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Effects of geometry and specimen size on out-of-plane tensile strength of aligned CFRP determined by direct tensile method
Composites: Part A 41 (2010) 1425–1433
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Composites: Part A journal homepage: www.elsevier.com/locate/compositesa
Effects of geometry and specimen size on out-of-plane tensile strength of aligned CFRP determined by direct tensile method Eiichi Hara a,b,*, Tomohiro Yokozeki c, Hiroshi Hatta d, Takashi Ishikawa e, Yutaka Iwahori f a
Foundation for Promotion of Japanese Aerospace Technology (JAST), 6-13-1, Ohsawa, Mitaka-shi, Tokyo 181-0015, Japan Department of Space and Astronautical Science, The Graduate University for Advanced Studies, 3-1-1 Yoshinodai, Sagamihara-shi, Kanagawa 229-8510, Japan c Department of Aeronautics and Astronautics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan d Institute of Space and Astronautical Science (ISAS), Japan Aerospace Exploration Agency (JAXA), 3-1-1, Yoshinodai, Sagamihara, Kanagawa 229-8510, Japan e Japan Aerospace Exploration Agency (JAXA), 6-13-1, Ohsawa, Mitaka-shi, Tokyo 181-0015, Japan f Aerospace Research and Development Directorate (ARD), Japan Aerospace Exploration Agency (JAXA), 6-13-1, Ohsawa, Mitaka-shi, Tokyo 181-0015, Japan b
a r t i c l e
i n f o
Article history: Received 18 August 2009 Received in revised form 14 May 2010 Accepted 15 June 2010
Keywords: A. Polymer-matrix composites (PMCs) B. Strength C. Statistical properties/methods D. Mechanical testing
a b s t r a c t The out-of-plane tensile strength of CFRP laminate determined by the direct tensile method varies with specimen geometry and size. This effect was first experimentally observed using aligned CFRP. To explain the geometry and size effects from a mechanical point of view, an analytical model combining Weibull statistics, including the concept of effective volume, and a fracture criterion under multi-axial loading was constructed on the basis of stress distributions calculated using the finite element method. The predicted out-of-plane tensile strength of aligned CFRP was found to be consistent with experimental results. Thus, the present model is useful for reducing experimentally determined out-of-plane tensile strength under complex stress distributions to that under a uniaxial and uniform stress distribution. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction The out-of-plane tensile strength of laminates made of carbon fiber reinforced plastics (CFRPs) is a variable whose accurate value is increasingly important in the design of laminate structures, especially those with a thick cross-section that must bear multiaxial loading. However, tensile loading on a laminated CFRP in the thickness direction is generally not easy because sufficient length for tensile loading is usually difficult to secure. In order to overcome this difficulty, several researchers have attempted tensile loading by bonding the sample to the test fixture using, for example, the test configuration shown in Fig. 1 [1–4]. On the basis of these results, an out-of-plane test method (ASTM D 7291) was standardized by the ASTM D 30 Committee in 2007 [5]. A standardized test measuring the strength of a material generally requires a uniform and uniaxial stress state in the gage section of a specimen. In contrast, the above procedure for the determination of the out-of-plane tensile strength as a standardized test method generates the following undesirable stress distribution [3].
* Corresponding author at: Department of Space and Astronautical Science, The Graduate University for Advanced Studies, 3-1-1 Yoshinodai, Sagamihara-shi, Kanagawa 229-8510, Japan. Tel.: +81 422 40 1366; fax: +81 422 40 3549. E-mail address: [email protected] (E. Hara). 1359-835X/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.compositesa.2010.06.003
(1) Stress distribution is not uniform, and stress concentration is locally generated. This means that fracture load depends on the maximum stress induced at the point of the stress concentration. (2) The above stress distribution gives rise to fracture stress that varies with the geometry and size of the specimen. (3) In addition, multi-axial stress is induced in spite of singleaxial strain loading. Therefore, the meaning of the resulting fracture stress from an out-of-plane tensile test is unclear and it is difficult to use the data obtained in crafting a structural design. In order to evaluate the effects of size on strength, Weibull statistics has often been used with success for fiber reinforced plastics (FRPs) [6–8]. Weibull statistics can be useful not only for specimens in which stress is uniformly distributed but also for those in which stress is unevenly distributed, by introducing the concept of effective volume. This concept makes it possible to treat situations where stress concentration is mild. In the present paper, the following three factors were introduced to overcome the above-mentioned three difficulties: (1) Stress ratio RZ (the maximum out-of-plane stress rz-max divided by mean out-of-plane stress rz-mean) to evaluate stress concentration.
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load
Pivoting joint
Fig. 2. Configuration of spool test specimens with varying waist radius r, minimum diameter d, and thickness t.
Steel end tab strength tests. This CFRP was fabricated from a unidirectionally reinforced prepreg made of reinforcing fiber IM600 and epoxy matrix #133. UD-CFRP plates were fabricated by unidirectionally laminating 120 plies of the prepreg sheets and curing at 180 °C. The resulting UD-CFRP plates were approximately 17.4 mm thick.
Specimen
2.2. Specimens
load Fig. 1. Test setup for out-of-plane tensile test.
(2) Weibull statistics to evaluate the geometry and size effects. (3) Multi-axial strength criterion to evaluate the influence of a multi-axial stress state. By constructing a model combining these factors, experimentally determined out-of-plane tensile strengths under complex stress distributions can be made comparable to those under a uniaxial and uniform stress distribution. Thus, an attempt is made to derive a material fracture property independent of stress concentration, specimen size and geometry. The reason for defining stress ratio was for modifying difference between apparent strength and stress concentration. Only apparent strength can be evaluated experimentally. However, apparent strength is undervalued because of stress concentration.
Spool and cylinder configurations were selected for the out-ofplane tensile tests. Spool specimens (Fig. 2) were chosen to secure a fracture location near the specimen center. The spool specimens had a minimum diameter (d) at the mid-plane and a maximum diameter (D) of 25 mm. As shown in Table 1, 13 types of spool specimens were fabricated by changing d, waist radius r and thickness t to examine specimen size and geometry effects. In the fabrication of specimens with different thicknesses, the original plates, t = 17.4 mm, were ground symmetrically from both surfaces to place the fracture location at the center of the sample’s thickness. Eight types of cylindrical specimens were also prepared to examine the possibility of using a simple geometry for the outof-plane tests. The cylindrical specimens had a constant diameter of 25 mm and thicknesses of t = 1, 2, 3, 4, 5, 8, 12, and 16 mm. For later convenience, Cartesian and cylindrical coordinates were introduced as shown in Fig. 3. Both coordinates have their origin at the center of the specimen, the fiber direction coincides with the x axis, h = 0°, and the loading axis is parallel to the z axis. The mid-plane is defined as a symmetrical plane perpendicular to the loading axis.
2. Experiments 2.3. Experimental procedure 2.1. Material In order to simplify test conditions, a unidirectionally reinforced composite was chosen as the material for the out-of-plane tensile
The out-of-plane tensile tests were carried out on the basis of ASTM D 7291. Fig. 1 shows the setup of a specimen and test fixture. The tests were performed with the following procedure:
Table 1 Configurations of spool specimens and their statistical parameters. Thickness (t) (mm)
Minimum diameter (d) (mm)
Waisting radius (r) (mm)
Number of specimens
Weibull modulus (m) (–)
Effective volume (mz: 12.68) (Veff) (mm3)
Effective volume (mz: 14.87) (Veff) (mm3)
4 4 4 4 8 8 8 8 16 16 16 16 16
3 6.25 12.5 18.75 3 6.25 12.5 18.75 2 3 6.25 12.5 18.75
1 1 1 1 3 3 3 3 7 7 7 7 7
5 6 6 6 5 3 5 4 7 7 10 10 9
– 6.13 11.71 4.76 – – – – 23.79 11.35 13.88 15.5 10.94
0.232 0.495 0.995 1.344 0.741 2.124 3.536 5.458 1.711 2.396 9.393 18.10 21.95
0.580 1.611 2.384 3.416 1.396 1.832 6.985 12.38 14.07
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Cylindrical Tensile displacement
Fiber direction ( 0o)
Adhesive t = 0.1 mm Specimen
θ =90o
Thickness, t z x
y
θ
measurement location Mid-plane
θ =0o measurement location
Fig. 3. Coordinate system, showing origin at the center of the specimen, fiber direction and definition of the mid-plane.
