Ind. Eng. Chem. Prod. Res.
Dev. 1983, 22,643-652
843
New Composite Materials from Natural Hard Fibers. 2. Fatigue Studies and a Novel Fatigue Degradation Model Hector Belmares, Arnold0 Barrera, and Margarlta Monjaras Centro de Investigacion en Qulmica Aplicada, Saltillo, Coahuila, Mexico
Constant-amplitude-of-force flexural fatigue of eleven different samples has been statistically studied. The samples include natural hard fiber composites and glass fiber composites from one to three plies, ply combinations of both composite types, and different fiber contents. The studies were performed at a cyclic maximum stress within the proportional limit (initial and final) of the samples. A fatigue testing machine was designed capable of accommodating 70 specimens at a frequency range of 0.24-2.30 Hz. A novel mathematical model for fatigue degradation was developed and an exponential parameter b was statistically defined which is constant for all eleven samples studied and equal to 3.04 f 0.06 (error given at a 90% confidence level). Due to the constancy of b , the model has a good predictive value. The model was combined with the basic equation of linear elastic fracture mechanics to obtain new perspectives of the fatigue behavior.
Introduction Lately, the use of renewable resources for the obtention of industrial raw materials has been the subject of great interest. In the field of composite materials a recent paper by us (Belmares et al., 1981, with references therein) describes the use of natural hard fibers to develop composite materials with a favorable strength/price ratio. Those findings could help to alleviate the worldwide scarcity of low-cost building materials and the market slump in which the natural hard fibers have been for several decades. As a continuation of this work the present paper describes and compares the statistical results obtained under cyclic stress fatigue degradation of the natural hard fiber composites, glass fiber composites, and ply combinations of both. Studies of fatigue degradation of the composites become very important when the material in service will be subject to periodic stresses as is usually the case. Additionally, a novel mathematical model of fatigue is described. This model correlates and explains reasonably well the experimental data. Experimental Section Sample Composition and Preparation. Table I gives the composition of samples. Our previous paper (Belmares et al., 1981) describes the natural hard fibers obtainment, procedures and conditions for polyester (PER) laminates manufacture, and their testing to determine tensile properties. Additional Equipment and Scanning Electron Microscopy (SEM) Results. The SEM micrographs were taken with an AMR Model 900 high-resolution scanning electron microscope. The samples were coated with a gold-palladium alloy to avoid charging in the microscope. Figure 1shows the surface topology of the natural fiber. The surface is marked by its characteristic vestigal attachments of the parenchymatous cells in which the fiber is embedded in the leaf. Similar surface features were obtained for sisal (Barkakaty, 1976). To obtain the natural fiber cross section features, the fiber was submerged in liquid nitrogen and then fractured by stretching. Figure 2 shows the cross section fracture morphology in which filaments related to the fiber ultimate cells are observed. The ultimate cells are cemented together with lignin, hemicellulose, and other natural adhesives. In fact, the fiber is in itself a composite material. The physical and mechanical properties as well as the chemical composition 0196-432118311222-0643$01.50/0
of several natural hard fibers have already been published (Belmares et al., 1979 and 1981). A diagram of the fatigue testing machine is shown in Figure 3. It has a motor as a driver, its driving force being governed by its torque. The motor is provided with a speed controller which fixes the frequency of testing. A rotating motion of the motor’s shaft is converted to the reciprocating sinusoidal driving force motion through an adjustable joint (attached to the motor’s shaft) and a conrod-slider combination. The distance from the adjustable joint to the center of the rotating motor’s shaft can be easily adjusted to the desired stroke. Thii operation fixes the stroke of the machine’s oscillating beam. The oscillating beam applies a concentrated load to flexure the specimens through a sharp padded edge which is 0.7 cm above the surface. The specimens are held as a cantilever beam by the machine’s fixed vise. This vise can accommodate up to 70 specimens. The specimen deflection is small enough (discussed later in more detail) to make unnecessary the fastening of the oscillating beam to the specimen. Measurements and Test Methods. Figure 4 shows the geometry and dimensions of the test specimen. The thickness of each specimen was measured to the nearest 0.006 mm. The standard deviation was about *2% of the mean value for a given set of specimens. The width and test span of each specimen were measured to the nearest 0.2 and 0.5 mm, respectively. The test span was usually equal to 7.00 cm. The thickness range for all the samples studied was 0.70-2.39 mm. The maximum cyclic stress S in the outer fiber is calculated by the well-known elementary cantilever beam equation (ASTM D671-71 test method) after having the value for the maximum test load. Since our fatigue machine applies a maximum constant deflection to the specimens, the maximum test load (for a given maximum constant deflection) is found by placing several unfatigued specimens in the machine’s vise and then applying a load with calibrated weights which gives the sought specimen deflection. The number of specimens is such as to give an allowable error of rt3% for S, with a 90% confidence level. As will be demonstrated later on, the modulus of elasticity of the specimens is statistically constant from beginning to end of the fatigue tests; therefore we truly have a constant-amplitude-of-force flexural fatigue in addition to the constant amplitude-of-deflection flexural fatigue. This special stress-deflection relationship happens 0 1983 American Chemical Society
644
Ind. Eng. Chem. Rod. Res. De".. VoI. 22, No.
4, 1983
0.21
A-adjustable p i n t attached to the oscillating beam and slider 6-flexible joint
motor's shaft
Figure 1. Scanning electron micrograph of a palm fiber lateral surface (X200).
gum 3. Fatigue testing machine of the cantilever type provided th a sinusoidal driving force assembly and a motor with sufficient ,wingforce in the frequency range of 0.24-2.30 Hz. The dimensions are given in meters.
t- 2.504
Figure 4. Geometry and dimensions (in cm) of fatigue specimens. The shaded area corresponds to the specimen portion that is fastened to the fired beam of the fatigue machine (see Figure 1).
Figure 2. Scanning electron micrograph of a palm fiber cross seetion (X500).
only when the specimens of the present work are tested within the proportional limit from beginning to end of the test.
