Copyrighted Materials CpUPress o CHAPTER TEN Failure Criteria Failure of fiber-reinforced composites may be caused by fiber buckling,fiber breakage,matrix cracking,delamination,or by a combination of these factors (Fig.10.1).Local fiber buckling,or microbuckling,reduces the compressive stiff- ness and strength of the laminate.Microbuckling does not necessarily lead to immediate failure because the surrounding matrix supports the fibers.The prop- erties of the fibers and the matrix greatly affect the onset and magnitude of fiber buckling and the resulting losses in the compressive properties of the laminate. One of the main roles of the fibers is to carry tensile loads.When dry fibers (with no matrix surrounding them)break,they,of course,can no longer carry ten- sile loads.When the fibers are embedded in a matrix,the matrix acts as a bridge about the break and transmits the load across the gap created by the breakage as well from the broken to the adjacent fibers.Fiber bridging,as this phenomenon is called,is the main reason that the tensile strengths of unidirectional,continu- ous fiber-reinforced composites are higher than the tensile strengths of dry fiber bundles. Matrix cracking frequently occurs in composite laminates.In itself,matrix cracking generally does not result in ultimate failure of a laminate.Nonetheless, matrix cracks have many detrimental effects:they facilitate moisture absorbtion, reduce the matrix-dominated stiffnesses of the laminate and,last but not least,may propagate into the interface between adjacent layers,initiating delamination. Delamination is a separation of adjacent layers that may be introduced either during manufacture or subsequently by loads applied to the laminate.For example, loads due to transverse impact by an object on the laminate are a frequent cause of delamination.Delamination reduces the bending stiffness and strength as well as the load carrying capability of the laminate under compression.Significantly, under repeated loading the size of the delamination may increase to a critical point. Like the behavior of a crack in metal,once the critical size is reached,the growth of the delamination becomes unstable,leading to a rapid loss of compressive strength. 411
CHAPTER TEN Failure Criteria Failure of fiber-reinforced composites may be caused by fiber buckling, fiber breakage, matrix cracking, delamination, or by a combination of these factors (Fig. 10.1). Local fiber buckling, or microbuckling, reduces the compressive stiffness and strength of the laminate. Microbuckling does not necessarily lead to immediate failure because the surrounding matrix supports the fibers. The properties of the fibers and the matrix greatly affect the onset and magnitude of fiber buckling and the resulting losses in the compressive properties of the laminate. One of the main roles of the fibers is to carry tensile loads. When dry fibers (with no matrix surrounding them) break, they, of course, can no longer carry tensile loads. When the fibers are embedded in a matrix, the matrix acts as a bridge about the break and transmits the load across the gap created by the breakage as well from the broken to the adjacent fibers. Fiber bridging, as this phenomenon is called, is the main reason that the tensile strengths of unidirectional, continuous fiber-reinforced composites are higher than the tensile strengths of dry fiber bundles. Matrix cracking frequently occurs in composite laminates. In itself, matrix cracking generally does not result in ultimate failure of a laminate. Nonetheless, matrix cracks have many detrimental effects: they facilitate moisture absorbtion, reduce the matrix-dominated stiffnesses of the laminate and, last but not least, may propagate into the interface between adjacent layers, initiating delamination. Delamination is a separation of adjacent layers that may be introduced either during manufacture or subsequently by loads applied to the laminate. For example, loads due to transverse impact by an object on the laminate are a frequent cause of delamination. Delamination reduces the bending stiffness and strength as well as the load carrying capability of the laminate under compression. Significantly, under repeated loading the size of the delamination may increase to a critical point. Like the behavior of a crack in metal, once the critical size is reached, the growth of the delamination becomes unstable, leading to a rapid loss of compressive strength. 411
