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

Analytical methods for designing composite repairs 147 attraction effect; a load attraction factor Q L can be defined as the ratio of the plate stress just outside the patch to the remo

144 Advances in the bonded composite repair of metallic aircrafi structure

with a step change in stiffness from ~ p t p / ( 1 - v’,) to Eptp/( 1 - v’,) + E R t R / ( 1 - v i )

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 = o p ( x = 0), can be readily determined by integrating Eq (7.4),

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, v p = V R = v The prospective stress

in the plate along y = 0 within the reinforced region (1x1 I A ) is

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 0 0 , 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 7 3 a ) 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

Chapter 7 Analytical methods for designing composite repairs 147

attraction effect; a load attraction factor Q L 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 g o 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,

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,

148 Advances in the bonded composite repair of metallic aircraft structure

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 (= G o t p J ) ,

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),

Chapter 7 Analytical methods for designing composite repairs 149

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 z y , the adhesive begins to yield at the following stress,

G , = o06 - / b o d s = / 6doo + / 6doo = kEp cri [ P 3 + 3 P - 1

150 Advances in the bonded contposite repair of metallic aircraft structure

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 f o r designing composite repairs 151

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

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)

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 t t t t t t t t t l

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

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 u J x ) 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

154 Advances in the bonded composite repair of metallic aircraft structure

Crack length a (mm) (b)

Fig 7.12 Comparison between finite element solution and analytical predictions

which is shown in Figure 7.12(a) As compared to the exact solutions by the Keer formulation (Eq 7.37), the crack-bridging model (Eq 7.42) slightly over-estimates the reduction in stress-intensity factor for balanced repair (S= 1) in the short crack limit Both the two interpolating formulas, Eqs (7.37) and (7.42) recover the asymptotic solution of Eq (30) in the long crack limit as a + co

7.4.5 Finite element validation

To substantiate the theoretical solutions obtained so far, an extensive finite element analysis has been performed for various crack lengths [12] Due to symmetry only a quadrant of the repair shown in Figure 7.l(a) was modelled No

Chapter 7 Analytical methods for designing composite repairs I55 Table 7 I

Dimensions and material properties of a typical repair

Young’s modulus Thickness Layer ( G W Poisson’s ratio (mm)

k = 0.096 mm-’ From the finite element results, the stress-intensity factor is

calculated using Eq (7.41), with Ep being replaced by the plane-strain value

Figure 7.12(a) shows a comparison between the theoretical estimate and the finite element results for a long crack ( k a z lo), indicating an excellent agreement within

the mean stress-intensity factor through the plate thickness It is also clear that the stress-intensity factor at the outer surface away from the adhesive layer is somewhat higher than near the adhesive layer The asymptotic behaviour of the stress-intensity factor is shown in Figure 7.12(b) together with the two analytical estimates (37) and (42) The crack-bridging solution seems to slightly over-estimate the repair efficiency

E p / ( 1 - v2)

7.5 Shear mode

Although cracks that are likely to be encountered in practice are generally aligned in a direction perpendicular to the principal tensile stress (or strain), giving rise to mode I cracking, there are at least two circumstances where mixed mode cracking is a major concern in the context of bonded repairs Firstly, application of bonded reinforcements, which are frequently anisotropic, may alter the local stress- state near the crack region so that the maximum principal stress may no longer remain perpendicular to the crack plane Secondly, structures are frequently subjected to non-proportional loading in which the principal stress-strain axes rotate with time, thus cracks may experience a time-dependent mixed mode

loading If the bonded repair technique is used to repair mode I1 cracks one

important question that needs consideration is the effectiveness of repairs

For simplicity let us consider the particular case of an isotropic circular patch

( A / B = I ) with a Poisson’s ratio v = 1/3 In this case, the prospective stress in the plate after repair can be determined using the general solution for biaxial tension

156 Advances in the bonded romposite repair of metallic aircraft struriure

presented in Section 7.4.1, namely Eq (7.19),

1 + 0.2773 - 0.0712S2

Detailed solution of the stress-intensity factor K , for shear loading can be found in

[14] We shall not repeat here the intermediate details of the analysis but simply recall the results for the upper-bound and the interpolating function The upper- bound solution of Kr is given by an equation similar to that for tensile mode,

where the normalised shear spring constant kI1 is given by

(7.47)

For finite crack size, the stress-intensity factor Kr can also be expressed as [14]

with F ( x ) being given by Eq (7.37) or Eq (7.42)