The cantilever beam equation will yield precise results for materials whose: (a) stress-strain relationship is linear; (b) stress-strain curve in tension is identical with that in compression; (c) internal damping is small; (d) modulus of elasticity is relatively high. Most plastics have one or more of these characteristics. No generally satisfactory method of taking these factors into account is yet available (ASTM D671-71 test method). Although several of these factors will be discussed later in more detail, the following can be anticipated (a) The fatigue studies were carried out within the proportional limit of the specimens from
Ind. Eng. Chem. Prod. Res. Dev., Vol. 22, No. 4, 1983 645
Table I. Composition of Samples sample no.
PER/Sty formulationa
no. of plies in composite
fiber type per ply b*c
composite fiber content, wt %
21 22 23 24 25 26 27 28 29 30 31 32
A A B A A A A A A A B A
1 1 1 2 2 2 2 2 3 3 3 2
GM NFM NFM GM/GM NFM/NFM NFM/GM NFM/GM NFM/GWR GM/GM/GM NFM/NFM/NFM NFM/NFM/NFM NFM / N FM
38 33 29 45 36 15:22d 14:23d 20:24d 43 37 37 35
Mixing ratio in weight; formulation A, PER/Sty = 60:40; formulation B, PER/Sty/HEMA (2-hydroxyethyl methacrylate) GM, chopped glass strand mat of 458 g/m’ made with randomly distributed strands ( 5 cm long); NFM, natural fiber mat (palm fiber mat) of 450 g/m’ made with 9 cm long randomly distributed natural fibers; GWR, commercial glass woven roving of 500 glm’. Sample no. 25 has 12% (by wt, fiber basis of poly(viny1 acetate) as the NFM binder and interfacial agent. The rest of the samples that contain one or several plies of NFM have 6% (by wt, fiber basis) For example, sample no. 26 has 15% and 22% (by wt) of of polyvinyl alcohol as the NFM binder and interfacial agent. NFM and GM, respectively (that is, 63% of the sample weight corresponds t o the PER polymer matrix). = 60:35:5.
beginning to end of the tests; (b) although this factor was not verified, general practice in fiber glass polyester laminates shows that at least the tensile and the compressive strength are relatively close (Lubin, 1969); ( c ) thin specimens and a very low frequency of testing were used to have insignificant heat rise; (d) this factor is fulfilled. Due to some of these factors, the number of specimens tested for the present work is relatively large if the experimental errors are to be kept reasonably low. This is why a special fatigue machine had to be designed to decrease the time of testing at the low frequency used (1Hz). Test Conditions. The fatigue tests were conducted at 23 f 2 OC, 40 f 5 % relative humidity, with a minimum specimen conditioning of 60 h. Experimental Errors. The scatter observed for fatigue studies in general is quite large (Whittaker, 1974; Rieke et al., 1980) and if it is not taken into account could give place to misleading interpretations of the experimental data. The scatter in the data was analyzed by using either Weibull, normal, or Student-t distributions depending on the nature of the random variable studied. When we assume that a random variable has the Student-t distribution, strictly speaking this is true only when the population is normal. For practical purposes, it is a good enough approximation for nonnormal distributions (Chou and Croman, 1979). Results and Discussion Frequency Effects on Fatigue. In metallic materials the frequency of load application in a cyclic fatigue test does not significantly influence the fatigue limit over a wide range of frequencies. In these materials, hysteresis is extremely small in the elastic stress range. However, typical glass fiber reinforced composites exhibit a moderate hysteresis loop on a stress-strain diagram. The area enclosed in the hysteresis loop represents the energy loss per unit volume for each frequency cycle. The energy loss is converted to heat which must be dissipated continuously during the test interval. The energy to be dissipated at conventional frequencies of load application (30 Hz,ASTM D671-71 test method) produces a significant temperature increase at modest stress levels. The fatigue life of the glass fiber reinforced composites is lowered with increasing cyclic frequency (Dally and Broutman, 1967). In our work, the data of Dally and Broutman were used as the guidelines to determine the frequency and specimen thickness at which the temperature increase would be insignificant within the range of stresses employed.
Cyclic Stress Amplitude Effects on Fatigue. The fatigue tests of a given plastic usually involve the determination of the number of cycles for fatigue failure at different flexural stresses. Fatigue failure is said to occur when: (a) the specimen breaks in two pieces; (b) for some fiber filled composites, by formation of a single crack or general cracking; (c) by decay of the modulus of elasticity to 70% of the original modulus (ASTM D671-71 test method); (d) the specimen residual strength is reduced to the value of maximum cyclic stress S (Yang and Liu, 1977). In the present work, for samples no. 21-31, there is no fatigue failure of the types mentioned but only fatigue damage. Therefore, periodically, the fatigue test was stopped and some specimens were taken to measure their residual strength in the Instron tensile tester. This type of fatigue tests have been done before (Yang and Liu, 1977). On the laminate level, in which a ply is treated as one constituent phase, typical failures can conveniently be grouped into ply failure and delamination (Hahn and Kim, 1976). As the authors point out, this distinction is not appropriate on a microscopic scale. Subcritical failures such as microcracks can occur even during the first cycle; therefore it is appropriate to divide the discussion into two groups depending on whether the applied stress is higher or lower than the first ply-failure stress. In fact, Yang and Liu (1977) consider that the ideal experimental testing conditions for fatigue strength is by use of cyclic loading that is below the first ply-failure stress. In the present work, the “ideal” testing condition for fatigue strength is fulfilled when the maximum cyclic stress S is within the composite proportional limit from beginning to end of the fatigue test (Tables I1 and 111). Sample no. 32 is a special case in which purposely the cyclic stress S is above the proportional limit. This sample suffered first ply-failure at about 300 000 cycles when its residual strength was reduced to the value of the maximum cyclic stress (404.5 kg/cm2). The rest of the eleven samples did not suffer either first ply-failure or delamination from beginning to end of the fatigue test, keeping therefore the initial macroscopic identity and continuity. The fatigue degradation curves for all these samples will be discussed in the next section. A practical advantage of fatigue testing within the proportional limit of a material is that usually all working stresses on the material are within the proportional limit; otherwise i t would suffer undesirable deformations when the load is entirely removed.
Ind. Eng. Chem. Prod. Res. Dev., Vol. 22, No. 4, 1983
646
L
-*-
21 22 --0.-32
23
-0-
7
!
