- Email: [email protected]
Estimation of the local interfacial strength parameters of carbon nanotube fibers in an epoxy matrix from a microbond test data
CARBON
8 6 ( 2 0 1 5 ) 5 4 –5 7
Available at www.sciencedirect.com
ScienceDirect journal homepage: www.elsevier.com/locate/carbon
Letter to the Editor
Estimation of the local interfacial strength parameters of carbon nanotube fibers in an epoxy matrix from a microbond test data Serge Zhandarov a b c
a,b
, Edith Ma¨der
a,c,*
Leibniz-Institut fu¨r Polymerforschung Dresden e.V., Hohe Strasse 6, Dresden 01069, Germany ‘‘V.A. Bely’’ Metal-Polymer Research Institute, National Academy of Sciences of Belarus, Kirov Street 32a, Gomel 246050, Belarus Institute of Materials Science, Technische Universita¨t Dresden, Helmholtz Strasse 10, Dresden 01062, Germany
A R T I C L E I N F O
A B S T R A C T
Article history:
Local interfacial strength parameters (local interfacial shear strength (IFSS), critical energy
Received 26 November 2014
release rate) and interfacial frictional stress between continuous carbon nanotube fibers
Accepted 13 January 2015
and epoxy matrix have been estimated using published experimental data from a micro-
Available online 20 January 2015
bond test. The ‘indirect’ method (from the maximum recorded force as a function of the embedded length) and the calculation from an individual force–displacement curve yielded very similar results. The estimated local IFSS value (about 50 MPa) is much greater than the effective IFSS reported for this fiber–matrix pair (14.4 MPa). Ó 2015 Elsevier Ltd. All rights reserved.
Zu et al. [1] determined the effective interfacial shear strength of continuous carbon nanotube (CNT) fibers in DER 353 epoxy resin by means of a microdroplet (microbond) test. They recorded 10 force–displacement curves for specimens in which interfacial debonding occurred and plotted the maximum fiber axial force, Fmax, recorded during the test, as a function of the embedment area. (In Fig. 1, we replotted their data as Fmax versus the embedded length, le). Then these data were fitted to the linear function passing through the origin (curve 1), and the average effective interfacial shear strength (IFSS) was obtained from the slope of the fitting line. The authors found the effective (apparent) IFSS value (sapp = 14. 4 MPa) to be comparable to those of glass fiber/epoxy and carbon fiber/epoxy composites. However, as is known, interfacial failure in a microbond specimen occurs not simultaneously over the whole embedded
length, but gradually, through interfacial crack propagation, which is governed by a local interfacial strength parameter. There are two groups of models which describe the laws of crack initiation and propagation. In the stress-based approach, it is assumed that interfacial debonding starts when the shear stress at some point at the interface reaches its critical value, sd, which is called the local interfacial shear strength, and during the crack propagation, the shear stress near the crack tip is close to sd. The energy-based approach based on fracture mechanics considers the critical energy release rate, or interfacial toughness, Gic, as a failure criterion which must be nearly constant during crack initiation and propagation. In contrast to the apparent IFSS, which depends on the specimen geometry and, first of all, on the embedded length, the sd and Gic parameters can be considered as the composite properties constant for a given fiber–matrix pair. Besides, interfacial friction plays
* Corresponding author at: Leibniz-Institut fu¨r Polymerforschung Dresden e.V., Hohe Strasse 6, Dresden 01069, Germany. E-mail address: [email protected] (E. Ma¨der). http://dx.doi.org/10.1016/j.carbon.2015.01.025 0008-6223/Ó 2015 Elsevier Ltd. All rights reserved.
