interface strength
Composite Structures 92 (2010) 150–154
Contents lists available at ScienceDirect
Composite Structures journal homepage: www.elsevier.com/locate/compstruct
Time dependency of carbon/epoxy interface strength Jun Koyanagi a,*, Satoru Yoneyama b, Katsuya Eri b, Pranav D. Shah c a
Institute of Space and Astronautical Science, Japan Aerospace Exploration Agency (ISAS/JAXA), 3-1-1 Yoshinodai, Sagamihara, Kanagawa 229-8510, Japan Department of Mechanical Engineering, Aoyama Gakuin University, 5-10-1 Fuchinobe, Sagamihara, Kanagawa 229-8558, Japan c Department of Aeronautics and Astronautics, Stanford University, Durand Building, 496 Lomita Mall Stanford, CA 94305, USA b
a r t i c l e
i n f o
Article history: Available online 16 July 2009 Keywords: Interface Time dependency Strength Unidirectional composite Transverse tensile test
a b s t r a c t This paper describes time-dependent fiber/matrix interfacial strength of carbon fiber reinforced polymeric composite. The time-dependent interfacial strength is extracted from the results of transverse tensile tests with various loading rates and their fractography in the unidirectional composite. The results show that the material failure is dominated by interface failure under relatively high-loading rate whereas matrix failure is dominant under relatively low-loading rate. In light of the results, it is concluded that the time dependency of the interfacial strength might be neglected or at least could be less significant than that of matrix strength. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction To guarantee a long-term reliability for composite materials, the time dependency of their constituents, fiber, matrix and interface, should be understood [1,2]. However, the number of studies conducted to understand these behaviors, especially in interface, has been extremely limited. This scarcity in the number of works related to interfacial time dependency is due to the fact that it is hard to evaluate the interfacial properties even in static condition. Based on a number of works done in the field of interfacial properties [3–14], the use of a single-fiber composite has been common in methods such as the fiber fragmentation test, micro droplet test, push-out test, pull-out test, Broutman test and transverse tensile test. The first four tests, which have been widely implemented [3–9], are primarily used for the evaluation of interfacial shear properties. The remaining two tests are applicable to the evaluation of the tensile properties of the interface [10–14]. These literatures do not deal with a time dependency of interface. Regarding the time dependency, although a viscoelastic behavior near the interface (e.g., interphase) has been studied [15–19], the time dependency of the interfacial strength itself has been rarely studied. There exist some works on the time dependency of interface strength [2,20–22] but the values presented are quite dependent upon the assumptions made in the derivation. Thus, a proper representation of the time dependency of interface strength is still under discussion.
* Corresponding author. Tel.: +81 427598695; fax: +81 427598461. E-mail addresses: [email protected] (J. Koyanagi), [email protected]. ac.jp (S. Yoneyama), [email protected] (P.D. Shah). 0263-8223/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2009.07.016
In this study, the time dependency of the interfacial strength was investigated based on a simplified scheme presented by Ha et al. [1]. In the scheme, they showed that the transverse strength of a unidirectional composite is determined by weaker strength of matrix or interface. Following, this observation, here, transverse tensile test was performed for a unidirectional carbon fiber reinforced polymeric composite with various loading speeds. The fractured surface of the specimen was observed using a scanning electron microscope to distinguish between the interfacial failure and the matrix failure. It is assumed that the failure mode, whether interface dominant or matrix dominant, will vary with the loading rate, provided that the time dependencies of the matrix and interface are different. In other words, there will be a transition point at which the two strengths order will change with the loading rate. Otherwise, i.e., if the two time dependencies are identical, the failure mode should remain same for all loading rates, e.g., the weaker component always dominates the transverse composite failure. Based on Ha et al.’s work [1], the weaker strength can be obtained by multiplying the stress-concentration factor at the tip of interface (critical point) by the specimen’s macro stress. It should be noted that the stress-concentration factor may vary with time. Hence, in this work, viscoelastic finite element analysis was performed to determine the stress-concentration factor as a function of time. Similar to their work, a unit-cell model was used to perform the finite element analysis. The viscoelastic parameter of matrix was determined by fitting stress–strain curves to experimental results. By employing this viscoelastic parameter, the time-dependent stress-concentration factor was calculated. By comparing the results from the experiments, fractography and finite element analysis, the time dependency of the interfacial strength was extracted.
