13 Open-Hole Tensile and Compressive Strengths of Laminates Experiments have shown that the tensile and compressive strengths of a composite laminate containing a hole or notch depend on hole or notch size. Because of the complexity of the fracture process in notched laminates,most strength models are semiempirical.In this chapter some of the more com- monly accepted and computationally simple strength models,i.e.,the point and average stress criteria developed by Whitney and Nuismer [1]will be discussed.In addition,a modification of the point stress criterion,proposed by Pipes et al.[2],will be introduced. Reasons for the substantial tensile and compressive strength reductions of composites because of holes and notches are the brittleness of the material and the large stress concentration factors brought about by the anisotropy of the material.These strength reductions are not necessarily the same for tensile and compressive loading because the failure modes are typically different. As discussed in Chapter 2,the stress concentration factor for a plate con- taining a circular hole of radius,R(Figure 13.1)is K=9(R,0) (13.1) where R is the hole radius,and is the average normal stress applied on the horizontal boundaries of the plate(Figure 13.1).For an infinite plate,i.e., where L,w→∞，Lekhnitski[3]derived the following expression K=1+2E./B,-vy+E/(2G》 (13.2) where EEy,vyx,and Gxy are the effective engineering constants of the plate. Note that the x-axis is oriented along the loading direction,and the y-axis is oriented transverse to the loading direction. It is observed from Equation(13.2)that the stress concentration factor for an infinite plate is independent of hole radius.For an ideally brittle infinite plate,the notched strength would thus be ©2003 by CRC Press LLC
13 Open-Hole Tensile and Compressive Strengths of Laminates Experiments have shown that the tensile and compressive strengths of a composite laminate containing a hole or notch depend on hole or notch size. Because of the complexity of the fracture process in notched laminates, most strength models are semiempirical. In this chapter some of the more commonly accepted and computationally simple strength models, i.e., the point and average stress criteria developed by Whitney and Nuismer [1] will be discussed. In addition, a modification of the point stress criterion, proposed by Pipes et al. [2], will be introduced. Reasons for the substantial tensile and compressive strength reductions of composites because of holes and notches are the brittleness of the material and the large stress concentration factors brought about by the anisotropy of the material. These strength reductions are not necessarily the same for tensile and compressive loading because the failure modes are typically different. As discussed in Chapter 2, the stress concentration factor for a plate containing a circular hole of radius, R (Figure 13.1) is (13.1) where R is the hole radius, and σx is the average normal stress applied on the horizontal boundaries of the plate (Figure 13.1). For an infinite plate, i.e., where L,w → ∞, Lekhnitski [3] derived the following expression (13.2) where Ex, Ey, νyx, and Gxy are the effective engineering constants of the plate. Note that the x-axis is oriented along the loading direction, and the y-axis is oriented transverse to the loading direction. It is observed from Equation (13.2) that the stress concentration factor for an infinite plate is independent of hole radius. For an ideally brittle infinite plate, the notched strength would thus be K x R x = σ ( ) σ , 0 Κ∞ =+ − + 12 2 ( ) EE v E G x y xy x xy ( ) TX001_ch13_Frame Page 169 Saturday, September 21, 2002 5:07 AM © 2003 by CRC Press LLC
