Advances in the Bonded Composite Repair o f Metallic Aircraft Structure phần 4 docx

144 Advances in the bonded composite repair of metallic aircrafi structure with a step change in stiffness from ~ptp/( 1 - v’,) to Eptp/( 1 - v’,) + ERtR/( 1 - vi) over a central potion ly( 5 B - b, as indicated in Figure 7.4(e), with b given by (7.13) This equivalence will be exploited in Section 7.4 to assess the redistribution of stress due to a bonded reinforcement. The prospective stress in the plate directly underneath the reinforcing strip, 00 = op(x = 0), can be readily determined by integrating Eq. (7.4), 1 1 b = - tanh j?B- (BB< 1) B “P ’ with S denoting the stiffness ratio given below, (7.14) (7.15) which is an important non-dimensional parameter characterising a repair. As will be shown in the following section the actual prospective stress 00 is somewhat higher than that given by Eq. (7.14). This under-estimation is primarily due to the ignorance of the “load attraction” effect in a 2D plate associated with reinforcing. 7.4. Symmetric repairs We return to the solution of the problem formulated in Section 7.2, assuming that the repaired structure is supported against out-of-plane bending or the cracked plate is repaired with two patched bonded on the two sides. The analysis will be divided into two stages as indicated in Section 7.2. 7.4.1. Stage I: Inclusion analogy Consider first the re-distribution of stress in an uncracked plate due to the local stiffening produced by the bonded reinforcement. As illustrated in Figure 7.2(a), the reinforced region will attract more load due to the increased stiffness, leading to a higher prospective stress than that given by Eq. (7.14). The 1D theory of bonded joints (Section 7.3) provides an estimate of the load-transfer length j?-’ for load transfer from the plate to the reinforcement. If that transfer length is much less than the in-plane dimensions A, B of the reinforcement, we may view the reinforced region as an inclusion of higher stiffness than the surrounding plate, and proceed in the following three steps. 1. Determine the elastic constants of the equivalent inclusion in terms of those of 2. Determine the stress in the equivalent inclusion. the plate and the reinforcing patch. Chapter 7. Analytical methodsfor designing composite repairs I45 3. Determine how the load which is transmitted through the inclusion is shared between the plate and the reinforcement, from which the prospective stress 00 can be calculated. Step (2) is greatly facilitated by the known results of ellipsoidal inclusions [3]: the stress and strain within an ellipsoidal inclusion is uniform. as indicated schematically in Figure 7.2(a). The uniform stress state can be determined analytically with the help of imaginary cutting, straining and welding operations. The results are derived in [SI for the case where both the plate and the reinforcing patch are taken to be orthotropic, with their principal axes parallel to the s - y axes. We shall not repeat here the intermediate details of the analysis but simply recall the results for the particular case where both the plate and the reinforcement are isotropic and have the same Poisson's ratio, vp = VR = v. The prospective stress in the plate along y = 0 within the reinforced region (1x1 I A) is 60 = &T= ~ (7.16) where (7.17) BA 4~- 4+2-+2-+S Z '[ A B with 2 = 3( 1 + S)2 + 2( 1 + S)(B/A + A/B + vS) + 1 - v2S2 (7.18) It is clear that the stress-reduction factor 4 depends on three non-dimensional parameters: (i) the stiffness ratio S, (ii) the aspect ratio B/A, (iii) the applied stress biaxiality i The parameters characterising the adhesive layer do not affect 00, but we recall that the idealisation used to derive Eq. (7.17) relies on B-' < A, B, and p-' is of course dependent on adhesive parameters. To illustrate the important features of Eq. (7.17), we show in Figure 73a) the variation of the stress-reduction factor 4 with aspect ratio for two loading configurations: (i) uniaxial tension (i = 0), and (ii) equal biaxial tension corresponding to pure shear (A = -l), setting S = 1 and v = 1/3 for both cases. It can be seen that there is little variation for aspects ratio ranging from B/A = 0 (horizontal strip) to B/A = 1 (circular patch), so that for preliminary design calculations, one can conveniently assume the patch to be circular, to reduce the number of independent parameters. It is also noted from Eq. (7.17) that for v = 1/3 and a circular patch (A/B = I), the stress-reduction factor cp becomes independent of the biaxiality ratio A. As illustrated in Figure 7.5(a) the curves for 1 = 0 and ;L = -1 cross over for B/A = 1, indicating that, for a circular patch, the transverse stress gFX does not contribute to the prospective stress, so that this parameter can also be ignored in preliminary design estimates. In this particular case, the stress- reduction factor 4 depends on the stiffness ratio S only, as depicted in Figure 7.5(b), together with the first-order approximation given by Eq. (7.14). It can be seen that the first-order solution ignoring the load attraction effect of composite 146 Advances in the bonded composite repair of metallic aircraft structure Uniaxial tension(X= 0) Pure shear(X=-1) 0.9 ". . 