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Interfacial toughness evaluation from the single-fiber fragmentation test
ELSEVIER
PII:
SO266-3538(96)00096-6
Composites Science and Technology 56 (1996) 1105-1109 0 1996 Elsevier Science Limited Printed in Northern Ireland. All rights reserved 0266-3538/96/$15.00
Short Communication
INTERFACIAL TOUGHNESS EVALUATION FROM THE SINGLE-FIBER FRAGMENTATION TEST J. Varna,
R. Joffe & L. A. Berglund
Department of Materials and Production Engineering, LuleB University of Technology, S-971 87 LuleB, Sweden
(Received 20 June 1996; accepted 11 July 1996) debonding events then need to be considered as separate processes during loading. Weibull statistics may be used to describe the probability of fiber fragmentation lengths and linear elastic fracture mechanics may be used for a description of debond growth. In order to determine the interfacial toughness, a stress analysis is needed for calculation of the changes in strain energy as a function of applied strain and increasing debond length. If the dependence of debond length on strain is known from experiments, such an analysis could be used in order to calculate the fracture toughness (critical strain energy release rate), G,, of the interface. The present results were presented orally at a recent conference although the related conference paper was more limited in scope.’
Abstract The single-fiber fragmentation test is widely used to determine the interfacial shear strength of polymer composites. In the present study, HTA carbon fibers in an epoxy matrix are studied for the two cases of strong and weak interfaces. The average fiber length is reported as a function of strain, both plotted on logarithmic scales. As deviation from linearity is observed, interfacial debond cracks extend to significant length. Debond crack lengths at given strains are determined from experimental data. A finiteelement stress analysis in combination with a fracture mechanics criterion is used to determine values of G,, the interfacial fracture toughness. G, is independent of debond length, supporting the idea that GC is a valid fracture criterion. 0 1996 Elsevier Science Limited Keywords:
interfacial toughness, single-fiber fragmentation test, finite element analysis, fracture mechanics criterion
2 EXPERIMENTAL Single-fiber fragmentation tests were performed on single HTA carbon fibers embedded in diglycidyl ether of bisphenol A/diethylene amine epoxy cured at 102°C. Specimens were cut with a diamond saw and polished to facilitate optical microscopy observations. Specimens were loaded in tension in the axial direction of the fibers. A Minimat tensile testing machine placed under the optical microscope was used, and images were recorded with a video camera. Two sets of specimen with different fiber treatments were considered (‘weak’ and ‘strong’ interface in the following discussion). The strong interface material was based on the commercially available HTA fiber. To obtain the weak interface, the same HTA fiber was coated with a polymer so that the interface was weakened. The total number of specimens in each group of material was 1.5. All specimens had the same fibers and matrix.
1 INTRODUCTION
The single-fiber fragmentation test is perhaps the most frequently used micromechanical method for characterization of fiber/matrix interface properties. Usually, the data reduction procedure is based on a simplified stress analysis and the critical property considered is the interfacial shear strength. Since the stress distribution at the interface is highly non-uniform, the use of a strength criterion is questionable. However, the interfacial strength approach may be sufficient for the ranking of effects from different fiber surface treatments or sizings. In the present study the objective is to improve the understanding of the single-fiber fragmentation test through the application of a fracture mechanics analysis. On the basis of experimental observations, in the present study the interface was considered to fail by the formation of an interfacial debond crack. In this debond growth is the phenomenon of context, interest. Fiber fracture (fragmentation) and interfacial
3 STRESS
ANALYSIS
The commercial finite element program ANSYS, version 5.OA, was used for stress analysis of a single 1105
J. Varna, R. Joffe, L. A. Berglund
1106
elements, N,, increased for larger debond lengths. In the fine mesh, N, was 1600 for the debond length a = 5r, and N, was 13500 for a = 15r,, where r, is the fiber radius. In our calculations we increased the dimensions of the matrix region until the stresses in the matrix were undisturbed by the presence of the fiber: this was at a distance of llrr. Calculations were performed with the assumption of linear elastic behavior of fiber and matrix. Material data are given in Table 1. 4 RESULTS AND DISCUSSION
El
Matrix
-
Debond
AAAAAAAAAAA x Fig. 1. Geometry and boundary conditions for an axially strained single-carbon-fiber composite with a debond crack. U, and uY are the displacements, a is the length of the debond crack and a,, is the global strain applied to the matrix.
