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 m<1 and C >0.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