0.5 mm. The thickness of the adhesive layer elements was set to 0.1 mm. In addition to the specimens prepared for the experiments, cylindrical models with specimen thicknesses of 25 and 50 mm were also produced and their stress distributions were calculated. Most of the elastic moduli of the CFRP used for finite element calculations were obtained on the basis of American testing standards as shown in Table 2. However, the in-plane shear modulus (Gxy) was taken from the Advanced Composites Database System published by the Japan Aerospace Exploration Agency [10]. In addition, the out-of-plane elastic modulus Ez was assumed to be equal to the in-plane transverse modulus Ey. The properties of the adhesive layer were also determined by tensile tests using coupon specimens made of only adhesive material fabricated by ourselves; Young’s modulus and Poisson’s ratio were found to be 2.75 GPa and 0.33, respectively. 3.2. Effective volume
(1) Bonding a specimen to two end-tab-blocks using a film adhesive of a 120 °C curing type, AF163-2K (film thickness: 0.1 mm) supplied by 3 M, USA. (2) Assembling specimen/end-tab-blocks (axial adjustment fixture) and end-plates with pivoting joint. (3) Attaching the assembled specimen to two grips of a testing machine (INSTRON 8500) and (4) Tensile loading at a constant crosshead displacement rate of 0.1 mm/min.
The effective volumes under Weibull statistics were estimated by the following procedure. Experimental results were assumed to obey the cumulative failure probability Q(r) [9].
m r ; QðrÞ ¼ 1 exp
ð1Þ
r0
r0 ¼ r0
1=m V eff ; V0
ð2Þ
In some of the tests on cylindrical specimens, eight strain gages were glued on the side-surface at the mid-plane with a spacing of 45°, and strain distribution was measured under loading. Apparent out-of-plane tensile strengths Faps were calculated from the fracture force divided by the area in the mid-plane, and Weibull modulus m was estimated from the variation of Faps. The estimation of m requires the cumulative failure probability Q(r). The median-rank method and the least-squares method were adopted for the calculation of Q(r). The detail procedure of the estimation of m is given, for example, in reference [9]. For adoption of a fracture criterion effective for multi-axial stress fields, two tensile strengths among the tri-axial strengths (other than the strength in the loading direction) were required. Experiments for these estimations are documented in Appendix A.
where C is the gamma function. In Eqs. (2) and (3), Veff and V0 are volumes subjected to uniform stress and showing average strengths equal to rm and r0, respectively. The effective volume Veff is useful for estimation of failure probability Q when r is not uniform. Davies defined Veff with Eq. (4) for a structure possessing a non-uniform stress distribution [11].
3. Analysis
V eff ¼
3.1. Finite element analysis (FEA) The stress distributions generated in specimens were estimated using a commercial finite element analysis program, ABAQUS 6, under the assumption of a linear elastic response. The finite element models include a half of the specimen with an adhesive layer, while the mid-plane of the specimen is constrained in the z-direction, considering the symmetry of the loading condition and specimen geometry. The end-tab-blocks were assumed to be a rigid body. Thus, uniform out-of-plane tensile displacements were given on the surface of the adhesive layer. Four-node tetrahedral elements were used for the models of the spool specimens. The thickness of the elements of the spool specimens was 0.145 mm, and in-plane edge lengths of elements were about 0.3 mm. Eight-node hexahedral elements with reduced integration were used for finite element models of the cylindrical specimens. The reduced integration generates a representative stress tensor for every element; in the present case, average values were given. The thickness of the elements of the cylindrical specimens was 0.145 mm, and the in-plane edge lengths were about
where Veff and V0 are volumes subjected to uniform stress and showing average strengths equal to rm and r0, respectively, and r0 and m are the scale parameter and Weibull modulus, respectively. The average strength rm obeying the Weibull distribution is given by,
rm ¼ r0 ðV eff =V 0 Þ1=m Cð1 þ 1=mÞ;
Z V
r rmax
m
ð3Þ
dV;
ð4Þ
where rmax is the maximum stress. When stress r is uniformly distributed, r/rmax is unity. Then, Veff is the specimen volume itself (cf. Eq. (A1) in Appendix A). Veff is given by a simple equation when Table 2 Elastic moduli of IM600/#133 CFRP and their test methods. Property
IM600/ #133
Test method
Ex (GPa) EY (GPa) GXY (GPa)
152 8.21 4.36
SACMA SRM 4R SACMA SRM 4R SACMA SRM 7R
7 6 5
mXY
0.334 8.21
7 – 10 10
GXZ (GPa) Gyz (GPa)
3.993
SACMA SRM 4R Assumption: Ez = Ey ASTM D 5379
2.516
ASTM D 5379
mXZ mYZ
0.346 0.536
SACMA SRM 4R SACMA SRM 4R
Ez (GPa)
Specimens
7 6
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rmax and r are given by simple equations. For example, in the case of 3-point-bending, Veff is given by Eq. (A2). For the spool and cylindrical specimens, Veff was estimated from Eq. (4) using the stress distribution obtained by finite element analysis. For this purpose, Eq. (4) was discretized as Eq. (5) in accordance with the finite element model. V eff ¼
n X
rzi rz- max
i¼1
m
DV i ;
ð5Þ
where n represents element numbers in the finite element model of the specimen, rzi is out-of-plane normal-stress in the ith element, rz-max is the maximum of rzi in the whole model, and DVi is a volume of the ith element. For two specimens consisting of the same material having different effective volumes Veff-1 and Veff-2 showing respective strengths of F1 and F2, the relation of Eq. (6) can be derived from Eq. (4).
F 1 ¼ F 2 ðV eff-1 =V eff-2 Þ1=m :
ð6Þ
3.3. Failure criterion for the multi-axial stress condition A multi-axial stress state should be generated in the out-ofplane test specimens under tensile loading. In order to estimate the influence of multi-axial stress on strength, the criterion expressed by Eq. (7) was examined.
rx
2
þ
Fx
ry
2
þ
Fy
rz
2
Fz
þ
sxy
2
F xy
þ
syz
2
F yz
þ
szx
F zx
2
¼ 1;
ð7Þ
where normal stresses in the fiber axis, in-plane transverse, and thickness directions are denoted by rx, ry, and rz, respectively, shear stresses by sxy, syz, and szx, tensile strengths under the uniaxial normal stress by Fx Fy, and Fz, and pure shear strengths by Fxy, Fyz, and Fzx. This equation was chosen due to simplicity of the following calculations. Since the composite examined in this study is unidirectionally reinforced, the first term of Eq. (7) is negligibly small, because Fx (=2700 MPa) is more than 10 times as large as other strengths Fy, Fz (60 MPa) and Fxy, Fyz, Fzx (50 MPa) in the present experiments. In addition, the terms related to shear stresses were neglected because the ratios of the shear stresses, sxy, syz, and szx, to respective shear strengths near the area appearing maximum stress rz-max were found to be sufficiently small. Thus, Eq. (7) reduces to
ry Fy
2
þ
rz Fz
2 ¼ 1:
ð8Þ
Then, Eq. (8) is rearranged to give
rz
F z ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 1 ðry =F y Þ2
when the direction coincides with the fiber axis. As this figure shows, ez varies with h. The curves in the figure are calculated results from the finite element method. The curves are consistent with experimental values. This result shows that tensile load is appropriately applied to a specimen in the present tests. Fig. 5a and b shows contours of out-of-plane tensile stress rz obtained by finite element analyses under an out-of-plane tensile load applied to spool and cylindrical specimens. In this figure, red regions represent maximum stresses.1 Thus, the maximum stress rz-max of a spool specimen appears at the h = 0° position on the side-surface of minimum diameter, and that of a cylindrical specimen at the h = 90° position near the adhesive layer. In order to evaluate stress concentration, the stress ratio Rz, maximum stress rz-max divided by mean stress rz-mean, was calculated. Figs. 6 and 7 show Rzs of the spool and cylindrical specimens. It follows from these figures that rz-max is relaxed by selecting smaller d for a spool specimen and thinner thickness t for a cylindrical specimen. In the tensile test shown in Fig. 1, multi-axial stresses were induced in spite of uniaxial loading. Hence, the fracture criterion based on multi-axial stresses should be used in this study. Tables 3 and 4 show maxima of stress components other than the loading stress rz divided by rz-max of the spool and cylindrical specimens. Table 3 shows that a near uniaxial stress state is attained for a spool specimen, especially when minimum diameter d is small. In contrast, Table 4 shows that a multi-axial stress state due to a high value of rx is notable for a cylindrical specimen. This tendency is especially noteworthy for thinner cylindrical specimens.