carbon fiber with a debond crack. The problem is presented in Fig. 1. Uniform strain is applied to the matrix in the upper part of Fig. 1. The upper end of the fiber is free to move; the lower end of the fiber and matrix are fixed although the material is free to move along one edge perpendicular to the loading direction. For reasons of symmetry, only one quarter of the specimen is considered. Since the problem is axisymmetric, plane 83 axisymmetric-harmonic-structural solid elements with three degrees of freedom per node were used. For the debonded zone in contact with the resin, contact 48 2-D point to surface contact elements were used to represent contact and sliding between two surfaces-two degrees of freedom for each node. Contact occurs when the contact node penetrates the target line interface. The purpose of using contact elements was to avoid penetration of the matrix into the fiber. Two different meshes were used in calculations‘fine’ with approximately 4600 plane 83 elements and ‘crude’ with 1200 elements; 20% of the elements were in the fiber. The mesh was mixed with rectangular elements in the bulk part and triangles at the interface crack tip. The size ratio between smaller and larger elements was 15 or higher. The number of contact
It is well known that fiber strength may be characterized by a two-parameter Weibull distribution. Since both the fiber fracture strains and the volumetric terms (fragment length) are present in the distribution, it has a dual nature. Usually it is considered as a failure strength distribution at a given fiber length. Then the volume effect is used to recalculate the distribution for different fiber lengths. However, as first demonstrated by Henstenburg and Phoenix’ for the single-fiber fragmentation test, from a mathematical point of view this distribution also characterizes the fiber length distribution at a given applied load. Certainly this distribution may be recalculated later for different loads. On a logarithmic scale, fibers which may be described by a two-parameter Weibull distribution show a linear relationship between the average fragment length and the applied strain. These experimental data can be obtained from the single-fiber fragmentation test and used to determine the Weibull parameters of the fiber strength distribution.’ Other studies confirm the applicability of this method?’ In the present study, single-fiber fragmentation experiments were performed on two types of HTA carbon fibers embedded in epoxy. The average fragment lengths were recorded as a function of strain. Results are presented in Fig. 2. Initially, data are Table 1. Data used in finite element Longitudinal fiber modulus (E,) Transverse fiber modulus (,&IL Fiber shear modulus (G&r) Fiber shear modulus (Cm) Poisson’s ratio (u-r) Poisson’s ratio (u,) Longitudinal fiber expansion coefficient (ol,) Transverse fiber expansion coefficient ([r,) Matrix modulus (E,) Matrix Poisson’s ratio (u,) Matrix thermal expansion coefficient (01,) Fiber radius (rJ Radius of matrix cylinder (R,)
calculations 300 GPa 20 GPa 20 GPa 8.33 GPa 0.2 0.2
-0.7
x 10h K-’
8.0 x 10-h K-’ 3.5 GPa 0.3 60 x 10 ~‘K-’ 4p.m 44 pm
Toughness evolution from fragmentation
2-
l-
O-
-1 -
1.0
0.5
0.0
1.5
2.0
Applied Strain, Ine Fig. 2. Average fiber length as a function of applied for single-fiber carbon-fiber (HTA)/epoxy composites different interface toughnesses.
strain with
dominated by the fiber fragmentation process and we obtain a linear relationship. As already discussed, the slope of that line represents the characteristics of the fiber. Since the behavior in the linear region is similar for both sets of specimens, we conclude that the coating of the weak interface fibers does not affect fiber strength. In this linear region, we only observe 3 \
2
I
I
I
I
I1 I
III
0
1
0
-1
-2 0.4
0.8
1.2
1.6
Applied Strain, InE Fig. 3. Illustration of the three characteristic regions in a plot of average fiber length versus strain. Region I, fiber fracture region; Region II, non-linear region; Region III, saturation region.