-X-3'
-E
-0-
- 0- 27
2 1500 k
I " -
r w 3
1
1
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1
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15
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0
.
cg
,
1
1
1
1
,
1
4
SC h x 13
1
1
1
1
1
135 I
1
260.5 292.9 328.6 430.9 254.3 425.5 438.8 313.5 31 5.9 275.6 339.8 404.5
I
I
/
,
,
,
l
"
-0-
t
25
I
I
0.92
n 0
45
135
90 Nx
180
cycles
Figure 7. Least-squares fit of residual strength vs. fatigue cycles for samples no. 24, 25, and 28.
Fatigue damage in reinforced plastics is progressive. Before fatigue failure there is what could be called incipient damage due to fiber debonding and resin cracking (micro and macro). Indeed, fatigue diagrams have been reported for one-ply chopped glass strand mat-polyester resin laminates (Sines, 1972) for the various stages of damage, debonding, resin cracking, and final specimen rupture.
These diagrams show that a laminate subject to a cyclic maximum stress equivalent to 25% and 44% of its ultimate (static) strength can go at least 2 million and 600 OOO cycles, respectively, with only fiber debonding and resin cracking. Table I11 shows that samples no. 21, 24,26, and 29 have an S / X ( O ) ratio that falls near the ratio of the reported fatigue diagrams. The four samples were fatigued at a total of 1888 800,378000,586800, and 644 400 cycles, respectively. Therefore, it can be stated that the four samples had only fiber debonding and resin cracking as fatigue damage. Since the rest of the samples (with the exception of sample no. 32 explained before) did not show first ply-failure or delamination under fatigue testing, the conclusion is that these samples also had only fiber de-
Table 111. Results of Static (Unfatigued Samples) and Fatigue Tests Obtained from Stress-Strain Curves
25
26 27 28 29 30 31 32
x(o),a
kg/cmz 948 482 496 1567 64 3 975 938 1223 1547 766 736 577
S/PL(0 ) b 0.43 0.92 0.68 0.82 0.57 0.86 0.96 0.68 0.86 0.73 0.95 1.17
,
-e-24 -x-2a
iri-
a The experimental error was i 3% at a 90% confidence level.
sample no. 21 22 23 24
,
60
45
Figure 6. Least-squares fit of residual strength vs. fatigue cycles for samples no. 23, 27, and 31.
Table 11. Maximum Cyclic Stress S for Fatigue Tests. The Minimum Cyclic Stress and the Cyclic Stress Frequency Were Equal to 0.0 kg/cmZ and 1.0 Hz, Respectively, for All Samples S,a kg/cmz
,
N x IO.', c y c i e s
cycles
21 22 23 24 25 26 27 28 29 30 31 32
1
1
180
Figure 5. Least-squares fit of residual strength X(N)vs. fatigue cycles N for samples no. 21 and 22. Sample 32 is a special case (maximum cyclic stress is above proportional limit, see text). For these three samples and for the samples listed in Figures 6,7, _and 8, the experimental error is *IO% (90% confidence level) for X ( 0 ) and an average of f15% (90% confidence level) for the rest of the X ( N ) data points.
sample no.
1
30
S/PL(N) 0.72 1.09 1.00 1.04 0.73 1.01 1.03 0.96 0.80 0.77 1.08 ND e
S / X (0 ) 0.27 0.61 0.66 0.27 0.40 0.44 0.47 0.26 0.20 0.36 0.46 0.70
s / X ( N )d 0.38 0.89 0.70 0.50
0.51 0.83 1.00 0.38 0.41 0.49 0.85 ND e
ply of max tensile stress
GM NFM NFM GM NFM GM NFM NFM GM NFM NFM NFM
Ultimate (that is, static samples) arithmetic mean strength. The experimental error was r 1 0 % at a 90% confidence level. Compare the closeness of values with the ultimate Weibull mean strength and its experimental error (Table V). PL(0)is the value of the proportional limit for unfatigued (static) samples. P L ( N ) is the value of the proportional limit for samples fatigued at N cycles; for this case, N is equal t o the total number of cycles a t which the fatigue test was discontinued (see corresponding figure for each sample). X ( N ) is the arithmetic mean strength for samples fatigued at N cycles, N being equal to the total number of cycles at which the fatigue test was discontinued. e ND, not determined, because of first ply-failure at about 300 000 cycles.
Ind. Eng. Chem. Prod. Res. Dev., Vol. 22, No. 4, 1983 647 N
1
29
-0-
Table IV. Comparison of E ( 0 ) and E. Experimental Errors Were Calculated for a 90% Confidence Level sample no.
I
0 : -
L
N
i .
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.
I
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,
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,
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r-;,,;;-,
3 m
1
I
b-x-x-
w
,
,
, , , ,
,
,
45
90
135
180
N x IO-', c y c l e s
Figure 9. Least-squares fit of residual tensile modulus vs. fatigue cycles for samples no. 21, 25, and 29.