6 ↑↑↑↑↑↑↑↑ ↓↓↓↓↓↓↓ 一w FIGURE 13.1 Finite-size plate containing a hole of diameter D=2R subject to uniaxial tension. ON Go/K (13.3) where co is the strength of the plate without a hole,i.e.,the unnotched strength.Experiments,however,show that the strength of composite plates containing large holes is much less than that observed for small holes [1,2]. Such a difference for large plates cannot be explained by a net area reduction. Consequently,there must be factors other than the stress concentration factor controlling the notched strength.Consideration of the normal stress distri- bution across the ligaments of the plate adjacent to the hole reveals some interesting features.The approximate stress distribution in an infinite plate containing a circular hole is [4] 。.g,0)=@2+2+35-K.-35-7飞】 (13.4) 2[ where =y/R,and o,()is the far-field normal stress.Figure 13.2 shows the stress,o(y,0)/o(),across a ligament for isotropic plates containing holes of two sizes(R/Ro=0.1 and 1.0),where Ro is a reference radius.It is observed that the volume of material subject to a high stress is much more localized for the plate with a smaller hole,thus leading to a greater oppor- tunity for stress redistribution to occur,explaining the increased notched strength with decreased hole size. 13.1 Point and Average Stress Criteria The point and average stress criteria [1]incorporate the hole size effect in computationally simple fracture criteria where failure of the notched lami- nate is assumed to occur when the stress,o,at a certain distance do ahead ©2003 by CRC Press LLC
σN = σ0/K∞ (13.3) where σ0 is the strength of the plate without a hole, i.e., the unnotched strength. Experiments, however, show that the strength of composite plates containing large holes is much less than that observed for small holes [1,2]. Such a difference for large plates cannot be explained by a net area reduction. Consequently, there must be factors other than the stress concentration factor controlling the notched strength. Consideration of the normal stress distribution across the ligaments of the plate adjacent to the hole reveals some interesting features. The approximate stress distribution in an infinite plate containing a circular hole is [4] (13.4) where ξ = y/R, and σx(∞) is the far-field normal stress. Figure 13.2 shows the stress, σx(y,0)/σx(∞), across a ligament for isotropic plates containing holes of two sizes (R/R0 = 0.1 and 1.0), where R0 is a reference radius. It is observed that the volume of material subject to a high stress is much more localized for the plate with a smaller hole, thus leading to a greater opportunity for stress redistribution to occur, explaining the increased notched strength with decreased hole size. 13.1 Point and Average Stress Criteria The point and average stress criteria [1] incorporate the hole size effect in computationally simple fracture criteria where failure of the notched laminate is assumed to occur when the stress, σx, at a certain distance d0 ahead FIGURE 13.1 Finite-size plate containing a hole of diameter D = 2R subject to uniaxial tension. σ σ ξξ ξξ x x y,0 2 2 3 35 7 24 68 ( ) = ( ) ∞ [ ] ++ − − ( )( ) − Κ∞ TX001_ch13_Frame Page 170 Saturday, September 21, 2002 5:07 AM © 2003 by CRC Press LLC
3 R/Ro=1.0 R/R。=0.1 00.51.01.52.02.5 Distance ahead of Hole Edge,y/Ro FIGURE 13.2 Normal stress distributions ahead of the hole edge for isotropic plates containing holes of two sizes. of the notch reaches the unnotched strength,oo(point stress criterion [PSC]), or the stress,o,averaged over a certain distance across the ligament reaches the unnotched strength (average stress criterion [ASC]).Mathematically, these criteria can be expressed as PSC:(R do,0)=o (13.5a) R+ao ASC: ox(y,0dy=o。 (13.5b) ao 13.1.1 Point Stress Criterion (PSC) Combination of the PSC(Equation(13.5a))and the expression for the stress distribution(Equation(13.4))yields 6N二 2 (13.6) 002+2+31-(K-356-718） where R (13.7) R+do Note that for very large holes,do is small compared with R,and Equation (13.6)gives ON=1K. (13.8) 00 Consequently,the notched strength ratio for a large hole is given by the inverse of the stress concentration factor.Furthermore,a notch-insensitive ©2003 by CRC Press LLC