J. Koyanagi et al. / Composite Structures 92 (2010) 150–154
2. Experiments 2.1. Experimental procedure As shown in Fig. 1, a rectangular strip specimen with 250 mm in length, including 50 mm in tabs, 20 mm in width and 2 mm in thickness was prepared. Reinforcement was T300 carbon fiber, and matrix was 180 °C curing type epoxy resin (Hyej17HX1, Mitsubishi). Fiber volume fraction of the specimen was 55%. Various loading rates, between approximately 0.002 and 10 mm/min, were applied to the specimen thus varying the strain rates. The tests were conducted at a constant temperature of 24 °C. Strain gages were used to measure the strains on the specimen surface. Strain gages used were manufactured by Kyowa Dengyo and the model of the testing machine used was Shimazu AUTO GRAPH. The fractured surfaces were observed by SEM, which was followed by platinum-surface treatment on them. 2.2. Experimental results and fractography Fig. 2 shows the experimental results obtained under various strain rates and failure modes based on the fractography results as shown in Fig. 3. It is typically observed that the matrix failure
151
dominant mode is observed under relatively low-loading rate conditions (Fig. 3a), whereas relatively high-loading rate conditions are dominated by the interface failure (Fig. 3c). In intermediate strain speed (Fig. 3b), combined failure of interface and matrix is observed. This implies that the matrix becomes weaker than interface in lower strain rate range. Because the interfacial and matrix stresses in loading direction at the location where the stress-concentration factor is maximum (critical point as shown in Fig. 4) remain equal to each other independently of any viscoelastic behavior. If we assume both of the interface and matrix have typical time dependencies, i.e., the strength decreases with a decrease in tensile testing speed, it can be suggested here that the time dependency of matrix is more pronounced than that of the interface. However, the stress-concentration factor is still unknown and thus, it is still impossible to estimate the time dependency of the interfacial strength quantitatively. Hence, a viscoelastic finite element analysis was performed to determine the time-dependent stress-concentration factor as explained in the following section.
3. Finite element analysis In order to extract the interfacial strength from the experimental results, determining time-dependent stress-concentration factor is indispensable. Based on Ha et al.’s [1] work, a viscoelastic analysis was performed employing a square unit-cell model as shown in Fig. 4. In the model the left and bottom edges were constrained symmetrically and the right edge was constrained to remain straight and vertical. A tensile displacement was applied to the upper edge. The analysis was performed using commercial software ABAQUS 6.7.1 and solid elements were used to model the problem. The range of the applied strain rate was from 2.0 107 to 4.0 104 s1. In this analysis, the fiber diameter considered was 6 lm and the length of each side of the square was determined with a fiber volume fraction 55%. The fiber is assumed to be elastic and isotropic in this two-dimensional plain; the elastic modulus which corresponds to fiber radial modulus of 19 GPa and Poisson’s ratio of 0.4 were used. For viscoelastic matrix, the instantaneous elastic modulus 4500 MPa and we specified the following viscoelastic relaxation modulus based on Power law compliance model was specified, which was used in Ref. [2].
EðtÞ ¼
Fig. 1. Specimen geometry and dimensions.
45 40
Failure stress MPa
35 30 25
Combination of Matrix and interface failure
20 15 10
Matrix failure dominant
Interfacial failure dominant
5 0 0.01
0.1
1 10 Strain rate 10-6/sec
100
Fig. 2. Results of transverse tensile failure stress for various strain rates and discussion.
1 J 0 f1 þ ðt=T 0 Þn g
ð1Þ
Here, J0 is the instantaneous (initial) compliance, T0 is the relaxation time and n is Power law component. In the present study, J0 = (1/4.5=) 0.222 GPa1, T0 = 500,000 s and n = 0.3 are employed. These viscoelastic parameters, T0 = 500,000 s and n = 0.3, were determined by fitting the analytical stress–strain curves to those of the experimental results as shown in Fig. 5. Fig. 6 shows the stress-concentration factor as a function of time, which was obtained by the interface (matrix) stress at the critical point divided by average unit-cell stress. This result is for 4.0 107 strain rate test. This analysis performed is corresponding to the linear viscoelastic behavior. Hence, the stress-concentration factor as a function of time do not depend on the strain rate; that of every strain rate test is represented by this identical curve. Although there is a difference between stress–strain curves corresponding to various strain rates, the time-dependent stress-concentration factor is almost constant in the range of this analysis. In the present study, it is assumed that the stress-concentration factor is always 1.4 and that the time-dependent variation of the stress-concentration factor is neglected. In other words, the product values of the specimen stresses in Fig. 2 and 1.4 approximately correspond to the weaker strength of either the matrix or interface.