ð9Þ
Using this equation, out-of-plane strength Fz under uniaxial loading can be estimated, provided that rz and ry at failure point and inplane transverse strength Fy are given. 4. Experimental results 4.1. Stress distributions and fracture surfaces 4.1.1. Stress distributions Fig. 4 shows distributions of tensile strain ez along the line on which the mid-plane and the side-surface intersect under average tensile stresses rz-means of 5, 15, and 30 MPa. The strains in this figure were measured using strain gages 2 mm long, and the horizontal axis represents the measuring location where h was set to be 0°
4.1.2. Fracture surfaces All the specimens fractured instantaneously in a brittle manner, and the spool specimens fractured near their minimum cross-section as was expected. In contrast, the cylindrical specimens were broken near adhesive lamina, so the fracture surfaces of the cylindrical specimens were examined. Fig. 8 shows typical fracture patterns of the cylindrical specimens. Fig. 8b and c schematically shows two typical fracture patterns, (b) the entire fracture surface of a specimen and (c) a portion of the specimen fracture surface and part of the interface between the adhesive layer and specimen. Fig. 8d and e are photographs of these fracture modes, where the white arrows indicate fiber direction. In these fracture surfaces, fracture propagated inside the specimen in the black regions, and along the interface in the pink colored region. The fracture mode of Fig. 8c was also judged to be a successful test result, because the fracture of a cylindrical specimen should initiate at the h = 90° position near the adhesive layer where the maximum out-of-plane tensile stress rz-max was generated, and the fracture did occur near the 90° position within the specimen (black region).
4.2. Effect of specimen geometry and size on the apparent strength Fig. 9 shows experimentally determined out-of-plane apparent strengths Faps (fracture load divided by minimum cross-sectional area) of the spool specimens as a function of minimum diameter d. It is clear from this figure that Fap increases with decreasing d. This tendency is caused by the relaxation of stress concentration (Table 3) and the decrease in effective volume (Table 1). Fig. 10 represents Faps of the cylindrical specimens as a function of specimen thickness t. As shown in Fig. 7, a thin cylindrical specimen induces negligible stress concentration. Nevertheless thin 1 For interpretation of color in Figs. 3–5, 7, 8, 10, 11, 13, 14 the reader is referred to the web version of this article.
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2.0 -mean=30MPa
Spool
3000
1000
0
-mean=15MPa
σ 0
90
-mean=5MPa
180
270
1.0
0.5
Results of Finite Element Aanlysis
360
Measurement position ( o ) Fig. 4. Distributions of tensile strain ez on a cylindrical specimen made of unidirectionally reinforced composite 12 mm thick along the periphery on the mid-plane, under the indicated uniform loads where measurement position shows the angle to the fiber orientation h. The curves represent calculated results using the finite element method.
0.0
0
6.25
12.5
18.75
Minimum diameter, d (mm) Fig. 6. The stress ratio Rz (=rz-max/rz-mean) of spool specimens (r = 7 mm and t = 16 mm) as a function of their minimum diameter d.
2.5
cylindrical specimens had low Fap. This is likely to be due to a notable multi-axial stress state (Table 4). Comparison of Figs. 9 and 10 indicates that the Faps of the spool specimens are greater than those of the cylindrical specimens. This is mainly due to the stronger stress concentrations and stronger multi-axial stress state of the cylindrical specimens compared with the spool specimens. In the following sections, we try to explain these tendencies shown in Figs. 9 and 10 in terms of three factors: effective volume, stress concentrations, and multi-axial stress state.
1000
Cylindrical
Stress ratio Rz
2.0 100 1.5 1.0 10 0.5
Results of Finite Element Aanlysis
0.0 0
5. Discussion
3
σ
Veff (mm )
2000
1.5
Effective volume
σ
Stress ratio, Rz
ε Z (micro strains)
4000
10
20
30
40
50
60
1
Specimen thickness t (mm)
5.1. Procedure to determine out-of-plane tensile strength An out-of-plane tensile strength Fz having clear physical meaning for the spool specimens was estimated from Fap by the procedure shown in Fig. 11. First, the effective volume Veffs was determined using Eq. (5) and the results of finite element analyses. For this calculation, the Weibull modulus mz was required; mz was estimated from experimental results of the spool specimens. Fig. 12 represents examples of the Weibull plot of Fap for spool specimens, the number of which was larger than 6. From the slopes of Weibull plots of Fig. 12, values for mz were determined and the results are included in Table 1. As this data shows, mz scattered widely. The reason for this scattering may be due to the low number of tested specimens. For example, Khalili et al. suggested that minimum 30 specimens are required for precise estimation of
Fig. 7. The stress ratio Rz and effective volume Veff of cylindrical specimens as a function of specimen thickness t.
Weibull modulus [12]. In this paper, mz was determined using weighted average over various specimens, because mz should be independent of specimen type. Thus, mz equal to 12.68 was used in the calculation of Fz. Then, Veff was calculated by substituting mz and the results of finite element analyses into Eq. (5). Thus calculated Veffs are shown in Table 1. The next step is the determination of in-plane transverse strength Fy. For this, the Weibull modulus my and scale parameter Fy0 at unit effective volume (Veff = 1 mm3) of in-plane transverse strength was at first experimentally determined by in-plane transverse tensile tests of the UD CFRP. Details of this experiment are
Tensile displacement σ z (MPa)
Tensile displacement σ z (MPa)
Fiber direction (0 )
Maximum stress
(a) Spool specimen
Fiber direction (0 )
Maximum stress (90odirection)
(b) Cylindrical specimen
Fig. 5. Stress contours of out-of-plane normal stress (rz) of (a) spool specimen (d = 6.25 and t = 8 mm) and (b) cylindrical specimen (t = 8 mm) generated under mean tensile stresses rmean = 119.5 and 93.50 MPa, respectively, in which the highest stress regions are shown in red. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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Diameter of center (mm)
Stress ratio (r or s/rz-max) (%)
rxrz-max
ryl/rz-max
sxy/rz-max
sxz/rz-max
syz/rz-max
3.125 6.25 12.5 18.75
16 12 9.9 9.6
13.1 25.5 33 38.5
4.4 5.9 0.1 0
1.7 1.9 3.2 3.1
2.2 1.1 0.9 0.2
shown in Appendix A at the end of this paper. Using this result and Eq. (2), Fy was estimated. Actual stress rzi and ryi were predicted with experimental results Fap and FEA results rz-mean, ryi, rzi. Finally, out-of-plane tensile strength Fz was calculated using Eq. (9). We discussed other failure criterion (e.g. Hill yield criterion) including interactions between the terms. However, calculated strength Fz from Hill criterion was not consistent with tendency of size effect. We thought that the reason for inconsistent was presence of interactions between the terms. In this study, we made a choice of simple failure criterion for above-mentioned reason. We showed this procedure in order to verify effectiveness of this method, though the procedure was complicated. 5.2. Strength of spool specimen Fig. 13 shows Fz for the all spool specimens determined by the above-mentioned procedure as a function of Veff. In this figure, fracture probability curves are also drawn. These curves, corresponding to failure probabilities Qs of 5%, 50%, and 95%, were calculated from Eq. (1). Fig. 14 shows only the results using spool specimens having thicknesses of 8 mm and 16 mm; i.e., excluding specimens having a thickness of 4 mm. It is noted here that mz equal to 14.87, determined only using 16 mm thick specimens, was used for the determination of the curves in Fig. 14. In Fig. 14, predicted Fzs are consistent with experimental results, but in Fig. 13 not consistent. Thus, the discrepancy of experimentally determined Fzs from calculated values stems from that of
Apparent strength, Fap (MPa)
Table 3 Multi-axial stress states of 8 mm spool specimens.
80
Spool 60
t16mm t8mm
40
t4mm 20
0 0
6.25
12.5
18.75
25
Minimum diameter, d (mm) Fig. 9. Average the out-of-plane apparent strength Fap of spool specimens as a function of their minimum diameter d. The variation of d reflects the effect of stress concentrations.