of the notch reaches the unnotched strength, σ0 (point stress criterion [PSC]), or the stress, σx, averaged over a certain distance across the ligament reaches the unnotched strength (average stress criterion [ASC]). Mathematically, these criteria can be expressed as PSC: σx(R + d0,0) = σ0 (13.5a) (13.5b) 13.1.1 Point Stress Criterion (PSC) Combination of the PSC (Equation (13.5a)) and the expression for the stress distribution (Equation (13.4)) yields (13.6) where (13.7) Note that for very large holes, d0 is small compared with R, and Equation (13.6) gives (13.8) Consequently, the notched strength ratio for a large hole is given by the inverse of the stress concentration factor. Furthermore, a notch-insensitive FIGURE 13.2 Normal stress distributions ahead of the hole edge for isotropic plates containing holes of two sizes. ASC a y dy x R R a : , 1 0 0 0 0 σ σ ( ) = + ∫ σ σ λλ λλ N 0 24 68 2 2 3 35 7 = ++ − − ( )( ) − Κ∞ λ = + R R d0 σ σ N K 0 = 1 ∞ TX001_ch13_Frame Page 171 Saturday, September 21, 2002 5:07 AM © 2003 by CRC Press LLC
1.0 Boron/Aluminum [O] 0.8 ● ,d。=0.8mm 60 0.6 ● 0.4 02 2 4 6 Hole Radius,mm FIGURE 13.3 Experimental data on notched strength of a boron-aluminum composite and predictions based on the point stress criterion.(From R.F.Karlak,Proceedings of a Conference on Failure Models in Composites (III),American Society for Metals,Chicago,1977.With permission.) laminate is characterized by a large do in comparison to R.For that case, =0 in Equation (13.6)and ON/0o=1.0. The PSC thus contains two parameters(do,oo)that have to be determined by experiment.Having established do and oo,the PSC allows for strength predictions of laminates containing holes of arbitrary size.Figure 13.3 shows oN/0o plotted vs.hole size for a unidirectional [O]boron/aluminum composite [3].Reasonable agreement with experimental data is observed. 13.1.2 Average Stress Criterion (ASC) Substitution of the stress distribution (Equation (13.4))into the ASC (Equation (13.5b))yields,after integration,the following expression for the notched laminate strength 2 (13.9) 。(1+δ2+82+(K-3)8 with R 6= (13.10) R+ao Figure 13.4 shows experimental strength data for a [0,/+45]carbon/epoxy laminate [5].Experimental results are in good agreement with the ASC with ao =5 mm. ©2003 by CRC Press LLC
laminate is characterized by a large d0 in comparison to R. For that case, λ ≈ 0 in Equation (13.6) and σN/σ0 ≈ 1.0. The PSC thus contains two parameters (d0, σ0) that have to be determined by experiment. Having established d0 and σ0, the PSC allows for strength predictions of laminates containing holes of arbitrary size. Figure 13.3 shows σN/σ0 plotted vs. hole size for a unidirectional [0]n boron/aluminum composite [3]. Reasonable agreement with experimental data is observed. 13.1.2 Average Stress Criterion (ASC) Substitution of the stress distribution (Equation (13.4)) into the ASC (Equation (13.5b)) yields, after integration, the following expression for the notched laminate strength (13.9) with (13.10) Figure 13.4 shows experimental strength data for a [02/±45]s carbon/epoxy laminate [5]. Experimental results are in good agreement with the ASC with a0 = 5 mm. FIGURE 13.3 Experimental data on notched strength of a boron–aluminum composite and predictions based on the point stress criterion. (From R.F. Karlak, Proceedings of a Conference on Failure Models in Composites (III), American Society for Metals, Chicago, 1977. With permission.) σ σ δδ δ N 0 K 2 6 2 12 3 = ( ) + ( ) ++ − ( ) ∞ δ = + R R a0 TX001_ch13_Frame Page 172 Saturday, September 21, 2002 5:07 AM © 2003 by CRC Press LLC
