energetic balance in the debonding of a reinforcing ... · energetic balance in the debonding of a...

Acceptedunder review

Energetic Balance in the Debonding of a Reinforcing

Stringer: Effect of the Substrate Elasticity

Annalisa Francoa, Gianni Royer-Carfagnib,∗

aDepartment of Civil and Industrial Engineering University of Pisa, Via Diotisalvi 2, I56126 Pisa, Italy

bDepartment of Civil-Environmental Engineering and Architecture, University of Parma,Parco Area delle Scienze 181/A, I 43100 Parma, Italy

Abstract

An effective way to strengthen deteriorated concrete or masonry struc-tures is to glue to them, at critical regions, strips or plates made of FiberReinforced Polymers (FRP). The reliability of this technique depends uponinterfacial adhesion, whose performance is usually evaluated through an ener-getic balance, assuming that the support is rigid. The present study analyzesthe contact problem between reinforcement and substrate, both assumed tobe linear elastic. The solution of the resulting integral equations is expressedin terms of Chebyshev polynomials. A generalization to this problem of theCrack Closure Integral Method developed by Irwin allows to calculate theenergy release rate associated with the debonding of the stiffener. Ener-getic balance a la Griffith emphasizes the role played by the length of thestiffener and the deformation of the substrate, predicting load vs. displace-ment curves that, in agreement with experimental measurements, exhibit asnap-back phase.

Keywords: Elastic stiffener, Fiber Reinforced Polymer (FRP), elasticcontact, Chebyshev polynomials, Crack Closure Integral method, energyrelease rate.

∗Corresponding Author. Tel.: +39 0521 905917; fax: +39 0521 905924Email address: [email protected] (Gianni Royer-Carfagni)

Preprint submitted to International Journal of Solids and Structures January 17, 2013


1. Introduction

There has been in the last decade a constantly increasing interest in thestrengthening of existing structures by confining them with Fiber ReinforcedPolymers (FRP). In this technology, plates or sheets made of carbon or glass-fiber reinforced polymers are bonded to the surface of the support throughepoxy adhesives, improving both structural stiffness and strength. Clearly,the performance of stress transfer between FRP and substrate depends uponthe bond that can be attained between the two materials.

A key issue in the design of an effective retrofitting system using externallybonded reinforcement is the evaluation of the strength of the FRP-substratebond. Delamination is the most frequent failure mechanism, which has tobe carefully considered because of its brittle nature. As a result, in orderto understand the interfacial debonding failure, extensive research has beencarried out by means of different in type experimental tests, including sheartests (Taljsten, 1997; Mazzotti et al., 2008; Carrara et al., 2011; Yao et al.,2005), double shear tests (Maeda et al., 1997) and modified beam tests (De-Lorenzis et al., 2001), for which an extensive list of references can be found in(Yao et al., 2005). In the case of a concrete support, the most recent studiessuggest that the main failure mode is the cracking of concrete under shear,occurring commonly a few millimeters below the adhesive-concrete interface.Therefore the bond strength, i.e. the maximum load that can be transmit-ted, depends significantly upon the concrete toughness, associated with itsspecific fracture energy.

Many researchers have developed models in fracture mechanics in orderto predict the theoretical load response for debonding failure mode. Taljsten(Taljsten, 1996) estimated the maximum transmissible load by consideringan energetic balance a la Griffith for the fracturing surface between stiffenerand substrate, both considered within the framework of beam theory. How-ever, the great majority of the models are mainly based upon an assumedconstitutive law for the interface supposed to be cohesive in type, i.e., a re-lationship between the interfacial shear stress and the relative displacement(slip) between substrate and FRP. The relevant literature is so wide that anyattempt of synthesis cannot avoid to be partial. Yuan (Yuan et al., 2001),for example, studied the influence of the shape of the interfacial constitutivelaw on the load capacity of FRP bonded to concrete. Wu et al. (Wu andNiu, 2000) proposed a theory to predict the initiation of debonding using anassumed material model. Experimental investigations have aimed at deter-

2


mining the interfacial constitutive law, usually by measuring strains in thestiffener and substrate with resistance strain gages (DeLorenzis et al., 2001;Savoia et al., 2003).

The aim of this study is to evaluate the influence of the substrate de-formability with reference to the solution of a contact problem in plane linearelasticity between an elastic stiffener and an elastic substrate, supposed ingeneralized plane stress. From the application point of view, the problem canbe categorized in two main groups: stiffeners or cover plates mainly used inaircraft structures (Melan, 1932; Benscoter, 1949; Bufler, 1961; Brown, 1957;Koiter, 1955; Reissner, 1940; Arutiunian, 1968) and thin films used in micro-electronics, sensors and actuators (Alaca et al., 2002; Hu, 1979; Shield andKim, 1992). In both fields, the primary interest is the evaluation of stressconcentrations or singularities near the edges of the film or the stiffener inorder to deepen the question of crack initiation and propagation in the sub-strate or along the interface. This aspect seems to have been only partiallyconsidered for the specific case of civil applications through the use of fiberreinforced polymer composites.

The stress transfer between an elastic stiffener and an elastic plate wasfirstly introduced by Melan in 1932 (Melan, 1932). By supposing perfectbond between the bodies, both considered infinite, and by treating the fiberas a one dimensional stringer loaded at one end by a longitudinal force,he was able to obtain a closed-form solution. An important result was theunboundedness of the interface tangential stress in the neighborhood of theforce application point. This work was then considered and extended bydifferent authors. The problem of a finite stiffener on an infinite plate wasthen treated by Benscoter (Benscoter, 1949). He considered the problemof stress transfer under symmetric and anti-symmetric loading and reducedthe governing integro-differential equation to a system of linear algebraicequations.