152
J. Koyanagi et al. / Composite Structures 92 (2010) 150–154
Fig. 3. Failure surfaces for various tensile strain rates.
50 2.0E-07/sec Experimental 4.0E-07/sec 4.0E-06/sec 4.0E-05/sec 4.0E-04/sec 2.0E-07/sec Analytical 4.0E-07/sec 4.0E-06/sec 4.0E-05/sec 4.0E-04/sec
45
Critical point
40
Matrix Fiber
Stress [MPa]
35 30 25 20 15 10 5 0 0
Fig. 4. Geometry and boundary condition of numerical model.
0.001
0.002
0.003 strain
0.004
0.005
Fig. 5. Stress–strain curves of experiments and numerical results.
0.006
J. Koyanagi et al. / Composite Structures 92 (2010) 150–154
2 Stress concentration factor
1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0
0
5000
10000
15000
20000
25000
Time-to -failure sec Fig. 6. Stress-concentration factor at interface with time.
4. Discussion The findings from this study are summarized in Fig. 7. In this figure a bi-linear curve represents the strength, either that of matrix or interface, as a function of the strain rate based on following discussion. First of all, since the stress-concentration factor at the critical point is assumed to have no-time dependency as mentioned in the previous section, specimen stress multiplied by 1.4 corresponds to the approximated weaker strength of either interface or matrix. There is a transition point which separates the interfacial failure dominant and matrix failure dominant region around the strain rate value of 4.0 107 s1 as discussed above with the fractography results. In Fig. 7, the right hand side of the transition indicates the interfacial failure dominant region and the left hand side is matrix failure dominant region. In the matrix failure dominant region, since a polymer material generally shows the tendency that the strength decreases with a decrease in tensile strain rate, and this experimental results shows qualitatively similar tendency, the left side curve is consistent with this phenomenon. For the region at the right hand side of the transition point, a horizontal curve is drawn. From the experimental results, it can not be observed that the strength increase with an increase in the strain rate. Instead, the experimental results look slightly decreasing with strain rate, which is a behavior completely opposite of that of general viscoelastic materials. However, since the
153
scattering is relatively large, we assume the experimental results can be approximately represented by the horizontal curve in the present study. Thus, based on the range of the strain rates considered, this result indicates that the interface has no considerable time dependency. The test employed in the present study is expected to be inherently scattering; it is hard to conclude more sharply by this test. Even if there is a time dependency for the interfacial strength, since the failure mode of the specimen changes with a decrease in the strain rate, the time dependency of the interface is verified to be less remarkable than that of the matrix. In other words, the interface degrades with time less remarkably than the matrix. This indication leads that when discussing the long-term durability of the transverse strength for unidirectional composite, which almost corresponds to ‘‘Firstply-failure” for composite materials, we can typically focus on only time-dependent strength of matrix. Also, it can be suggested for any other unidirectional CFRP that a relatively rapid test can potentially lead to interface failure dominant failure and a relatively slower test can cause matrix failure dominant failure. Taniguchi et al. presented dynamic strength of a unidirectional CFRP [23]. In their work, it has been reported that there is no remarkable strengths difference between static and dynamic tests for 90° specimen (transverse strength), and interfacial failure dominant mode was observed in both cases. This indicates that the time dependency of the interface strength is not remarkable also under a dynamic condition. 5. Conclusions In the present study, a time dependency of interface strength in carbon reinforced polymeric composite is extracted from timedependent transverse strength of the unidirectional composite. We carried out the tensile tests with various tensile test speeds and observation of the failure surfaces, and then the time-dependent stress-concentration factor at a critical point is determined by a viscoelastic numerical analysis employing a unit-cell model. The viscoelastic parameters are determined by fitting the analytical stress–strain curves to the experimental those. Based on the experimental, fractographic and analytical results, following finding is derived: the time dependency of the interfacial strength might be neglected or could be less remarkable than that of the matrix strength. Acknowledgements This work is partially supported by The Kurata Memorial Hitachi Science and Technology Foundation and Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Young Scientist (B). References
Fig. 7. Approximated strengths of matrix and interface as a function of strain rate with transition of specimen-failure mode.