1.0 Experimental dota 0.8 Average stress criterion (ao=5mm) 0.6 ON Carbon/Epoxy [02/±451s 0.2 0 0.2 0.40.60.81.01.21.4 Hole radius,mm FIGURE 13.4 Notched strength data and predictions based on the average stress criterion for a notched [02/45. carbon/epoxy laminate [4]. 13.1.3 Modification of PSC To improve the accuracy of notched strength predictions using the PSC,Pipes et al.[2],following Karlak's modification [3],let the characteristic distance, do(Equation(13.6))become a power function of hole radius do=(R/Ro)m/C (13.11) where m is an exponential parameter,Ro is a reference radius,and C is the notch sensitivity factor.In essence,this model adds one more parameter(the exponential parameter)to the PSC.The reference radius may arbitrarily be chosen as Ro=1 mm.The parameter A(Equation(13.7))then becomes 入=1/(1+Rm-1C-1) (13.12) Figures 13.5 and 13.6 display the influences on notched strength,oN/oo, of the parameters m and C.Figure 13.5 shows that the exponential parameter affects the slope of the notch sensitivity curve,while Figure 13.6 shows that the notch sensitivity factor shifts the curves along the log R axis without affecting the shape of the curves.The admissible ranges for the parameters are 0 s m0.A notch-insensitive laminate is characterized by a large do in comparison to R.This corresponds to m-1 and C->0. Figure 13.7 shows notched strength vs.hole radius for two quasi-isotropic carbon/epoxy laminates with [+45/0/90],and [90/0/+45],lay-ups and the magnitudes of the corresponding fitting parameters m and C determined as outlined in Section 13.3. ©2003 by CRC Press LLC
13.1.3 Modification of PSC To improve the accuracy of notched strength predictions using the PSC, Pipes et al. [2], following Karlak’s modification [3], let the characteristic distance, d0 (Equation (13.6)) become a power function of hole radius d0 = (R/R0)m/C (13.11) where m is an exponential parameter, R0 is a reference radius, and C is the notch sensitivity factor. In essence, this model adds one more parameter (the exponential parameter) to the PSC. The reference radius may arbitrarily be chosen as R0 = 1 mm. The parameter λ (Equation (13.7)) then becomes λ = 1/(1 + Rm–1C–1) (13.12) Figures 13.5 and 13.6 display the influences on notched strength, σN/σ0, of the parameters m and C. Figure 13.5 shows that the exponential parameter affects the slope of the notch sensitivity curve, while Figure 13.6 shows that the notch sensitivity factor shifts the curves along the log R axis without affecting the shape of the curves. The admissible ranges for the parameters are 0 ≤ m < 1 and C ≥ 0. A notch-insensitive laminate is characterized by a large d0 in comparison to R. This corresponds to m → 1 and C → 0. Figure 13.7 shows notched strength vs. hole radius for two quasi-isotropic carbon/epoxy laminates with [±45/0/90]s and [90/0/±45]s lay-ups and the magnitudes of the corresponding fitting parameters m and C determined as outlined in Section 13.3. FIGURE 13.4 Notched strength data and predictions based on the average stress criterion for a notched [02/±45]s carbon/epoxy laminate [4]. TX001_ch13_Frame Page 173 Saturday, September 21, 2002 5:07 AM © 2003 by CRC Press LLC
1.00 m=0 0.75 y 04 0.50 0.6 0.8 1.0 0.25 1/K -2 0 Log R(mm) FIGURE 13.5 Influence of exponential parameter on notched strength,C=10.0 mm-[2]. 1.00 C0.0 C-0.02 0.75 C-0.2 ON 0.50 C=2.0 C=20 0.25 1/K 0 0.01.02.03.04.0 Log R (mm) FIGURE 13.6 Influence of notch sensitivity factor on notched strength,m =0.5,unit of C is mm-1[2]. 1.00 m=0.475 T300/934 Carbon/Epoxy C=1.37mm 0.75 。±45/0/901s a[90/0/±45s N0.50 m=0.40 00 C=1.65mm 0.25 VK. 0 -0.5 0 0.51.0 1.52.0 Log R (mm) FIGURE 13.7 Notched strength data for [+45/0/90]s and [90/0/+45].carbon/epoxy laminates [2]. 13.2 Test Specimen Preparation Although any laminate configuration can be used,most commonly a [0/+45/90]s(quasi-isotropic)laminate is selected.Laminates with higher ©2003 by CRC Press LLC