test
1107
debond cracks which are very short as compared with the total fragment length. The straight line represents the Weibull fit to the data. For the specimens in the present study, the length of the debond cracks increases at higher strains. The strain at which the length of the short debond cracks starts to increase coincides with the strain at which non-linearity commences. For the early deviations from linearity, some fragmentation events still occur. At higher strains, the fiber fragmentation process stops and the average fragment length is unaffected by strain. This is termed the saturation region. It is apparent from Fig. 2 that the average fiber length for the weak interface is the longest in this region. In typical analyses for the fragmentation test based on the original Kelly-Tyson paper,6 this fiber length and the strength of the fiber at that length is used to calculate an interfacial shear strength. In the saturation region we observed increased debond crack lengths with increased strain. Observations of details of failure mechanisms for the materials studied may be summarized for the three characteristic regions illustrated in Fig. 3. Region I: fiber fracture region. Fiber fractures occur and at the same time very short debond cracks are formed. These short debonds do not grow in Region I and appear to form from dynamic effects associated with fiber fracture. Region I is the linear region in Fig. 3. Region II: non-linear region. In the non-linear region, the rate of fragmentation events decreases and for the present materials the short debond cracks formed in Region I begin to grow. Region III: saturation region. In the saturation region, fragmentation events no longer occur. In the materials investigated, debond growth occurs with increased strain. One may note that for fiber/matrix combinations other than those in the present study, saturation may occur without any debonding. The observations of interfacial debond crack growth suggest a fracture mechanics approach in the analysis of the fragmentation test data. For such an analysis we need a stress analysis and a suitable fracture criterion. Related approaches for the fragmentation test have been suggested by Liu et al.? and Wagner et al.’ The analysis of Liu et al. was performed with a different goal in mind as compared with the present study. Their purpose was to simulate the development of debond length and fragment length based on interfacial toughness as an input parameter. The stress analysis was highly simplified and it is not clear how the value for interfacial toughness was obtained. Wagner et af.8 had an objective similar to that of the present study. The major difference is in principles for stress analysis and in how the debond length data are obtained. Wagner et al. considered existing debond
J. Varna, R. Jo,ffe, L. A. Berglund
1108
cracks of constant length formed in Region I whereas we consider debond length data as a function of strain in Region II. A shear-lag analysis was used by Wagner et al. to determine the stress state whereas we use finite element analysis. The approximate nature of the shear-lag analysis originates in the simplified assumptions. Equilibrium equations are only satisfied in integral form, not at each point in the material. As the analysis is applied to the fragmented fiber problem, the choice of radius of the material cylinder considered also appears to be arbitrary. The finite element stress analysis must be combined with a criterion for debond growth. We decided to use a criterion based on a critical value for the total strain energy release rate. This rests on the assumption that the released strain energy during debond propagation is equal to the energy required to form the crack. We calculated the strain energy in the whole volume, see Fig. 1, in order to avoid problems with stress singularities at the debond crack tip. Another alternative is to use a crack closure technique. However, the stress and displacement field calculations close to the tip of the debond crack then need to be very accurate. Numerical solutions by finite element analysis in the contact zone and close to the crack tip are not feasible. The fracture criterion is given by:
-au ~ = 2mfG,
(u = const)
aa
i
1 .oo
$ 5 5 .-C Q
t% (3
0.90
Ai++
0
qn
0
AT=-80°C,p=0.8
0
AT=-80"C,p=O
0
AT=O'C,p=O
+
q
+
+
0
q
0 0
Z
0
N ._ z
+
0
5
(1)
0 0
0.80
E
0
8
0
z
0
0.70
t
0
5
10
15
Debond Length a/r, Fig. 4. Calculated normalized strain energy as a function of normalized debond length. AT is the temperature difference between the strain-free temperature and room temperature and is proportional to the residual strain. p is the coefficient of friction. The debond length is expressed as the ratio between debond crack length, a, and fiber radius, r,.