[1] Ha SK, Jin KK, Huang Y. Micro-mechanics of failure (MMF) for continuous fiber reinforced composites. J Compos Mater 2008;42:1873–95. [2] Koyanagi J, Ogawa F, Kawada H, Hatta H. Time-dependent reduction of tensile strength caused by interfacial degradation under constant strain duration in UD-CFRP. J Compos Mater 2007;41:3007–26. [3] Chua PS, Piggott MR. The glass fibre–polymer interface: II. Work of fracture and shear stresses. Compos Sci Technol 1985;22:107–19. [4] Keran RJ, Parthasarathy TA. Theoretical analysis of the fiber pull-out and pushout tests. J Am Ceram Soc 1991;74:1585–96. [5] Zhandarov SF, Pisanova EV. The local bond strength and its determination by fragmentation and pull-out tests. Compos Sci Technol 1997;57:957–64. [6] Zhou XF, Nairn JA, Wagner HD. Fiber–matrix adhesion from the single-fiber composite test: nucleation of interfacial debonding. Composites: Part A 1999;30:1387–400. [7] Kessler H, Schuller T, Beckert W, Lauke B. A fracture-mechanics model of the microbond test with interface friction. Compos Sci Technol 1999;59:2231–42. [8] Hsueh CH. Strength versus energy debonding criteria at fiber/matrix interfaces. Compos Eng 1992;2(8):665–75.
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[9] Kimura S, Koyanagi J, Kawada H. Evaluation of initiation of the interfacial debonding in single fiber composites (energy balance method considering an energy dissipation of the plastic deformation). JSME Int J Ser A 2006;49(3): 451–7. [10] Ageorges C, Friedrich K, Schüller T, Lauke B. Single-fibre Broutman test (fiber– matrix interface transverse debonding). Composites: Part A 1999;30:1423–34. [11] Schuller T, Beckert W, Lauke B, Ageorges C, Friedrich K. Single fibre transverse debonding (stress analysis of the Broutman test). Composites: Part A 2000;31:661–70. [12] Tandon GP, Kim RY, Bechel VT. Evaluation of interfacial normal strength in a SCS-0/epoxy composite with cruciform specimens. Compos Sci Technol 2000;60:2281–95. [13] Ogihara S, Sakamoto Y, Koyanagi J. Evaluation of tensile strength in glass fiber/ epoxy resin interface using the cruciform specimen method. Trans Jpn Soc Mech Eng, Ser A 2009;75:49–55 [in Japanese]. [14] Koyanagi J, Shah PD, Kimura S, Ha SK, Kawada H. Mixed-mode interfacial debonding simulation in single fiber composite under transverse load. J Solid Mech Mater Eng 2009;3:796–806. [15] Fisher FT, Brinson LC. Viscoelastic interphases in polymer–matrix composites: theoretical models and finite-element analysis. Compos Sci Technol 2001;61: 731–48.
[16] Fink BK, McCullough RL. Interphase research issues. Composites: Part A 1999; 30:1–2. [17] Dzenis YA. Effective thermo-viscoelastic properties of fibrous composite with fractal interfaces and an interphase. Compos Sci Technol 1997;57:1057–63. [18] Gosz M, Moran B, Achenbach JD. Effect of a viscoelastic interface on the transverse behavior of fiber-reinforced composites. Int J Solids Struct 1991;27: 1757–71. [19] Hashin Z. Composite materials with viscoelastic interphase: creep and relaxation. Mech Mater 1991;11:135–48. [20] Beyerlein IJ, Zhou CH, Schadler LS. A time dependent micro-mechanical fiber composite model for inelastic zone growth in viscoelastic matrices. Int J Solids Struct 2003;40:1–24. [21] Zhou CH, Schadler LS, Beyerlein IJ. Time-dependent micromechanical behavior in graphite/epoxy composites under constant load: a combined experimental and theoretical study. Acta Mater 2002;50:365–77. [22] Zhou CH, Beyerlein IJ, Schadler LS. Time-dependent micromechanical behavior in graphite/epoxy composites under constant load at elevated temperatures. J Mater Sci 2003;38:877–84. [23] Taniguchi N, Nishiwaki T, Kawada H. Experimental characterization of dynamic tensile strength in unidirectional carbon/epoxy composite. Adv Compos Mater 2008;17:139–56.