An important implication arising from the difference in the spring constants is that when strongly anisotropic reinforcements with low in-plane shear modulus, such as unidirectional plastic reinforced composites, are used to repair a crack under shear loading (with the fibres being perpendicular to the crack), the repair efficiency will be much lower than that could be expected on the basis of mode I analysis It should of course be mentioned that under remote shear loading, the crack would be aligned perpendicular to the maximum tensile stress, hence aligning the fibres perpendicular to the crack is still the optimal configuration

Chapter 7 Analytical methods f o r 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, or one-sided repairs to fully supported structures For the case of un-supported one-sided repairs, it is again convenient to divide the analysis into two stages In Section 6.1 the stress reduction due to stage I will be anaiysed within the framework of geometrically linear elasticity [ 121, whereas a geometrically non-linear analysis [ 171 will be presented in Section 7.6.2 These two solutions will provide an upper and lower bound to the actual stress distribution In both cases the geometrically linear analysis is all that needed for stage 11 For stage I we shall consider the particular case where the reinforcement covers the entire cracked plate, ignoring the load attraction effect

7.6 I Geometrically linear analysis

Consider first the effect of one-sided reinforcement on an un-cracked plate which

is subjected to a uniaxial tension Assuming that the reinforcement is far greater than the shear stress transfer length, we treat the reinforced region as a composite plate with a rigid bondline The stress 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 portion satisfies the usual kinetic condition that plane sections remain plane The position of the neutral plane of the composite plate consisting of the base plate and rigidly-bonded reinforcement is denoted by F, referring to Figure 7.13,

The stress distribution in the patched plate is assumed to be linear in the thickness

direction, so that it can be specified in terms of the membrane force NO and a

bending moment Mo per unit length in the x-direction, as depicted in Figure 7.13

158 Advances in the bonded composite repair of merallic aircraft structure

neutral axis of

c composite section

(b) Fig 7.13 Stress distribution in an un-cracked plate reinforced with a patch (a) composite plate

subjected to uniaxial tension; (b) stress distribution in the plate

(see [12,17] for more details),

In stage 11, analysis of the crack-tip deformation requires the use of the shear deformation theory, which yields that the stress intensity factor varies linearly through the plate thickness,

Chapter I Analytical methods f o r designing composite repairs 159

No

(b)

Fig 7.14 (a) Single strap joint representing one-sided repairs subjected to membrane tension and

bending moment, and (b) notations and boundary conditions

where K,,,, and Kb denote respectively the membrane and bending stress intensity

factors The strain-energy release rate can be determined following the method outlined in Section 7.4.2, except that the change in the potential energy now consists of two terms: work done by the membrane force and the bending moment

160 Advances in the bonded composite repair of metallic aircraft structure

G, = - [CIIN; + (Ci2 + ~21)NoMo + C22M;]

which can be simplified to become

Chapter 7 Anal.vtica1 method7 f o r designing composite repairs 161

where F, is given by Eq (7.37) Figure 7.15 shows a comparison between Eq (7.70)

and the results of 3D finite element analyses The repair configuration being

considered is the same as that analysed in Section 7.4.4 The same problem has been

analysed using two different finite element codes, namely ABAUQS [26] and

PAFEC [25]; both yielded approximately the same result It can be seen that the

above formula is in good correlation with the finite element results It is also worth

noting that the results confirm that the stress-intensity factor Krms for a one-sided

repair is much higher than that for an equivalent two-sided repair, indicating the importance of out-of-plane bending

162 Advances in the bonded composite repair of metallic aircraft structure

+

Fig 7.16 A plate with a through crack reinforced with tension and bending springs

The root-mean-square stress-intensity factor Kms is related to the membrane and bending stress intensity factors [12],

(7.71)

Although the root-mean-square of the stress intensity factor has been derived, the maximum and minimum stress intensity factors still remain unresolved It is apparent that the energy method alone is insufficient determine the membrane and bending stress intensity factors, as an additional equation is required to partition

K,, into membrane and bending components To this end, let us now briefly discuss a crack-bridging model which is capable of analysing the combined tensile stretching and bending of one-sided repairs

7.6.2 Crack bridging model

The perturbation problem of stage I1 for a one-sided repair can be reduced by representing the patch by distributed springs bridging the crack faces [ 171, as illustrated in Figure 7.16 The springs have both tension and bending resistances; their stiffness constants are determined from a ID analysis for a single strap joint, representative of the load transfer from the cracked plate to the bonded patch The spring constants are given by Eq (7.57) For the purpose of parametric investigation, we introduce the following non-dimensional variables,

