FAILURE FAILURE ELASTIC-PLASTIC MODEL TRUE CHARACTERISTIC ADHESIVE BI-LINEAR MODEL SHEAR SHEAR EQUAL AREAS STRESS ELASTIC-PLASTIC MODEL STRESS FOR PARTIAL LOAD LEVEL JOINING OF COMPOSITE STRUCTURES ADHESIVE SHEAR STRAIN Y (+Y) Y。 ADHESIVE SHEAR STRAIN Y (+Yp) Fig.9.4 Models for representing the shear stress/strain behavior in an adhesive.Taken from Ref.11. 品
JOINING OF COMPOSITE STRUCTURES 299 ,i w =, i ....... w ::c ::z:: ~_ L, + ,. < O
300 COMPOSITE MATERIALS FOR AIRCRAFT STRUCTURES The ideally elastic/plastic models,(Fig.9.4),greatly simplify the analysis, allowing closed-form solutions to be developed for a wide range of joints.It is shown that the requirement for the elastic/ideally plastic model is that it has the same shear strain energy (area under the curve)as the actual curve and intersects it at the required level of shear stress.Thus,as indicated in Figure 9.4,the effective shear modulus G and shear yield stress used in the model vary with the strain level. In most joint designs,it is sufficient to undertake a simple elastic analysis to check that,for most of the operation of the joint (below limit load),the adhesive will not deform plastically and then,using the effective elastic/plastic parameters,assess the load-carrying capacity of the joint. For the strength analysis to be conservative,the hot/wet shear yield strength should be used to assess the likelihood of fatigue damage,then the low- temperature stress/strain behavior of the adhesive used to estimate static strength (because the area under the stress-strain curve is then a minimum). 9.3.4 Load Transfer Mechanisms in Overlap Joints The skin/doubler joint shown in Figure 9.5 provides a simple illustration of the main features of load transfer in a lap joint.The overlap length is assumed to be semi-infinite,which means that it is very much larger than the load-transfer length based on the exponent B. Loading of the outer adherend occurs by the development of surface shear forces,which arise as the adhesive layer resists the shear displacement between the inner,directly loaded adherend and the outer,initially unloaded,adherend. Load transfer by the shear forces produces an increasing axial strain in the outer (reinforcing)adherend and a reducing strain in the inner adherend until,at some point,the strains in the two adherends become equal;the shear strain in the adhesive is then zero. 9.3.4.1 Elastic Model for the Adhesive.The analysis assuming elastic behavior is outlined in Figure 9.5,and the outcome is illustrated in Figure 9.6.It is assumed here that failure occurs when Tax=Tp for the adhesive.Bending effects,for example,due to joint rotation,are not considered in this analysis.It therefore corresponds to a symmetric double-lap joint or symmetric doubler configuration,or a single-lap/single-sided doubler configuration in which bending is reacted by other supporting structure. The main analytical results from this model are as follows: Shear stress and strain distributions are given by: T=Tpe-Bx (9.1) y=Ype-Br (9.2)
300 COMPOSITE MATERIALS FOR AIRCRAFT STRUCTURES The ideally elastic/plastic models, (Fig. 9.4), greatly simplify the analysis, allowing closed-form solutions to be developed for a wide range of joints. It is shown that the requirement for the elastic/ideally plastic model is that it has the same shear strain energy (area under the curve) as the actual curve and intersects it at the required level of shear stress. Thus, as indicated in Figure 9.4, the effective shear modulus GA and shear yield stress used in the model vary with the strain level. In most joint designs, it is sufficient to undertake a simple elastic analysis to check that, for most of the operation of the joint (below limit load), the adhesive will not deform plastically and then, using the effective elastic/plastic parameters, assess the load-carrying capacity of the joint. For the strength analysis to be conservative, the hot/wet shear yield strength should be used to assess the likelihood of fatigue damage, then the lowtemperature stress/strain behavior of the adhesive used to estimate static strength (because the area under the stress-strain curve is then a minimum). 