CARBON
55
8 6 (2 0 15 ) 5 4–57
0.08
Maximum force, N
Maximum force, N
0.08
0.06
0.04
2 0.02
4 1
0.00 0.00
0.06
2
3
0.04
0.02
1
3 0.05 0.10 0.15 Embedded length, mm
0.20
Fig. 1 – Maximum force versus embedded length in the microbond test on CNT fibers in DER 353 epoxy matrix. Filled circles denote experimental points from [1]. Curve 1 is the best ‘‘proportional’’ fit [1] (sapp = 14.4 MPa); curve 2 is the best ‘‘non-proportional’’ linear fit estimating interfacial frictional stress; curve 3 is the best-fit stress-based theoretical curve (sd ¼ 54:8 MPa, sf ¼ 0); and curve 4 is the best-fit energy-based theoretical curve (Gic ¼ 9:10 J/m2, sf ¼ 4:39 MPa). (A colour version of this figure can be viewed online.)
an important part in fiber/matrix debonding; the recorded applied force is the sum of the adhesion force developing in the intact part of the interface and the frictional force in already debonded regions. As a result, the Fmax and sapp values are determined not by solely adhesion but by an intricate combination of adhesional and frictional contributions. It seems to be very interesting to separate these contributions and estimate both sd (or Gic) and the frictional stress in the debonded zones. Fortunately, almost all information required for this estimation is available in [1]. In our analysis, we will use our own models described in detail elsewhere [2–4], in which the maximum force, Fmax, for a microbond test is derived as a function of the embedded length, interfacial adhesion and friction, and some other parameters. In the stress-based approach this can be expressed as Fmax ¼ Fmax le ; sd ; sf ; sT ; b; Vf ; ::: ; ð1Þ where sf is the frictional interfacial stress in debonded areas, which is assumed to be constant [5]; sT is a term having dimensions of stress, which appears due to residual thermal stresses; b is the shear-lag parameter as defined by Nayfeh [6]; Vf is the fiber volume fraction within the specimen, and the ellipsis designates mechanical properties of the fiber and the matrix (elastic moduli, Poisson’s ratios, etc.) We have shown that a typical plot of Fmax as a function of le has the form shown in Fig. 2. For small embedded lengths, the Fmax value is determined predominantly by interfacial adhesion; however, for large le’s, when the ultimate specimen failure occurs at the crack length close to the whole embedded length, the plot tends asymptotically to a straight line whose slope is proportional to sf. The experimental points usually fall somewhere between these extreme positions. The authors of [1] have chosen the fitting line such that Fmax / le ;
0.00 0.00
0.05 0.10 0.15 Embedded length, mm
0.20
Fig. 2 – Theoretical plot of Fmax as a function of the embedded length (1) and its asymptote as le ! 1 (2) determined by interfacial friction. (3) denotes the region where experimental points are assumed to fall. (A colour version of this figure can be viewed online.)
this would be correct if the whole interface failed simultaneously (e.g., through plastic deformation or pure friction). However, if we admit that the fitting line may not pass through the origin, we obtain curve 2 (see Fig. 1), whose slope corresponds to the interfacial stress of 7.68 MPa. In our approach, this value should be considered as the upper estimate for the interfacial frictional stress, sf. Indeed, this line is parallel to the tangent drawn to the theoretical curve in the region with real experimental points, and its slope is always greater than the slope of the asymptote. And the fact that line 2 intersects the vertical axis at the point where Fmax > 0 clearly indicates that interfacial interaction between the fiber and the matrix involves an essential adhesional contribution. This contribution is determined by the local adhesion strength parameter (sd or Gic) which can be estimated in several ways. First, we can fit experimental data by a theoretical curve (Eq. (1)) whose explicit form has been obtained by us in [4], using a non-linear least-square method with sd and sf as fitting parameters. The problem for a microbond test is that, though both sd and sf are assumed to be constant for all specimens, sT and b are functions of the fiber volume fraction, Vf, which, in turn, depends on the droplet shape. In order to calculate Vf, the droplet shape can be approximated by an ellipsoid. However, to do this, we must know at least the droplet diameter, D. The authors of [1] gave no information about the transverse size of the droplets; fortunately, we can calculate the diameter of the droplet on a fiber using a theory [7] which relates it to the droplet length (embedded fiber length), provided that the wetting angle at the matrix–fiber interface is known. We measured the wetting angle from the photograph in Fig. 1 in [1]; it was approximately # 51. Then, using this value, we calculated the diameters of all 10 droplets involved in the experiment. With good accuracy, these appeared to be linearly related to the embedded lengths: D ¼ 0:906 le 5:89 (lm), similarly to the results reported by Scheer and Nairn [8] for glass/epoxy and Kevlar/epoxy systems. Having calculated the fiber volume fractions, as proposed in [2,4], and taken the mechanical properties of
56
CARBON
8 6 ( 2 0 1 5 ) 5 4 –5 7
the fiber and the matrix from [1] and the Dow Chemical datasheet, we found the best fit to experimental data. The map of the sum of least squares is shown in Fig. 3a; the best-fit values are sd ¼ 54:8 MPa and sf 0. The latter value is obviously wrong; in [9,10] we showed that this ‘indirect’ approach may substantially underestimate interfacial friction. However, as was demonstrated in [9], the true (sd , sf ) point which characterizes the fiber/matrix system must fall somewhere in the narrow valley in the least squares map. Analyzing this map, we can estimate that the sd value for the CNT fiber/DER 353 epoxy is between approximately 40 and 55 MPa, while sf falls between 0 and 9 MPa. The upper estimate for sf is indirectly confirmed by the slope of curve 2 in Fig. 1. In any case, the variations in sf have little effect on the sd value, so that the estimation 40 MPa < sd < 55 MPa should be considered as rather reliable. Similar approach can be used with (Gic , sf ) as the pair of fitting parameters [3]. The corresponding least squares map is presented in Fig. 3b. The best-fit values Gic ¼ 9:10 J/m2 and sf ¼ 4:39 MPa look quite reasonable; however, taking into account that the map in this case is also in the shape of a narrow valley and the minimum is very shallow, we can only estimate 4 J=m2 < Gic < 17 J=m2 and 0 < sf < 9 MPa. Determining interfacial parameters from individual specimens usually yields better results than the ‘indirect’ approach [9,10]. In [1], only one force–displacement curve is available in Fig. 3b (Fig. 4 in this paper). The traditional way of estimating interfacial parameters from force–displacement curves is using the ‘kink’ force in the ascending part of the curve to determine the local IFSS, sd , and then calculating sf from sd and Fmax [9]. However, due to a relatively large free fiber length resulting in high system compliance, the ‘kink’ in the curve cannot be discerned; therefore, we used our new approach [10], in which sf is first determined from the postdebonding part of the curve (Fb value in Fig. 4), and then sf and Fmax are used to calculate sd . The droplet diameter, calculated from the le value (99.5 lm) given by the authors, was 84.2 lm. Using experimental values Fb = 16.6 mN and
Fig. 4 – Force–displacement curve for droplet debonding (as shown in [1]) and three important force values required for the calculation of interfacial parameters: Fd, the ‘kink’ force at which interfacial debonding starts (cannot be reliably determined from this curve); Fmax, the maximum recorded force, and Fb, the force immediately after debonding completion. (A colour version of this figure can be viewed online.)
Fmax = 45.8 mN, we calculated the interfacial parameters for this specimen: sd ¼ 53:7 MPa and sf ¼ 5:56 MPa. Similar calculation using the energy-based approach gives Gic ¼ 7:76 J/m2 (and the same sf ¼ 5:56 MPa, since it was calculated from the same Fb value). Both estimates are in agreement with the values determined above indirectly for the whole set of 10 specimens. Thus, local interfacial strength parameters in CNT fiber/ epoxy matrix system have been successfully estimated using the published experimental data from a microbond test. The estimated local IFSS value (about 50 MPa) is much greater than the effective IFSS reported for this fiber–matrix pair (14.4 MPa). Our approach also allowed us to estimate the interfacial frictional stress.
Fig. 3 – The maps of the sum of least squares as functions of sd and sf (left) and Gic and sf (right) (see text for details). (A colour version of this figure can be viewed online.)