There are two types of approaches to study the problem of debonding fromthe theoretical treatment standpoint. The first deals with crack initiation byassuming a preexisting crack (Yu et al., 2001) ; the second assumes thatthe edge delamination occurs due to stress singularities at the edges of thefilm (Alaca et al., 2002; Erdogan and Gupta, 1971; Shield and Kim, 1992;Guler, 2008; Guler et al., 2012). Erdogan and Gupta (Erdogan and Gupta,1971) provided one of the earliest and most relevant contributions to thinfilms, where they solved the problem of an elastic stiffener bonded to a halfplane using the membrane assumption. Later, Shield and Kim (Shield and

3


Kim, 1992) extended this analysis using the plate assumption for the film, inorder to take into account the bending stiffness and the effect of peel stressesespecially near the edges of the film. It was demonstrated that the membraneassumption is still valid when the stiffener thickness is ”small” compared tothe other dimensions in the system. Freund and Suresh (Freund and Suresh,2008) gave a qualitative indication for the thickness of the stiffener, whichhas to be at least 20 times smaller than the other dimensions to assure amembrane behavior.

In this work, the contact problem of an elastic finite stiffener bonded tothe boundary of a semi-infinite plate and loaded at one end by a longitu-dinal concentrated force is considered. A compatibility equation is writtenthat automatically furnishes the integral equation in terms of the tangentialstresses between stiffener and plate. An approximate solution is then ob-tained in term of Chebyshev polynomial, following the approach proposedby Grigolyuk (Grigolyuk and Tolkachev, 1987), tentatively pursued by Vil-laggio (Villaggio, 2001, 2003) and probably firstly introduced by Benscoter(Benscoter, 1949).

We do not consider here the variety of responses that can be obtainedunder the assumption of cohesive shear fractures a la Barenblatt, regulatedby an assumed shear stress vs. slip constitutive law. Being interested in theeffect of the substrate elasticity, we limit at this stage to consider the minimalmodel, in which the debonding process is assumed to begin and continue assoon as the energy release rate due to an infinitesimal crack growth equals theinterfacial fracture energy (Griffith balance). The evaluation of the energyrelease rate due to a propagating interface crack does not seem to have beencorrectly considered by previous contributions (Villaggio, 2003). This is whywe analyze here in detail the extension to this particular problem of the CrackClosure Integral Method developed by Irwin (Irwin, 1957). This energeticbalance is then used to derive the maximum load as a function of the bondlength provided that specific fracture energy is known. Moreover, one canreproduce a pull out test, following step by step the corresponding interface-crack path.

A parametric study has been conducted in order to evaluate the load vs.displacement curves predicted by this model, which are compared with care-ful experimental data obtained from recent direct tensile tests (Carrara et al.,2011). Despite the simplicity of the Griffith energetic balance, the analyticalresults are in good agreement with the experimental pull-out curves for highbond length, being able to reproduce, at least at the qualitative level, their

4


typical trend. This is characterized by a plateau, during which debondingoccurs, followed by a snap-back phase, related to the release of the strainenergy stored by the FRP stringer during the delamination process. Thelatter was obtained with a closed loop control of the crack opening in thedetaching stringer (Carrara et al., 2011).

2. Load transfer from an elastic stiffener to a semi-infinite plate

Suppose that an elastic stiffener of constant width bs and (small) thicknessts is bonded to the boundary of an elastic semi-infinite plate in generalizedplane stress over the interval [0, l], considered with respect to the ξ-axis of theCartesian system shown in Figure 1. At one end, the stiffener is loaded by alongitudinal force P , uniformly distributed on the cross sectional area of thestiffener. Since ts is small, the bending strength of the stiffener is negligible,so that its normal component of the contact stress with the semi-plane maybe neglected. The state of stress in the stiffener is then uni-axial, due to Pand the tangential contact stresses transmitted by the plate.

Figure 1: A finite stiffener bonded to the boundary of a semi-infinite plate.

Equilibrium for that part of the stiffener comprised between the originand a section ξ = x allows to write the axial force Ns(x) in the form

Ns(x) = P −∫ x

0

q(ξ) dξ , (2.1)

5


where q(ξ) is the contact tangential force per unity length.By Hooke’s law, the stiffener strain reads

εs(x) =Ns(x)

EsAs

=1

EsAs

[P −

∫ x

0

q(ξ) dξ

], (2.2)

where Es is its elastic modulus and As its cross sectional area. Besides,on the boundary of the semi-plane, the strain in the interval [0, l] due tothe tangential contact stress may be written in the form (Grigolyuk andTolkachev, 1987)

εp(x) = − 2

πEpbp

∫ l

0

q(ξ)

ξ − xdξ , (2.3)

where Ep is the elastic modulus of the plate and bp its width. Since thestrains must be equal over the interval of contact, equating (2.2) and (2.3)one obtains the singular integral equation

1

EsAs

[P −

∫ x

0

q(ξ) dξ

]= − 2

πEpbp

∫ l

0

q(ξ)

ξ − xdξ. (2.4)

Introducing the rigidity parameter λ, defined as

λ =2

π

Epbpl

EsAs

. (2.5)

and the dimensionless coordinate τ = ξ/l, equation (2.4) can be written inthe form ∫ 1

0

q(τ)

τ − τ0dτ = −π2λ

4

[P

l−∫ τ0

0

q(τ) dτ

], (2.6)

which has to be solved under the equilibrium condition

l

∫ 1

0

q(τ) dτ = P. (2.7)