9.3.4 Load Transfer Mechanisms M Overlap Joints The skin/doubler joint shown in Figure 9.5 provides a simple illustration of the main features of load transfer in a lap joint. The overlap length is assumed to be semi-infinite, which means that it is very much larger than the load-transfer length based on the exponent/3. Loading of the outer adherend occurs by the development of surface shear forces, which arise as the adhesive layer resists the shear displacement between the inner, directly loaded adherend and the outer, initially unloaded, adherend. Load transfer by the shear forces produces an increasing axial strain in the outer (reinforcing) adherend and a reducing strain in the inner adherend until, at some point, the strains in the two adherends become equal; the shear strain in the adhesive is then zero. 9.3.4. 1 Elastic Model for the Adhesive. The analysis assuming elastic behavior is outlined in Figure 9.5, and the outcome is illustrated in Figure 9.6. It is assumed here that failure occurs when rmax = ~'p for the adhesive. Bending effects, for example, due to joint rotation, are not considered in this analysis. It therefore corresponds to a symmetric double-lap joint or symmetric doubler configuration, or a single-lap/single-sided doubler configuration in which bending is reacted by other supporting structure. The main analytical results from this model are as follows: Shear stress and strain distributions are given by: "1" = 'Tpe -~x (9.1) 3/-- ype -~x (9.2)
JOINING OF COMPOSITE STRUCTURES 301 0 Ax E.t, E,0: G..t..t..YY. P Displacement Horizontal +()ar T. +（ar 五+(a Ax Ax Equilibrium: dT1 Material: T=GAY -T=0 d g=Ee→ du d2+=0 t dx dx T2=E2 u2 t dx Compatibility: Y=41-u2 t=G,飞-2 and dr_Gad」 du, kt4（E4,El2 ..t=Tme where B2= G1+1 Fig.9.5 Configuration and analysis of a single-lap joint. where B-G1+ (9.3) tA Et E2 Because of the low shear moduli of polymer-matrix composites,a modification2 of this equation is required to estimate B.This can be done (assuming a linear shear lag across the thickness)by replacing tA/GA with the
JOINING OF COMPOSITE STRUCTURES 301 0 x p < Displacement g~ Ax J Ax ,~t~,O~ p,~.,% Horizontal < XAX t~ Ax P Equilibrium: Compatibility: ~ = -- dT1 - z = 0 dx dT2 ~x=0 dx Material: z=GA~/ or= E8 H 1 --U 2 tA .'. "r = G A u, - u 2 and dr G A ( T 1 ] 2" "" dx 2 t A ~ E~tt E2tz dx t~ k dx dx ) G~( 1 .'. r = rx=o e-~ where /~2 ~-- .,l q. [ tA [.E, tl E2t2 ) "I"1_ E du, tl - {--~- T._3_2 = E 2 duz t2 dx Fig. 9.5 Configuration and analysis of a single-lap joint. where .= OAF , = ~ LEltl E2t2/ (9.3) Because of the low shear moduli of polymer-matrix composites, a modification 12 of this equation is required to estimate /3. This can be done (assuming a linear shear lag across the thickness) by replacing tA/GA with the
302 COMPOSITE MATERIALS FOR AIRCRAFT STRUCTURES Shear stress/strain distribution at adhesive yleld t=tet y=Yet Load transfer 95%T T Fig.9.6 Outcome of the analysis of the skin-doubler joint,assuming elastic behavior in the adhesive. effective value: +2+3 (9.4) eff GAG28GT where Gi and G2 are,respectively,the shear moduli of the outer and inner adherend. The maximum load that can be transferred from the inner to the outer adherend per unit width of joint before adhesive yield is given by: Timax Tp e-Br dx =Tp (9.5) Jo which is the area under the shear-stress/length curve. The distance to transfer most(95%)of the load,the load transfer length,is given by: 3 Lanin= (9.6) If the transfer length is less than 3/B,it is not possible to obtain full load sharing between the adherends. In the absence of residual stress due to thermal-expansion,mismatch between adherends,and a differential between operating and cure temperatures,the maximum load per unit width(Pmax)that can be applied at the end of adherend 2
302 COMPOSITE MATERIALS FOR AIRCRAFT STRUCTURES Shear stress/strain distribution at adhesive yield ~'~l ~e x---~ X "U ='[ee -Ik ~,=%e -~ Load transfer 95%T; L T, r,- L x ~_ Fig. 9.6 Outcome of the analysis of the skin-doubler joint, assuming elastic behavior in the adhesive. effective value: tA ~ ta t2 3tl )ee = + + (9.4) where G~ and G2 are, respectively, the shear moduli of the outer and inner adherend. The maximum load that can be transferred from the inner to the outer adherend per unit width of joint before adhesive yield is given by: oo re (9.5) T1 max = Zp e-OXdx = which is the area under the shear-stress/length curve. The distance to transfer most (95%) of the load, the load transfer length, is given by: 3 Lmin = ~ (9.6) If the transfer length is less than 3/fl, it is not possible to obtain full load sharing between the adherends. In the absence of residual stress due to thermal-expansion, mismatch between adherends, and a differential between operating and cure temperatures, the maximum load per unit width (Pmax) that can be applied at the end of adherend 2