bonding and resin cracking as fatigue damage. Fatigue Degradation Curves and Constancy of the Tensile Modulus. Figures 5-8 show that the shape of strength degradation for samples no. 21-31 can be represented by straight lines reasonably well. Therefore the residual strength X ( N ) after N fatigue cycles is related to the ultimate strength, X ( 0 ) by X ( N ) = X ( 0 ) - mN (1) in which (-m)is the slope of the straight line. Notice that m always has a positive value. Experimental fatigue data (Yang and Liu, 1977) on unnotched composite laminates indicates that the residual strength after N fatigue cycles is a monotonically decreasing function of N in which eq l would be a limit for the set of curves. Sample no. 32 (Figure 5) has a strength degradation shape that cannot be represented by a simple curve. As stated before, at about 300 O00 cycles the specimens suffer first ply-failure. At this point the modulus of elasticity falls rapidly to about 40% of its initial value. However, for the other eleven samples (Table IV and Figure 9) the modulus of elasticity is statistically constant. The 90% confidence intervals of the population mean values of E(0) (static specimens) and E (all the specimens, unfatigued and fatigued) are joint. This implies that the two mean values come from undistinguishable populations. The 95 % confidence intervals would overlap even more. Point Estimation, Weibull Distribution. It is wellknown (Chou and Croman, 1979; Park, 1979, with references therein) that the strength and fatigue life of materials can be best characterized by the Weibull distribution. In this study, the two parameter Weibull distribution F ( x ) = 1 - exp[-(x/P)"]
(x
> 0)
4265 1912 3089 4520 2855 3922 3810 1603 4513 2524 1338 2850
44046 i 25641 i 27 525 i 47 078 i 34 1 5 1 + 4 0 0 7 0 ?r 45490 i 44404 i 47 992 + 30649 + 32459 i
1193 922 2163 2281 1782 3117 1387 901 1896 1097 1298
Table V. Two-Parameter Weibull Distribution. Shape Parameter a ( O ) , Characteristic Ultimate Strength (Scale Parameter) p(O), Coefficient of Variation COV, Ultimate Mean Strength p ( 0 ) . All Data Are for Unfatigued (Static) Samples
sample
20 0
44 654 i 27 315 f 30 192 2 47079 f 33967 + 42970 + 42330 i 45492 i 48426 i 32 115 f 35015 i 31 039 +
kg/cm2
(I is the aril metic mean (average value) of the mo-ulus of elasticity. It includes all the specimens (unfatigued and fatigued) from beginning to end of a fatigue test for a given sample. This sample has a steep decrease of modulus of elasticity at about 300 000 cycles. At 360 000 cycles its modulus is only 1 2 700 kg/cm2.
-X-29 - 0-21 -0-25
0
iI
,
80-
? ,.,
w
21 22 23 24 25 26 27 28 29 30 31 32
E +error:
E ( 0 ) i error, kg/cm2
(2)
2lC 22 23 24c 25 26 27 28 2gC 3OC 31 32
p(O),'"
confidence,b
a(O)(I . ,
kg/cm*
%
COV
kg/cmz
7.43 4.81 25.15 8.79 9.09 6.63 7.87 9.43 5.07 11.52 15.96 3.80
1007 524 498 1652 674 1036 992 1289 1674 794 764 641
95 90 99 95 90 90 95 90 95 95 99 95
0.16 0.23 0.05 0.14 0.13 0.18 0.15 0.12 0.22 0.11 0.09 0.30
946 480 487 1563 639 967 932 1223 1538 7 54 739 57 7
P(0)td
Parameter obtained by the method of linear regression This is for an allowable (Chou and Croman, 1979). error of + l o %for p(0) in all samples. Since the larger the value of a(0)the smaller the value of the sample size estimate, for the present work we used the statistical study already published (Park, 1979). For these samples (chosen at random), the fitted Weibull distribution was acceptable at a significance level of 5% when the Kolmogorov-Smirnov goodness of fit test was used. For the Weibull distribution, the mean is obtained by the expression p(0) = p ( O ) r [ ( a ( O )+ l ) / a ( O ) ] . The error of the ultimate mean strength and its confidence level can be shown that are identical with the corresponding ones of the scale parameter (Park, 1979) by cancellation of r which denotes the gamma function. Table VI. Weibull Parameters for Samples Fatigued at N Cycles p(N) i error, kg/cm2
sample no. a(N) 21a 4.96 952 30a 3.15 561
i i
143 112
confidence, %
95 90
COV 0.23 0.52
N, cycles 200000 590400
The fitted Weibull distribution was acceptable at a significance level of 5% when the Kolmogorov-Smirnov goodness of fit test was used.
where (Y and P are the shape and scale parameters respectively, is used to represent our results. Having collected sample points x , the parameters are then estimated
648
Ind. Eng. Chem. Prod. Res. Dev., Vol. 22, No. 4, 1983 Table VII. Experimental Slope m and Exponential Parameter b. The Errors Were Calculated for a 90% Confidence Level
-
(m r error) X l o 4 , kg/cm* cycle
sample no.
c
c
c:
205
400
603
8EC
1000
1.3 i 2.3 i 4.6 i 16.1 i 1.3 r 8.7 r 13.6 i 3.8 i 10.3 i 5.4 * 5.1 i
21 22 23 24 25 26 27 28 29 30 31
1230
b
i
3.08 3.23 3.07 2.95 3.16 3.03 2.95 3.02 2.84 2.98 3.07
0.27 0.58 4.6 5.96 0.56 1.72 1.72 0.42 0.92 1.96 0.60
error i ?
i t i
*
?
i i t i
0.14 0.15 0.33 0.18 0.18 0.14 0.13 0.08 0.11 0.17 0.09
TtbSILE STRENGTP, < g / c r r *
Figure 10. Weibull distribution fit of: (a) static (ultimate) strength for samples no. 23, 27 (solid line); (b) residual strength for fatigued (590400 cycles) sample no. 30 (dashed line). Each experimental point represents one specimen test.
E ( O i x l O - ' , kg/cm2 Figure 11. Least-squares fit of p(0) vs. E(0). Sample number is shown adjacent to the data point. The confidence level for the experimental error of each data point is given in Table V.