Table 4 Multi-axial stress states of cylindrical specimens. Thickness (mm)
1 2 3 4 5 8 12 16 25 50
Stress ratio (r or s/rz-max) (%)
rx/rz-max
ry/rz-max
sxy/rz-max
sxz/rz-max
syz/rz-max
49.8 38.7 25.9 32.3 33.5 34.2 33.7 33.3 32.8 32.6
54.5 54.2 53.5 8.7 8.5 8.1 7.9 7.8 7.7 7.7
0 0 0 2.1 0.3 0.3 0.3 0.3 0.3 0.3
0 0 0 1.9 0.3 0.3 0.3 0.3 0.3 0.3
0 0 0 17.4 17.3 16.7 16.1 15.8 15.5 15.5
the 4 mm-thick specimens. As Table 1 shows, the 4 mm-thick specimens are featured by small curvature of r (1 mm) and small Veff. This fact indicates that the observed low Fzs in Fig. 13 are most likely due to sharp stress concentration of the 4 mm-thick specimens. Thus the procedure to predict Fzs shown in this paper might be effective only for the cases of mild stress concentration.
Fig. 8. Two types of fracture mode observed after out-of-plane tensile tests: (a) Specimen, (b) Single mode, (c) Mixed mode, (d) single mode, (e) mixed mode.
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2.0
dip
Cylindrical
Ln (Ln-(1-P))
Apparent strength, Fap (MPa)
60 50
Q =95% 40
Q =50% 30
Q =5%
1.0
Spool
t16d3.0 0.0 -1.0 m =11.35
-2.0 -3.0 3.8
20
m=13.88
t16d6.25
4.0
4.2
4.4
Ln (Fap )
10
Fig. 12. Typical Weibull plots of apparent out-of-plane tensile strength.
0 0
4
8
12
16
Specimen thickness, t (mm) Fig. 10. Apparent strength Fap and curves of predicted strength under failure probabilities Q = 5%, 50%, 95% as a function of specimen thickness (mz: 14.87).
5.3. Apparent strength of cylindrical specimen In order to calculate the apparent strength Fap of the cylindrical specimens, let us introduce stress ratios Rz and Ry defined by Rz = rz-max/Fap and Ry = ry-zmax/Fap, respectively. Here, rz-max is the maximum tensile stress when the specimen fractures, and ry-zmax is ry in the element rz-max appears. These ratios are easily obtained from the finite element calculation, because the calculation assumed linear elastic response. When these definitions are substituted in Eq. (8), we obtain;
1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi F ap ¼ r : Ry Fy
2
þ
Rz Fz
Fig. 13. Apparent strength Fap, Calculated strength Fz of spool specimens, and curves of predicted strength under failure probabilities of Q = 5%, 50%, 95% as a function of effective volume. (mz: 12.68).
ð10Þ
2
In this equation, Fy and Fz have already been obtained in Appendix A and in the results for the spool specimens as a function of effective volume Veff, respectively.
In Fig. 10, the predicted Faps of the cylindrical specimens are compared with those experimentally determined. As this figure
FEA (spool specimen) σz-mean (D1) Coupon specimen
σx-i, σy-i, σz-i, σz-max (: max. of σz-i), τxy-i, τyz-i, τzx-i , (i : number of element) (D2)
my, Fy0, V0
Experimental (spool specimen)
(D2)
Fap (D3) Veff
mz (D4) mz
⎛ σ ⎞ Veff = ∑ ⎜⎜ z −i ⎟⎟ ΔVi σ i =1 ⎝ z − max ⎠ n
Fy
Fy = Fy 0 (Veff / V0 )
−1 / m
(D4)
Γ(1 + 1 / m)
(D3) (D1) (D2)
eq.(10).
Fz −i =
σ z −i 1 − (σ y −i / Fy ) 2
Fz is maximum Fz-i Fig. 11. A flow chart explaining calculation procedure for out-of-plane tensile strength.
E. Hara et al. / Composites: Part A 41 (2010) 1425–1433
Calculated out-of-plane strength Fz (MPa)
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Acknowledgements
120
Spool
Suggestions for this paper by reviewers are gratefully acknowledged.
100 80
Q =95%
60 40
Appendix A. Estimation of Weibull parameters for transverse tensile strength
Q =5%
20 0 0.1
1.0
10.0
100.0
1000.0
Effective volume V eff (mm3)
Fig. 14. Calculated strength Fz of spool specimens and curves of predicted strength under failure probabilities of Q = 5%, 50%, 95% as a function of effective volume (mz: 14.87).
shows, the experimental values of Faps are consistent with predicted results for a failure probability Q of 50% except for the 1 mm thick specimen. The range of Veff of the spool specimens was 0.6–14 mm3. In contrast, that of cylindrical specimens was estimated from 2 mm3 to 900 mm3. Despite such large differences in Veff, the present model explained experimental results of the spool and cylinder specimens well. This means that the present model is effective for treating experimental results of out-of-plane tensile tests. In Fig. 10, the curves of predicting strength show slight dips at around a thickness t of 3 mm. This tendency is explanted in terms of the thickness dependency of the effective volume Veff and stress ratio Rz shown in Fig. 7. As we already discussed, Fap decreases with the increases in Veff and Rz. The dip appeared due to steep maximum of Veff at thickness t = 3 mm. In contrast, overall tendency of the decrease in Fap with increasing t is mainly affected by the monotonic increase of Rz with t. 6. Concluding remarks
Three-point-bending and transverse tensile tests were carried out in order to obtain Weibull parameters for the in-plane transverse tensile strength Fy of the UD-CFRP, IM600/#133. For these tests, a 16-ply laminated UD-CFRP prepreg was chosen. Table A1 shows the configurations used for the tests and the resulting average strength. The effective volume Veff of the tensile test specimen should be same as the volume of a gage section of the tensile specimen, and thus given by
V eff ¼ LBt;
where the length of the gage section is denoted by L, the width by B, and the thickness by t. In contrast, Veff of a 3-point-bending test was calculated using Eq. (A2) based on the assumption that tensile stress is linearly distributed.
V eff ¼
Ls Bt 2ðm þ 1Þ2
ðA2Þ
;
where Ls is the span of the bending test. The Weibull parameters were calculated from experimental results partly given in Table A1. The cumulative failure probability Q was estimated using the median-rank method. Then, Weibull modulus my was estimated from Weibull plots adopting the leastsquares method. The resulting scale parameter r0 when Veff = 0.588 mm3 and Q = 50% was estimated to be 138 MPa, and my to be 16.4. Fig. A1 shows tensile and bending strength as a function of Veff.
The effects of size and geometry on the out-of-plane tensile strength of UD CFRP were discussed on the basis of the experimental and calculated results for spool and cylindrical specimens. Using Weibull statistics and fracture criteria appropriate for multi-axial loading conditions, an analytical model was proposed to predict the effects of size and geometry on out-of-plane tensile strength. This model was shown to be effective. Thus, out-of-plane tensile strength obtained from a tensile test using spool and cylindrical specimens can be applied quantitatively when designing actual structures. On the basis of the present study, the authors recommend the following conditions for obtaining the out-of-plane tensile strength of CFRP laminates: (1) Use a spool specimen rather than a cylindrical specimen. (2) Choose a thick specimen with a large waist radius, and (3) Obtain an accurate Weibull modulus.
ðA1Þ
Strength, FY (MPa)
160
120
Q =95%
80
Q =5%
Q =50%
40
0 0.1
10.0
1000.0
100000.0
Effective volume, V eff (mm3) Fig. A1. Transverse tensile strengths of unidirectional composites as a function of effective volume.
Table A1 Configurations of bending and tensile tests.
3-pointbending 3-pointbending Tension
Span (Ls) (mm)
Length (L) (mm)
Width (B) (mm)
Thickness (t) (mm)
Number of specimens
Average strength (MPa)
Effective volume (mm3)
15
–
10
2.35
41
134.1
0.59
80
–
10
2.35
20
115
3.12
–
150
10
2.35
35
81.0
3557
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References [1] Abot JL, Daniel IM. Through-thickness mechanical characterization of woven fabric composites. J Compos Mater 2004;38(7):543–53. [2] Mespoulet S, Hodgkinson JM, Matthews FL, Hitchings D, Robinson P. Design, development and implementation of test methods for determination of through thickness properties of laminated composites. Plast Rubber Compos 2000;29(9):496–502. [3] Lodeiro MJ, Broughton WR, Sims GD. Understanding the limitations of throughthicknesss test methods. In: Proceedings of the 4th European conference on composites (ECCM/4), Lisbon; 31 August–2 September 1998. p. 80–90. [4] Daniel IM, Luo J-J, Schubel PM. Three-dimensional characterization of textile composites. Composites: Part B 2008;39:13–6. [5] ASTM D 7291. Standard test method for through-thickness ‘‘flatwise” tensile strength and elastic modulus of a fiber-reinforced polymer matrix composite material. American Society for Testing and Materials; 2007.