An approximate solution for (2.6) can be obtained by expressing the con-tact force q in term of a series of Chebyshev polynomials (Grigolyuk andTolkachev, 1987; Erdogan and Gupta, 1971, 1972). Chebyshev terms are or-thogonal in the interval [−1, 1], so that it is convenient to make the changeof variable

6


t = 2τ − 1 ,

so that conditions (2.6) and (2.7) become, respectively,∫ 1

−1

q(t)

t− t0dt = −π2λ

8

[2P

l−

∫ t0

−1

q(t) dt

], (2.8)

l

∫ 1

−1

q(t) dt = 2P . (2.9)

The approximate solution of (2.8) can be sought in the form of an expan-sion in Chebyshev polynomials of the first kind Ts(t) defined as

q(t) =2P

πl√1− t2

n∑s=0

XsTs(t) , (2.10)

where Xs are constants to be determined. Observe that there is a square-root singularity in the solution at both ends of the reinforcement, which istypical of most contact problems in linear elasticity theory; the strength of thesingularity is determined by all terms of the series. Substituting (2.10) intocondition (2.9) and recalling the orthogonality conditions of the Chebyshevpolynomials of the second kind (see Appendix, eq. (A.5)), one obtains that

X0 = 1.

Moreover, substitution of the expansion (2.10) in (2.8) allows to determine,after integration, the other constants Xs by means of the Bubnov method(Grigolyuk and Tolkachev, 1987). The final result is a set of algebraic equa-tions for Xj of the type

Xj +λ

4

n∑s=1

Xsajs = −λ

4bj , for j = 1, 2, ..., n (2.11)

where {ajs = − 4j

[(j+s)2−1][(j−s)2−1], for even j − s,

ajs = 0, for odd j − s,

and

7


b1 =

π2

4,

bj = − 4j(j2−1)2

, for even j,

bj = 0, for odd j = 1.

Solving the system of algebraic equations (2.11), it is immediate to de-termine the Xj and hence q(t). It may be seen that in the neighborhood oft = ±1, the contact problem for the stiffener/plate gives a singularity analo-gous to a crack problem under pure Mode II loading conditions. Therefore,one can define the Mode II stress intensity factor at ξ = 0 (t = −1) in theform

KII = limξ→0

q(ξ)√

2πξ. (2.12)

Substitution of the contact stress (2.10) into (2.12) gives the expression

KII =2P√2πl

n∑s=0

Xs(−1)s , (2.13)

which represents the governing parameter for the problem at hand.

3. Energetic balance

Linear elastic fracture mechanics (LEFM) is based upon an energeticbalance a la Griffith between the strain energy release rate and the increasein surface energy.

3.1. Generalization of the Crack Closure Integral Method by Irwin

For the problem at hand, let us consider the case of an elastic stringerbonded for a length l to an elastic plate in generalized plane stress. Thestringer is pulled by a force P in the configurations sketched in Figure 2,referred to as the sound state.

Let us consider another configuration, i.e., the debonded state representedin Figure 3, in which delamination has occurred over a portion of length c.A reference system (ξ, η) is introduced with the origin on the left-hand-sideborder of the stringer, so that the bonded portion is c ≤ ξ ≤ l. The compositebody is loaded by two system of forces. System I is the force P I appendedat the stringer left-hand-side border, while system II is composed of forcesper-unit-length qII(ξ), representing a mutual interaction stress between plate

8


Figure 2: Sound state: stiffener bonded for a length l upon an elastic plate.

and stringer (Figure 3). Let uIs(ξ) (u

IIs (ξ)) represent the displacement of the

stringer in the positive ξ−axes direction of the stringer due to system I(II) of forces, and let uI

p(ξ) (uIIp (ξ)) be the corresponding displacement of

the plate, again associated with system I (II). In the following, quantitiesreferred to system I or II will be labeled with the I or II apex, respectively.

Figure 3: Debonded state, where delamination has occurred on a portion of length c.

By Clapeyron theorem, the elastic strain energy U I due to the action ofsystem I reads

9


U I = −1

2P IuI

s(0) . (3.14)

The strain energy U (I+II), associated with system I + II, is of the form

U I+II = −1

2P IuI

s(0) +1

2

∫ c

0

qII(ξ)[uIIs (ξ)− uII

p (ξ)]dξ +

∫ c

0

qII(ξ)[uIs(ξ)− uI

p(ξ)]dξ

= −1

2P IuI

s(0) +1

2

∫ c

0

qII(ξ){[uIs(ξ) + uII

s (ξ)]− [uIp(ξ) + uII

p (ξ)]}dξ (3.15)

+1

2

∫ c

0


p(ξ)]dξ .

Let us then assume that P I = P and that qII(ξ) represent the contactbonding forces for the sound state of Figure 2. Since in this case the portion0 ≤ ξ ≤ c is perfectly bonded, one has that

[uIs(ξ) + uII

s (ξ)]− [uIp(ξ) + uII

p (ξ)] = 0 , (3.16)

and consequently, from (3.14) and (3.15), one finds

∆U = U I+II − U I =1

2

∫ c

0


p(ξ)]dξ . (3.17)

Here ∆U represents the difference of the strain energy between the soundstate and the debonded one. Obviously, the variation of the total energy ∆Eequals −∆U . The latest expression represents the extension to this case ofthe Crack Closure Integral Method developed by Irwin (Irwin, 1957).