0 0.2 0.4 0.6 0.8 Aspect ratio of patch B/(A+B) (a) 1 .o 0.9 0.8 0.7 8 0.6 2 0.5 2 0.4 2 0.3 0.2 0.1 0 a - o Exactsolution - += (1+0.277S-O.O712S2)/(l+S) - - -&= 1/(1+S) 1 .o 0 0.5 1 .o 1.5 2.0 Stiffness ratio S (b) Fig. 7.5. Variation of reduced stress with (a) aspect ratio for an elliptical patch of serni-axes A and B under uniaxial tension and biaxial tension equivalent to pure shear; (b) stiffness ratio S for a circular patch. patch overestimates the reduction in plate stress. An improved solution can be obtained by constructing an interpolating function based on the exact solution, , (7.19) 1 + 0.277s - 0.0712S2 l+s $= which is shown by solid curve in Figure 7.5(b), indicating a very good fit to the exact solution. The inclusion analogy also gives, as a natural by-product, the stress in the plate outside the reinforced region. The stress at the point x = 0, y = B+ is of particular interest, because this stress represents the increased stress due to the so-called load Chapter 7. Analytical methods for designing composite repairs 147 attraction effect; a load attraction factor QL can be defined as the ratio of the plate stress just outside the patch to the remote applied stress (7.20) It is clear from Figure 7.5(a) that for the case of a balanced patch (S = 1) under uniaxial tension, this load attraction factor ranges between 1 for patch of infinite width to 2 for patch of zero width. For the typical case of circular patch, the load attraction factor is approximately 1.2. 7.4.2. Stage 11: Stress intensity factor Once the stress at the prospective crack location is known, one can proceed to the second stage of the analysis in which the plate is cut along the line segment (1x1 5 a, y = 0), and a pressure equal to go is applied internally to the faces of this cut to make these faces stress-free. Provided that the load transfer to the reinforcement during this second stage takes place in the immediate neighbourhood of the crack, the reinforcement may be assumed to be of infinite extent. Thus the problem at this stage is to determine the stress intensity factor K, for the configuration shown in Figure 7.3(a). Without the reinforcement, the stress-intensity factor would have the value KO given by the well-known formula, KO = ~ofi (7.21) This provides an upper bound for K,, since the restraining action of the patch would reduce the stress-intensity factor. However, KO increases indefinitely as the crack length increases, whereas the crucial property of the reinforced plate of Figure 7.3(a) is that K, does not increase beyond a limiting value, denoted by K,, as will be confirmed later. That limiting value is the value of the stress intensity factor for a semi-infinite crack. It can be determined by deriving first the corresponding strain-energy release rate as follows. Before we proceed, let us first determine the deformation of the reinforced strips shown in Figure 7.3(b). The adhesive shear stress ZA is governed by the differential Eq. (7.9), which has the following solution for the particular case of semi-infinite strip, ~A(Y) = ZA,rnaxe-’.’ , (7.22) where rmax can be determined from the simple equilibrium condition, QOtf = so“ ZA (YPY, ZA,max = P~P~o (7.23) Recalling Eq. (7.4), the opening displacement of the plate at y = 0 can be readily 148 Advances in the bonded composite repair of metallic aircraft structure determined, (7.24) Let us denote the total opening as 6 = 224,. The above equation can be rewritten as, with (7.25) (7.26) Consider the configuration shown in Figure 7.6. If the semi-infinite crack extends by a distance da, the stress and displacement fields are simply shifted to the right by da. The change in the strain energy UE is that involved in converting a strip of width da from the state shown as section AA' in Figure 7.6 to that shown in section BB', as depicted in Figure 7.7. Consequently the change in the potential energy for a crack advancement ha, which is defined as the difference between the strain energy change UE (= 1/2ootp6) and the work performed by the external load W (= GotpJ), 1 n=uE-w= rJ OtPd The crack extension force, Le. the strain-energy release rate G, is given by which can be re-written as, recalling Eq. (7.25), +Z A A B B4 B' B'+ * A' A' Fig 7 6 A patched crack subjected to internal pressure (7.27) (7.28) (7.29) Chapter 7. Analytical methods for designing composite repairs 149 6 s (a) (b) Fig. 7.7. Illustration of the interpretation of G, as a complementary energy. (a) Elastic adhesive and (b) elastic-plastic adhesive. From the above equation, assuming that the usual relation holds between the strain- energy release rate G and the stress-intensity factor K [22], we obtain, 00 K "-& (7.30) It is clear from this derivation that K, is an upper-bound for K,. The validity of this formula will be substantiated by an independent finite element analysis to be discussed later. 