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[6] Wisnom MR. Size effects in the testing of fiber-composite materials. Compos Sci Technol 1999;59:1937–57. [7] Weil NA, Daniel IM. Analysis of fracture probabilities in nonuniformly stressed brittle materials. J Am Ceram Soc 1964;47:268–74. [8] Wisnom MR, Jones MI. Size effects in interlaminar tensile and shear strength of unidirectional glass fiber/epoxy. J Reinf Plast Compos 1996;15: 2–15. [9] O’Brien TK, Chawan AD, DeMorco K, Paris I. Influence of specimen preparation and specimen size on composite transverse tensile strength and scatter. NASA/ Technical Memorandum-2001-211030 ARL-TR-2540; 2001. [10] Advanced composites database system: JAXA-ACDB, Ver.06-1.
Contents lists available at ScienceDirect
Composites: Part A journal homepage: www.elsevier.com/locate/compositesa
Effects of geometry and specimen size on out-of-plane tensile strength of aligned CFRP determined by direct tensile method Eiichi Hara a,b,*, Tomohiro Yokozeki c, Hiroshi Hatta d, Takashi Ishikawa e, Yutaka Iwahori f a
Foundation for Promotion of Japanese Aerospace Technology (JAST), 6-13-1, Ohsawa, Mitaka-shi, Tokyo 181-0015, Japan Department of Space and Astronautical Science, The Graduate University for Advanced Studies, 3-1-1 Yoshinodai, Sagamihara-shi, Kanagawa 229-8510, Japan c Department of Aeronautics and Astronautics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan d Institute of Space and Astronautical Science (ISAS), Japan Aerospace Exploration Agency (JAXA), 3-1-1, Yoshinodai, Sagamihara, Kanagawa 229-8510, Japan e Japan Aerospace Exploration Agency (JAXA), 6-13-1, Ohsawa, Mitaka-shi, Tokyo 181-0015, Japan f Aerospace Research and Development Directorate (ARD), Japan Aerospace Exploration Agency (JAXA), 6-13-1, Ohsawa, Mitaka-shi, Tokyo 181-0015, Japan b
a r t i c l e
i n f o
Article history: Received 18 August 2009 Received in revised form 14 May 2010 Accepted 15 June 2010
Keywords: A. Polymer-matrix composites (PMCs) B. Strength C. Statistical properties/methods D. Mechanical testing
a b s t r a c t The out-of-plane tensile strength of CFRP laminate determined by the direct tensile method varies with specimen geometry and size. This effect was first experimentally observed using aligned CFRP. To explain the geometry and size effects from a mechanical point of view, an analytical model combining Weibull statistics, including the concept of effective volume, and a fracture criterion under multi-axial loading was constructed on the basis of stress distributions calculated using the finite element method. The predicted out-of-plane tensile strength of aligned CFRP was found to be consistent with experimental results. Thus, the present model is useful for reducing experimentally determined out-of-plane tensile strength under complex stress distributions to that under a uniaxial and uniform stress distribution. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction The out-of-plane tensile strength of laminates made of carbon fiber reinforced plastics (CFRPs) is a variable whose accurate value is increasingly important in the design of laminate structures, especially those with a thick cross-section that must bear multiaxial loading. However, tensile loading on a laminated CFRP in the thickness direction is generally not easy because sufficient length for tensile loading is usually difficult to secure. In order to overcome this difficulty, several researchers have attempted tensile loading by bonding the sample to the test fixture using, for example, the test configuration shown in Fig. 1 [1–4]. On the basis of these results, an out-of-plane test method (ASTM D 7291) was standardized by the ASTM D 30 Committee in 2007 [5]. A standardized test measuring the strength of a material generally requires a uniform and uniaxial stress state in the gage section of a specimen. In contrast, the above procedure for the determination of the out-of-plane tensile strength as a standardized test method generates the following undesirable stress distribution [3].
* Corresponding author at: Department of Space and Astronautical Science, The Graduate University for Advanced Studies, 3-1-1 Yoshinodai, Sagamihara-shi, Kanagawa 229-8510, Japan. Tel.: +81 422 40 1366; fax: +81 422 40 3549. E-mail address: [email protected] (E. Hara). 1359-835X/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.compositesa.2010.06.003
(1) Stress distribution is not uniform, and stress concentration is locally generated. This means that fracture load depends on the maximum stress induced at the point of the stress concentration. (2) The above stress distribution gives rise to fracture stress that varies with the geometry and size of the specimen. (3) In addition, multi-axial stress is induced in spite of singleaxial strain loading. Therefore, the meaning of the resulting fracture stress from an out-of-plane tensile test is unclear and it is difficult to use the data obtained in crafting a structural design. In order to evaluate the effects of size on strength, Weibull statistics has often been used with success for fiber reinforced plastics (FRPs) [6–8]. Weibull statistics can be useful not only for specimens in which stress is uniformly distributed but also for those in which stress is unevenly distributed, by introducing the concept of effective volume. This concept makes it possible to treat situations where stress concentration is mild. In the present paper, the following three factors were introduced to overcome the above-mentioned three difficulties: (1) Stress ratio RZ (the maximum out-of-plane stress rz-max divided by mean out-of-plane stress rz-mean) to evaluate stress concentration.
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load
Pivoting joint
Fig. 2. Configuration of spool test specimens with varying waist radius r, minimum diameter d, and thickness t.
Steel end tab strength tests. This CFRP was fabricated from a unidirectionally reinforced prepreg made of reinforcing fiber IM600 and epoxy matrix #133. UD-CFRP plates were fabricated by unidirectionally laminating 120 plies of the prepreg sheets and curing at 180 °C. The resulting UD-CFRP plates were approximately 17.4 mm thick.
Specimen
2.2. Specimens
load Fig. 1. Test setup for out-of-plane tensile test.
(2) Weibull statistics to evaluate the geometry and size effects. (3) Multi-axial strength criterion to evaluate the influence of a multi-axial stress state. By constructing a model combining these factors, experimentally determined out-of-plane tensile strengths under complex stress distributions can be made comparable to those under a uniaxial and uniform stress distribution. Thus, an attempt is made to derive a material fracture property independent of stress concentration, specimen size and geometry. The reason for defining stress ratio was for modifying difference between apparent strength and stress concentration. Only apparent strength can be evaluated experimentally. However, apparent strength is undervalued because of stress concentration.
Spool and cylinder configurations were selected for the out-ofplane tensile tests. Spool specimens (Fig. 2) were chosen to secure a fracture location near the specimen center. The spool specimens had a minimum diameter (d) at the mid-plane and a maximum diameter (D) of 25 mm. As shown in Table 1, 13 types of spool specimens were fabricated by changing d, waist radius r and thickness t to examine specimen size and geometry effects. In the fabrication of specimens with different thicknesses, the original plates, t = 17.4 mm, were ground symmetrically from both surfaces to place the fracture location at the center of the sample’s thickness. Eight types of cylindrical specimens were also prepared to examine the possibility of using a simple geometry for the outof-plane tests. The cylindrical specimens had a constant diameter of 25 mm and thicknesses of t = 1, 2, 3, 4, 5, 8, 12, and 16 mm. For later convenience, Cartesian and cylindrical coordinates were introduced as shown in Fig. 3. Both coordinates have their origin at the center of the specimen, the fiber direction coincides with the x axis, h = 0°, and the loading axis is parallel to the z axis. The mid-plane is defined as a symmetrical plane perpendicular to the loading axis.
2. Experiments 2.3. Experimental procedure 2.1. Material In order to simplify test conditions, a unidirectionally reinforced composite was chosen as the material for the out-of-plane tensile
The out-of-plane tensile tests were carried out on the basis of ASTM D 7291. Fig. 1 shows the setup of a specimen and test fixture. The tests were performed with the following procedure:
Table 1 Configurations of spool specimens and their statistical parameters. Thickness (t) (mm)
Minimum diameter (d) (mm)
Waisting radius (r) (mm)
Number of specimens
Weibull modulus (m) (–)
Effective volume (mz: 12.68) (Veff) (mm3)
Effective volume (mz: 14.87) (Veff) (mm3)
4 4 4 4 8 8 8 8 16 16 16 16 16
3 6.25 12.5 18.75 3 6.25 12.5 18.75 2 3 6.25 12.5 18.75
1 1 1 1 3 3 3 3 7 7 7 7 7
5 6 6 6 5 3 5 4 7 7 10 10 9
– 6.13 11.71 4.76 – – – – 23.79 11.35 13.88 15.5 10.94
0.232 0.495 0.995 1.344 0.741 2.124 3.536 5.458 1.711 2.396 9.393 18.10 21.95
0.580 1.611 2.384 3.416 1.396 1.832 6.985 12.38 14.07
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Cylindrical Tensile displacement
Fiber direction ( 0o)
Adhesive t = 0.1 mm Specimen
θ =90o
Thickness, t z x
y
θ
measurement location Mid-plane
θ =0o measurement location
Fig. 3. Coordinate system, showing origin at the center of the specimen, fiber direction and definition of the mid-plane.