3.2. Strain energy release rate

With the same notation of Section 2, indicating with Γ the surface fractureenergy and with bs the width of the stiffener, energetic balance states that

Γ bs = limc→0

d

dc∆U = − lim

c→0

d

dc∆E = G , (3.18)

where G denotes the strain energy release rate.Substituting the preceding expressions in the relation (3.18), the problemreduces to the evaluation of G, i.e.,

10


G = limc→0

d

dc

[1

2

∫ c

0

qII(ξ) {uIs(ξ)− uI

p(ξ)}dξ

]= lim

c→0

d

dc

[1

2

∫ c

0

qII(ξ)uIrel(ξ)dξ

],

(3.19)where we have defined uI

rel = uIs − uI

p. By using Leibniz’s rule for differenti-ation under the integral sign, the preceding expression becomes

G = limc→0

{1

2qII(c)uI

rel(c) +1

2

∫ c

0

qII(ξ)∂

∂cuIrel(ξ) dξ

}. (3.20)

The first term is null because there is no relative displacement for ξ = c,since in this point the stiffener is still bonded to the plate. As regards to thesecond term, denoting with εIs the axial strain in the stiffener, and with εIpthe normal strain component in the ξ direction of the plate, observe that

∂

∂cuIrel(ξ) =

∂

∂c

[∫ c

ξ

εIs(ζ) dζ

]− ∂

∂c

[∫ c

ξ

εIp(ζ) dζ

]

= εIs(c)−∂

∂c

[∫ c

ξ

εIp(ζ) dζ

].

(3.21)

Consider first the term containing εIp, i.e., the one associated with thestrain in the plate. Referring to fig. 3, the strain needs to be evaluated atpoints that are external to the interval [c, l], where stiffener and plate arebonded. The elastic solution for a plate reinforced by a stringer of lengthl − c can be obtained with the same method described in Section 2. Withreference to eq. (2.3), let us introduce the new variable

t =2ξ − l − c

l − c.

Solving the elastic problem in terms of the new variable t, from eq. (2.3),one obtains

εIp(t0) = − 4P

π2Epbp(l − c)

n∑s=0

Xs

∫ 1

−1

Ts(t)√1− t2(t− t0)

dt , (3.22)

11


where t0 = (2ξ0 − l − c)/(l − c). The integral can be evaluated by using theproperty of Chebyshev polynomials reported in Appendix (eq. (A.6)), withreference to the case |t0| > 1. The final result is

εIp(t0) = − 4P

πEpbp(l − c)

n∑s=0

Xs(t0 +

√t20 − 1)s√

t20 − 1, (3.23)

and therefore the displacement reduces to

uIp(t0) =

∫ −1

t0

εIp(t)dt = − 4P

πEpbp(l − c)

n∑s=0

Xs

∫ −1

t0

(t+√t2 − 1)s√

t2 − 1

l − c

2dt

= − 2P

πEpbp

n∑s=0

Xs

s

[(−1)s −

(t0 +

√t20 − 1

)s].

(3.24)

Written in term of ξ, using a Taylor expansion in a neighborhood of ξ = c,(3.24) reads

uIp(ξ) = − 2P

πEpbp

n∑s=0

Xs(−1)s2

√c− ξ

l − c. (3.25)

Consequently, the derivative of the displacement respect to the interfacialcrack length c is

∂

∂cuIp(ξ) = − 2P

πEpbp

n∑s=0

Xs(−1)s(l − ξ)

(l − c)√

(c− ξ)(l − c). (3.26)

The contact stresses qII(ξ) are given by (2.10) and can also be expanded inTaylor’s series in neighborhood of ξ = 0 to obtain

qII(ξ) =P

π

n∑s=0

Xs cos (πs)

[1

√ξ√l− 2s2

√ξ

l√l

]. (3.27)

Therefore, the strain energy release rate G can be evaluated substituting(3.26) in (3.21) and the result, together with (3.27), in the second term of(3.20). After integration, one obtains the expression

12


G = limc→0

1

2εIs(c)

∫ c

0

qII(ξ) dξ +P 2

πEpbpl

[n∑

s=0

Xs(−1)s

]2

. (3.28)

But the first term of (3.28) is null, because the contact stress qII(ξ) of (3.27)has a square-root singularity in a neighborhood of ξ = 0 so that for c → 0the integral vanishes1. Consequently, one finds the general expression for theenergy release rate G in the form

G =P 2

πEpbpl

[n∑

s=0

Xs(−1)s

]2

. (3.29)

Recalling the expression of Mode II stress intensity factor given by (2.13),the expression (3.29) can be re-written in the form

G =K2

II

2Epbp. (3.30)

Equation (3.30) plays a key role since it bridges the energetic approach withthe stress analysis. Remarkably, it is similar to Irwin’s relationship betweenthe strain energy release rate and the stress intensity factor. To the authors’knowledge, the method used to derive the strain energy release rate in thecontext of plane elasticity has never been stated up to now. As a matter offact, common ways to evaluate G are based on the J-integral (Cherepanovet al., 1979).

The expression (3.30) is particularly important because the stress inten-sity factor KII can also be evaluated numerically2, without resorting to theChebyshev expansion. The energetic balance detailed in Section 3.3 thusallows to calculate the maximum tensile load P once the fracture energy ofthe bond is known.

1Indeed, one can demonstrate that when Ep → ∞ (rigid substrate) qII(ξ) tends tobecome a Dirac distribution centered at ξ = 0, so that the integral does not vanish whenc → 0. Here we consider the elastic solution for Ep < ∞ and will show later on that whenEp → ∞ the second term of (3.28) tends to the energy release rate associated with theproblem of an elastic stiffener on a rigid substrate. This fact does not seem to have beenrecognized in Villaggio (2003), where the expression proposed is not correct.