(Tables V and VI with Figure 10). From these results we can conclude that the strength of the static and fatigued samples follow the Weibull distribution. The expected decrease in the value of the shape parameter (after the samples were fatigued) is also observed. Notice that the mean values X ( 0 ) (Table 111) and ~ ( 0(Table ) V) and their respective errors are practically identical. E(0)vs. B(0) Relationship. Figure 11shows that, for static samples, the tensile modulus and the scale parameter relationship can be represented by straight lines. Section A (samples no. 22, 23, 30, 25, 31) represents data points that come only from natural hard fiber laminates, while the corresponding laminates of section B (samples no. 27, 26,21,28,24,29) incorporate at least one ply of glass fibers. Another way to express the mentioned relationship approximately (which will be useful later on) is for section
A E(0)1.52
--
- 10290 kg0.52cm-l.04 P(0) = 4.06 x 1 0 6 NO.52 m-1.04
(3)
m = (3(0)KSb
= 5.595 x 1018 kg3.6938 cm-7.3875
P(0) = 1.530 x 1037 N3.6938 m-7.3875
(4)
Residual Strength Degradation Model. It was mentioned before that eq 1 is a limit for the set of curves which are obtained when the residual strength is a monotonically decreasing function of load cycles N . From statistical considerations, Yang and Liu (1977) derived the equation RC(W= Rc(0)- Bc(0)KSbN (5)
(6)
Since we want to correlate eleven totally different samples, it is necessary to find an expression that easily predicts m (Table VII) in terms of easily defined parameters, preferably related to sample bulk properties and experimental conditions, and if posible, by means of eq 6. The route of micromechanics was not chosen because the inclusion of all the possible variables that are present in actual composites in the solution of the problem would constitute a monumental, if not impossible task. Simplifying assumptions have to be made even for the case of internal stresses in fiber composites, and even so the approximate solutions are somewhat cumbersome (Greszczuk, 1971). After many attempts that will not be described here, eq 6 was transformed into the expression
m=
[
h ] b P ( 0 ) (per fatigue cycle)
(7)
by making K = [l/E(0)lb. Since the maximum cyclic stress is within the composite proportional limit, eq 7 can also be written as
m = tb@(0) (per fatigue cycle)
and for section B E(o)4.6938
This equation expresses the residual strength (per specimen), at N cycles, R(N),in terms of the ultimate strength (per specimen), R(O),the number of load cycles, N , the scale parameter, P(O), and the maximum cyclic stress, S. K , c , and b are constants to be determined from experimental data by the method of moments. Yang and Liu experimentally verified eq 5 by testing specimens from a single panel of graphite/epoxy laminate. They verified the equation by correlating the predicted Weibull distribution of R(0) (from eq 5 ) with the experimental one (static specimens). The agreement was qualified as outstanding. For eq 5 when c = 1 we obtain an expression similar to eq 1 even when in the latter expression arithmetic mean values are involved rather than specimen single values as in eq 5. This implies that for eq 1
(8)
in which t is the deformation (Hooke's law) in the direction of the load S. By use of eq 7 the mean value of b was then calculated for each sample. The respective mean values and errors for the exponential parameter b are given in Table VII. The mean value range for the eleven samples is relatively small. As an example, let us consider what happens with the exponential parameter when eq 7 is changed to m=
[
&]'P(o)
(per fatigue cycle)
(9)
Ind. Eng. Chem. Prod. Res. Dev., Vol. 22, No. 4, 1983 649
Table VIII. Exponential Parameter B . The Error Was Calculated for a 90%Confidence Level sample B f error no. 11.7 f 1.40 25.2 i 6.24 33.4 i 11.70 10.3 i 1.28 15.9 i 2.34 15.7 i 2.51 16.6 i 2.92 10.6 i 1.05 8.6 ? 0.67 13.4 i 1.92 17.6 i 3.00
21 22 23 24 25 26 27 28 29 30 31
c,
$E 550
ULL
-fitted normal distribution functlon
1, 2.50
e x p e c t e d c u r i e at S = 300 k g / c m 2 b = 3.04: 3 06
-t -o5 I
E
0 30
20
40 EIOI, 1 d k g / c m z
50
Figure 13. Tensile modulus dependence of m. Sample number is shown adjacent to the data point. 10
I
__
expected curve a t 5.300 k g / c m z b=3.04:0.06
j
I
21 23
p" a28
, , ,
, J,, , , , , ,
2.75 3.0 0 EXPONENTIAL PARAMETER
~
3.25
b Figure 12. Normal distribution fit of the exponential parameter b. Sample number is shown adjacent to the data point.
Table VI11 gives the respective mean values and errors for B. The mean value range for the eleven samples is relatively large. Additionally, the exponential parameter b (eq 7) has a normal distribution (Figure 12). The KolmogorovSmirnov and x2 goodness of fit tests were used to see how well the normal distribution described the data. The fitted normal distribution was acceptable at significance levels of 1, 5, 10, and 25% for both tests. This implies that the exponential parameter b has a random distribution most probably due to experimental error and a mean value given by b = 3.04 f 0.06 (10) with the error computed at a 90% confidence level. This means that the exponential parameter b is a constant for samples no. 21-31. This is a striking feature when we consider that the samples differ widely in fiber type and content, in number of plies, in applied cyclic stress, in some cases, in resin formulation, in other cases, they have glass and natural fiber ply combinations (Table I). The constancy of b allows us to predict the fatigue behavior of the samples on the basis of some of their bulk properties for a given maximum cyclic stress. This is done by means of eq 1 and 7. Conversely, the prediction of the fatigue behavior of eleven widely different samples proves that the mathematical model for fatigue given by eq 1 and 7 is correct, or at least reasonable. Figures 13 and 14 show the tensile modulus and scale parameter dependence of m. The two curves were obtained by means of eq 7 with the help of the least-squares fit of Figure 11. Each curve can be divided into two sections at about a tensile modulus of 40000 kg/cm2 or a scale parameter of 900 kg/cm2. The first section (samples no. 22, 23, 30, 25, 31) represents data points that come only from natural hard fiber laminates, and its slope is negative. The second section (samples no. 27,26,21, 28,24,29) have laminates that incorporate at least one ply of glass fibers, and its slope is slightly positive. These special correlations can be rationalized by the following facts and considera-
500
1000
1500
p@), k g / c m z Figure 14. Scale parameter dependence of m. Sample number is shown adjacent to the data point.