Contents lists available at ScienceDirect
Composite Structures journal homepage: www.elsevier.com/locate/compstruct
Time dependency of carbon/epoxy interface strength Jun Koyanagi a,*, Satoru Yoneyama b, Katsuya Eri b, Pranav D. Shah c a
Institute of Space and Astronautical Science, Japan Aerospace Exploration Agency (ISAS/JAXA), 3-1-1 Yoshinodai, Sagamihara, Kanagawa 229-8510, Japan Department of Mechanical Engineering, Aoyama Gakuin University, 5-10-1 Fuchinobe, Sagamihara, Kanagawa 229-8558, Japan c Department of Aeronautics and Astronautics, Stanford University, Durand Building, 496 Lomita Mall Stanford, CA 94305, USA b
a r t i c l e
i n f o
Article history: Available online 16 July 2009 Keywords: Interface Time dependency Strength Unidirectional composite Transverse tensile test
a b s t r a c t This paper describes time-dependent fiber/matrix interfacial strength of carbon fiber reinforced polymeric composite. The time-dependent interfacial strength is extracted from the results of transverse tensile tests with various loading rates and their fractography in the unidirectional composite. The results show that the material failure is dominated by interface failure under relatively high-loading rate whereas matrix failure is dominant under relatively low-loading rate. In light of the results, it is concluded that the time dependency of the interfacial strength might be neglected or at least could be less significant than that of matrix strength. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction To guarantee a long-term reliability for composite materials, the time dependency of their constituents, fiber, matrix and interface, should be understood [1,2]. However, the number of studies conducted to understand these behaviors, especially in interface, has been extremely limited. This scarcity in the number of works related to interfacial time dependency is due to the fact that it is hard to evaluate the interfacial properties even in static condition. Based on a number of works done in the field of interfacial properties [3–14], the use of a single-fiber composite has been common in methods such as the fiber fragmentation test, micro droplet test, push-out test, pull-out test, Broutman test and transverse tensile test. The first four tests, which have been widely implemented [3–9], are primarily used for the evaluation of interfacial shear properties. The remaining two tests are applicable to the evaluation of the tensile properties of the interface [10–14]. These literatures do not deal with a time dependency of interface. Regarding the time dependency, although a viscoelastic behavior near the interface (e.g., interphase) has been studied [15–19], the time dependency of the interfacial strength itself has been rarely studied. There exist some works on the time dependency of interface strength [2,20–22] but the values presented are quite dependent upon the assumptions made in the derivation. Thus, a proper representation of the time dependency of interface strength is still under discussion.
* Corresponding author. Tel.: +81 427598695; fax: +81 427598461. E-mail addresses: [email protected] (J. Koyanagi), [email protected]. ac.jp (S. Yoneyama), [email protected] (P.D. Shah). 0263-8223/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2009.07.016
In this study, the time dependency of the interfacial strength was investigated based on a simplified scheme presented by Ha et al. [1]. In the scheme, they showed that the transverse strength of a unidirectional composite is determined by weaker strength of matrix or interface. Following, this observation, here, transverse tensile test was performed for a unidirectional carbon fiber reinforced polymeric composite with various loading speeds. The fractured surface of the specimen was observed using a scanning electron microscope to distinguish between the interfacial failure and the matrix failure. It is assumed that the failure mode, whether interface dominant or matrix dominant, will vary with the loading rate, provided that the time dependencies of the matrix and interface are different. In other words, there will be a transition point at which the two strengths order will change with the loading rate. Otherwise, i.e., if the two time dependencies are identical, the failure mode should remain same for all loading rates, e.g., the weaker component always dominates the transverse composite failure. Based on Ha et al.’s work [1], the weaker strength can be obtained by multiplying the stress-concentration factor at the tip of interface (critical point) by the specimen’s macro stress. It should be noted that the stress-concentration factor may vary with time. Hence, in this work, viscoelastic finite element analysis was performed to determine the stress-concentration factor as a function of time. Similar to their work, a unit-cell model was used to perform the finite element analysis. The viscoelastic parameter of matrix was determined by fitting stress–strain curves to experimental results. By employing this viscoelastic parameter, the time-dependent stress-concentration factor was calculated. By comparing the results from the experiments, fractography and finite element analysis, the time dependency of the interfacial strength was extracted.