2Most numerical codes evaluate the stress intensity factor using the J-integral.

13


3.3. Energetic balance

Suppose that the toughness of the bonded joint is defined by the fractureenergy per unit area ΓF . Then, energetic balance a la Griffith implies thatthe crack propagates when

G = ΓF bs , (3.31)

where bs is the width of the stiffener. Then, from (3.29), one finds that thecritical value Pcr of P reads

Pcr =

√ΓF bs

πEpbpl

[∑n

s=0 Xs(−1)s]2 . (3.32)

Apparently Pcr depends upon the elasticity of the substrate only, because theelasticity of the stiffener is not explicitly involved in the expression (3.29) ofG. But it should be noticed that the terms of the Chebyshev expansionstrongly depend upon the mechanical properties of the stiffener through therigidity parameter λ, defined in (2.5).

To illustrate, it is useful to consider directly the limit condition Ep = ∞,i.e., the case of a rigid substrate. A simple calculation indicates that theenergy release rate takes the form

Gr =P 2

2EsAs

, (3.33)

which is the same expression derived by Taljsten in (Taljsten, 1996), fora general linear and non-linear interface law with reference to a pure shearbond-slip model, and by Wu et al. in (Wu et al., 2002), for a bilinear interfacelaw.

Figure 4 shows the ratio G/Gr, with G evaluated through (3.30) andGr through (3.33), as a function of the bond length l for values of Ep/Es

ranging from 0.01 to 100. Notice that G → Gr as l → ∞, and the limit valueis attained more quickly as Ep/Es increases, i.e., as the substrate tends tobecome rigid.

Moreover, as shown more in detail in Figure 5, for short bond lengths thevalue of the energy release rate may be much higher than the value associatedwith the case of rigid substrate. From (3.31), this means that short stiffenersmay detach at much lower load levels than long stiffeners. This effect isentirely due to the elasticity of the substrate, because if the substrate is rigidthen Gr is given by (3.33), which is independent of the length of the stringer.

14


0 5 10 15 20 25 30 35 40 45 500

2

4

6

8

10

12

14

16

18

Bond length, [mm]

G /

Gr

Ep/E

s=0.01

Ep/E

s=0.1

Ep/E

s=1

Ep/E

s=10

Ep/E

s=100

l

Figure 4: Normalized strain energy release rate G/Gr for different values of the ratioEp/Es.

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 30

2

4

6

8

10

12

14

16

18

Bond length, [mm]

G /

Gr

Ep/E

s=0.01

Ep/E

s=0.1

Ep/E

s=1

Ep/E

s=10

Ep/E

s=100

l

Figure 5: Normalized strain energy release rate G/Gr for different values of the ratioEp/Es. Detail for small bond lengths l.

15


It is important at this point to quantify the meaning of “short” and“long” stiffeners. Recall that terms Xs defining the Chebyshev expansiononly depend upon the non-dimensional parameter λ of (2.5). Figure 6 showsthe ratio G/Gr now as a function of λ: obviously the graphs obtained inFigures 4 and 5 for varying Ep/Es collapse into one curve (for convenienceof representation, the scale for λ is now logarithmic). It is then quite evidentthat the transition between the case of a soft elastic substrate to the case ofa rigid substrate is marked by a value λ = λ∗ that can be estimated of theorder λ∗ ≃ 101. But since the stringer length l enters in the definition (2.5)of λ, the “rigidity” of the substrate does not depend upon its elastic modulusonly. In other words, it is λ that represents the similarity parameter: thecase λ ≫ λ∗ (λ ≪ λ∗) is associated with long (short) stiffeners and rigid(soft) substrates.

10−1

100

101

102

103

104

100

Rigidity parameter, λ

G /

Gr

Ep/E

s=0.01

Ep/E

s=0.1

Ep/E

s=1

Ep/E

s=10

Figure 6: Normalized strain energy release rate G/Gr as function of the rigidity parameterλ for different values of the ratio Ep/Es.

The presence of a step change in the distribution of contact stress alongthe stiffener bond length is also evident in the logarithmic plot of figure 7,where ξ denotes again the distance from the stringer edge where the loadP is applied. As ξ → 0, the slope of the curves equals to −1/2 because ofthe typical square root singularity. The graphs tend to a vertical asymp-

16


tote when approaching the second edge of the stringer, where another stresssingularity occurs (the various graphs refer to different bond lengths). Theslope of the graphs changes for a value of ξ comprised between 100 and 101.This transition value should not be confused with the anchorage length, i.e.,the minimum length assuring maximum anchoring force. In fact, there arestress singularities at both edges of the stiffener, so that the axial strain inthe stiffener is never zero. This is a characteristic feature (and perhaps alimitation) of this model.

Figure 8 represents, as a function of ξ, the normalized axial load Ns/Pcalculated as per (2.1), for two different value of the substrate elastic modulusEp. The continuous lines may be associated with a typical reinforcement ona concrete support, whereas the dashed lines refer to the case of a substrateten times more deformable (elastic modulus one tenth of the previous one).From the graphs it is evident that the softer the substrate, the higher is thelength that is necessary to transfer the load from the stringer.

10−4

10−3

10−2

10−1

100

101

102

103

10−4

10−3

10−2

10−1

100

101

102

103

ξ [mm]

τ [N

/mm

2 ]

l = 1 mm

l = 2 mm

l = 5 mm

l = 10 mm

l = 20 mm

l = 40 mm

l = 100 mm

l = 200 mm

l = 300 mm

1

2

Figure 7: Interfacial shear stress along the stiffener for different bond lengths l.