tions. In the next subsection they will be discussed again with the help of linear elastic fracture mechanics. (A) The natural fibers used in this study have a diameter of about 200 pm and a rugged lateral surface (Figures 1 and 2). This surface favors mechanical bonding with the matrix, that is, the type of interlocking that occurs as a consequence of the geometrical shape of the surface. To make the composites, the fibers are coated with a binder (Table I) which is also an interface agent (Belmares et al., 1981). The glass fibers have a diameter of about 10-12 pm and a smooth surface (White and Czarnecki, 1980) coated with silanes to promote chemical bonding with the matrix. The natural fibers, having a larger diameter than the glass fibers, have less space coverage in the composite at a given fiber volume. This means that the natural fibers leave larger spaces of matrix resin in which a crack propagates before finding a fiber. However, as the tensile modulus or the scale parameter increase (by either increasing the number of plies, the fiber content, or by favorably changing the matrix resin composition) the effect of the fiber-free matrix resin space on fatigue strength tends to decrease. The negative slope of the curves in Figures 13 and 14 is probably due in part to this effect. At equal natural fiber content and matrix resin composition, the effect is particularly a function of the thickness (number of plies) of the specimen. The influence of the matrix resin mobility (decrease) by the intersected fibers (at an acute angle) has been published (Kodama, 1976). (B) A t a tensile modulus of about 40000 kg/cm2 or a scale parameter equal to 900 kg/cm2 the natural fiber laminates and the glass fiber containing laminates are practically equivalent in fatigue strength in such a way that their differences dissappear or cancel out. (C) In principle, fracture toughness (K,) of fibrous composites is improved by large diameter fibers as was shown in a consolidated and improved theoretical treatment of fracture toughness mechanisms (Piggott, 1970). As a first approximationwe can say that fracture toughness and fatigue strength are directly proportional; that is, the larger the value of fracture toughness the smaller the value of m. Therefore the natural fiber large diameter and rugged surface help to explain in part that the palm fiber
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laminates and the glass fiber containing laminates become eventually equivalent in fatigue strength even when the hydrophilic natural fiber is expected to have very little affinity for the hydrophobic polyester resin. In one extreme, in the absence of bonding, the fibers would act as voids and therefore would be stress concentrations, thereby greatly diminishing the tensile and fatigue strength of the composite. In the other extreme, for very high bond strengths, the composite material will behave as a homogeneous brittle solid. Apparently for all the samples of the present work, we are between the two extremes in bond strength. (D) Above a tensile modulus of 40000 kg/cm2 or a scale parameter of 900 kg/cm2, the fatigue strength curve of the composites becomes slightly positive. This effect is probably due in part to a decrease of the fracture toughness with increasing beam thickness in fiber glass reinforced polyester composites. The effect has been reported to occur in the study of interlaminar fracture energy for the mentioned type of composites especially for thin plates (less than 0.5 in. height). As the plate becomes thicker, the viscoelastic-plastic bending beam energy component of the total interlaminar fracture energy becomes smaller with the consequent decrease of the latter. Width tapered double cantilever beam specimens were used for the study (Han and Koutsky, 1981), with references therein). It is also well known that the fracture toughness of a metal is higher if the measurement is made on a thin plate than if it is made on a thick plate (Bucknall, 1978). As an interesting point, for samples no. 26 and 27, which have approximately the same composition (Table I); it makes no difference in fatigue strength (Figures 13 and 14) which ply (GM or NFM, Table 111) is subject to tensile stress. This shows that a t a given S , the fatigue strength is only a function of the tensile modulus and the scale parameter (eq 7) even for such a complex case as this. (E) For a wide variety of fiber glass contents in polyester or epoxy matrices (edge notched specimens, square lay-up) McGarry and Mandell (1970) found that the fracture toughness increases as the strength and the modulus increase. This is in agreement with the results shown in Figures 13 and 14 for the section of curves that have negative slope. Combination of Linear Elastic Fracture Mechanics (LEFM) and the New Fatigue Degradation Model. Due to the mechanical properties of the composite materials of the present work, the use of LEFM in an appropriate manner is well justified. Moreover, using the basis of LEFM for width tapered double cantilever beam specimens (glass fiber reinforced polyester laminates), Han and Koutsky (1981) developed the equations relating the strain energy release rate (fracture energy GIc) as a function of the critical load for crack propagation in a specimen of constant beam height but of varying width. The energy release rate or fracture energy was defined as the rate of change of the stored strain energy of the beam with a change in the fractured bond area. For a wide and relatively thick plate containing a central crack of length a , the Griffith energy-balance criterion (including plastic work) predicts catastrophic fracture when the stress o reaches a critical value (r, given by CT,' = E ( O ) G I c / r a ( I- J J ~ ) (in plane strain) (11) where E(0)is tensile modulus, Y is Poisson's ratio, and GI, is the critical strain energy release rate, generally measured in joules per square meter of the crack (Bucknall, 1978). This equation is the basis of LEFM. In the present work, the specimens are fatigued under an applied stress that causes debonding and cracking
(macro and micro) of the composite material. The phenomenon to a degree is related to the early stages of the failure of polymeric materials before catastrophic fracture occurs. Basic positions of the micromechanics of the latter process have been formulated (Frolov, 1981). Underlying the failure is the multiple accumulation of microcracks. On reaching those concentrations of microcracks for which the distance between them becomes commensurate with the dimensions of the cracks themselves, conditions arise for their interaction and fusion; this results in the appearance of larger cracks. Appearance of the latter in the final stage of failure has been confirmed experimentally. In the present work after the composites are fatigued, they are tested to determine their mechanical properties. It is assumed that at this stage, eq 11 is a good approximation where g, = X ( N ) , and a is the crack length or its equivalent at which the material suffers catastrophic failure at a stress X ( N ) . Substituting eq 11 into eq 1, we obtain
The larger the value of the fracture energy the smaller the value of m, that is, fracture toughness and fatigue strength are directly proportional as it was assumed in the previous subsection. In this fatigue test, the maximum cyclic stress S is kept constant for a given sample. By transposing N to the left side of eq 12, substituting eq 7 into eq 12, finding the derivative of a with respect to N , and then by making use of eq 3, eq 4 (in SI units), and eq 10, we obtain for samples below a tensile modulus of 3.92 X lo9 N/m2 (40000 kg/cm2) (Figures 11 and 13)
and for samples above a tensile modulus of 3.92 N/m2 (Figures 11 and 13)
X
lo9
where d a / W is the crack intensity of growth per cycle during the fatigue test of the mentioned samples. A comparison of eq 13 and 14 shows that the tensile modulus dependence of the crack intensity of growth is different. The equations also show that when the value of the fracture energy decreases, then the value of the crack intensity of growth increases. All this helps one to understand quantitatively the statements made in the previous subsection when Figures 13 and 14 were discussed. At the transition, eq 13 and 14 become equal to each other. At this point, the calculated transitional tensile modulus is about 4.41 X lo9 N/m2 (45000 kg/cm2). Other Correlations of the New Fatigue Model. (a) In several metals such as stainless steel, copper, aluminum, nickel, titanium, and others, the fatigue life under high stress is expressed by (Guy, 1980) N = Ccp-"
(15)
where cp is the plastic deformation, n is an exponent that has a value of 2, and C is a constant that increases with increasing values of metal ductility. The fatigue test is carried until the specimen fails (100% tensile loss). For the present work, if the fatigue tests are always stopped at a fixed tensile loss, let us say, as an example, 40% of the ultimate tensile strength X ( O ) , or better, of the
Ind. Eng. Chem. Prod. Res. Dev., Vol. 22, No. 4, 1983 651
100
I
than fiber flass laminates of equivalent thickness. At greater thickness, the natural hard fiber laminates become equivalent to the glass fiber laminates in fatigue strength degradation. The new mathematical model for fatigue strength degrdation has a good predictive value. Acknowledgment
____ 22 21 6
-2
I
O
150
300 S,
450
kg/cm’
Figure 15.
