6.1.2.2 Ultimate Failure of a Laminate Steps for the ultimate failure prediction of a laminate are as follows. 1.Calculate stresses and strains in each lamina using the lamination theory 2.Apply an appropriate failure theory to predict which lamina failed first 3.Assign reduced stiffness and strength to the failed lamina 4.Recalculate stresses and strains in each of the remaining laminas using the lamination theory 5.Follow through steps 2 and 3 to predict the next lamina failure 6.Repeat steps 2-4 until ultimate failure of the laminate occurs Following the procedure outlined earlier,it is possible to generate failure envelopes describing the FPF as well as the ultimate failure of the laminate. In practice,a series of failure envelopes is drawn in a two-dimensional normal stress space in which the coordinate axes represent the average laminate stresses N/h and Ny/h.The area bounded by each failure envelope represents the safe design space for a constant average laminate shear stress Nx/h (Figure 6.4). Experimental verification for the laminate failure prediction methods requires the use of biaxial tests in which both normal stresses and shear stresses are present.Thin-walled large-diameter tubes subjected to various combin- ations of internal and external pressures,longitudinal loads,and torsional loads are the most suitable specimens for this purpose [6].From the limited number of experimental results reported in the literature,it can be concluded Nylh 50 Nxy/h=0 10 25 -100 -75 75 -50-25 25 50 100 Na/h -25 20 -50 20 10 FIGURE 6.4 Theoretical failure envelopes for a carbon fiber-epoxy [0/90+45]s laminate (note that the in-plane loads per unit laminate thickness are in ksi). 2007 by Taylor Francis Group,LLC
6.1.2.2 Ultimate Failure of a Laminate Steps for the ultimate failure prediction of a laminate are as follows. 1. Calculate stresses and strains in each lamina using the lamination theory 2. Apply an appropriate failure theory to predict which lamina failed first 3. Assign reduced stiffness and strength to the failed lamina 4. Recalculate stresses and strains in each of the remaining laminas using the lamination theory 5. Follow through steps 2 and 3 to predict the next lamina failure 6. Repeat steps 2–4 until ultimate failure of the laminate occurs Following the procedure outlined earlier, it is possible to generate failure envelopes describing the FPF as well as the ultimate failure of the laminate. In practice, a series of failure envelopes is drawn in a two-dimensional normal stress space in which the coordinate axes represent the average laminate stresses Nxx=h and Nyy=h. The area bounded by each failure envelope represents the safe design space for a constant average laminate shear stress Nxy=h (Figure 6.4). Experimental verification for the laminate failure prediction methods requires the use of biaxial tests in which both normal stresses and shear stresses are present. Thin-walled large-diameter tubes subjected to various combinations of internal and external pressures, longitudinal loads, and torsional loads are the most suitable specimens for this purpose [6]. From the limited number of experimental results reported in the literature, it can be concluded Nyy /h Nxx /h Nxy /h=0 −100 −75 75 10 10 100 20 20 50 50 −50 −50 25 25 −25 −25 FIGURE 6.4 Theoretical failure envelopes for a carbon fiber–epoxy [0=90±45]S laminate (note that the in-plane loads per unit laminate thickness are in ksi). 2007 by Taylor & Francis Group, LLC