CARBON
8 6 (2 0 15 ) 5 4–57
R E F E R E N C E S
[1] Zu M, Li Q, Zhu Y, Dey M, Wang G, Lu W, et al. The effective interfacial shear strength of carbon nanotube fibers in an epoxy matrix characterized by a microdroplet test. Carbon 2012;50(3):1271–9. http://dx.doi.org/10.1016/ j.carbon.2011.10.047. [2] Zhandarov SF, Ma¨der E, Yurkevich OR. Indirect estimation of fiber/polymer bond strength and interfacial friction from maximum load values recorded in the microbond and pullout tests. Part I: Local bond strength. J Adhes Sci Technol 2002;16:1171–200. http://dx.doi.org/10.1163/ 156856102320256837. [3] Zhandarov S, Ma¨der E. Indirect estimation of fiber/polymer bond strength and interfacial friction from maximum load values recorded in the microbond and pull-out tests. Part II: Critical energy release rate. J Adhes Sci Technol 2003;17(7):967–80. http://dx.doi.org/10.1163/ 156856103322112879. [4] Zhandarov S, Ma¨der E. Peak force as function of the embedded length in pull-out and microbond tests: effect of specimen geometry. J Adhes Sci Technol 2005;19(10):817–55. http://dx.doi.org/10.1163/1568561054929937.
57
[5] Nairn JA. Analytical fracture mechanics analysis of the pullout test including the effects of friction and thermal stresses. Adv Compos Lett 2000;9(6):373–83. [6] Nayfeh AH. Thermomechanically induced interfacial stresses in fibrous composites. Fibre Sci Technol 1977;10(3):195–209. http://dx.doi.org/10.1016/0015-0568(77)90020-3. [7] Wu XF, Dzenis YA. Droplet on a fiber: geometrical shape and contact angle. Acta Mech 2006;185:212–25. http://dx.doi.org/ 10.1007/s00707-006-0349-0. [8] Scheer RJ, Nairn JA. A comparison of several fracture mechanics methods for measuring interfacial toughness with microbond tests. J Adhes 1995;53:45–68. http:// dx.doi.org/10.1080/00218469508014371. [9] Zhandarov S, Ma¨der E. Characterization of fiber/matrix interface strength: applicability of different tests, approaches and parameters. Compos Sci Technol 2005;65:149–60. http:// dx.doi.org/10.1016/j.compscitech.2004.07.003. [10] Zhandarov S, Ma¨der E. An alternative method of determining the local interfacial shear strength from force–displacement curves in the pull-out and microbond tests. Int J Adhes Adhes 2014;55:37–42. http://dx.doi.org/10.1016/ j.ijadhadh.2014.07.006.
8 6 ( 2 0 1 5 ) 5 4 –5 7
Available at www.sciencedirect.com
ScienceDirect journal homepage: www.elsevier.com/locate/carbon
Letter to the Editor
Estimation of the local interfacial strength parameters of carbon nanotube fibers in an epoxy matrix from a microbond test data Serge Zhandarov a b c
a,b
, Edith Ma¨der
a,c,*
Leibniz-Institut fu¨r Polymerforschung Dresden e.V., Hohe Strasse 6, Dresden 01069, Germany ‘‘V.A. Bely’’ Metal-Polymer Research Institute, National Academy of Sciences of Belarus, Kirov Street 32a, Gomel 246050, Belarus Institute of Materials Science, Technische Universita¨t Dresden, Helmholtz Strasse 10, Dresden 01062, Germany
A R T I C L E I N F O
A B S T R A C T
Article history:
Local interfacial strength parameters (local interfacial shear strength (IFSS), critical energy
Received 26 November 2014
release rate) and interfacial frictional stress between continuous carbon nanotube fibers
Accepted 13 January 2015
and epoxy matrix have been estimated using published experimental data from a micro-
Available online 20 January 2015
bond test. The ‘indirect’ method (from the maximum recorded force as a function of the embedded length) and the calculation from an individual force–displacement curve yielded very similar results. The estimated local IFSS value (about 50 MPa) is much greater than the effective IFSS reported for this fiber–matrix pair (14.4 MPa). Ó 2015 Elsevier Ltd. All rights reserved.