13.2 Test Specimen Preparation Although any laminate configuration can be used, most commonly a [0/±45/90]ns (quasi-isotropic) laminate is selected. Laminates with higher FIGURE 13.5 Influence of exponential parameter on notched strength, C = 10.0 mm–1 [2]. FIGURE 13.6 Influence of notch sensitivity factor on notched strength, m = 0.5, unit of C is mm–1 [2]. FIGURE 13.7 Notched strength data for [±45/0/90]s and [90/0/±45]s carbon/epoxy laminates [2]. TX001_ch13_Frame Page 174 Saturday, September 21, 2002 5:07 AM © 2003 by CRC Press LLC
percentages of 0 plies are tested when of specific interest to the intended design application. Often the same specimen configuration is used for both tensile and com- pressive open-hole tests.One commonly used specimen size is 305 mm long and 38 mm wide.Another standard open-hole compression test method uti- lizes a specimen only 75 mm long and 25 mm wide,as will be discussed later. If the test facilities permit,it is strongly recommended that wide specimens be used to accommodate a large range of hole sizes and better approximate an infinitely wide specimen.Daniel [5],for example,used 127-mm-wide lami- nates and hole diameters ranging from 6.4 to 25.4 mm.Specimen thickness is not critical and is somewhat dependent upon the specific laminate configura- tion to be tested.Aspecimen thickness on the order of 2.5 to 5 mm is commonly used.The diameter of the hole in the specimen,which is to be centered at the midlength of the specimen,can also be arbitrarily selected.However,as discussed at the beginning of this chapter,the ratio of specimen width to hole diameter influences the magnitude of the stress concentration induced.A hole diameter of 6.4 mm has become a commonly used size. Unless a laminate with a high percentage of 0 plies is to be tested,tabs are not usually necessary.If aggressively serrated tensile wedge grips are used it may be necessary to protect the open-hole tension specimen surfaces with one or more layers of emery cloth,an(unbonded)layer of plastic sheet material(approximately 1 to 2 mm thick),or similar padding material.The open-hole compression test methods typically involve the use of some type of special fixture to prevent specimen buckling,as will be discussed.These fixtures are usually designed for use with an untabbed specimen. Measure the cross-sectional dimensions (average six measurements)and check for parallelism of the edges and of the end-tab surfaces if used(see Chapter 4 for typical specimen tolerances). If a series of tests are to be conducted for various hole sizes,divide the specimens into groups by hole size.Note also that one of these groups should be specimens without holes to determine the unnotched strength,0o.At least three specimens should be assigned to each group,although a minimum of five specimens is more common.At least three hole diameters should be investigated;for example,D=3,5,and 7 mm.Machine the holes as specified in Section 4.2. 13.3 Tensile Test Procedure and Data Reduction The specimens should be mounted and tested in a properly aligned and calibrated testing machine with mechanical wedge action or hydraulic grips. Set the crosshead rate at about 0.5 to 1 mm/min.Record the load vs.cross- head displacement to detect the ultimate load and any anomalous load- displacement behavior.If a strain gage is used,place it midway between the ©2003 by CRC Press LLC
percentages of 0° plies are tested when of specific interest to the intended design application. Often the same specimen configuration is used for both tensile and compressive open-hole tests. One commonly used specimen size is 305 mm long and 38 mm wide. Another standard open-hole compression test method utilizes a specimen only 75 mm long and 25 mm wide, as will be discussed later. If the test facilities permit, it is strongly recommended that wide specimens be used to accommodate a large range of hole sizes and better approximate an infinitely wide specimen. Daniel [5], for example, used 127-mm-wide laminates and hole diameters ranging from 