that no single failure theory represents all laminates equally well.Among the various deficiencies in the theoretical prediction methods are the absence of interlaminar stresses and nonlinear effects.The assumption regarding the load transfer between the failed laminas and the active laminas can also introduce errors in the theoretical analyses. The failure load prediction for a laminate depends strongly on the lamina failure theory selected [7].In the composite material industry,there is little agreement on which lamina failure theory works best,although the maximum strain theory is more commonly used than the others [8].Recently,the Tsai-Wu failure theory is finding more applications in the academic field. 6.1.3 FAILURE PREDICTION IN RANDOM FIBER LAMINATES There are two different approaches for predicting failure in laminates contain- ing randomly oriented discontinuous fibers. In the Hahn's approach [9],which is a simple approach,failure is predicted when the maximum tensile stress in the laminate equals the following strength averaged over all possible fiber orientation angles: (6.6 where S =strength of the random fiber laminate SLt=longitudinal strength of a 0 laminate Sr=transverse strength of a 0 laminate In the Halpin-Kardos approach [10],the random fiber laminate is modeled as a quasi-isotropic [0/+45/90]s laminate containing discontinuous fibers in the 0°，±45°，and90°orientations.The Halpin--Tsai equations,Equations3.49 through 3.53,are used to calculate the basic elastic properties,namely,E1, E22,v12,and Gi2,of the 0 discontinuous fiber laminas.The ultimate strain allowables for the 0 and 90 laminas are estimated from the continuous fiber allowables using the Halpin-Kardos equations: -0.87 ELt(d +0.50 for le le (6.7) and eTdd=emn(1-1.21v249） The procedure followed by Halpin and Kardos [10]for estimating the ultimate strength of random fiber laminates is the same as the ply-by-ply analysis used for continuous fiber quasi-isotropic [0/+45/90]s laminates. 2007 by Taylor&Francis Group.LLC
that no single failure theory represents all laminates equally well. Among the various deficiencies in the theoretical prediction methods are the absence of interlaminar stresses and nonlinear effects. The assumption regarding the load transfer between the failed laminas and the active laminas can also introduce errors in the theoretical analyses. The failure load prediction for a laminate depends strongly on the lamina failure theory selected [7]. In the composite material industry, there is little agreement on which lamina failure theory works best, although the maximum strain theory is more commonly used than the others [8]. Recently, the Tsai–Wu failure theory is finding more applications in the academic field. 6.1.3 FAILURE PREDICTION IN RANDOM FIBER LAMINATES There are two different approaches for predicting failure in laminates containing randomly oriented discontinuous fibers. In the Hahn’s approach [9], which is a simple approach, failure is predicted when the maximum tensile stress in the laminate equals the following strength averaged over all possible fiber orientation angles: Sr ¼ 4 p ffiffiffiffiffiffiffiffiffiffiffiffi SLtSTt p , (6:6) where Sr ¼ strength of the random fiber laminate SLt ¼ longitudinal strength of a 08 laminate STt ¼ transverse strength of a 08 laminate In the Halpin–Kardos approach [10], the random fiber laminate is modeled as a quasi-isotropic [0=±45=90]S laminate containing discontinuous fibers in the 08, ±458, and 908 orientations. The Halpin–Tsai equations, Equations 3.49 through 3.53, are used to calculate the basic elastic properties, namely, E11, E22, n12, and G12, of the 08 discontinuous fiber laminas. The ultimate strain allowables for the 08 and 908 laminas are estimated from the continuous fiber allowables using the Halpin–Kardos equations: «Lt(d) ¼ «Lt Ef Em 0:87 þ 0:50 " # for lf > lc (6:7) and «Td(d) ¼ «Tt 1 1:21v 2=3 f : The procedure followed by Halpin and Kardos [10] for estimating the ultimate strength of random fiber laminates is the same as the ply-by-ply analysis used for continuous fiber quasi-isotropic [0=±45=90]S laminates. 2007 by Taylor & Francis Group, LLC.