P r e d i c t e d relationship between t h e maximum cyclic stress and t h e number o f fatigue cycles to b r i n g t h e following tensile losses: sample no. 21, 26.6%;sample no. 22, 37.1%; sample n o 26, 48.0%. T h e respective values f o r t h e exponential parameter were t a k e n f r o m T a b l e VII.
Weibull ultimate mean strength p ( 0 ) (Tables I11 and V), then from eq 1, mN = 0.4 ~(0).Substituting the latter expression into eq 8, and making use of the Weibull expression for ~ ( 0(Table ) V), we obtain
N = 0.4r[l/a(O)
+l]~-~
(16)
which is of the same form as eq 15. The constant in eq 16 depends on the shape parameter a(O),which is directly related to the scattering of the specimens ultimate tensile strength. A related type of dependence on the shape parameter has already been reported (Chou and Croman, 1979). In principle, it should be possible to find an expression for the constant C of eq 15 by using eq 16 as a starting point and then introducing an appropriate ductivility function. For eq 15, typically C = 4-7. For eq 16 the constant has a value of about 0.4. (b) Fatigue tests were conducted on thin-walled tubular specimens of graphite/epoxy in combinations of axial, torsional, and internal pressure loading (Francis et al., 1977). The best fit of data points gave an equation of the form (tubular specimens notched in the center) S = AN-T
(17)
where A and T are constants, S is the maximum cyclic stress, and N is the number of cycles to failure. Typical values of T are in the range 0.041-0.060. In the present work, if the expression for E is used (eq 7), we obtain for a given sample (from eq 16)
s = O . ~ ~ / ~ E ( O ) [ ~ (+I /1)]14v-1/b ~(O)
(18)
in which l / b = 0.3289. This expression is similar in form to eq 17. Interestingly, in the fatigue of unnotched graphite/epoxy laminates l / b = 0.056 (eq 5) (Yang and Liu, 1977), in the range of T (eq 17). (c) Finally, Figure 15 shows the predicted relationship between the maximum cyclic stress and the number of fatigue cycles. Notice that for low values of applied maximum cyclic stress there is what is called a fatigue limit and the samples are in a safety range, practically away from fatigue degradation. This is a common phenomenon for many materials. Thus, for a glass fiber/phenolic resin composite, the fatigue limit is at an applied stress equivalent to about 20% of the ultimate tensile strength (Guy, 1980).
Conclusions Thin laminates (less than 2.4 mm) of natural hard fibers present a fatigue strength degradation somewhat lower
The authors wish to thank Prof. James L. White and Mr. Hernan Menendez of the Polymer Engineering Department, The University of Tennessee, for the scanning electron micrographs. We also thank the Consejo Nacional de Ciencia y Tecnologia (CONACYT) and the Comision Nacional de Zonas Aridas (CONAZA) for the grants that supported this work. Nomenclature N = number of fatigue cycles S_ = maximum cyclic stress X ( 0 ) = ultimate (that is, static samples) arithmetic mean strength X ( N ) = arithmetic mean strength for samples fatigued at N cycles PL(0) = value of the proportional limit for static samples PL(N) = value of the proportional limit for samples fatigued at N cycles E(0) = modulus of elasticity, arithmetic mean for static samples E = modulus of elasticity, arithmetic mean that includes all the specimens (fatigued and unfatigued) from beginning to end of a fatigue test, for a given sample b, B , c , T, n = exponential parameters K , C, A = constants -m = slope of the strength degradation straight lines R(0) = ultimate strength per specimen R(N) = strength per specimen fatigued at N cycles a = crack length da/dN = crack intensity of growth per cycle (crack growth rate) K , = fracture toughness GIc = fracture energy in joules per square meter of the crack (energy release rate) Greek Letters ~ ( 0= ) Weibull ultimate mean strength a(0) = shape parameter for static samples @(O) = scale parameter or Weibull characteristic ultimate strength a ( N ) = shape parameter for samples fatigued at N cycles @ ( N )= scale parameter for samples fatigueed at N cycles t = deformation (Hooke’s law) in the direction of the load S tp = plastic deformation for metals 7r = constant equal to 3.1416 v = Poisson’s ratio r = gamma function a, = critical stress value for catastrophic failure in the Griffith energy-balance criterion Literature Cited Barkakaty, 8. C. J. Appl. Polym. Sci. 1976, 2 0 , 2921. Belmares, H.;Castlllo, E.; Barrera, A. Text. Res. J. 1979, 49, 619. Belmares, H.; Barrera, A.; Castlllo, E.; Verheugen, E.; Monjaras, M.; Patfoort, G.A.; Bucquoye, E. N. I d . Eng. Chem. Rod. Res. Dev. 1961, 20, 555. Bucknall, C. B. “Fracture Phenomena in Polymer Blends”; I n Paul, D. R.; Newman, S. “Polymer Blends”; Vol. 2; Academic Press: New York, 1978; Chapter 14. Chou, P. Ch.; Croman, R. J. Compos. Mater. 1979, 13, 178. Dally, J. W.; Broutman, L. J.; J. Compos. Mater. 1967, I , 424. Francis, P. H.; Walrath, D. E.; Sims, D. F.; Weed, D. N. J. Compos. Mater. 1977, 11, 446. Frolov, D. I.; Kil’keev, R. Sh.; Kuksenko, V. S. Me&. Compos. Mater. 1981, 17(1), 104. Greszczuk, L. B. AIAA J. 1971, 9(7), 1274. Guy, A. G. “Fundamentos de Clencia de Materiales” (Translation from “Essentials of Material Science”), McGraw-Hill: New York, 1980; Chapters 4 and 8. Hahn, H. T.; Kim, R. Y.; J. Compos. Mater. 1978, 10. 156. Han, K. S.;Koutsky, J. J. Compos. Mater. 1981, 15, 371. Kodama, M. J. Appl. Polym. Sci. 1976, 2 0 , 2165.