0.5 mm. The thickness of the adhesive layer elements was set to 0.1 mm. In addition to the specimens prepared for the experiments, cylindrical models with specimen thicknesses of 25 and 50 mm were also produced and their stress distributions were calculated. Most of the elastic moduli of the CFRP used for finite element calculations were obtained on the basis of American testing standards as shown in Table 2. However, the in-plane shear modulus (Gxy) was taken from the Advanced Composites Database System published by the Japan Aerospace Exploration Agency [10]. In addition, the out-of-plane elastic modulus Ez was assumed to be equal to the in-plane transverse modulus Ey. The properties of the adhesive layer were also determined by tensile tests using coupon specimens made of only adhesive material fabricated by ourselves; Young’s modulus and Poisson’s ratio were found to be 2.75 GPa and 0.33, respectively. 3.2. Effective volume
(1) Bonding a specimen to two end-tab-blocks using a film adhesive of a 120 °C curing type, AF163-2K (film thickness: 0.1 mm) supplied by 3 M, USA. (2) Assembling specimen/end-tab-blocks (axial adjustment fixture) and end-plates with pivoting joint. (3) Attaching the assembled specimen to two grips of a testing machine (INSTRON 8500) and (4) Tensile loading at a constant crosshead displacement rate of 0.1 mm/min.
The effective volumes under Weibull statistics were estimated by the following procedure. Experimental results were assumed to obey the cumulative failure probability Q(r) [9].
m r ; QðrÞ ¼ 1 exp
ð1Þ
r0
r0 ¼ r0
1=m V eff ; V0
ð2Þ
In some of the tests on cylindrical specimens, eight strain gages were glued on the side-surface at the mid-plane with a spacing of 45°, and strain distribution was measured under loading. Apparent out-of-plane tensile strengths Faps were calculated from the fracture force divided by the area in the mid-plane, and Weibull modulus m was estimated from the variation of Faps. The estimation of m requires the cumulative failure probability Q(r). The median-rank method and the least-squares method were adopted for the calculation of Q(r). The detail procedure of the estimation of m is given, for example, in reference [9]. For adoption of a fracture criterion effective for multi-axial stress fields, two tensile strengths among the tri-axial strengths (other than the strength in the loading direction) were required. Experiments for these estimations are documented in Appendix A.
where C is the gamma function. In Eqs. (2) and (3), Veff and V0 are volumes subjected to uniform stress and showing average strengths equal to rm and r0, respectively. The effective volume Veff is useful for estimation of failure probability Q when r is not uniform. Davies defined Veff with Eq. (4) for a structure possessing a non-uniform stress distribution [11].
3. Analysis
V eff ¼
3.1. Finite element analysis (FEA) The stress distributions generated in specimens were estimated using a commercial finite element analysis program, ABAQUS 6, under the assumption of a linear elastic response. The finite element models include a half of the specimen with an adhesive layer, while the mid-plane of the specimen is constrained in the z-direction, considering the symmetry of the loading condition and specimen geometry. The end-tab-blocks were assumed to be a rigid body. Thus, uniform out-of-plane tensile displacements were given on the surface of the adhesive layer. Four-node tetrahedral elements were used for the models of the spool specimens. The thickness of the elements of the spool specimens was 0.145 mm, and in-plane edge lengths of elements were about 0.3 mm. Eight-node hexahedral elements with reduced integration were used for finite element models of the cylindrical specimens. The reduced integration generates a representative stress tensor for every element; in the present case, average values were given. The thickness of the elements of the cylindrical specimens was 0.145 mm, and the in-plane edge lengths were about
where Veff and V0 are volumes subjected to uniform stress and showing average strengths equal to rm and r0, respectively, and r0 and m are the scale parameter and Weibull modulus, respectively. The average strength rm obeying the Weibull distribution is given by,
rm ¼ r0 ðV eff =V 0 Þ1=m Cð1 þ 1=mÞ;
Z V
r rmax
m
ð3Þ
dV;
ð4Þ
where rmax is the maximum stress. When stress r is uniformly distributed, r/rmax is unity. Then, Veff is the specimen volume itself (cf. Eq. (A1) in Appendix A). Veff is given by a simple equation when Table 2 Elastic moduli of IM600/#133 CFRP and their test methods. Property
IM600/ #133
Test method
Ex (GPa) EY (GPa) GXY (GPa)
152 8.21 4.36
SACMA SRM 4R SACMA SRM 4R SACMA SRM 7R
7 6 5
mXY
0.334 8.21
7 – 10 10
GXZ (GPa) Gyz (GPa)
3.993
SACMA SRM 4R Assumption: Ez = Ey ASTM D 5379
2.516
ASTM D 5379
mXZ mYZ
0.346 0.536
SACMA SRM 4R SACMA SRM 4R
Ez (GPa)
Specimens
7 6
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rmax and r are given by simple equations. For example, in the case of 3-point-bending, Veff is given by Eq. (A2). For the spool and cylindrical specimens, Veff was estimated from Eq. (4) using the stress distribution obtained by finite element analysis. For this purpose, Eq. (4) was discretized as Eq. (5) in accordance with the finite element model. V eff ¼
n X
rzi rz- max
i¼1
m
DV i ;
ð5Þ
where n represents element numbers in the finite element model of the specimen, rzi is out-of-plane normal-stress in the ith element, rz-max is the maximum of rzi in the whole model, and DVi is a volume of the ith element. For two specimens consisting of the same material having different effective volumes Veff-1 and Veff-2 showing respective strengths of F1 and F2, the relation of Eq. (6) can be derived from Eq. (4).
F 1 ¼ F 2 ðV eff-1 =V eff-2 Þ1=m :
ð6Þ
3.3. Failure criterion for the multi-axial stress condition A multi-axial stress state should be generated in the out-ofplane test specimens under tensile loading. In order to estimate the influence of multi-axial stress on strength, the criterion expressed by Eq. (7) was examined.
rx
2
þ
Fx
ry
2
þ
Fy
rz
2
Fz
þ
sxy
2
F xy
þ
syz
2
F yz
þ
szx
F zx
2
¼ 1;
ð7Þ
where normal stresses in the fiber axis, in-plane transverse, and thickness directions are denoted by rx, ry, and rz, respectively, shear stresses by sxy, syz, and szx, tensile strengths under the uniaxial normal stress by Fx Fy, and Fz, and pure shear strengths by Fxy, Fyz, and Fzx. This equation was chosen due to simplicity of the following calculations. Since the composite examined in this study is unidirectionally reinforced, the first term of Eq. (7) is negligibly small, because Fx (=2700 MPa) is more than 10 times as large as other strengths Fy, Fz (60 MPa) and Fxy, Fyz, Fzx (50 MPa) in the present experiments. In addition, the terms related to shear stresses were neglected because the ratios of the shear stresses, sxy, syz, and szx, to respective shear strengths near the area appearing maximum stress rz-max were found to be sufficiently small. Thus, Eq. (7) reduces to
ry Fy
2
þ
rz Fz
2 ¼ 1:
ð8Þ
Then, Eq. (8) is rearranged to give
rz
F z ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 1 ðry =F y Þ2
when the direction coincides with the fiber axis. As this figure shows, ez varies with h. The curves in the figure are calculated results from the finite element method. The curves are consistent with experimental values. This result shows that tensile load is appropriately applied to a specimen in the present tests. Fig. 5a and b shows contours of out-of-plane tensile stress rz obtained by finite element analyses under an out-of-plane tensile load applied to spool and cylindrical specimens. In this figure, red regions represent maximum stresses.1 Thus, the maximum stress rz-max of a spool specimen appears at the h = 0° position on the side-surface of minimum diameter, and that of a cylindrical specimen at the h = 90° position near the adhesive layer. In order to evaluate stress concentration, the stress ratio Rz, maximum stress rz-max divided by mean stress rz-mean, was calculated. Figs. 6 and 7 show Rzs of the spool and cylindrical specimens. It follows from these figures that rz-max is relaxed by selecting smaller d for a spool specimen and thinner thickness t for a cylindrical specimen. In the tensile test shown in Fig. 1, multi-axial stresses were induced in spite of uniaxial loading. Hence, the fracture criterion based on multi-axial stresses should be used in this study. Tables 3 and 4 show maxima of stress components other than the loading stress rz divided by rz-max of the spool and cylindrical specimens. Table 3 shows that a near uniaxial stress state is attained for a spool specimen, especially when minimum diameter d is small. In contrast, Table 4 shows that a multi-axial stress state due to a high value of rx is notable for a cylindrical specimen. This tendency is especially noteworthy for thinner cylindrical specimens.