6.1.4 FAILURE PREDICTION IN NOTCHED LAMINATES 6.1.4.1 Stress Concentration Factor It is well known that the presence of a notch in a stressed member creates highly localized stresses at the root of the notch.The ratio of the maximum stress at the notch root to the nominal stress is called the stress concentration factor. Consider,for example,the presence of a small circular hole in an infinitely wide plate(i.e.,w>R,Figure 6.5).The plate is subjected to a uniaxial tensile stress far from the hole.The tangential stress o at the two ends of the horizontal diameter of the hole is much higher than the nominal stress o.In this case,the hole stress concentration factor Kr is defined as KT =(R.0) For an infinitely wide isotropic plate,the hole stress concentration factor is 3. For a symmetric laminated plate with orthotropic in-plane stiffness properties, the hole stress concentration factor is given by 2 Kr=1+ A11A2-A2 VA22 VA11A22-A12+ 2A66 (6.8) where A,A12 422,and 466 are the in-plane stiffnesses defined in Chapter 3. W>>R FIGURE 6.5 A uniaxially loaded infinite plate with a circular hole. 2007 by Taylor Francis Group,LLC
6.1.4 FAILURE PREDICTION IN NOTCHED LAMINATES 6.1.4.1 Stress Concentration Factor It is well known that the presence of a notch in a stressed member creates highly localized stresses at the root of the notch. The ratio of the maximum stress at the notch root to the nominal stress is called the stress concentration factor. Consider, for example, the presence of a small circular hole in an infinitely wide plate (i.e., w >> R, Figure 6.5). The plate is subjected to a uniaxial tensile stress s far from the hole. The tangential stress syy at the two ends of the horizontal diameter of the hole is much higher than the nominal stress s. In this case, the hole stress concentration factor KT is defined as KT ¼ syy(R,0) s : For an infinitely wide isotropic plate, the hole stress concentration factor is 3. For a symmetric laminated plate with orthotropic in-plane stiffness properties, the hole stress concentration factor is given by KT ¼ 1 þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 A22 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi A11A22 p A12 þ A11A22 A2 12 2A66 s , (6:8) where A11, A12,, A22, and A66 are the in-plane stiffnesses defined in Chapter 3. s s w >> R y x 2R FIGURE 6.5 A uniaxially loaded infinite plate with a circular hole. 2007 by Taylor & Francis Group, LLC.
TABLE 6.2 Circular Hole Stress Concentration Factors Circular Hole Stress Material Laminate Concentration Factor(KT) Isotropic material 3 S-glass-epoxy 0 [02/±45s 3.313 [0/90/±451s 3 [±45]s 2.382 T-300 carbon-epoxy 0 6.863 [04/±45s 4.126 [0/90/±45s 3 [0/±45s 2.979 [±45s 1.984 Note that,for an infinitely wide plate,the hole stress concentration factor Kr is independent of the hole size.However,for a finite width plate,Kr increases with increasing ratio of hole diameter to plate width.No closed- form solutions are available for the hole stress concentration factors in finite width orthotropic plates.They are determined either by finite element methods [11,12]or by experimental techniques,such as strain gaging,moire interfero- metry,and birefringent coating,among other techniques [13].Appendix A.7 gives the finite width correction factor for isotropic plates,which can be used for approximate calculation of hole stress concentration factors for orthotropic plates of finite width. Table 6.2 lists values of hole stress concentration factors for a number of symmetric laminates.For each material,the highest value of Kr is observed with 0 fiber orientation.However,Kr decreases with increasing proportions of +45 layers in the laminate.It is interesting to note that a [0/90/+45]s laminate has the same Kr value as an isotropic material and a [+45]s laminate has a much lower Kr than an isotropic material. 6.1.4.2 Hole Size Effect on Strength The hole stress concentration factor in wide plates containing very small holes (R w)is constant,yet experimental results show that the tensile strength of many laminates is influenced by the hole diameter instead of remaining constant.This hole size effect has been explained by Waddoups et al.[14]on the basis of intense energy regions on each side of the hole.These energy regions were modeled as incipient cracks extending symmetrically from the hole boundary perpendicular to the loading direction.Later,Whitney and Nuismer [15,16]proposed two stress criteria to predict the strength of notched composites.These two failure criteria are discussed next. 2007 by Taylor Francis Group.LLC