6.4 to 25.4 mm. Specimen thickness is not critical and is somewhat dependent upon the specific laminate configuration to be tested.Aspecimen thickness on the order of 2.5 to 5 mm is commonly used. The diameter of the hole in the specimen, which is to be centered at the midlength of the specimen, can also be arbitrarily selected. However, as discussed at the beginning of this chapter, the ratio of specimen width to hole diameter influences the magnitude of the stress concentration induced. A hole diameter of 6.4 mm has become a commonly used size. Unless a laminate with a high percentage of 0° plies is to be tested, tabs are not usually necessary. If aggressively serrated tensile wedge grips are used it may be necessary to protect the open-hole tension specimen surfaces with one or more layers of emery cloth, an (unbonded) layer of plastic sheet material (approximately 1 to 2 mm thick), or similar padding material. The open-hole compression test methods typically involve the use of some type of special fixture to prevent specimen buckling, as will be discussed. These fixtures are usually designed for use with an untabbed specimen. Measure the cross-sectional dimensions (average six measurements) and check for parallelism of the edges and of the end-tab surfaces if used (see Chapter 4 for typical specimen tolerances). If a series of tests are to be conducted for various hole sizes, divide the specimens into groups by hole size. Note also that one of these groups should be specimens without holes to determine the unnotched strength, σ0. At least three specimens should be assigned to each group, although a minimum of five specimens is more common. At least three hole diameters should be investigated; for example, D = 3, 5, and 7 mm. Machine the holes as specified in Section 4.2. 13.3 Tensile Test Procedure and Data Reduction The specimens should be mounted and tested in a properly aligned and calibrated testing machine with mechanical wedge action or hydraulic grips. Set the crosshead rate at about 0.5 to 1 mm/min. Record the load vs. crosshead displacement to detect the ultimate load and any anomalous loaddisplacement behavior. If a strain gage is used, place it midway between the TX001_ch13_Frame Page 175 Saturday, September 21, 2002 5:07 AM © 2003 by CRC Press LLC
FIGURE 13.8 Carbon-epoxy open-hole tensile specimen tested to failure. TABLE 13.1 Unnotched and Notched Strength Data for [0/+45/90] Carbon/Epoxy Coupons Notch Radius Strength (mm) (MPa) 0 607(c) 1.6 437 2.5 376 3.3 348 hole and the end tab.Make sure eyes are protected in the test area.Load all specimens to failure.Figure 13.8 shows an open-hole tension carbon/epoxy specimen after testing. Notched strength,oN,is calculated based on the gross cross-sectional area (A=wh).A typical set of unnotched and notched strength data for [0/45/ 90]carbon/epoxy coupons is given in Table 13.1.Because the strength model discussed here is restricted to a plate with an infinite width-to-hole diameter ratio,a comparison between experimental data and the notch strength model requires correction for the finite width of the specimen.A common way to correct the data is to multiply the experimental notched strength with a correction factor,K/K,where K is the stress concentration factor for an orthotropic plate of finite width;i.e., oN(∞)=x(w K (13.13) where oN(w)is the experimental strength for a plate of width w,and oN() is the corresponding strength for an infinite plate.A closed-form expression ©2003 by CRC Press LLC