7.4.3. Plastic adhesive The stress-intensity factor solution derived in the previous section is valid only if the adhesive remains elastic. If the maximum adhesive shear stress does exceed the shear yield-stress, the relationship between 00 and the crack-opening displacement 6 will become non-linear, as illustrated in Figure 7.7(b), which also shows the correct area corresponding to G,. For an adhesive that is elastic-perfectly plastic with a shear yield-stress zy, the adhesive begins to yield at the following stress, ZY boy = - BtP (7.31) It can be shown that for 00 2 boy the crack opening-displacement 6 is given by, ZYtA 1 + ~ =- PA [ (;y)2] =E [I + ($))*I > (00 2 boy) (7.32) Following the method outlined in the previous section, the strain-energy release rate G, can be determined, s 00 Y 60 G, = o06 - /bods = / 6doo + / 6doo = kEp [ cri P3+3P-1 3p2 ] ' (7.33) 0 0 00 Y 150 Advances in the bonded contposite repair of metallic aircraft structure 1.4 4 \ hi 1.3 8 'I 1.2 d b * ." A D 1.1 2 LI 0 CI 1.0 where c0 p=- 00 Y Then, the stress-intensity factor for P 2 1 can be expressed as (7.34) (7.35) where K,,el denotes the value which would be obtained from Eq. (7.30) for the stress a0 ignoring the plastic yielding in the adhesive. As can be seen from Eq. (7.35), the increase in Km due to adhesive yielding depends only on the plasticity ratio P defined by Eq. (7.34), as shown in Figure 7.8. 7.4.4. Finite crack size Adhesive plasticity ratio P Fig. 7.8. Increase in stress-intensity factor due to adhesive yielding. Chapter 7. Analytical methods for designing composite repairs 151 h g 1.00 5 3 >.I ) .3 2 0.75 s E C .3 (I) rn Y 0.50 .e C 0 0 S -m .e Y 2 0.25 0.01 Nomalised crack length ku Fig. 7.9. Reduction in stress-intensity factor for various patch configurations. Symbols denote the exact solutions by the Keer method, solid curves denote the interpolating function, and dashed curve denotes the solution of crack bridging model. reduction factor Fdepends strongly on the parameter k given by Eq. (7.26) and to a lesser extent on the stiffness ratio S, as shown by the symbols in Figure 7.9. Based on the solutions of the integral equation [ 141, the following interpolating function can be constructed, 112 F(ku) = [nLu - tanh (1 :zku)] ’ (7.37) where constant B has been determined by curve fitting the numerical solution of the integral equation, which gives B = 0.3 for balanced repairs (S = 1 .O) and B = 0.1 for infinitely-rigid patch (S + a). A simple yet more versatile method of determining the reduction in stress- intensity factor after repair is the crack bridging model [lo], which has been recently extended to analyse the coupled in-plane stretching and out-of-plane bending of one-sided repairs [17]. From the previous analysis it is clear that the essential reinforcing action at the second stage is the restraint on the crack opening by the bonded reinforcements. The basic idea underlying the crack bridging model is that this restraining action can be represented by a continuous distribution of springs acting between the crack faces, as illustrated in Figure 7.10. This idealisation reduces the problem at stage TI to two parts: (i) determine the appropriate constitutive relation (i.e. stress-displacement relation) for the springs, and (ii) solve a one-dimensional integral equation for the crack opening, 6(x) = ulr(x,y + Of) - .,’(x,y + 0-) = 2u;(x,y + O+), 1x1 5 a (7.38) 0.1 1 It is assumed that distributed linear springs act between the crack faces over the 152 Advances in the bonded composite repair of metallic aircraft structure go t tt t tt ttt tl X Fig. 7.10. Schematic representation of a centre-crack reinforced by distributec jprings. crack region so that the boundary conditions on y = 0 are u,,(x) = 0, 1x1 2 a , (7.39b) where k denotes a normalised spring constant which has dimension length-’. It is worth noting that this normalised spring constant k has already been determined in Section 7.4.2 and is given by Eq. (7.26). With these boundary conditions, the problem of determining the crack opening displacement u,(x) can be reduced to that of solving the following integral equation [lo, 171, (7.40) The integral in the above equation is interpreted as a Hadamard finite part [24], which can be viewed as the derivative a Cauchy principal value integral. The above equation can be efficiently solved using either Galerkin’s method or collocation methods. Once the crack-opening displacement uJx) is determined, the stress- intensity factor K, can be calculated by (7.41) Detailed numerical results for K, are available in reference [lo], which also provided the following interpolating function constructed based on the numerical Chapter I. Analytical methods for designing composite repairs , patch - adhesive ,plate (b) Fig. 7.11. Finite element mesh (a) quarter model and (b) mesh near crack tip. results, 1 + 2.23ka 1 + 4.776ka + 7(k~)~ F(ka) = 153 (7.42) [...]