It should also be mentioned that, in order to achieve a good approxima-tion, the number n of Chebyshev terms that are needed in the series (3.29)to define G, strongly increases as Ep/Es increases, i.e., as the substrate be-comes stiffer and stiffer. This fact is shown in Figure 9, which refers to cases

17


10−4

10−3

10−2

10−1

100

101

102

103

10−4

10−3

10−2

10−1

100

101

ξ [mm]

Ns/P

l = 1 mml = 2 mm

l = 5 mml = 10 mm

l = 40 mm

l = 150 mm

l = 300 mm

Soft Substrate

Hard Substrate

Figure 8: Distribution of the normalized longitudinal force in the stiffener, Ns, for twodifferent values of the substrate elasticity and different values of the bond length l. Foreach pairs of curves, the continuous line is for a typical concrete substrate, whereas andthe dashed line is for a substrate ten times softer.

18


when λ ≫ λ∗ (rigid substrate) and represents the ratio G/Gr as a functionof n for varying Ep/Es. Observe that when Ep/Es = 0.1 just a few terms aresufficient to obtain a good approximation, but when Ep/Es = 1000, morethan one thousands terms are necessary. This remark is useful to indicatea suitable value for n in the case of a typical concrete/FRP stiffness ratio.Since for this case Ep/Es ≃ 0.1 ÷ 0.2, one finds in figure 9 that the curveof interest lays between the curves Ep/Es = 0.1 and Ep/Es = 1, for whichn ≃ 100 can be considered appropriate.

0 500 1000 15000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Number of terms of Chebyshev series,

G /

Gr

Ep/E

s=0.1

Ep/E

s=1

Ep/E

s=10

Ep/E

s=100

Ep/E

s=1000

n

Figure 9: Case λ ≫ λ∗ (rigid substrate). Normalized strain energy release rate G/Gr as afunction of the number n of terms in the Chebyshev series for different values of the ratioEp/Es.

4. Comparison with experiments

Expression (3.32) allows to calculate the critical tensile load P = Pcr inthe stiffener as a function of the geometric and mechanical parameters, inparticular the fracture energy ΓF . In general, there may be two distinct fail-ure mechanisms: i) failure in the thin glue layer or ii) failure in neighboringlayer of the substrate. In the first case, ΓF represents the fracture energy ofthe glued interface, whereas in the second case it is the (mode II) fractureenergy of the substrate.

19


One of the most common applications certainly consists in the strength-ening of concrete with Carbon Fiber Reinforced Polymers (CFRP). In mostof the tests recorded in the technical literature, fracture occurs through theshearing of a thin concrete layer underneath the CFRP plate. Thus, onecan assume that ΓF is the concrete fracture energy, for which the relationproposed by Italian technical recommendations (CNR-DT/200, 2004), alsoaccepted at the European Community level, is of the form

ΓF =1

2sfκaκb

√fckfctm . (4.1)

Here sf is the maximum slip, associated with an assumed bilinear shear-stressvs. relative-slip constitutive relationship, usually taken equal to 0.2 mm; fckand fctm are the characteristic compression strength and the mean tensilestrength of concrete; κa is a value calculated on the basis of a statisticalanalysis of experimental data, for which 0.64 represents an average value;κb is a geometric parameter that depends upon the stiffener width bs andsubstrate width bp, that takes the form

κb =

√√√√ 2− bsbp

1 + bs400[mm]

≥ 1 , (4.2)

when bs/bp ≥ 0.33 (when bs/bp < 0.33, assume bs/bp = 0.33).In this study the results of a series of pull tests on CFRP-to-concrete

bonded joints collected from the existing literature are considered. The fun-damental problem is the evaluation of the critical load which can be trans-mitted to the reinforcement before debonding occurs.

Experimental evidence suggests that, in general, crack propagation dueto debonding occurs approximately at a constant load. The model predictsthis response in the case of “long” strips. In fact, when the parameter λ of(2.5) exceeds the threshold value λ∗ ≃ 101, Figure 6 shows that the energyrelease rate G is almost constant and equal to the value Gr of (3.33) for therigid support. The energetic balance (3.31) thus furnishes the value

Pcr,r =√2EsAsΓF bs = bs

√2EstsΓF , (4.3)

which coincides with the expression suggested by most technical standards.Debonding of the stiffener occurs when P ≃ Pcr,r = const. as long as λ ≫ λ∗,i.e., when the bonding length l is sufficiently high. When λ ≪ λ∗, one

20


understands from Figure 6 that the energy release rate becomes much higherthan Gr and consequently Pcr results much lower than Pcr,r.

In summary, “long” stiffeners progressively detach from the support, untilthe bond length becomes so small that equilibrium can only be attainedprovided that the pull out load P is decreased. This decay provokes anelastic release in that part of the stiffener that has already debonded fromthe substrate and is strained by P . The main consequence of this is that,after a plateau, pull out tests on long strip should exhibit a snap-back phase.

Most of the pull-out tests considered in the technical literature are strain-driven tests that cannot capture any snap back response. An exception is theexperimental campaign recently performed in the laboratories of the Univer-sity of Parma by Carrara et al. (2011), who used a closed-loop tensometer tocontrol the pull-out-force P from the output of LVDT transducers, placed atthe non-loaded end of the stringer, i.e., at point ξ = l in the scheme of Figure2. Among other tests, recorded in Carrara et al. (2011), concrete prisms of150 × 90 × 300 mm nominal size were reinforced by pultred CFRP plates30 mm wide and 1.3 mm thick. The measured mechanical properties of thematerials used in the tests are reported in Table 1.