Chapter I Analytical methods for designing composite repairs 163

By using Reissner’s plate theory, the normalised crack face displacement hl and normalised crack face rotation h2 are solutions of the following coupled integral

with KO and K2 are the modified Bessel functions of the second kind The integral

equations can be readily solved by expanding the unknowns using Chebyshev polynomials of the second kind The membrane and bending stress intensity factors

K, and Kb are directly related to the values of hl and h2 at q + 1

With the prospective membrane force NO and bending moment MO are given by

Eqs (7.53) and (7.54), respectively, the membrane and bending stress intensity factors can be solved For the repair configuration specified by Table 7.1, the results are shown in Figure 7.17 together with the finite element results Considering the approximate nature of the crack bridging model and the finite element method, the reasonably good correlation between the predictions and the finite element results confirms the validity of the above theoretical model

7.6.3 Geometrically non-linear analysis

The geometrically linear analysis presented in the preceding section is strictly speaking applicable only when the out-of-plane deflection is negligible relative to

plate thickness, i.e when the applied stress om is very low Otherwise the

geometrically non-linear deformation, as indicated in Figure 7.18, has to be taken

164 Advances in the bonded composite repair of metallic aircraft structure

Fig 7.17 Theoretical predictions and finite element results for a typical one-sided repair assuming

geometrically linear deformation

Fig 7.18 Geometrically non-linear deformation of a single strap joint representing one-sided repairs

into account We shall use the rigid-bond approximation and denote the deflection

of the plate as w The governing equation for the deflection of the beam inside and outside the repaired region is,

EpIt - d2 w = ~ " O t p ( w + 2) , IyI 5 B , (7.77a) dY2

Chapter 7 Analytical methods for designing composite repairs 165

where

(7.80)

and constants C1, CZ, C3, and C, can be determined from the boundary condition

(78) and the following continuity and symmetry conditions,

the joint y = 0 are shown in Figure 7.19(a) together with the analytical prediction

(85), indicating the accuracy of the beam theory solution Similarly the finite element results for the deflection along the joint and Jhe analytical solution are shown in Figure 7.19(b), together with the analytical solution (79), indicating a good agreement From the above analytical solution it is clear that the displacement w at the centre of the strap joint w depends on three non-dimensional

parameters, xB, x p / x , and LIB It is easy to show that w approaches -2 as

xpB + x, provided LIBZO This limiting case corresponds to when the neutral

axis of the patched region is aligned perfectly with the path of the applied load

166 Advances in the bonded composite repair of metallic aircrafi structure

Normalised overlap length ZB

Coordinate y (mm)

(b)

Fig 7.19 Deflection of a one-sided strap joint accounting for geometrically non-linear: deformation: (a)

centre of overlap, and (b) along the joint

Since the bending moment at the centre of the overlap is

Chapter I Analytical methods for designing composite repairs 167

With the prospective force and bending moment given by Eqs (7.87), the coupled Eqs (7.73) can then be solved numerically The superposition principle used in the previous section to reduce the problem of a one-sided repair subjected to remote tension to a simple perturbation problem where the crack is internally pressurised

is, strictly speaking, not valid should the structure undergo geometrically nonlinear deformation However, an upper bound solution can be obtained by a hybrid method, in which the prospective stress distribution is solved using geometrically

nonlinear elasticity theory, the stage I1 analysis is carried out using the

geometrically linear theory, i.e the crack bridging method developed in Section 7.5.2 A proof that this hybrid method will provide a conservative prediction of the stress intensity factors can be found in [17]; a validation using the geometrically non-linear finite element method will be presented later

For a given repair configuration, the prospective membrane force (Eq (87a)) increases with the load P while the bending moment (Eq (87b)) decreases, resulting in a net increase in the stress intensity factors, although the rate of increase is slower than that expected from geometrical linear considerations This is illustrated in Figure 7.20(a) for the case of half crack length a of 20mm It is seen that the minimum stress-intensity factor &in determined by the hybrid method correlates very well with the finite element results However the hybrid method over-predicts the maximum stress-intensity factor, Kmax, confirming that the hybrid

method is an upper-bound solution [17] A similar trend can be observed in Figure 7.20(b) which shows the asymptotic behaviour of the stress-intensity factors

as the crack length increases The remote applied stress oJ: is kept to be 400 MPa Again the minimum stress-intensity factor Kmin determined by the hybrid method correlates very well with the finite element results, whereas the maximum stress- intensity factor Kmax determined by the hybrid method is greater than that obtained from finite element analysis