J. Koyanagi et al. / Composite Structures 92 (2010) 150–154
2. Experiments 2.1. Experimental procedure As shown in Fig. 1, a rectangular strip specimen with 250 mm in length, including 50 mm in tabs, 20 mm in width and 2 mm in thickness was prepared. Reinforcement was T300 carbon fiber, and matrix was 180 °C curing type epoxy resin (Hyej17HX1, Mitsubishi). Fiber volume fraction of the specimen was 55%. Various loading rates, between approximately 0.002 and 10 mm/min, were applied to the specimen thus varying the strain rates. The tests were conducted at a constant temperature of 24 °C. Strain gages were used to measure the strains on the specimen surface. Strain gages used were manufactured by Kyowa Dengyo and the model of the testing machine used was Shimazu AUTO GRAPH. The fractured surfaces were observed by SEM, which was followed by platinum-surface treatment on them. 2.2. Experimental results and fractography Fig. 2 shows the experimental results obtained under various strain rates and failure modes based on the fractography results as shown in Fig. 3. It is typically observed that the matrix failure
151
dominant mode is observed under relatively low-loading rate conditions (Fig. 3a), whereas relatively high-loading rate conditions are dominated by the interface failure (Fig. 3c). In intermediate strain speed (Fig. 3b), combined failure of interface and matrix is observed. This implies that the matrix becomes weaker than interface in lower strain rate range. Because the interfacial and matrix stresses in loading direction at the location where the stress-concentration factor is maximum (critical point as shown in Fig. 4) remain equal to each other independently of any viscoelastic behavior. If we assume both of the interface and matrix have typical time dependencies, i.e., the strength decreases with a decrease in tensile testing speed, it can be suggested here that the time dependency of matrix is more pronounced than that of the interface. However, the stress-concentration factor is still unknown and thus, it is still impossible to estimate the time dependency of the interfacial strength quantitatively. Hence, a viscoelastic finite element analysis was performed to determine the time-dependent stress-concentration factor as explained in the following section.
3. Finite element analysis In order to extract the interfacial strength from the experimental results, determining time-dependent stress-concentration factor is indispensable. Based on Ha et al.’s [1] work, a viscoelastic analysis was performed employing a square unit-cell model as shown in Fig. 4. In the model the left and bottom edges were constrained symmetrically and the right edge was constrained to remain straight and vertical. A tensile displacement was applied to the upper edge. The analysis was performed using commercial software ABAQUS 6.7.1 and solid elements were used to model the problem. The range of the applied strain rate was from 2.0 107 to 4.0 104 s1. In this analysis, the fiber diameter considered was 6 lm and the length of each side of the square was determined with a fiber volume fraction 55%. The fiber is assumed to be elastic and isotropic in this two-dimensional plain; the elastic modulus which corresponds to fiber radial modulus of 19 GPa and Poisson’s ratio of 0.4 were used. For viscoelastic matrix, the instantaneous elastic modulus 4500 MPa and we specified the following viscoelastic relaxation modulus based on Power law compliance model was specified, which was used in Ref. [2].
EðtÞ ¼
Fig. 1. Specimen geometry and dimensions.
45 40
Failure stress MPa
35 30 25
Combination of Matrix and interface failure
20 15 10
Matrix failure dominant
Interfacial failure dominant
5 0 0.01
0.1
1 10 Strain rate 10-6/sec
100
Fig. 2. Results of transverse tensile failure stress for various strain rates and discussion.
1 J 0 f1 þ ðt=T 0 Þn g
ð1Þ
Here, J0 is the instantaneous (initial) compliance, T0 is the relaxation time and n is Power law component. In the present study, J0 = (1/4.5=) 0.222 GPa1, T0 = 500,000 s and n = 0.3 are employed. These viscoelastic parameters, T0 = 500,000 s and n = 0.3, were determined by fitting the analytical stress–strain curves to those of the experimental results as shown in Fig. 5. Fig. 6 shows the stress-concentration factor as a function of time, which was obtained by the interface (matrix) stress at the critical point divided by average unit-cell stress. This result is for 4.0 107 strain rate test. This analysis performed is corresponding to the linear viscoelastic behavior. Hence, the stress-concentration factor as a function of time do not depend on the strain rate; that of every strain rate test is represented by this identical curve. Although there is a difference between stress–strain curves corresponding to various strain rates, the time-dependent stress-concentration factor is almost constant in the range of this analysis. In the present study, it is assumed that the stress-concentration factor is always 1.4 and that the time-dependent variation of the stress-concentration factor is neglected. In other words, the product values of the specimen stresses in Fig. 2 and 1.4 approximately correspond to the weaker strength of either the matrix or interface.