... orthotropic patch, the thermal residual stress 00" in the plate is well approximated by the solution for an isotropic patch, which takes the major properties of the orthotropic patch [IS] Therefore in the following we will present 172 Advances in the bonded composite repair of metallic aircraft structure Fig 7.22 Spring representation for simulating finite size effect only the solution for an isotropic... along they axis The crack extends for a total distance of 2a in the x direction, as shown in Figure 8.1, symmetrically about the y axis The brilliance of Rose’s analysis is that, even though the load transfer through the bonded joint around the periphery of the patch is not instantaneous in the manner of the inclusion model, it is so close to it that the differences have no effect in the vicinity of... coefficient of thermal expansion a ~ , (7.99) The above solution applies strictly for an infinite plate In practice, however, structures to be reinforced may be finite in size or constrained by the surrounding structure To quantify the effect of this constraint, consider the configuration shown in Figure 7.22, in which an un-cracked circular plate of radius Ri is reinforced by a concentric patch of... optimal configuration Chapter 7 Analytical methods for designing composite repairs 157 7.6 One-sided repairs So far we have ignored the tendency for out-of-plane bending that would result from bonding a reinforcing patch to only one face of an un-supported plate, so that, strictly speaking, the preceding analysis is more appropriate for the case of two-sided reinforcement, with patches bonded to both faces,... formula is in good correlation with the finite element results It is also worth noting that the results confirm that the stress-intensity factor Krmsfor a one-sided repair is much higher than that for an equivalent two-sided repair, indicating the importance of out-of-plane bending Advances in the bonded composite repair of metallic aircraft structure 162 + Fig 7.16 A plate with a through crack reinforced... impact of Rose’s novel use of the inclusion model to generate closed-form solutions for the stresses in and under bonded patches will ever be surpassed by later work.) 8.1.1 Rose’s use o the inclusion model to establish stressjields f The issue of interest here, in the simple form analyzed by Rose, in reference [5],is a local perturbation in an otherwise uniformly loaded infinite plate, as shown in Figure... dimensions of isotropic reinforcement Material Young's modulus GPa) Poisson's ratio Thickness (mm) Thermal coefficient Plate Reinforcement 71 156 0.3 0.3 I o 0.5 23 x 6. 24 x Advances in the bonded composite repair of metallic aircraft structure 1 74 Table 7.3 Properties and dimensions of orthotropic reinforcement Material Plate Reinforcement Young's modulus (GW 71 El = 156 E2 = 29.7 V" Poisson's ratio 0.3... to design local reinforcement to be applied at the time of initial manufacture, with the goal ofpreventing the initiation of cracks in service The methods can even quantify the severe stress concentrations associated with poorly designed stiffener run-outs, which are often terminated short of the ends of integrally stiffened panels in order to simplify the splice plates In short, Rose’s original analyses... stiffness of the patched region and the lower coefficient of thermal expansion of the composite 168 Advances in the bonded composite repair of metaIlic uircraft structure Geometrically ,I linear solution ,' \ , ,,/ u ii l 2l rn n i, 0 100 200 300 Applied stress nm(MPa) (a) < e4 40 0 4 High crack length a (mm) (b) Fig 7.20 Geometrically non-linear deformation of a single strap joint representing one-sided... change in the potential energy now consists of two terms: work done by the membrane force and the bending moment [12], referring to Figure 7. 14, tpG: = Nouo + MoOo , (7.56) where the superscript "* " refers to the strain-energy release rate for one-sided repair, and uo and 190 denote the opening displacement and the angle of rotation of the crack faces, which are related to the membrane force NO and MO . with the first-order approximation given by Eq. (7. 14) . It can be seen that the first-order solution ignoring the load attraction effect of composite 146 Advances in the bonded composite repair. lesser extent on the stiffness ratio S, as shown by the symbols in Figure 7.9. Based on the solutions of the integral equation [ 141 , the following interpolating function can be constructed,. distribution in the plate and the reinforcement can be determined using the conventional theory of cylindrical bending of plates, i.e. we shall assume that the bending deformation of the reinforced