where Ii is the strain energy, a is the debond length, rf is the fiber radius, G, is the fracture toughness (critical strain energy release rate) and u is the displacement. Some computational results are presented in Fig. 4. Normalized strain energy is presented as a function of the debond length normalized with respect to the fiber radius. The strain energy is normalized with respect to the strain energy in the system of Fig. 1 without a debond crack. The lowest curve represents results from a calculation where no residual thermal strains are included. It is interesting to note the immediate drop in strain energy as a short debond is formed. This indicates the presence of a large stress peak of singular nature at the fiber end. Residual thermal strains are taken into account for the two upper curves. Although the value of the strain energy is strongly affected, G, for a given debond length is weakly influenced since it is proportional only to the slope of the strain energy versus debond length curve. Finally, the effect of friction at the debonded fiber/matrix interface is illustrated for a friction coefficient of p = 0.8. If the contribution of friction to the strain energy is considered in the calculations, G, is reduced since the slope in Fig. 4 is reduced for p = 0.8. Since we do not know the coefficient of friction, frictional effects were not considered and are therefore included in our fracture toughness values. In order to determine G, we need data for strain versus debond length. For each data set (strain and debond length) we determine the strain energy for two debond crack lengths, a - ha and a + ha, in order to determine aU/aa. Aa was chosen as the fiber radius. G, can then be calculated from eqn (1). From optical microscopy observations we obtained data for global strain and corresponding debond crack lengths for the weak interface. During collection of the data we noted that individual debond lengths showed scatter for similar fragment lengths, indicating local differences in interfacial fracture toughness. We had some difficulties in determining the debond length with accuracy for short debond lengths, especially for the strong interface material and a graphical procedure for determination of debond lengths was therefore developed. We considered a fiber fragment of total length l2 with a total debond length of 2a (one debond at each fiber end). The central part of the same fiber is still perfectly bonded to the matrix and of length 1,. We therefore have l2 - I, = 2a. Let us again consider Fig. 3. The initial straight part of the curve is related to the Weibull distribution of fiber strains to failure. We can use this to predict continued fragmentation of the fiber without debonding taking place. It then follows that the difference between this line and the actual fragmentation data should be related to the debond length 2a. In the present analysis, as a first approximation we assumed that this difference, as illustrated by the arrows in Fig. 3, is the
Toughness evolution from fragmentation Table 2. Data for strain and debond length used in finite element calculations E
Strong interface
(%)
2.80 2.88 2.95 2.65 2.87 3.07
Weak interface
a b-4
a/h
58 74 80 79 111 149
-14 218 20 -20 ~28 -37
debond length. The data used in the calculations for global strain and debond crack length were obtained from this graphical procedure and are given in Table 2. Debond lengths were determined by two techniques: the graphical procedure suggested, and direct microscopical observation. The debond length determined from microscopy was compared with the debond length determined by the graphical procedure. A linear relationship was found between the two. If this relationship is known, we can use the graphical procedure without the necessity of direct microscopy measurements of debond lengths. However, the hoped-for 1:l relationship between microscopy data and data from the graphical procedure was met only at the lowest strain. It is possible that this discrepancy
‘OOe 250 -
1109
test
could be explained from more accurate finite element calculations of fiber stress distributions. Figure 5 shows the results for total G, versus debond length for the two materials. G, is independent of debond length. This supports the applicability of a fracture mechanics criterion. Values are 250 J/m2 for the strong interface and 200 J/m2 for the weak interface. Since no friction coefficient is known we cannot exclude the friction contribution from our calculated G, values. The calculated G, values are in the same range as reported by Wagner et al.’ for glass-fiber/urethane-acrylate materials (183 and 264 J/m2). 5 CONCLUSIONS Three different regions were observed for the single-fiber fragmentation behavior as average fragment length was plotted versus strain on logarithmic scales for HTA carbon fiber in epoxy. In Region I, fiber fracture and short debonds were observed. In the non-linear Region II at higher strains, debond crack growth and some fragmentation were observed. Region III was the saturation region where only debond growth was observed for the investigated materials. A finite element stress analysis was used in combination with a fracture mechanics criterion to determine values for interfacial fracture toughnesses of two carbon-fiber/epoxy materials. Fracture toughnesses, G,, were independent of debond length. Provided experimental observations confirm interfacial debond crack growth, interfacial fracture toughness can be determined from a stress analysis and fragmentation data in the non-linear Region II. REFERENCES
0
0
0
1. Joffe, R., Varna, J. & Berglund, Conf
-i Y 6
+
+
on
Deformation
and
L. A., in Proc. 3rd Int. Fracture
of
Composites,
University of Surrey, Guildford, UK, 27-29 March 1995. The Institute of Materials, 1995, p. 126. 2. Henstenburg, R. B. & Phoenix, S. L., Polym. Comp., 10 (1989) 389. 3. Yavin, B., Gallis, H. E., Scherf, J., Eitan, A. & Wagner, H. D., Polym. Comp., 12 (1991) 436. 4. Tamusz, V., Korabelnikov, Y., Rashkovan, I., Karklins, A., Gorbatkina, Y. & Zaharova, T., Mech. Comp. Mater. (in Russian), 27 (1991) 413.
Andersons,
J. & Tamusz,
V., Comp. Sci. Technol.,
48
(1993) 57.
Kelly, A. & Tyson, W. R., J. Mech. Phys. Solids, 13 (1965) 329. Liu, H. Y., Mai, Y. W., Zhou, L. M. & Ye, L., Comp. Fig. 5. Interfacial G, versus debond length for two different single-fiber composite materials based on HTA carbon fiber in epoxy.
Sci. Technol., 52 (1994) 253.