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Lubin, G. "Handbook of Fiberglass and Advanced Composites"; Society of Plastics Engineers: Van Nostrand-Reinhold Co.: New York, 1969;Appendix, p 873. McGarry, F. J.; Mandeli, J. F. "Fracture Toughness of Fiber Reinforced Composites", M.I.T. Civil Eng. Report R70-79,Dec 1970. Park, W. J. J. Compos. Mater. 1979, 73, 219. Piggott, M. R. J. Mater. Sd.1970, 5 , 669. Rieke, J.; Bhateja, S.; Andrews, E. Ind. Eng. Chem. prod. Res. Dev. 1980,
Sines, G. "Recent Advances in Composite Materials": UCLA course 853.15, Mar 1972. WhRe, J. L.: Czarnecki, L. J. Appl. Pdym. S d . 1980, 25, 1217. Whittaker, R. E. J . Appl. folym. Sd.1974, 78, 2339. Yang, J. N.; Liu, M. D. J. Compos. Mater. 1977, 1 7 , 176.
Received for reuieu, December 14, 1982 Accepted June 6, 1983
19, 601.
New Composite Materials from Natural Hard Fibers. 3. Biodeterioration Kinetics and Mechanism Hector Belmares, Arnold0 Barrera, and Margarlta MonJaras Centro de Investigacion en Quimica Aplicada, Saltillo, Coahuila, Mexico
The effect of soil burial biodeterioration on natural hard fibers, natural hard fiber composites, and glass fiber composites was studied. First-order kinetics of the tensile strength degradation rate was found. A comparison of tensile strength half-life values was made for a variety of composites. An inverse relation between the tensile strength half-life values and the hydrophilicity of the matrix was found. The mechanism of biodeterioration involves water transport to the natural hard fiber, fiber swelling, slight matrix cracking, and finally, massive microorganism penetration to the fiber. Means to avoid water penetration and to protect the fiber from biodegradation were developed. Therefore, important disadvantages of the natural hard fiber composites have been overcome. Additionally, the wide angle X-ray diffraction pattern of palm fibers is essentially due to a crystalline lattice of cellulose I, in agreement with reported work on other lignocellulosic vegetable fibers.
Introduction Renewable resources are regaining the interest of many nations in a programmed search to find economically useful industrial applications. Natural hard fibers are one example of such a programmed search (Belmares et al., 1979a,b). A recent paper by us (Belmares et al., 1981, with references therein) describes the use of such fibers to develop composite materials with a favorable strength/price ratio. The studies were done with palm fibers (Yucca carnerosana) which were demonstrated to be similar in physical, mechanical, and chemical properties to other natural hard fibers in the world. Those findings could help to alleviate the worldwide scarcity of low-cost building materials and the market slump in which the natural hard fibers have been for several decades. The present paper describes the biodeterioration of natural hard fiber composites, its kinetics and mechanism, and ways to avoid it. This work is a continuation of our programmed studies of the mentioned composite materials. The behavior of the composites under cyclic stress fatigue degradation has already been reported (Belmares et al., 1983). Experimental Section Sample Composition and Preparation. Table I shows the composition of samples. A recent paper (Belmares et al., 1981) describes the natural hard fibers obtainment, procedures and conditions for polyester (PER) laminates manufacture, and their testing to determine tensile properties. Additional Equipment and Wide Angle X-ray Diffraction (WAXS) Pattern. The WAXS palm fiber patterns were made with nickel filtered Cu K a radiation. The diffraction patterns were recorded on flat film using a Rigaku-General Electric rotating anode X-ray generator. The sample-to-film distance was 3.00 cm. The X-ray unit was operated at 30 kV and 20 mA. Figure 1shows a typical
WAXS palm fiber pattern. In agreement with reported work on other lignocellulosicvegetable fibers including sisal (Barkakaty, 1976, with references therein), the X-ray pattern of the palm fiber is essentially due to a crystalline lattice of cellulose I. The pattern includes the (101), (lOT), and (002) reflections (Ray, 1968; Ray and Montague, 1977). Measurements and Test Methods. The soil burial test is a severe test; it exposes the materials to a broad spectrum of destructive organisms and it is a reasonable simulation of the field use (Turner, 1972). This test was performed by burying the samples (composite materials or fibers) in a matured compost of loam and grass clippings. The soil was always kept at 80% (by wt) of its water saturation point. This was done by weighing the soil each week and adding any missing water. Incubation conditions were 30 f 2 "C and 80-90% relative humidity. Before the start of a burial test, the matured compost was statistically "calibrated" in its biodeteriorating effects by the use of palm fibers. Figure 2 shows a typical result. Less tensile strength than the lowest point makes the fibers unmanageable. The effectivity of the compost is comparable to other more complex preparations that include manure (Kulkarni, 1963). We believe that the present soil preparation minimizes variations in the repeatability of the burial test. Turner (1972) has shown statistically the importance of controlling moisture content and apparent density of the soil bed. Considerable local variations in both parameters will give place to local intensification or abatement of attack, thus damaging the repeatability of the soil burial test. For the present work, the samples used for the burial test were 30 X 30 cm laminates with the edges coated with a resin formulation PER/Sty = 60:40. The coating was dried for 5 days and the test laminates were then immersed in the soil with care being taken that they did not touch each other. At each specified time, one laminate per
0196-4321/83/1222-0652$01.50/00 1983 American Chemical Society