hole and the end tab. Make sure eyes are protected in the test area. Load all specimens to failure. Figure 13.8 shows an open-hole tension carbon/epoxy specimen after testing. Notched strength, σN, is calculated based on the gross cross-sectional area (A = wh). A typical set of unnotched and notched strength data for [0/45/ 90]s carbon/epoxy coupons is given in Table 13.1. Because the strength model discussed here is restricted to a plate with an infinite width-to-hole diameter ratio, a comparison between experimental data and the notch strength model requires correction for the finite width of the specimen. A common way to correct the data is to multiply the experimental notched strength with a correction factor, K/K∞, where K is the stress concentration factor for an orthotropic plate of finite width; i.e., (13.13) where σN(w) is the experimental strength for a plate of width w, and σN(∞) is the corresponding strength for an infinite plate. A closed-form expression FIGURE 13.8 Carbon–epoxy open-hole tensile specimen tested to failure. TABLE 13.1 Unnotched and Notched Strength Data for [0/±45/90]s Carbon/Epoxy Coupons Notch Radius (mm) Strength (MPa) 0 1.6 2.5 3.3 607 (σ0) 437 376 348 σ σ N N w K K ( ) ∞ = ( ) ∞ TX001_ch13_Frame Page 176 Saturday, September 21, 2002 5:07 AM © 2003 by CRC Press LLC
TABLE 13.2 Finite Width Correction Factors for Various Carbon/Epoxy(AS4/3501-6) Lay-ups [8]E1=125 GPa,E2 =9.9 GPa,Vi2=0.28,and Gu2 =5.5 GPa Lay-Up K w/D=2 6 8 10 [0/±45/901. 3.00 1.4340 1.1495 1.0736 1.0260 1.0107 1.0037 [02/±451. 3.48 1.3725 1.1291 1.0632 1.0216 1.0093 1.0031 [0/±45l. 4.07 1.3226 1.1109 1.0577 1.0172 1.0095 1.0041 [0./±451. 4.44 1.2992 1.1006 1.0472 1.0152 1.0102 1.0051 [±451l 2.06 1.6425 1.2379 1.1215 1.0442 1.0180 1.0062 Equation (13.14) 1.417 1.148 1.076 1.031 1.017 1.011 Determined from Equation(13.2). for K,however,does not exist,and K has to be determined using the boundary collocation method [6]or the finite element method [7].Table 13.2 gives finite width correction factors as a function of width-to-hole diameter ratio(w/D) for various carbon/epoxy lay-ups [8].Note that K/K is >1,which means that finite-width specimens exhibit larger stress concentrations than infinitely wide specimens(w/D28).Consequently,it is expected that a hole in a finite- width specimen will be more detrimental in terms of strength than a hole in an infinitely wide specimen.A common approximation,which is also reasonably accurate for composite laminates with w/D>4 [8](see also Table 13.2)is to use an isotropic expression [9,10]for K/K. K_2+1-(D/w) K31-(D/w)） (13.14) To enable comparison of the data with the PSC(Equation(13.6)),first correct the experimental data(Table 13.1)for the finite size,using the approximate expression for the stress concentration factor(Equation(13.14)).Then solve for the parameter A in Equation(13.6)using an interactive method such as Muller's method [11]or Newton-Raphson's method [12,13].From the defini- tion of A(Equation (13.12))it is observed that only the root between zero and one is required.For illustrative purposes,the notched strength data listed in Table 13.1 were corrected according to Equation(13.14),and the corresponding A values were determined with the Newton-Raphson method, and are listed in Table 13.3. To obtain the parameters m and C,Equation(13.12)may be written as -log(1/-1)=log C+(1-m)log R (13.15) By plotting-log(1/-1)vs.log R,the slope and the intercept at log R=0 can be obtained by the least-squares method.The slope is equal to 1-m, and the intercept is equal to log C.Figure 13.9 shows -log (1/A-1)plotted vs.log R for the data of Table 13.3. ©2003 by CRC Press LLC
for K, however, does not exist, and K has to be determined using the boundary collocation method [6] or the finite element method [7]. Table 13.2 gives finite width correction factors as a function of width-to-hole diameter ratio (w/D) for various carbon/epoxy lay-ups [8]. Note that K/K∞ is >1, which means that finite-width specimens exhibit larger stress concentrations than infinitely wide specimens (w/D ≥ 8). Consequently, it is expected that a hole in a finitewidth specimen will be more detrimental in terms of strength than a hole in an infinitely wide specimen. A common approximation, which is also reasonably accurate for