7.7 Residual thermal stress due to adhesive curing

The process of adhesive bonding using high-strength structural adhesives (thermal-plastics) generally requires curing the adhesive above the ambient temperature For instance, in a typical repair applied to aircraft structures the reinforced region is initially heated to a temperature of 120 "C, under pressure, for approximately one hour (the precise curing cycle depends on the adhesive being used) Upon cooling the fully cured, patched structure to the ambient temperature, thermal stress will inevitably develop in both the plate and the reinforcement, due

to cooling a locally stiffened structure, especially when the reinforcing patch has a lower coefficient of thermal expansion than the plate being repaired Thermal stresses may also arise when the patch structure experiences thermal cycling in service Therefore thermal residual stresses represent a major concern to the repair efficiency of a repair This is because the resulting thermal residual stresses post cure in the metal plate are inevitably tensile, owing to the increase in the stiffness of the patched region and the lower coefficient of thermal expansion of the composite

168 Advances the bonded composite repair of metaIlic uircraft structure

patches This tensile residual stress will increase the maximum stress-intensity factor of the crack after repair, hence may enhance fatigue crack growth rate (see Chapters 11 and 12)

7.7.1 Temperature distribution

Solutions of the thermal residual stresses in symmetric repairs and one-sided repairs have been developed in [ 181 and [ 191, respectively In the following, only the results pertaining to symmetric repairs will be presented; details of solution for one- sided repairs can be found in reference [19] Consider the configuration shown in Figure 7.21(a), in which an isotropic plate is reinforced by a circular patch of radius

Ri The coordinate system xy is chosen so that the principal axes of the orthotropic

Chapter I Analytical methods for designing composite repairs 169

plate

Fig 7.21 An infinite plate reinforced with a circular composite patch (a) Configuration and (b)

temperature distribution during heating and cooling

patch are aligned and parallel to the x, y axes, with the major direction along the y -

axis The objective here is to determine the thermal stresses in the patch, in the plate both inside the bonded region and just outside the patch

During the first step of bonding, suppose that the inner portion ( r 5 Ri) is heated

to a temperature Ti during the curing process, with the usual convention that the

170 Advances in the bonded composite repair of meiaIlic aircraft structure

ambient temperature is taken as the zero of temperature The temperature field satisfies the Laplacian Eq (7.88) [21],

7.7.2 Residual stress due localised heating

Due to this non-uniform temperature distribution given by Eq (7.89), thermal stresses (equal biaxial) develop in the plate, which can be readily derived [21],

(7.90)

1

2

c# = - -apEpAT ,

where AT = - To, u p and Ep denote the thermal expansion coefficient and

Young's modulus of the plate Since Ti - To > 0 during heating, the above thermal initial stress is compressive It should be noted that this thermal stress arises only in the case of localised heating of a large structure; for the case of a finite size specimen being uniformly heated to Ti, no thermal stress will develop This stress distribution serves as the initial stress that will be added to the thermal stress induced by cooling the patched region down to the ambient temperature

Now consider the case of a circular plate of radius Ro whose outer edge r = R, is constrained by a continuous distribution of springs of stiffness 0 according to the following relation,

The cases of free edge and a clamped edge r = R, can be recovered by setting 0 = 0

and 0 -+ co, respectively In this case, the thermal stress can be determined from

Eq (7.96), to be discussed in the next section, by setting S = 0,

1 - e / a p

l + I + h J p - v V p '

Chapter 7 Analytical methods for designing composite repairs 171

7.7.3 Residual stresses after cooling from cure

For the second step of adhesive bonding we assume that there is no shear stress

in the adhesive layer during curing, so that the reinforcing patch expands freely without developing any stresses After the adhesive is fully cured, the patched plate

is then cooled down to the ambient temperature In other words, the temperature change over the entire patched plate is subjected to the following temperature field, referring to Figure 7.21(c),

where the superscript C denotes the temperature change corresponding to the

second step: cooling Now the problem is to determine the thermal stress 00' on cooling to ambient temperature after curing The final stress in the plate

(7.98) During this cooling process, it is assumed that the adhesive bond between the composite patch and the metal plate is absolutely rigid, so that the same strain-state prevails in both the patch and the plate directly beneath the patch Details of a general solution for orthotropic patch are given in [18] It is interesting to note that even for an 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