ð9Þ
Using this equation, out-of-plane strength Fz under uniaxial loading can be estimated, provided that rz and ry at failure point and inplane transverse strength Fy are given. 4. Experimental results 4.1. Stress distributions and fracture surfaces 4.1.1. Stress distributions Fig. 4 shows distributions of tensile strain ez along the line on which the mid-plane and the side-surface intersect under average tensile stresses rz-means of 5, 15, and 30 MPa. The strains in this figure were measured using strain gages 2 mm long, and the horizontal axis represents the measuring location where h was set to be 0°
4.1.2. Fracture surfaces All the specimens fractured instantaneously in a brittle manner, and the spool specimens fractured near their minimum cross-section as was expected. In contrast, the cylindrical specimens were broken near adhesive lamina, so the fracture surfaces of the cylindrical specimens were examined. Fig. 8 shows typical fracture patterns of the cylindrical specimens. Fig. 8b and c schematically shows two typical fracture patterns, (b) the entire fracture surface of a specimen and (c) a portion of the specimen fracture surface and part of the interface between the adhesive layer and specimen. Fig. 8d and e are photographs of these fracture modes, where the white arrows indicate fiber direction. In these fracture surfaces, fracture propagated inside the specimen in the black regions, and along the interface in the pink colored region. The fracture mode of Fig. 8c was also judged to be a successful test result, because the fracture of a cylindrical specimen should initiate at the h = 90° position near the adhesive layer where the maximum out-of-plane tensile stress rz-max was generated, and the fracture did occur near the 90° position within the specimen (black region).
4.2. Effect of specimen geometry and size on the apparent strength Fig. 9 shows experimentally determined out-of-plane apparent strengths Faps (fracture load divided by minimum cross-sectional area) of the spool specimens as a function of minimum diameter d. It is clear from this figure that Fap increases with decreasing d. This tendency is caused by the relaxation of stress concentration (Table 3) and the decrease in effective volume (Table 1). Fig. 10 represents Faps of the cylindrical specimens as a function of specimen thickness t. As shown in Fig. 7, a thin cylindrical specimen induces negligible stress concentration. Nevertheless thin 1 For interpretation of color in Figs. 3–5, 7, 8, 10, 11, 13, 14 the reader is referred to the web version of this article.
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2.0 -mean=30MPa
Spool
3000
1000
0
-mean=15MPa
σ 0
90
-mean=5MPa
180
270
1.0
0.5
Results of Finite Element Aanlysis
360
Measurement position ( o ) Fig. 4. Distributions of tensile strain ez on a cylindrical specimen made of unidirectionally reinforced composite 12 mm thick along the periphery on the mid-plane, under the indicated uniform loads where measurement position shows the angle to the fiber orientation h. The curves represent calculated results using the finite element method.
0.0
0
6.25
12.5
18.75
Minimum diameter, d (mm) Fig. 6. The stress ratio Rz (=rz-max/rz-mean) of spool specimens (r = 7 mm and t = 16 mm) as a function of their minimum diameter d.
2.5
cylindrical specimens had low Fap. This is likely to be due to a notable multi-axial stress state (Table 4). Comparison of Figs. 9 and 10 indicates that the Faps of the spool specimens are greater than those of the cylindrical specimens. This is mainly due to the stronger stress concentrations and stronger multi-axial stress state of the cylindrical specimens compared with the spool specimens. In the following sections, we try to explain these tendencies shown in Figs. 9 and 10 in terms of three factors: effective volume, stress concentrations, and multi-axial stress state.
1000
Cylindrical
Stress ratio Rz
2.0 100 1.5 1.0 10 0.5
Results of Finite Element Aanlysis
0.0 0
5. Discussion
3
σ
Veff (mm )
2000
1.5
Effective volume
σ
Stress ratio, Rz
ε Z (micro strains)
4000
10
20
30
40
50
60
1
Specimen thickness t (mm)
5.1. Procedure to determine out-of-plane tensile strength An out-of-plane tensile strength Fz having clear physical meaning for the spool specimens was estimated from Fap by the procedure shown in Fig. 11. First, the effective volume Veffs was determined using Eq. (5) and the results of finite element analyses. For this calculation, the Weibull modulus mz was required; mz was estimated from experimental results of the spool specimens. Fig. 12 represents examples of the Weibull plot of Fap for spool specimens, the number of which was larger than 6. From the slopes of Weibull plots of Fig. 12, values for mz were determined and the results are included in Table 1. As this data shows, mz scattered widely. The reason for this scattering may be due to the low number of tested specimens. For example, Khalili et al. suggested that minimum 30 specimens are required for precise estimation of
Fig. 7. The stress ratio Rz and effective volume Veff of cylindrical specimens as a function of specimen thickness t.
Weibull modulus [12]. In this paper, mz was determined using weighted average over various specimens, because mz should be independent of specimen type. Thus, mz equal to 12.68 was used in the calculation of Fz. Then, Veff was calculated by substituting mz and the results of finite element analyses into Eq. (5). Thus calculated Veffs are shown in Table 1. The next step is the determination of in-plane transverse strength Fy. For this, the Weibull modulus my and scale parameter Fy0 at unit effective volume (Veff = 1 mm3) of in-plane transverse strength was at first experimentally determined by in-plane transverse tensile tests of the UD CFRP. Details of this experiment are
Tensile displacement σ z (MPa)
Tensile displacement σ z (MPa)
Fiber direction (0 )
Maximum stress
(a) Spool specimen
Fiber direction (0 )
Maximum stress (90odirection)
(b) Cylindrical specimen
Fig. 5. Stress contours of out-of-plane normal stress (rz) of (a) spool specimen (d = 6.25 and t = 8 mm) and (b) cylindrical specimen (t = 8 mm) generated under mean tensile stresses rmean = 119.5 and 93.50 MPa, respectively, in which the highest stress regions are shown in red. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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Diameter of center (mm)
Stress ratio (r or s/rz-max) (%)
rxrz-max
ryl/rz-max
sxy/rz-max
sxz/rz-max
syz/rz-max
3.125 6.25 12.5 18.75
16 12 9.9 9.6
13.1 25.5 33 38.5
4.4 5.9 0.1 0
1.7 1.9 3.2 3.1
2.2 1.1 0.9 0.2
shown in Appendix A at the end of this paper. Using this result and Eq. (2), Fy was estimated. Actual stress rzi and ryi were predicted with experimental results Fap and FEA results rz-mean, ryi, rzi. Finally, out-of-plane tensile strength Fz was calculated using Eq. (9). We discussed other failure criterion (e.g. Hill yield criterion) including interactions between the terms. However, calculated strength Fz from Hill criterion was not consistent with tendency of size effect. We thought that the reason for inconsistent was presence of interactions between the terms. In this study, we made a choice of simple failure criterion for above-mentioned reason. We showed this procedure in order to verify effectiveness of this method, though the procedure was complicated. 5.2. Strength of spool specimen Fig. 13 shows Fz for the all spool specimens determined by the above-mentioned procedure as a function of Veff. In this figure, fracture probability curves are also drawn. These curves, corresponding to failure probabilities Qs of 5%, 50%, and 95%, were calculated from Eq. (1). Fig. 14 shows only the results using spool specimens having thicknesses of 8 mm and 16 mm; i.e., excluding specimens having a thickness of 4 mm. It is noted here that mz equal to 14.87, determined only using 16 mm thick specimens, was used for the determination of the curves in Fig. 14. In Fig. 14, predicted Fzs are consistent with experimental results, but in Fig. 13 not consistent. Thus, the discrepancy of experimentally determined Fzs from calculated values stems from that of
Apparent strength, Fap (MPa)
Table 3 Multi-axial stress states of 8 mm spool specimens.