Table 1: Mechanical properties of materials used for the tests of Carrara et al. (2011)

Concrete FRP Adhesive

Young’s Modulus, E [MPa] 28700 168500 3517.3

Poisson’s Ratio, ν 0.2 0.248 0.315

Tensile Strength, ft [MPa] 3.2 - 12.01

Average Compression Strength, fc [MPa] 37.2 - -

The results of the pull-out experiments are summarized in the graphsof Figure 10, reporting the load P as a function of ∆, i.e., the measureddisplacement at the point of application of P . What should be noticed hereis the marked snap-back response, which occurs approximately when ∆ =0.30÷ 0.35 mm.

21


−0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

2

4

6

8

10

12

14

16

18

20

Displacement, ∆ [mm]

Load

, P [k

N]

Test 150 A

Test 150 B

Test 150 C

Figure 10: Load P vs. displacement ∆ curves for the pull-out tests of Carrara et al.(2011). Initial bond length l = 150 mm.

In order to compare this results according to the prediction of the pro-posed model, parameter calibration has to be performed. The critical loadis evaluated through (3.32), where the Chebyshev coefficients Xs dependupon the parameter λ of (2.5). Material parameters are taken from Table 1.The geometry of the stiffener is known, but attention should be paid in theevaluation of bp. The proposed model is two-dimensional and consequentlyis accurate only when bp/bs ≃ 1. For the case at hand bp/bs = 5 and thehypothesis of plate in generalized plane stress is questionable. A technicalsolution can be found through the following argument. Recalling from Fig-ure 6 that the decrease of load P occurs at λ = λ∗ ≃ 101, one can measurefrom experiments (Carrara et al., 2011) what is the bond length l∗ that isassociated with the beginning of the decay of the tensile strength. By using(2.5), the effective width b∗p can be evaluated as

b∗p =π

2λ∗Estsbs

Ep(l∗)= α∗ bs. (4.4)

For the experiments of Figure 10 the value l∗ ≃ 60 mm has been measured(Carrara et al., 2011), from which α∗ ≃ 2.0 and b∗p ≃= 60 mm.

The results are shown in figure 11, which represents the experimental forcevs. displacement curves juxtaposed with that obtained through the model.

22


There is a good estimate of the plateau associated with stable debonding.Moreover, the model can also predict the snap-back phenomenon: that partof the CFRP stiffener already detached from the substrate is strained by theapplied load that, when released, causes its contraction. In the theoreticalcurve, the bond length calculated through the model are evidenced by la-beled dots: bigger circles are at multiples of 10 mm, whereas smaller dotsare for lengths multiple of 1 mm. Notice that material softening starts ap-proximately in the fourth quarter of the plateau, when the bond strength isabout 60 mm, even if the decay is just appreciable at the scale of resolution ofthe graph. Remarkably, when the snap-back branch starts, the bond lengthrapidly diminishes. This is a phase governed by an abrupt phenomenon,whose experimental evaluation needs appropriate feed-back controls. It mustalso be mentioned that the value of the fracture energy ΓF that has been usedin the relevant expressions is that obtained by integrating the P −∆ curvesof Figure 10, i.e., ΓF ≃ 0.57 N/mm. Such a value is much lower that thatobtainable with the expression (4.1), which would give ΓF = 0.77 N/mm.

−0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

2

4

6

8

10

12

14

16

18

20

Displacement, ∆ [mm]

Load

, P [k

N]

l = 150 mm l = 100 mm l = 10 mm

l = 9 mm

l = 3 mm

l = 1 mm

Analytical result

Test 150 A

Test 150 B

Test 150 C

Bond Length (10 mm interval)

Bond Length (1 mm interval)

Figure 11: Load vs. displacement curves: comparison between theoretical and experimen-tal (Carrara et al., 2011) results. In the theoretical curve, bond lengths l are evidencedby dots (bigger dots for l multiple of 10 mm, smaller dots for l multiple of 1 mm).

There are however some aspects that the model is not able to capture,such as the strain-hardening trend evidenced by the experimental data. Thisfinding may be ascribed to an increase in surface toughness as the crack

23


propagates, a phenomenon observed in quasi-brittle materials such as con-crete. Quasi brittle materials exhibit an extensive microcracking in a limitedarea, known as the fracture process zone. Whereas in ductile fracture of met-als the fracture process zone is negligible in size when compared to the nonlinear plastic-hardening zone, in a quasi-brittle material the process zone islarger than the plastic hardening zone. Microcracking affects the behaviorof the material and results in an apparent increase of toughness, describedthrough the well-known rising R-curve (crack Resistance curve). Fractureenergy cannot be considered constant with crack growth as in the case of aflat R-curve typical of ideally brittle materials (Anderson, 2005): then, thedriving force due to P must increase to maintain crack growth.

Another aspect is that the predicted slope of the snap-back branch is lowerthan the one measured through experiments. There is little uncertainty aboutthis, because the occurrence of the snap-back phase is associated with therelease of elastic strain energy in the stringer whose geometry and mechanicalproperties are perfectly known. In the theoretical model the final deformationof the stiffener tends to the null value, because no detachment is assumedfrom the substrate matrix; on the other hand, the experimental curves ofFigure 10 highlight a permanent displacement of the stiffener. Consequently,other phenomena such as residual cohesion, inelastic slip, or friction betweenthe detached surfaces, must be considered for a deeper characterization ofthe phenomenon.