Note that, for an infinitely wide plate, the hole stress concentration factor KT is independent of the hole size. However, for a finite width plate, KT increases with increasing ratio of hole diameter to plate width. No closedform solutions are available for the hole stress concentration factors in finite width orthotropic plates. They are determined either by finite element methods [11,12] or by experimental techniques, such as strain gaging, moire interferometry, and birefringent coating, among other techniques [13]. Appendix A.7 gives the finite width correction factor for isotropic plates, which can be used for approximate calculation of hole stress concentration factors for orthotropic plates of finite width. Table 6.2 lists values of hole stress concentration factors for a number of symmetric laminates. For each material, the highest value of KT is observed with 08 fiber orientation. However, KT decreases with increasing proportions of ±458 layers in the laminate. It is interesting to note that a [0=90=±45]S laminate has the same KT value as an isotropic material and a [±458]S laminate has a much lower KT than an isotropic material. 6.1.4.2 Hole Size Effect on Strength The hole stress concentration factor in wide plates containing very small holes (R w) is constant, yet experimental results show that the tensile strength of many laminates is influenced by the hole diameter instead of remaining constant. This hole size effect has been explained by Waddoups et al. [14] on the basis of intense energy regions on each side of the hole. These energy regions were modeled as incipient cracks extending symmetrically from the hole boundary perpendicular to the loading direction. Later, Whitney and Nuismer [15,16] proposed two stress criteria to predict the strength of notched composites. These two failure criteria are discussed next. TABLE 6.2 Circular Hole Stress Concentration Factors Material Laminate Circular Hole Stress Concentration Factor (KT) Isotropic material — 3 S-glass–epoxy 0 4 [02=±45]S 3.313 [0=90=±45]S 3 [±45]S 2.382 T-300 carbon–epoxy 0 6.863 [04=±45]S 4.126 [0=90=±45]S 3 [0=±45]S 2.979 [±45]S 1.984 2007 by Taylor & Francis Group, LLC
2R FIGURE 6.6 Failure prediction in a notched laminate according to the point stress criterion. Point Stress Criterion:According to the point stress criterion,failure occurs when the stress over a distance do away from the notch (Figure 6.6)is equal to or greater than the strength of the unnotched laminate.This characteristic distance do is assumed to be a material property,independent of the laminate geometry as well as the stress distribution.It represents the distance over which the material must be critically stressed to find a flaw of sufficient length to initiate failure. To apply the point stress criterion,the stress field ahead of the notch root must be known.For an infinitely wide plate containing a circular hole ofradius R and subjected to a uniform tensile stress o away from the hole,the most significant stress is acting along the x axis on both sides of the hole edges. For an orthotropic plate,this normal stress component is approximated as [17]: (x,0) ++图---} (6.9) which is valid for x>R.In this equation,Kr is the hole stress concentration factor given in Equation 6.8. 2007 by Taylor Francis Group,LLC
Point Stress Criterion: According to the point stress criterion, failure occurs when the stress over a distance d0 away from the notch (Figure 6.6) is equal to or greater than the strength of the unnotched laminate. This characteristic distance d0 is assumed to be a material property, independent of the laminate geometry as well as the stress distribution. It represents the distance over which the material must be critically stressed to find a flaw of sufficient length to initiate failure. To apply the point stress criterion, the stress field ahead of the notch root must be known. For an infinitely wide plate containing a circular hole ofradius R and subjected to a uniform tensile stress s away from the hole, the most significant stress is syy acting along the x axis on both sides of the hole edges. For an orthotropic plate, this normal stress component is approximated as [17]: syy(x, 0) ¼ s 2 2 þ R x 2 þ 3 R x 4 (KT 3) 5 R x 6 7 R x 8 ( ) " # , (6:9) which is valid for x R. In this equation, KT is the hole stress concentration factor given in Equation 6.8. 2R d0 x y sUt syy s s FIGURE 6.6 Failure prediction in a notched laminate according to the point stress criterion. 2007 by Taylor & Francis Group, LLC.