Zu et al. [1] determined the effective interfacial shear strength of continuous carbon nanotube (CNT) fibers in DER 353 epoxy resin by means of a microdroplet (microbond) test. They recorded 10 force–displacement curves for specimens in which interfacial debonding occurred and plotted the maximum fiber axial force, Fmax, recorded during the test, as a function of the embedment area. (In Fig. 1, we replotted their data as Fmax versus the embedded length, le). Then these data were fitted to the linear function passing through the origin (curve 1), and the average effective interfacial shear strength (IFSS) was obtained from the slope of the fitting line. The authors found the effective (apparent) IFSS value (sapp = 14. 4 MPa) to be comparable to those of glass fiber/epoxy and carbon fiber/epoxy composites. However, as is known, interfacial failure in a microbond specimen occurs not simultaneously over the whole embedded
length, but gradually, through interfacial crack propagation, which is governed by a local interfacial strength parameter. There are two groups of models which describe the laws of crack initiation and propagation. In the stress-based approach, it is assumed that interfacial debonding starts when the shear stress at some point at the interface reaches its critical value, sd, which is called the local interfacial shear strength, and during the crack propagation, the shear stress near the crack tip is close to sd. The energy-based approach based on fracture mechanics considers the critical energy release rate, or interfacial toughness, Gic, as a failure criterion which must be nearly constant during crack initiation and propagation. In contrast to the apparent IFSS, which depends on the specimen geometry and, first of all, on the embedded length, the sd and Gic parameters can be considered as the composite properties constant for a given fiber–matrix pair. Besides, interfacial friction plays
* Corresponding author at: Leibniz-Institut fu¨r Polymerforschung Dresden e.V., Hohe Strasse 6, Dresden 01069, Germany. E-mail address: [email protected] (E. Ma¨der). http://dx.doi.org/10.1016/j.carbon.2015.01.025 0008-6223/Ó 2015 Elsevier Ltd. All rights reserved.
CARBON
55
8 6 (2 0 15 ) 5 4–57
0.08
Maximum force, N
Maximum force, N
0.08
0.06
0.04
2 0.02
4 1
0.00 0.00
0.06
2
3
0.04
0.02
1
3 0.05 0.10 0.15 Embedded length, mm
0.20
Fig. 1 – Maximum force versus embedded length in the microbond test on CNT fibers in DER 353 epoxy matrix. Filled circles denote experimental points from [1]. Curve 1 is the best ‘‘proportional’’ fit [1] (sapp = 14.4 MPa); curve 2 is the best ‘‘non-proportional’’ linear fit estimating interfacial frictional stress; curve 3 is the best-fit stress-based theoretical curve (sd ¼ 54:8 MPa, sf ¼ 0); and curve 4 is the best-fit energy-based theoretical curve (Gic ¼ 9:10 J/m2, sf ¼ 4:39 MPa). (A colour version of this figure can be viewed online.)
an important part in fiber/matrix debonding; the recorded applied force is the sum of the adhesion force developing in the intact part of the interface and the frictional force in already debonded regions. As a result, the Fmax and sapp values are determined not by solely adhesion but by an intricate combination of adhesional and frictional contributions. It seems to be very interesting to separate these contributions and estimate both sd (or Gic) and the frictional stress in the debonded zones. Fortunately, almost all information required for this estimation is available in [1]. In our analysis, we will use our own models described in detail elsewhere [2–4], in which the maximum force, Fmax, for a microbond test is derived as a function of the embedded length, interfacial adhesion and friction, and some other parameters. In the stress-based approach this can be expressed as Fmax ¼ Fmax le ; sd ; sf ; sT ; b; Vf ; ::: ; ð1Þ where sf is the frictional interfacial stress in debonded areas, which is assumed to be constant [5]; sT is a term having dimensions of stress, which appears due to residual thermal stresses; b is the shear-lag parameter as defined by Nayfeh [6]; Vf is the fiber volume fraction within the specimen, and the ellipsis designates mechanical properties of the fiber and the matrix (elastic moduli, Poisson’s ratios, etc.) We have shown that a typical plot of Fmax as a function of le has the form shown in Fig. 2. For small embedded lengths, the Fmax value is determined predominantly by interfacial adhesion; however, for large le’s, when the ultimate specimen failure occurs at the crack length close to the whole embedded length, the plot tends asymptotically to a straight line whose slope is proportional to sf. The experimental points usually fall somewhere between these extreme positions. The authors of [1] have chosen the fitting line such that Fmax / le ;
0.00 0.00
0.05 0.10 0.15 Embedded length, mm
0.20
Fig. 2 – Theoretical plot of Fmax as a function of the embedded length (1) and its asymptote as le ! 1 (2) determined by interfacial friction. (3) denotes the region where experimental points are assumed to fall. (A colour version of this figure can be viewed online.)