80
Spool 60
t16mm t8mm
40
t4mm 20
0 0
6.25
12.5
18.75
25
Minimum diameter, d (mm) Fig. 9. Average the out-of-plane apparent strength Fap of spool specimens as a function of their minimum diameter d. The variation of d reflects the effect of stress concentrations.
Table 4 Multi-axial stress states of cylindrical specimens. Thickness (mm)
1 2 3 4 5 8 12 16 25 50
Stress ratio (r or s/rz-max) (%)
rx/rz-max
ry/rz-max
sxy/rz-max
sxz/rz-max
syz/rz-max
49.8 38.7 25.9 32.3 33.5 34.2 33.7 33.3 32.8 32.6
54.5 54.2 53.5 8.7 8.5 8.1 7.9 7.8 7.7 7.7
0 0 0 2.1 0.3 0.3 0.3 0.3 0.3 0.3
0 0 0 1.9 0.3 0.3 0.3 0.3 0.3 0.3
0 0 0 17.4 17.3 16.7 16.1 15.8 15.5 15.5
the 4 mm-thick specimens. As Table 1 shows, the 4 mm-thick specimens are featured by small curvature of r (1 mm) and small Veff. This fact indicates that the observed low Fzs in Fig. 13 are most likely due to sharp stress concentration of the 4 mm-thick specimens. Thus the procedure to predict Fzs shown in this paper might be effective only for the cases of mild stress concentration.
Fig. 8. Two types of fracture mode observed after out-of-plane tensile tests: (a) Specimen, (b) Single mode, (c) Mixed mode, (d) single mode, (e) mixed mode.
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E. Hara et al. / Composites: Part A 41 (2010) 1425–1433
2.0
dip
Cylindrical
Ln (Ln-(1-P))
Apparent strength, Fap (MPa)
60 50
Q =95% 40
Q =50% 30
Q =5%
1.0
Spool
t16d3.0 0.0 -1.0 m =11.35
-2.0 -3.0 3.8
20
m=13.88
t16d6.25
4.0
4.2
4.4
Ln (Fap )
10
Fig. 12. Typical Weibull plots of apparent out-of-plane tensile strength.
0 0
4
8
12
16
Specimen thickness, t (mm) Fig. 10. Apparent strength Fap and curves of predicted strength under failure probabilities Q = 5%, 50%, 95% as a function of specimen thickness (mz: 14.87).
5.3. Apparent strength of cylindrical specimen In order to calculate the apparent strength Fap of the cylindrical specimens, let us introduce stress ratios Rz and Ry defined by Rz = rz-max/Fap and Ry = ry-zmax/Fap, respectively. Here, rz-max is the maximum tensile stress when the specimen fractures, and ry-zmax is ry in the element rz-max appears. These ratios are easily obtained from the finite element calculation, because the calculation assumed linear elastic response. When these definitions are substituted in Eq. (8), we obtain;
1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi F ap ¼ r : Ry Fy
2
þ
Rz Fz
Fig. 13. Apparent strength Fap, Calculated strength Fz of spool specimens, and curves of predicted strength under failure probabilities of Q = 5%, 50%, 95% as a function of effective volume. (mz: 12.68).
ð10Þ
2
In this equation, Fy and Fz have already been obtained in Appendix A and in the results for the spool specimens as a function of effective volume Veff, respectively.
In Fig. 10, the predicted Faps of the cylindrical specimens are compared with those experimentally determined. As this figure
FEA (spool specimen) σz-mean (D1) Coupon specimen
σx-i, σy-i, σz-i, σz-max (: max. of σz-i), τxy-i, τyz-i, τzx-i , (i : number of element) (D2)
my, Fy0, V0
Experimental (spool specimen)
(D2)
Fap (D3) Veff
mz (D4) mz
⎛ σ ⎞ Veff = ∑ ⎜⎜ z −i ⎟⎟ ΔVi σ i =1 ⎝ z − max ⎠ n
Fy
Fy = Fy 0 (Veff / V0 )
−1 / m
(D4)
Γ(1 + 1 / m)
(D3) (D1) (D2)
eq.(10).
Fz −i =
σ z −i 1 − (σ y −i / Fy ) 2
Fz is maximum Fz-i Fig. 11. A flow chart explaining calculation procedure for out-of-plane tensile strength.
E. Hara et al. / Composites: Part A 41 (2010) 1425–1433
Calculated out-of-plane strength Fz (MPa)
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Acknowledgements
120
Spool
Suggestions for this paper by reviewers are gratefully acknowledged.
100 80
Q =95%
60 40
Appendix A. Estimation of Weibull parameters for transverse tensile strength
Q =5%
20 0 0.1
1.0
10.0
100.0
1000.0
Effective volume V eff (mm3)
Fig. 14. Calculated strength Fz of spool specimens and curves of predicted strength under failure probabilities of Q = 5%, 50%, 95% as a function of effective volume (mz: 14.87).
shows, the experimental values of Faps are consistent with predicted results for a failure probability Q of 50% except for the 1 mm thick specimen. The range of Veff of the spool specimens was 0.6–14 mm3. In contrast, that of cylindrical specimens was estimated from 2 mm3 to 900 mm3. Despite such large differences in Veff, the present model explained experimental results of the spool and cylinder specimens well. This means that the present model is effective for treating experimental results of out-of-plane tensile tests. In Fig. 10, the curves of predicting strength show slight dips at around a thickness t of 3 mm. This tendency is explanted in terms of the thickness dependency of the effective volume Veff and stress ratio Rz shown in Fig. 7. As we already discussed, Fap decreases with the increases in Veff and Rz. The dip appeared due to steep maximum of Veff at thickness t = 3 mm. In contrast, overall tendency of the decrease in Fap with increasing t is mainly affected by the monotonic increase of Rz with t. 6. Concluding remarks
Three-point-bending and transverse tensile tests were carried out in order to obtain Weibull parameters for the in-plane transverse tensile strength Fy of the UD-CFRP, IM600/#133. For these tests, a 16-ply laminated UD-CFRP prepreg was chosen. Table A1 shows the configurations used for the tests and the resulting average strength. The effective volume Veff of the tensile test specimen should be same as the volume of a gage section of the tensile specimen, and thus given by
V eff ¼ LBt;
where the length of the gage section is denoted by L, the width by B, and the thickness by t. In contrast, Veff of a 3-point-bending test was calculated using Eq. (A2) based on the assumption that tensile stress is linearly distributed.
V eff ¼
Ls Bt 2ðm þ 1Þ2
ðA2Þ
;
where Ls is the span of the bending test. The Weibull parameters were calculated from experimental results partly given in Table A1. The cumulative failure probability Q was estimated using the median-rank method. Then, Weibull modulus my was estimated from Weibull plots adopting the leastsquares method. The resulting scale parameter r0 when Veff = 0.588 mm3 and Q = 50% was estimated to be 138 MPa, and my to be 16.4. Fig. A1 shows tensile and bending strength as a function of Veff.
The effects of size and geometry on the out-of-plane tensile strength of UD CFRP were discussed on the basis of the experimental and calculated results for spool and cylindrical specimens. Using Weibull statistics and fracture criteria appropriate for multi-axial loading conditions, an analytical model was proposed to predict the effects of size and geometry on out-of-plane tensile strength. This model was shown to be effective. Thus, out-of-plane tensile strength obtained from a tensile test using spool and cylindrical specimens can be applied quantitatively when designing actual structures. On the basis of the present study, the authors recommend the following conditions for obtaining the out-of-plane tensile strength of CFRP laminates: (1) Use a spool specimen rather than a cylindrical specimen. (2) Choose a thick specimen with a large waist radius, and (3) Obtain an accurate Weibull modulus.
ðA1Þ
Strength, FY (MPa)
160
120
Q =95%
80
Q =5%
Q =50%
40
0 0.1
10.0
1000.0
100000.0
Effective volume, V eff (mm3) Fig. A1. Transverse tensile strengths of unidirectional composites as a function of effective volume.
Table A1 Configurations of bending and tensile tests.
3-pointbending 3-pointbending Tension
Span (Ls) (mm)
Length (L) (mm)
Width (B) (mm)
Thickness (t) (mm)
Number of specimens
Average strength (MPa)
Effective volume (mm3)
15
–
10
2.35
41
134.1
0.59
80
–
10
2.35
20
115
3.12
–
150
10
2.35
35
81.0
3557
E. Hara et al. / Composites: Part A 41 (2010) 1425–1433
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