Wagner, H. D., Nairn, J. A. & Detassis, M., Proc. R. Sot. A (submitted).
PII:
SO266-3538(96)00096-6
Composites Science and Technology 56 (1996) 1105-1109 0 1996 Elsevier Science Limited Printed in Northern Ireland. All rights reserved 0266-3538/96/$15.00
Short Communication
INTERFACIAL TOUGHNESS EVALUATION FROM THE SINGLE-FIBER FRAGMENTATION TEST J. Varna,
R. Joffe & L. A. Berglund
Department of Materials and Production Engineering, LuleB University of Technology, S-971 87 LuleB, Sweden
(Received 20 June 1996; accepted 11 July 1996) debonding events then need to be considered as separate processes during loading. Weibull statistics may be used to describe the probability of fiber fragmentation lengths and linear elastic fracture mechanics may be used for a description of debond growth. In order to determine the interfacial toughness, a stress analysis is needed for calculation of the changes in strain energy as a function of applied strain and increasing debond length. If the dependence of debond length on strain is known from experiments, such an analysis could be used in order to calculate the fracture toughness (critical strain energy release rate), G,, of the interface. The present results were presented orally at a recent conference although the related conference paper was more limited in scope.’
Abstract The single-fiber fragmentation test is widely used to determine the interfacial shear strength of polymer composites. In the present study, HTA carbon fibers in an epoxy matrix are studied for the two cases of strong and weak interfaces. The average fiber length is reported as a function of strain, both plotted on logarithmic scales. As deviation from linearity is observed, interfacial debond cracks extend to significant length. Debond crack lengths at given strains are determined from experimental data. A finiteelement stress analysis in combination with a fracture mechanics criterion is used to determine values of G,, the interfacial fracture toughness. G, is independent of debond length, supporting the idea that GC is a valid fracture criterion. 0 1996 Elsevier Science Limited Keywords:
interfacial toughness, single-fiber fragmentation test, finite element analysis, fracture mechanics criterion
2 EXPERIMENTAL Single-fiber fragmentation tests were performed on single HTA carbon fibers embedded in diglycidyl ether of bisphenol A/diethylene amine epoxy cured at 102°C. Specimens were cut with a diamond saw and polished to facilitate optical microscopy observations. Specimens were loaded in tension in the axial direction of the fibers. A Minimat tensile testing machine placed under the optical microscope was used, and images were recorded with a video camera. Two sets of specimen with different fiber treatments were considered (‘weak’ and ‘strong’ interface in the following discussion). The strong interface material was based on the commercially available HTA fiber. To obtain the weak interface, the same HTA fiber was coated with a polymer so that the interface was weakened. The total number of specimens in each group of material was 1.5. All specimens had the same fibers and matrix.
1 INTRODUCTION
The single-fiber fragmentation test is perhaps the most frequently used micromechanical method for characterization of fiber/matrix interface properties. Usually, the data reduction procedure is based on a simplified stress analysis and the critical property considered is the interfacial shear strength. Since the stress distribution at the interface is highly non-uniform, the use of a strength criterion is questionable. However, the interfacial strength approach may be sufficient for the ranking of effects from different fiber surface treatments or sizings. In the present study the objective is to improve the understanding of the single-fiber fragmentation test through the application of a fracture mechanics analysis. On the basis of experimental observations, in the present study the interface was considered to fail by the formation of an interfacial debond crack. In this debond growth is the phenomenon of context, interest. Fiber fracture (fragmentation) and interfacial
3 STRESS
ANALYSIS
The commercial finite element program ANSYS, version 5.OA, was used for stress analysis of a single 1105
J. Varna, R. Joffe, L. A. Berglund
1106
elements, N,, increased for larger debond lengths. In the fine mesh, N, was 1600 for the debond length a = 5r, and N, was 13500 for a = 15r,, where r, is the fiber radius. In our calculations we increased the dimensions of the matrix region until the stresses in the matrix were undisturbed by the presence of the fiber: this was at a distance of llrr. Calculations were performed with the assumption of linear elastic behavior of fiber and matrix. Material data are given in Table 1. 4 RESULTS AND DISCUSSION
El
Matrix
-
Debond
AAAAAAAAAAA x Fig. 1. Geometry and boundary conditions for an axially strained single-carbon-fiber composite with a debond crack. U, and uY are the displacements, a is the length of the debond crack and a,, is the global strain applied to the matrix.