this would be correct if the whole interface failed simultaneously (e.g., through plastic deformation or pure friction). However, if we admit that the fitting line may not pass through the origin, we obtain curve 2 (see Fig. 1), whose slope corresponds to the interfacial stress of 7.68 MPa. In our approach, this value should be considered as the upper estimate for the interfacial frictional stress, sf. Indeed, this line is parallel to the tangent drawn to the theoretical curve in the region with real experimental points, and its slope is always greater than the slope of the asymptote. And the fact that line 2 intersects the vertical axis at the point where Fmax > 0 clearly indicates that interfacial interaction between the fiber and the matrix involves an essential adhesional contribution. This contribution is determined by the local adhesion strength parameter (sd or Gic) which can be estimated in several ways. First, we can fit experimental data by a theoretical curve (Eq. (1)) whose explicit form has been obtained by us in [4], using a non-linear least-square method with sd and sf as fitting parameters. The problem for a microbond test is that, though both sd and sf are assumed to be constant for all specimens, sT and b are functions of the fiber volume fraction, Vf, which, in turn, depends on the droplet shape. In order to calculate Vf, the droplet shape can be approximated by an ellipsoid. However, to do this, we must know at least the droplet diameter, D. The authors of [1] gave no information about the transverse size of the droplets; fortunately, we can calculate the diameter of the droplet on a fiber using a theory [7] which relates it to the droplet length (embedded fiber length), provided that the wetting angle at the matrix–fiber interface is known. We measured the wetting angle from the photograph in Fig. 1 in [1]; it was approximately # 51. Then, using this value, we calculated the diameters of all 10 droplets involved in the experiment. With good accuracy, these appeared to be linearly related to the embedded lengths: D ¼ 0:906 le 5:89 (lm), similarly to the results reported by Scheer and Nairn [8] for glass/epoxy and Kevlar/epoxy systems. Having calculated the fiber volume fractions, as proposed in [2,4], and taken the mechanical properties of
56
CARBON
8 6 ( 2 0 1 5 ) 5 4 –5 7
the fiber and the matrix from [1] and the Dow Chemical datasheet, we found the best fit to experimental data. The map of the sum of least squares is shown in Fig. 3a; the best-fit values are sd ¼ 54:8 MPa and sf 0. The latter value is obviously wrong; in [9,10] we showed that this ‘indirect’ approach may substantially underestimate interfacial friction. However, as was demonstrated in [9], the true (sd , sf ) point which characterizes the fiber/matrix system must fall somewhere in the narrow valley in the least squares map. Analyzing this map, we can estimate that the sd value for the CNT fiber/DER 353 epoxy is between approximately 40 and 55 MPa, while sf falls between 0 and 9 MPa. The upper estimate for sf is indirectly confirmed by the slope of curve 2 in Fig. 1. In any case, the variations in sf have little effect on the sd value, so that the estimation 40 MPa < sd < 55 MPa should be considered as rather reliable. Similar approach can be used with (Gic , sf ) as the pair of fitting parameters [3]. The corresponding least squares map is presented in Fig. 3b. The best-fit values Gic ¼ 9:10 J/m2 and sf ¼ 4:39 MPa look quite reasonable; however, taking into account that the map in this case is also in the shape of a narrow valley and the minimum is very shallow, we can only estimate 4 J=m2 < Gic < 17 J=m2 and 0 < sf < 9 MPa. Determining interfacial parameters from individual specimens usually yields better results than the ‘indirect’ approach [9,10]. In [1], only one force–displacement curve is available in Fig. 3b (Fig. 4 in this paper). The traditional way of estimating interfacial parameters from force–displacement curves is using the ‘kink’ force in the ascending part of the curve to determine the local IFSS, sd , and then calculating sf from sd and Fmax [9]. However, due to a relatively large free fiber length resulting in high system compliance, the ‘kink’ in the curve cannot be discerned; therefore, we used our new approach [10], in which sf is first determined from the postdebonding part of the curve (Fb value in Fig. 4), and then sf and Fmax are used to calculate sd . The droplet diameter, calculated from the le value (99.5 lm) given by the authors, was 84.2 lm. Using experimental values Fb = 16.6 mN and
Fig. 4 – Force–displacement curve for droplet debonding (as shown in [1]) and three important force values required for the calculation of interfacial parameters: Fd, the ‘kink’ force at which interfacial debonding starts (cannot be reliably determined from this curve); Fmax, the maximum recorded force, and Fb, the force immediately after debonding completion. (A colour version of this figure can be viewed online.)