412 FAILURE CRITERIA Fiber buckling Fiber breakage Matrix cracking Delamination Figure 10.1:Typical failure modes of composites. Designers would be well served by mechanism-based (physical)theories that would indicate the load at which failure occurs as well as the mode of failure. Although such theories have been proposed,12 none is as yet at the stage where it could be applied in practical engineering design.Instead,frequently,ply-stress- based failure theories are used.3.4 According to these theories the criterion for failure in any one of the plies is 1 no failure f(o1,02,3,23,t13,t12,F,F, -1 failure limit, (10.1) >1 failure where o1,...,12 are the stresses in the ply and F1,F2,...are strength parameters. The criterion expressed by Eq.(10.1)is established in every ply,and failure is taken to occur when any one of the plies fails (first-ply failure). Here,we present three failure criteria for composites based on the aforemen- tioned concept:the quadratic,the maximum stress,and the maximum strain failure criteria.These criteria offer results that are sufficiently accurate for many(but by no means all)problems of practical interest.For this reason,in spite of their short- comings,they are relevant to the engineer.Nonetheless,the reader is warned to be cognizant of the following significant limits of the criteria listed above: Each criterion provides only the load at which first-ply failure occurs,that is, the load at which the linear load-displacement curve first changes(Fig.10.2). Under the load set that causes first-ply failure,the laminate does not necessarily fail because other undamaged plies can still carry load.As the applied loads 1 R.F.Gibson,Principles of Composite Material Mechanics.McGraw-Hill,New York,1994,pp.114- 126.244-249,and356-367. 2 S.R.Swanson,Advanced Composite Materials.Prentice-Hall,Upper Saddle River.New Jersey.1997. pp.91-120.123-147. 3 R.E.Rowlands,Strength(Failure)Theories and Their Experimental Correlation.In:Handbook of Composites,Vol.3.G.C.Sih and A.M.Skudra,eds.,Elsevier,Amsterdam,1985,pp.71-125. 4 M.N.Nahas,Survey of Failure and Post-Failure Theories of Laminated Fiber Reinforced Composites. Journal of Composites Technology and Research,Vol.8,138-153,1986
412 FAILURE CRITERIA Fiber buckling Fiber breakage Matrix cracking Delamination Figure 10.1: Typical failure modes of composites. Designers would be well served by mechanism-based (physical) theories that would indicate the load at which failure occurs as well as the mode of failure. Although such theories have been proposed,1,2 none is as yet at the stage where it could be applied in practical engineering design. Instead, frequently, ply–stressbased failure theories are used.3,4 According to these theories the criterion for failure in any one of the plies is f (σ1, σ2, σ3, τ23, τ13, τ12, F1, F2,...) < 1 no failure = 1 failure limit > 1 failure , (10.1) where σ1,...,τ12 are the stresses in the ply and F1, F2,... are strength parameters. The criterion expressed by Eq. (10.1) is established in every ply, and failure is taken to occur when any one of the plies fails (first-ply failure). Here, we present three failure criteria for composites based on the aforementioned concept: the quadratic, the maximum stress, and the maximum strain failure criteria. These criteria offer results that are sufficiently accurate for many (but by no means all) problems of practical interest. For this reason, in spite of their shortcomings, they are relevant to the engineer. Nonetheless, the reader is warned to be cognizant of the following significant limits of the criteria listed above: Each criterion provides only the load at which first-ply failure occurs, that is, the load at which the linear load-displacement curve first changes (Fig. 10.2). Under the load set that causes first-ply failure, the laminate does not necessarily fail because other undamaged plies can still carry load. As the applied loads 1 R. F. Gibson, Principles of Composite Material Mechanics. McGraw-Hill, New York, 1994, pp. 114– 126, 244–249, and 356–367. 2 S. R. Swanson, Advanced Composite Materials. Prentice-Hall, Upper Saddle River, New Jersey, 1997, pp. 91–120, 123–147. 3 R. E. Rowlands, Strength (Failure) Theories and Their Experimental Correlation. In: Handbook of Composites, Vol. 3. G. C. Sih and A. M. Skudra, eds., Elsevier, Amsterdam, 1985, pp. 71–125. 4 M. N. Nahas, Survey of Failure and Post-Failure Theories of Laminated Fiber Reinforced Composites. Journal of Composites Technology and Research, Vol. 8, 138–153, 1986