carbon fiber with a debond crack. The problem is presented in Fig. 1. Uniform strain is applied to the matrix in the upper part of Fig. 1. The upper end of the fiber is free to move; the lower end of the fiber and matrix are fixed although the material is free to move along one edge perpendicular to the loading direction. For reasons of symmetry, only one quarter of the specimen is considered. Since the problem is axisymmetric, plane 83 axisymmetric-harmonic-structural solid elements with three degrees of freedom per node were used. For the debonded zone in contact with the resin, contact 48 2-D point to surface contact elements were used to represent contact and sliding between two surfaces-two degrees of freedom for each node. Contact occurs when the contact node penetrates the target line interface. The purpose of using contact elements was to avoid penetration of the matrix into the fiber. Two different meshes were used in calculations‘fine’ with approximately 4600 plane 83 elements and ‘crude’ with 1200 elements; 20% of the elements were in the fiber. The mesh was mixed with rectangular elements in the bulk part and triangles at the interface crack tip. The size ratio between smaller and larger elements was 15 or higher. The number of contact
It is well known that fiber strength may be characterized by a two-parameter Weibull distribution. Since both the fiber fracture strains and the volumetric terms (fragment length) are present in the distribution, it has a dual nature. Usually it is considered as a failure strength distribution at a given fiber length. Then the volume effect is used to recalculate the distribution for different fiber lengths. However, as first demonstrated by Henstenburg and Phoenix’ for the single-fiber fragmentation test, from a mathematical point of view this distribution also characterizes the fiber length distribution at a given applied load. Certainly this distribution may be recalculated later for different loads. On a logarithmic scale, fibers which may be described by a two-parameter Weibull distribution show a linear relationship between the average fragment length and the applied strain. These experimental data can be obtained from the single-fiber fragmentation test and used to determine the Weibull parameters of the fiber strength distribution.’ Other studies confirm the applicability of this method?’ In the present study, single-fiber fragmentation experiments were performed on two types of HTA carbon fibers embedded in epoxy. The average fragment lengths were recorded as a function of strain. Results are presented in Fig. 2. Initially, data are Table 1. Data used in finite element Longitudinal fiber modulus (E,) Transverse fiber modulus (,&IL Fiber shear modulus (G&r) Fiber shear modulus (Cm) Poisson’s ratio (u-r) Poisson’s ratio (u,) Longitudinal fiber expansion coefficient (ol,) Transverse fiber expansion coefficient ([r,) Matrix modulus (E,) Matrix Poisson’s ratio (u,) Matrix thermal expansion coefficient (01,) Fiber radius (rJ Radius of matrix cylinder (R,)
calculations 300 GPa 20 GPa 20 GPa 8.33 GPa 0.2 0.2
-0.7
x 10h K-’
8.0 x 10-h K-’ 3.5 GPa 0.3 60 x 10 ~‘K-’ 4p.m 44 pm
Toughness evolution from fragmentation
2-
l-
O-
-1 -
1.0
0.5
0.0
1.5
2.0
Applied Strain, Ine Fig. 2. Average fiber length as a function of applied for single-fiber carbon-fiber (HTA)/epoxy composites different interface toughnesses.
strain with
dominated by the fiber fragmentation process and we obtain a linear relationship. As already discussed, the slope of that line represents the characteristics of the fiber. Since the behavior in the linear region is similar for both sets of specimens, we conclude that the coating of the weak interface fibers does not affect fiber strength. In this linear region, we only observe 3 \
2
I
I
I
I
I1 I
III
0
1
0
-1
-2 0.4
0.8
1.2
1.6
Applied Strain, InE Fig. 3. Illustration of the three characteristic regions in a plot of average fiber length versus strain. Region I, fiber fracture region; Region II, non-linear region; Region III, saturation region.