Fmax = 45.8 mN, we calculated the interfacial parameters for this specimen: sd ¼ 53:7 MPa and sf ¼ 5:56 MPa. Similar calculation using the energy-based approach gives Gic ¼ 7:76 J/m2 (and the same sf ¼ 5:56 MPa, since it was calculated from the same Fb value). Both estimates are in agreement with the values determined above indirectly for the whole set of 10 specimens. Thus, local interfacial strength parameters in CNT fiber/ epoxy matrix system have been successfully estimated using the published experimental data from a microbond test. The estimated local IFSS value (about 50 MPa) is much greater than the effective IFSS reported for this fiber–matrix pair (14.4 MPa). Our approach also allowed us to estimate the interfacial frictional stress.
Fig. 3 – The maps of the sum of least squares as functions of sd and sf (left) and Gic and sf (right) (see text for details). (A colour version of this figure can be viewed online.)
CARBON
8 6 (2 0 15 ) 5 4–57
R E F E R E N C E S
[1] Zu M, Li Q, Zhu Y, Dey M, Wang G, Lu W, et al. The effective interfacial shear strength of carbon nanotube fibers in an epoxy matrix characterized by a microdroplet test. Carbon 2012;50(3):1271–9. http://dx.doi.org/10.1016/ j.carbon.2011.10.047. [2] Zhandarov SF, Ma¨der E, Yurkevich OR. Indirect estimation of fiber/polymer bond strength and interfacial friction from maximum load values recorded in the microbond and pullout tests. Part I: Local bond strength. J Adhes Sci Technol 2002;16:1171–200. http://dx.doi.org/10.1163/ 156856102320256837. [3] Zhandarov S, Ma¨der E. Indirect estimation of fiber/polymer bond strength and interfacial friction from maximum load values recorded in the microbond and pull-out tests. Part II: Critical energy release rate. J Adhes Sci Technol 2003;17(7):967–80. http://dx.doi.org/10.1163/ 156856103322112879. [4] Zhandarov S, Ma¨der E. Peak force as function of the embedded length in pull-out and microbond tests: effect of specimen geometry. J Adhes Sci Technol 2005;19(10):817–55. http://dx.doi.org/10.1163/1568561054929937.
57
[5] Nairn JA. Analytical fracture mechanics analysis of the pullout test including the effects of friction and thermal stresses. Adv Compos Lett 2000;9(6):373–83. [6] Nayfeh AH. Thermomechanically induced interfacial stresses in fibrous composites. Fibre Sci Technol 1977;10(3):195–209. http://dx.doi.org/10.1016/0015-0568(77)90020-3. [7] Wu XF, Dzenis YA. Droplet on a fiber: geometrical shape and contact angle. Acta Mech 2006;185:212–25. http://dx.doi.org/ 10.1007/s00707-006-0349-0. [8] Scheer RJ, Nairn JA. A comparison of several fracture mechanics methods for measuring interfacial toughness with microbond tests. J Adhes 1995;53:45–68. http:// dx.doi.org/10.1080/00218469508014371. [9] Zhandarov S, Ma¨der E. Characterization of fiber/matrix interface strength: applicability of different tests, approaches and parameters. Compos Sci Technol 2005;65:149–60. http:// dx.doi.org/10.1016/j.compscitech.2004.07.003. [10] Zhandarov S, Ma¨der E. An alternative method of determining the local interfacial shear strength from force–displacement curves in the pull-out and microbond tests. Int J Adhes Adhes 2014;55:37–42. http://dx.doi.org/10.1016/ j.ijadhadh.2014.07.006.