5. Discussion and Conclusions

An analytical model has been presented for the description of the in-terfacial debonding failure of an elastic stiffener from a substrate, in viewto practical applications such as the characterization of reinforcements withFiber Reinforced Polymers (FRP). The contact problem is analyzed underthe hypotheses that the bending stiffness of the stringer is negligible andthe substrate is a linear elastic semi-infinite plate in generalized plane stress.Compatibility conditions for the relative displacement allow to obtain an in-tegral equation in terms of the tangential stresses (Grigolyuk and Tolkachev,1987). A solution with Chebyshev polynomials can then be used to estab-lish an energetic balance a la Griffith, providing the maximum transmissibleload. In order to determine the energy release rate, a generalization of theCrack Closure Integral Method developed by Irwin (Irwin, 1957) has beendetailed.

24


Results of the calculations show that the strain energy release rate stronglydepends upon the elasticity of the substrate, tending to the limit value for arigid substrate calculated by Taljsten (Taljsten, 1996) when the Young mod-ulus of the substrate, Ep, tends to ∞. In general, a soft substrate influencesthe fracture propagation process and, consequently, the diffusion of the load.The qualitative properties of the solution depend upon a coefficient λ, definedin (2.5), which represents a non-dimensional similarity parameter providinga synthesis of all those physical variables that influence the phenomenon,such as elastic moduli of stiffener and substrate, geometry and bond length.The substrate can be considered rigid when λ ≫ λ∗, where λ∗ is of the orderof 101. Clearly λ is directly proportional to the substrate modulus Ep, butremarkably λ also depends linearly upon the bond length l. Consequently,the substrate can be considered rigid when, left aside all the other materialproperties, the length of the stringer is sufficiently high.

In other words, “long” (“short”) stringers are those for which λ ≫ λ∗

(λ ≪ λ∗). In “long” stringers, the elasticity of the substrate does not influ-ence the strain energy release rate (case of rigid substrate), so that energeticbalance predicts a gradual detachment at approximately constant pull-outforce. In “short” stringers, the contribution from the substrate is important:the lower the bond length, the higher the strain energy release rate. Shortstringers thus exhibit a strain softening response.

In a load history when the relative displacement of the stringer is con-trolled in a closed loop testing machine, such as in the experiments of Carraraet al. (2011), the stringer gradually debonds from the substrate at approxi-mately constant load, until the bond length becomes so small that the equilib-rium load decreases. Release of strain energy in the elastic stringer results intypical load vs. displacement snap-back response, that has been experimen-tally verified. Results obtained through the model are in good quantitativeagreement with the experimental results of Carrara et al. (2011), providedthat the fracture energy considered in the formulas is the one experimentallymeasured through integration of load-displacement curve.

The model just presented may be considered minimal, because it onlyrelies upon an energetic Griffith balance for the description of the debondingphenomenon. One of the major drawbacks of this assumption is that thediffusion of the load from the stringer to the substrate only depends uponthe elasticity of the material: stress singularities occur at both ends of theadherent interface, so that it is difficult to give a consistent definition ofthe effective anchorage length. However, despite its simplicity, the model

25


is able to capture the maximum transmissible load, the progression of thedebonding phenomenon as well as the onset of a snap-back phase, remarkingthe important role played by the elasticity of the substrate, which is usuallyneglected in the practice. In order to provide a more accurate description,it would be necessary to slightly complicate the model, taking into accountfor the possibility of cohesive sliding before final detachment through theassumption of a proper shear-stress vs. slip constitute law at the interface.This is the subject of current work in progress.

Acknowledgements. The authors acknowledge the Italian MURST for partialsupport under the PRIN2008 program.

Appendix A. Appendix: Chebyshev Polynomials

The Chebyshev polynomials of the first kind are defined through thevariables

t = cos(φ) , φ = arccos(t) , (A.1)

by the relation (Abramowitz and Stegun, 1964)

Ts(t) = cos(sφ(t)) = cos(s arccos(t)) . (A.2)

The Chebyshev polynomial of the second kind are of the form

Us(t) =sin(s+ 1)φ(t)

sin(φ(t)). (A.3)

Both Ts and Us form a sequence of orthogonal polynomials. The polyno-mials of the first kind are orthogonal with respect to the weight 1/

√1− t2

on the interval [−1, 1], that is,

∫ 1

−1

Ts(t)Tm(t)√1− t2

dt =

0 , for m = s ,π2, for m = s = 0 ,

π , for m = s = 0 .

(A.4)

Similarly, the polynomials of the second kind are orthogonal with respectto the weight

√1− t2 on the interval [−1, 1], i.e.,∫ 1

−1

Us(t)Um(t)√1− t2dt =

{0 , for m = s ,π2, for m = s .

(A.5)

26


The following property is often useful:

∫ 1

−1

Ts(t)√1− t2(t− t0)

dt =

0 , for s = 0 and |t0| < 1 ,

πUs−1(t0) , for s > 0 and |t0| < 1 ,

−π(t0− |t0|

t0

√t20−1)s

|t0|t0

√t20−1

, for s ≥ 0 and |t0| > 1 .

(A.6)Another property of the Chebyshev polynomials is that, in the interval

−1 ≤ x ≤ 1, they attain the maximum and minimum values at the endpoints,given by

Ts(1) = 1 ,

Ts(−1) = (−1)s ,

Us(1) = s+ 1 ,

Us(−1) = (s+ 1)(−1)s .

(A.7)

This peculiarity is of help while estimating qualitative properties of the so-lution.

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