test
1107
debond cracks which are very short as compared with the total fragment length. The straight line represents the Weibull fit to the data. For the specimens in the present study, the length of the debond cracks increases at higher strains. The strain at which the length of the short debond cracks starts to increase coincides with the strain at which non-linearity commences. For the early deviations from linearity, some fragmentation events still occur. At higher strains, the fiber fragmentation process stops and the average fragment length is unaffected by strain. This is termed the saturation region. It is apparent from Fig. 2 that the average fiber length for the weak interface is the longest in this region. In typical analyses for the fragmentation test based on the original Kelly-Tyson paper,6 this fiber length and the strength of the fiber at that length is used to calculate an interfacial shear strength. In the saturation region we observed increased debond crack lengths with increased strain. Observations of details of failure mechanisms for the materials studied may be summarized for the three characteristic regions illustrated in Fig. 3. Region I: fiber fracture region. Fiber fractures occur and at the same time very short debond cracks are formed. These short debonds do not grow in Region I and appear to form from dynamic effects associated with fiber fracture. Region I is the linear region in Fig. 3. Region II: non-linear region. In the non-linear region, the rate of fragmentation events decreases and for the present materials the short debond cracks formed in Region I begin to grow. Region III: saturation region. In the saturation region, fragmentation events no longer occur. In the materials investigated, debond growth occurs with increased strain. One may note that for fiber/matrix combinations other than those in the present study, saturation may occur without any debonding. The observations of interfacial debond crack growth suggest a fracture mechanics approach in the analysis of the fragmentation test data. For such an analysis we need a stress analysis and a suitable fracture criterion. Related approaches for the fragmentation test have been suggested by Liu et al.? and Wagner et al.’ The analysis of Liu et al. was performed with a different goal in mind as compared with the present study. Their purpose was to simulate the development of debond length and fragment length based on interfacial toughness as an input parameter. The stress analysis was highly simplified and it is not clear how the value for interfacial toughness was obtained. Wagner et af.8 had an objective similar to that of the present study. The major difference is in principles for stress analysis and in how the debond length data are obtained. Wagner et al. considered existing debond
J. Varna, R. Jo,ffe, L. A. Berglund
1108
cracks of constant length formed in Region I whereas we consider debond length data as a function of strain in Region II. A shear-lag analysis was used by Wagner et al. to determine the stress state whereas we use finite element analysis. The approximate nature of the shear-lag analysis originates in the simplified assumptions. Equilibrium equations are only satisfied in integral form, not at each point in the material. As the analysis is applied to the fragmented fiber problem, the choice of radius of the material cylinder considered also appears to be arbitrary. The finite element stress analysis must be combined with a criterion for debond growth. We decided to use a criterion based on a critical value for the total strain energy release rate. This rests on the assumption that the released strain energy during debond propagation is equal to the energy required to form the crack. We calculated the strain energy in the whole volume, see Fig. 1, in order to avoid problems with stress singularities at the debond crack tip. Another alternative is to use a crack closure technique. However, the stress and displacement field calculations close to the tip of the debond crack then need to be very accurate. Numerical solutions by finite element analysis in the contact zone and close to the crack tip are not feasible. The fracture criterion is given by:
-au ~ = 2mfG,
(u = const)
aa
i
1 .oo
$ 5 5 .-C Q
t% (3
0.90
Ai++
0
qn
0
AT=-80°C,p=0.8
0
AT=-80"C,p=O
0
AT=O'C,p=O
+
q
+
+
0
q
0 0
Z
0
N ._ z
+
0
5
(1)
0 0
0.80
E
0
8
0
z
0
0.70
t
0
5
10
15
Debond Length a/r, Fig. 4. Calculated normalized strain energy as a function of normalized debond length. AT is the temperature difference between the strain-free temperature and room temperature and is proportional to the residual strain. p is the coefficient of friction. The debond length is expressed as the ratio between debond crack length, a, and fiber radius, r,.