JOINING OF COMPOSITE STRUCTURES 303 without yielding the adhesive is given by: Pmax (9.7) If residual stresses can arise as a result of the difference in expansion coefficients between,say,a metal panel and a composite doubler,the maximum load that can be applied without yielding the adhesive is given by: Pmax =Pmax +E2t2(a1 -@2)AT (9.8) In this equation,AT(service temperature-cure temperature)is always negative so that the influence of residual stresses on the maximum load transfer depends on the sign of Aa If,for example,adherend 1 is carbon/epoxy and adherend 2 aluminum,then Aa is negative and almost equal in magnitude to a for aluminum; taking aluminum as 23x 106C-1 and a for a quasi isotopic carbon/epoxy composite to be4×l0-6oC-】. The result is an increase in the maximum tensile load and a reduction in the maximum compressive load that can be carried by the joint after cooling from the cure temperature.This result is easily explained physically.The inner metallic adherend contracts as the joint is cooled from the cure temperature,producing shear stresses in the adhesive opposing those produced by the applied tensile load.Thus,a significant tensile load must be applied to the joint to overcome this contraction before adhesive starts to be sheared in the original direction.The converse occurs if the applied load is compressive.This topic of residual stress is discussed again later with respect to the double-overlap joint. 9.3.4.2 Elastic/Plastic Model for the Adhesive.The shear stress/joint length relationship in the adhesive,assuming elastic/plastic behavior,is shown in Figure 9.7.In this figure,it is assumed that the adhesive is strained to its full (y.+ye） plastic zone elastic zone Fig.9.7 Shear stress/length and shear strain/length distribution in the skin- doubler joint,assuming elastic/plastic behavior in the adhesive
JOINING OF COMPOSITE STRUCTURES 303 without yielding the adhesive is given by: E2t___~2") (9.7) Pmax:(fl)(l+Eltl / If residual stresses can arise as a result of the difference in expansion coefficients between, say, a metal panel and a composite doubler, the maximum load that can be applied without yielding the adhesive is given by: T Pmax : Pmax + E2t2(Otl -- a2)AT (9.8) In this equation, AT (service temperature--cure temperature) is always negative so that the influence of residual stresses on the maximum load transfer depends on the sign of Aa If, for example, adherend 1 is carbon/epoxy and adherend 2 aluminum, then Aa is negative and almost equal in magnitude to a for aluminum; taking aluminum as 23 x 10 -6 °C-1 and a for a quasi isotopic carbon/epoxy composite to be 4 x 10 -6 °C -1. The result is an increase in the maximum tensile load and a reduction in the maximum compressive load that can be carried by the joint after cooling from the cure temperature. This result is easily explained physically. The inner metallic adherend contracts as the joint is cooled from the cure temperature, producing shear stresses in the adhesive opposing those produced by the applied tensile load. Thus, a significant tensile load must be applied to the joint to overcome this contraction before adhesive starts to be sheared in the original direction. The converse occurs if the applied load is compressive. This topic of residual stress is discussed again later with respect to the double-overlap joint. 9.3.4.2 Elastic/Plastic Model for the Adhesive. The shear stress/joint length relationship in the adhesive, assuming elastic/plastic behavior, is shown in Figure 9.7. In this figure, it is assumed that the adhesive is strained to its full "Cp ,- 5 )lasti( ,.one ~zo ne x~ xp > x --~ (y~ + ye) X > Fig. 9.7 Shear stress/length and shear strain/length distribution in the skindoubler joint, assuming elastic/plastic behavior in the adhesive