10.1 QUADRATIC FAILURE CRITERION 413 First-ply failure Load个 Displacement Figure 10.2:Load-displacement curve of a composite part. are increased beyond those at which first-ply failure occurs,there will be a sequence of ply failures until the load set is reached at which every ply has failed.The loads at ultimate failure may be considerably higher than at first- ply failure.Therefore,criteria based on first-ply failure are conservative. 。 None of the criteria sheds light on the failure mechanism or indicates the mode of failure. None of the criteria provides acceptable results for every condition of practical interest. Each criterion requires data,some of which are difficult to measure. Each criterion applies in regions inside the composite away from discontinu- ities such as holes,cracks,and edges.(Criteria applicable to plates containing a hole or a notch are given in Section 10.4.) 10.1 Quadratic Failure Criterion The quadratic failure criterion includes stresses up to the second power.In its most general form the quadratic failure criterion states that no failure occurs when the inequality below (Eq.10.2)is satisfied.This criterion and some of its simplified forms are variously referred to as Tsai-Wu,Hill,or Tsai-Hill failure criterion. F101+F202+F303+F4t23+Fst3+F6712+ F1o7+F2o+F3o+F442+F5+F66+ 2(F20102+F3O13+F4O123+F5O1t13+F16O1T12+ F30203+F242t23+F252T13+F26O2t12+F3403t23+ F353t13+f363t12+F45t23t3+F46t23t12+F56t13T2)<1, (10.2) where o1,02,...,ti2 are the components (in the x1,x2,x3 coordinate system)of the stress at the point of interest,that is,the stress that results from the applied loads,and the F's are strength parameters that depend on the material.No failure occurs when the left-hand side of Eq.(10.2)is less then unity.This means that the resultant stress is inside the failure surface (Fig.10.3,left).On the failure surface (Fig.10.3,middle),where the stress components are denoted by of
10.1 QUADRATIC FAILURE CRITERION 413 Load First-ply failure Displacement Figure 10.2: Load-displacement curve of a composite part. are increased beyond those at which first-ply failure occurs, there will be a sequence of ply failures until the load set is reached at which every ply has failed. The loads at ultimate failure may be considerably higher than at firstply failure. Therefore, criteria based on first-ply failure are conservative. None of the criteria sheds light on the failure mechanism or indicates the mode of failure. None of the criteria provides acceptable results for every condition of practical interest. Each criterion requires data, some of which are difficult to measure. Each criterion applies in regions inside the composite away from discontinuities such as holes, cracks, and edges. (Criteria applicable to plates containing a hole or a notch are given in Section 10.4.) 10.1 Quadratic Failure Criterion The quadratic failure criterion includes stresses up to the second power. In its most general form the quadratic failure criterion states that no failure occurs when the inequality below (Eq. 10.2) is satisfied. This criterion and some of its simplified forms are variously referred to as Tsai-Wu, Hill, or Tsai–Hill failure criterion. F1σ1 + F2σ2 + F3σ3 + F4τ23 + F5τ13 + F6τ12 + F11σ2 1 + F22σ2 2 + F33σ2 3 + F44τ 2 23 + F55τ 2 13 + F66τ 2 12 + 2(F12σ1σ2 + F13σ1σ3 + F14σ1τ23 + F15σ1τ13 + F16σ1τ12 + F23σ2σ3 + F24σ2τ23 + F25σ2τ13 + F26σ2τ12 + F34σ3τ23 + F35σ3τ13 + F36σ3τ12 + F45τ23τ13 + F46τ23τ12 + F56τ13τ12) < 1, (10.2) where σ1, σ2, ... , τ12 are the components (in the x1, x2, x3 coordinate system) of the stress at the point of interest, that is, the stress that results from the applied loads, and the F’s are strength parameters that depend on the material. No failure occurs when the left-hand side of Eq. (10.2) is less then unity. This means that the resultant stress is inside the failure surface (Fig. 10.3, left). On the failure surface (Fig. 10.3, middle), where the stress components are denoted by σf 1, σf 2,....,τ f 12
414 FAILURE CRITERIA No failure Failure Failure 02 01 01=0 01 Figure 10.3:Representation of the failure surface when only o1 and o2 stresses are applied. Eq.(10.2)is E+F吲+F+E吆+喵+F6h+1(o)2+ F2(2+F3()2+F4()2+s()2+ F6(t2)2+2(F2oo+Fi3oo+…+F6t2)=1. (10.3) The strength parameters must be determined by tests.For generally anisotropic and monoclinic materials,27 and 17 types of tests are required,respectively.This makes the use of the quadratic failure criterion impractical for structures made of generally anisotropic or monoclinic materials.The criterion becomes more manageable when the material is orthotropic or transversely isotropic.Therefore, in the following the criterion is presented only for these two types of materials. 10.1.1 Orthotropic Material An orthotropic material has three planes of symmetry (Figs.2.11 and 2.12).We select the x1,x2.x3 coordinate system with axes perpendicular to these symmetry planes. First,we consider only the shear-stress component t23 acting in the plane of symmetry (Fig.10.4).When only t23 acts,and it is in the positive direction,then Plane of symmetry Figure 10.4:The positive and negative shear stresses at failure acting in an orthotropic material; x1,x2,and x3 are perpendicular to the orthotropy planes