where Ii is the strain energy, a is the debond length, rf is the fiber radius, G, is the fracture toughness (critical strain energy release rate) and u is the displacement. Some computational results are presented in Fig. 4. Normalized strain energy is presented as a function of the debond length normalized with respect to the fiber radius. The strain energy is normalized with respect to the strain energy in the system of Fig. 1 without a debond crack. The lowest curve represents results from a calculation where no residual thermal strains are included. It is interesting to note the immediate drop in strain energy as a short debond is formed. This indicates the presence of a large stress peak of singular nature at the fiber end. Residual thermal strains are taken into account for the two upper curves. Although the value of the strain energy is strongly affected, G, for a given debond length is weakly influenced since it is proportional only to the slope of the strain energy versus debond length curve. Finally, the effect of friction at the debonded fiber/matrix interface is illustrated for a friction coefficient of p = 0.8. If the contribution of friction to the strain energy is considered in the calculations, G, is reduced since the slope in Fig. 4 is reduced for p = 0.8. Since we do not know the coefficient of friction, frictional effects were not considered and are therefore included in our fracture toughness values. In order to determine G, we need data for strain versus debond length. For each data set (strain and debond length) we determine the strain energy for two debond crack lengths, a - ha and a + ha, in order to determine aU/aa. Aa was chosen as the fiber radius. G, can then be calculated from eqn (1). From optical microscopy observations we obtained data for global strain and corresponding debond crack lengths for the weak interface. During collection of the data we noted that individual debond lengths showed scatter for similar fragment lengths, indicating local differences in interfacial fracture toughness. We had some difficulties in determining the debond length with accuracy for short debond lengths, especially for the strong interface material and a graphical procedure for determination of debond lengths was therefore developed. We considered a fiber fragment of total length l2 with a total debond length of 2a (one debond at each fiber end). The central part of the same fiber is still perfectly bonded to the matrix and of length 1,. We therefore have l2 - I, = 2a. Let us again consider Fig. 3. The initial straight part of the curve is related to the Weibull distribution of fiber strains to failure. We can use this to predict continued fragmentation of the fiber without debonding taking place. It then follows that the difference between this line and the actual fragmentation data should be related to the debond length 2a. In the present analysis, as a first approximation we assumed that this difference, as illustrated by the arrows in Fig. 3, is the
Toughness evolution from fragmentation Table 2. Data for strain and debond length used in finite element calculations E
Strong interface
(%)
2.80 2.88 2.95 2.65 2.87 3.07
Weak interface
a b-4
a/h
58 74 80 79 111 149
-14 218 20 -20 ~28 -37
debond length. The data used in the calculations for global strain and debond crack length were obtained from this graphical procedure and are given in Table 2. Debond lengths were determined by two techniques: the graphical procedure suggested, and direct microscopical observation. The debond length determined from microscopy was compared with the debond length determined by the graphical procedure. A linear relationship was found between the two. If this relationship is known, we can use the graphical procedure without the necessity of direct microscopy measurements of debond lengths. However, the hoped-for 1:l relationship between microscopy data and data from the graphical procedure was met only at the lowest strain. It is possible that this discrepancy
‘OOe 250 -
1109
test
could be explained from more accurate finite element calculations of fiber stress distributions. Figure 5 shows the results for total G, versus debond length for the two materials. G, is independent of debond length. This supports the applicability of a fracture mechanics criterion. Values are 250 J/m2 for the strong interface and 200 J/m2 for the weak interface. Since no friction coefficient is known we cannot exclude the friction contribution from our calculated G, values. The calculated G, values are in the same range as reported by Wagner et al.’ for glass-fiber/urethane-acrylate materials (183 and 264 J/m2). 5 CONCLUSIONS Three different regions were observed for the single-fiber fragmentation behavior as average fragment length was plotted versus strain on logarithmic scales for HTA carbon fiber in epoxy. In Region I, fiber fracture and short debonds were observed. In the non-linear Region II at higher strains, debond crack growth and some fragmentation were observed. Region III was the saturation region where only debond growth was observed for the investigated materials. A finite element stress analysis was used in combination with a fracture mechanics criterion to determine values for interfacial fracture toughnesses of two carbon-fiber/epoxy materials. Fracture toughnesses, G,, were independent of debond length. Provided experimental observations confirm interfacial debond crack growth, interfacial fracture toughness can be determined from a stress analysis and fragmentation data in the non-linear Region II. REFERENCES
0
0
0
1. Joffe, R., Varna, J. & Berglund, Conf
-i Y 6
+
+
on
Deformation
and
L. A., in Proc. 3rd Int. Fracture
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University of Surrey, Guildford, UK, 27-29 March 1995. The Institute of Materials, 1995, p. 126. 2. Henstenburg, R. B. & Phoenix, S. L., Polym. Comp., 10 (1989) 389. 3. Yavin, B., Gallis, H. E., Scherf, J., Eitan, A. & Wagner, H. D., Polym. Comp., 12 (1991) 436. 4. Tamusz, V., Korabelnikov, Y., Rashkovan, I., Karklins, A., Gorbatkina, Y. & Zaharova, T., Mech. Comp. Mater. (in Russian), 27 (1991) 413.
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Kelly, A. & Tyson, W. R., J. Mech. Phys. Solids, 13 (1965) 329. Liu, H. Y., Mai, Y. W., Zhou, L. M. & Ye, L., Comp. Fig. 5. Interfacial G, versus debond length for two different single-fiber composite materials based on HTA carbon fiber in epoxy.
Sci. Technol., 52 (1994) 253.
Wagner, H. D., Nairn, J. A. & Detassis, M., Proc. R. Sot. A (submitted).