composite laminates with w/D > 4 [8] (see also Table 13.2) is to use an isotropic expression [9,10] for K/K∞ (13.14) To enable comparison of the data with the PSC (Equation (13.6)), first correct the experimental data (Table 13.1) for the finite size, using the approximate expression for the stress concentration factor (Equation (13.14)). Then solve for the parameter λ in Equation (13.6) using an interactive method such as Muller’s method [11] or Newton-Raphson’s method [12,13]. From the definition of λ (Equation (13.12)) it is observed that only the root between zero and one is required. For illustrative purposes, the notched strength data listed in Table 13.1 were corrected according to Equation (13.14), and the corresponding λ values were determined with the Newton-Raphson method, and are listed in Table 13.3. To obtain the parameters m and C, Equation (13.12) may be written as –log(1/λ – 1) = log C + (1 – m)log R (13.15) By plotting –log(1/λ – 1) vs. log R, the slope and the intercept at log R = 0 can be obtained by the least-squares method. The slope is equal to 1 – m, and the intercept is equal to log C. Figure 13.9 shows –log (1/λ – 1) plotted vs. log R for the data of Table 13.3. TABLE 13.2 Finite Width Correction Factors for Various Carbon/Epoxy (AS4/3501-6) Lay-ups [8] E1 = 125 GPa, E2 = 9.9 GPa, ν12 = 0.28, and G12 = 5.5 GPa Lay-Up K∞ a w/D = 2 3 4 6 8 10 [0/±45/90]s [02/±45]s [04/±45]s [06/±45]s [±45]s Equation (13.14) 3.00 3.48 4.07 4.44 2.06 1.4340 1.3725 1.3226 1.2992 1.6425 1.417 1.1495 1.1291 1.1109 1.1006 1.2379 1.148 1.0736 1.0632 1.0577 1.0472 1.1215 1.076 1.0260 1.0216 1.0172 1.0152 1.0442 1.031 1.0107 1.0093 1.0095 1.0102 1.0180 1.017 1.0037 1.0031 1.0041 1.0051 1.0062 1.011 a Determined from Equation (13.2). K K 2 Dw ∞ D w = + − ( ) ( ) ( ) − ( ) 1 3 1 3 TX001_ch13_Frame Page 177 Saturday, September 21, 2002 5:07 AM © 2003 by CRC Press LLC
TABLE 13.3 Corrected Notched Strength Data(Table 13.1) and Values of A Determined Using the Newton-Raphson Method R (mm) GNIGo 1.6 0.73 0.5998 2.5 0.64 0.6867 3.3 0.62 0.7052 Corrected using Equation(13.14). 0.8 Carbon/Epoxy [0/±45/901s 0.6 0.4 -m 0.2 -lg C 0 0.20.40.6 0.81.0 Log R FIGURE 13.9 Determination of the parameters m and C for [0/+45/90],carbon/epoxy specimens,m =0.36, and C=1.16 mm-i. 1.0 m=0.36 C=1.16mm 0.8 ON 0.6 00 0.4 IK 0.2 Carbon/Epoxy [0/生45/90]s -0.5 00.51.0 1.52.0 Log R (mm) FIGURE 13.10 Theoretical and experimental notch sensitivity for [0/+45/901,carbon/epoxy specimens. As an illustration of the goodness of the fit for the parameters m and C, the theoretical curve of oN/oo is plotted vs.log R along with the experimental values in Figure 13.10.Within the limited range of experimental data,excellent agreement is observed.Once the parameters m and C are established,it is possible to predict the notched strength for any hole size and coupon width (within reasonable limits)using the above methodology. 2003 by CRC Press LLC
As an illustration of the goodness of the fit for the parameters m and C, the theoretical curve of σN/σ0 is plotted vs. log R along with the experimental values in Figure 13.10. Within the limited range of experimental data, excellent agreement is observed. Once the parameters m and C are established, it is possible to predict the notched strength for any hole size and coupon width (within reasonable limits) using the above methodology. TABLE 13.3 Corrected Notched Strength Data (Table 13.1) and Values of λ Determined Using the Newton-Raphson Method R (mm) σN /σ0 a λ 1.6 2.5 3.3 0.73 0.64 0.62 0.5998 0.6867 0.7052 a Corrected using Equation (13.14). FIGURE 13.9 Determination of the parameters m and C for [0/±45/90]s carbon/epoxy specimens, m = 0.36, and C = 1.16 mm–1. FIGURE 13.10 Theoretical and experimental notch sensitivity for [0/±45/90]s carbon/epoxy specimens. TX001_ch13_Frame Page 178 Saturday, September 21, 2002 5:07 AM © 2003 by CRC Press LLC