414 FAILURE CRITERIA σ1 σ2 No failure Failure Failure σ1 σ2 f = σσ 22 f = σσ 11 Figure 10.3: Representation of the failure surface when only σ1 and σ2 stresses are applied. Eq. (10.2) is F1σf 1 + F2σf 2 + F3σf 3 + F4τ f 23 + F5τ f 13 + F6τ f 12 + F11 σf 1 2 + F22 σf 2 2 + F33 σf 3 2 + F44 τ f 232 + F55 τ f 132 + F66 τ f 122 + 2 F12σf 1σf 2 + F13σf 1σf 3 +···+ F56τ f 13τ f 12 = 1. (10.3) The strength parameters must be determined by tests. For generally anisotropic and monoclinic materials, 27 and 17 types of tests are required, respectively. This makes the use of the quadratic failure criterion impractical for structures made of generally anisotropic or monoclinic materials. The criterion becomes more manageable when the material is orthotropic or transversely isotropic. Therefore, in the following the criterion is presented only for these two types of materials. 10.1.1 Orthotropic Material An orthotropic material has three planes of symmetry (Figs. 2.11 and 2.12). We select the x1, x2, x3 coordinate system with axes perpendicular to these symmetry planes. First, we consider only the shear–stress component τ23 acting in the plane of symmetry (Fig. 10.4). When only τ23 acts, and it is in the positive direction, then x3 x1 x2 Plane of symmetry f + 23 τ − − f 23 τ Figure 10.4: The positive and negative shear stresses at failure acting in an orthotropic material; x1, x2, and x3 are perpendicular to the orthotropy planes
10.1 QUADRATIC FAILURE CRITERION 415 at failure(2=)the quadratic failure criterion yields(Eq.10.3) E站+F4()2=1. (10.4) When only 723 acts in the negative direction,then,at failure(23=)the quadratic failure criterion becomes (Eq.10.3) -F站+F4(站)2=1. (10.5) Because of symmetry,the failure stress for positive shear is the same as for negative shear(=).The two preceding equations satisfy this condition only if F4 is zero. By similar argument it can be shown that Fs,F are zero.Thus,we have F=F=F6=0. (10.6) Next,we apply the normal stresses a1,02,o3;the shear stresses ti2,Ti3;and either a positive or a negative shear stress 2(Fig.10.5).For a positive shear stress at failure,ts=t∴，and we have(Eq.l0.3) o+Fi+Fo+F1(o'+F2()2+ Fs(o)2+F4(塔)2+Fs(6)2+6()2+ 2(E2ooi+3oo+）+·+ +2(F4o+F4o+F4o+F45+F46)站=1 (10.7) For a negative shear stress at failure,=,and Eq.(10.3)gives Fof+F2o+F3os+Fu(of)2+Fz2 ()2+ F3(o)2+F4(站)2+Fs(c)2+F6()2+ 2(2o+F3oog+…）+…+ -2(F4o+F4o+F4o吲+F4s+F46b)点=1 (10.8) Because of symmetry,the failure stress for positive out-of-plane shear is the same as for negative out-of-plane shear().This condition,together with 12 Plane of symmetry Figure 10.5:The stresses at failure acting on an orthotropic material
10.1 QUADRATIC FAILURE CRITERION 415 at failure (τ23 = τ f+ 23 ) the quadratic failure criterion yields (Eq. 10.3) F4τ f+ 23 + F44 τ f+ 23 2 = 1. (10.4) When only τ23 acts in the negative direction, then, at failure (τ23 = −τ f− 23 ) the quadratic failure criterion becomes (Eq. 10.3) −F4τ f− 23 + F44 τ f− 23 2 = 1. (10.5) Because of symmetry, the failure stress for positive shear is the same as for negative shear (τ f+ 23 = τ f− 23 ). The two preceding equations satisfy this condition only if F4 is zero. By similar argument it can be shown that F5, F6 are zero. Thus, we have F4 = F5 = F6 = 0. (10.6) Next, we apply the normal stresses σ1, σ2, σ3; the shear stresses τ12, τ13; and either a positive or a negative shear stress τ23 (Fig. 10.5). For a positive shear stress at failure, τ23 = τ f+ 23 , and we have (Eq. 10.3) F1σf 1 + F2σf 2 + F3σf 3 + F11 σf 1 2 + F22 σf 2 2 + F33 σf 3 2 + F44 τ f+ 23 2 + F55 τ f 132 + F66 τ f 122 + 2 F12σf 1σf 2 + F13σf 1σf 3 +··· +····+ +2 F14σf 1 + F24σf 2 + F34σf 3 + F45τ f 13 + F46τ f 12 τ f+ 23 = 1. (10.7) For a negative shear stress at failure, τ23 = −τ f− 23 , and Eq. (10.3) gives F1σf 1 + F2σf 2 + F3σf 3 + F11 σf 1 2 + F22 σf 2 2 + F33 σf 3 2 + F44 τ f− 23 2 + F55 τ f 132 + F66 τ f 122 + 2 F12σf 1σf 2 + F13σf 1σf 3 +··· +····+ −2 F14σf 1 + F24σf 2 + F34σf 3 + F45τ f 13 + F46τ f 12 τ f− 23 = 1. (10.8) Because of symmetry, the failure stress for positive out-of-plane shear is the same as for negative out-of-plane shear (τ f+ 23 = τ f− 23 ). This condition, together with x3 x1 x2 Plane of symmetry f + 23 τ − − f 23 τ f σ3 f σ3 f 13 τ f 13 τ f σ1 f f σ1 12 τ f 12 τ f σ2 f σ2 Figure 10.5: The stresses at failure acting on an orthotropic material