152
J. Koyanagi et al. / Composite Structures 92 (2010) 150–154
Fig. 3. Failure surfaces for various tensile strain rates.
50 2.0E-07/sec Experimental 4.0E-07/sec 4.0E-06/sec 4.0E-05/sec 4.0E-04/sec 2.0E-07/sec Analytical 4.0E-07/sec 4.0E-06/sec 4.0E-05/sec 4.0E-04/sec
45
Critical point
40
Matrix Fiber
Stress [MPa]
35 30 25 20 15 10 5 0 0
Fig. 4. Geometry and boundary condition of numerical model.
0.001
0.002
0.003 strain
0.004
0.005
Fig. 5. Stress–strain curves of experiments and numerical results.
0.006
J. Koyanagi et al. / Composite Structures 92 (2010) 150–154
2 Stress concentration factor
1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0
0
5000
10000
15000
20000
25000
Time-to -failure sec Fig. 6. Stress-concentration factor at interface with time.
4. Discussion The findings from this study are summarized in Fig. 7. In this figure a bi-linear curve represents the strength, either that of matrix or interface, as a function of the strain rate based on following discussion. First of all, since the stress-concentration factor at the critical point is assumed to have no-time dependency as mentioned in the previous section, specimen stress multiplied by 1.4 corresponds to the approximated weaker strength of either interface or matrix. There is a transition point which separates the interfacial failure dominant and matrix failure dominant region around the strain rate value of 4.0 107 s1 as discussed above with the fractography results. In Fig. 7, the right hand side of the transition indicates the interfacial failure dominant region and the left hand side is matrix failure dominant region. In the matrix failure dominant region, since a polymer material generally shows the tendency that the strength decreases with a decrease in tensile strain rate, and this experimental results shows qualitatively similar tendency, the left side curve is consistent with this phenomenon. For the region at the right hand side of the transition point, a horizontal curve is drawn. From the experimental results, it can not be observed that the strength increase with an increase in the strain rate. Instead, the experimental results look slightly decreasing with strain rate, which is a behavior completely opposite of that of general viscoelastic materials. However, since the
153
scattering is relatively large, we assume the experimental results can be approximately represented by the horizontal curve in the present study. Thus, based on the range of the strain rates considered, this result indicates that the interface has no considerable time dependency. The test employed in the present study is expected to be inherently scattering; it is hard to conclude more sharply by this test. Even if there is a time dependency for the interfacial strength, since the failure mode of the specimen changes with a decrease in the strain rate, the time dependency of the interface is verified to be less remarkable than that of the matrix. In other words, the interface degrades with time less remarkably than the matrix. This indication leads that when discussing the long-term durability of the transverse strength for unidirectional composite, which almost corresponds to ‘‘Firstply-failure” for composite materials, we can typically focus on only time-dependent strength of matrix. Also, it can be suggested for any other unidirectional CFRP that a relatively rapid test can potentially lead to interface failure dominant failure and a relatively slower test can cause matrix failure dominant failure. Taniguchi et al. presented dynamic strength of a unidirectional CFRP [23]. In their work, it has been reported that there is no remarkable strengths difference between static and dynamic tests for 90° specimen (transverse strength), and interfacial failure dominant mode was observed in both cases. This indicates that the time dependency of the interface strength is not remarkable also under a dynamic condition. 5. Conclusions In the present study, a time dependency of interface strength in carbon reinforced polymeric composite is extracted from timedependent transverse strength of the unidirectional composite. We carried out the tensile tests with various tensile test speeds and observation of the failure surfaces, and then the time-dependent stress-concentration factor at a critical point is determined by a viscoelastic numerical analysis employing a unit-cell model. The viscoelastic parameters are determined by fitting the analytical stress–strain curves to the experimental those. Based on the experimental, fractographic and analytical results, following finding is derived: the time dependency of the interfacial strength might be neglected or could be less remarkable than that of the matrix strength. Acknowledgements This work is partially supported by The Kurata Memorial Hitachi Science and Technology Foundation and Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Young Scientist (B). References
Fig. 7. Approximated strengths of matrix and interface as a function of strain rate with transition of specimen-failure mode.
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