Minimum bond length and size effects in FRP–substrate bonded joints

Engineering Fracture Mechanics 76 (2009) 1957–1976

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Minimum bond length and size effects in FRP–substrate bonded joints Alessia Cottone, Giuseppe Giambanco * Department of Structural, Aerospace and Geotechnical Engineering, University of Palermo, Viale delle Scienze, 90128 Palermo, Italy

a r t i c l e

i n f o

Article history: Received 13 May 2008 Received in revised form 21 January 2009 Accepted 10 May 2009 Available online 18 May 2009 Keywords: Fibre reinforced plastics Bonded joints Interface

a b s t r a c t The load transfer mechanism between the fibre-reinforced polymer (FRP) materials and the substrate plays a crucial role in the overall response of retrofitted structural members. The FRP–support material interface can be studied by using pull tests in which a reinforcement plate is bonded to a prism and subjected to a tensile force. The experimental results obtained regard the assembly (FRP strip-support block), then a central problem is how to carry out the interface constitutive laws and the related parameters. The principal objective of the present paper is to contrive a procedure which, for a fixed interface constitutive law, permits to derive the interface mechanical parameters from the pull tests experimental data. This procedure has its basis on the size effect phenomena which the experimental tests evidence. Furthermore, the size effect laws show that the effective or minimum bond length coincides with the internal length of the interface and permits to distinguish small and large specimens in the sense of fracture mechanics.  2009 Elsevier Ltd. All rights reserved.

1. Introduction Nowadays a quite popular technique for structural strengthening is the application of external additional reinforcements made up of fibre reinforced polymers (FRP). This technology offers unique advantages with respect to the traditional strengthening techniques among which a good immunity to corrosion, low weight and excellent mechanical properties. Furthermore, the hand lay-up allows to adapt this reinforcement to the shape of any structural element. The load transfer mechanism between the reinforcement and the substrate material plays a crucial role in the reinforcement design and strongly influences the effectiveness and the durability of this strengthening technique. The FRP–support material interface constitutes for the reinforced structure a weakness zone where relevant stress concentrations develop leading to the debonding of the composite lamina from the substrate (delamination). The structure failure modes related to the delamination are generally of brittle type. Therefore, a good understanding of the FRP–substrate interface behaviour is an important requisite for a safe and apposite design of externally bonded FRP systems. In recent years, growing attention has been paid by researchers in structural mechanics to bonded joints in order to provide theoretical and numerical tools for better description of the interfacial bonding/debonding phenomena. The research efforts in this area regard the formulation of reliable bond–slip models based on experimental data coming from laboratory tests performed on small specimens [1–6]. Due to the fact that most failure modes occurs at the FRP–support material interface in presence of a pure shear stress state, the bond strength and other interesting engineering quantities are derived in laboratories by using pull tests in which the reinforcement plate is bonded to a prism and subjected to a tensile force. In this way, the adhesive joint is tested in pure shear mode (mode II in fracture mechanics jargon).

* Corresponding author. Tel.: +39 091 6568445; fax: +39 091 6568407. E-mail address: [email protected] (G. Giambanco). 0013-7944/$ - see front matter  2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2009.05.007

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A. Cottone, G. Giambanco / Engineering Fracture Mechanics 76 (2009) 1957–1976

Nomenclature

w we wi ue up n EI h D

s v Up a0 kp uE uF sI td tc G Gin Gc L b t N(x) t(x) KB

q q* qi qef FE FU Fmax

sn su

Helmholtz free energy density elastic energy density plastic component of the Helmholtz free energy density elastic displacement plastic displacement kinematic internal variable interface tangential elastic stiffness hardening/softening parameter intrinsic interface dissipation tangential traction static internal variable energetically conjugated to n plastic activation function adhesion of the virgin joint plastic multiplier elastic limit displacement elasto-plastic limit displacement interface softening modulus decohesion process starting time crack propagation starting time dissipated energy density energy density locked in the microstructure fracture energy bond length lamina or beam width lamina or beam thickness lamina or beam axial force interaction force per unit length beam axial stiffness decohesion length crack length maximum extension of the decohesion zone effective bond length elastic limit value of the applied force ultimate limit value of the applied force maximum value of the applied force nominal tangential traction at the interface ultimate tangential traction at the interface

Different set-ups have been proposed for the pull tests and a complete classification is reported in Chen et al. [7]. It is important to observe that, in the case of bonded joints, the interface constitutive parameters identification is complicated by the difficulty to carry out tests at the interface level. The available experimental results regard more a structure, formed by the adherends and the joint, rather than a material in a general sense. Thus, the local interface laws and the relative material parameters are commonly deduced from the pull tests by two different methodologies: by a direct way, from the axial strain of the FRP plate measured by closely spaced strain gages, or by an indirect way, from the load–displacement curve. By the first method it is possible to derive, in a particular location along the FRP–substrate material interface, the experimental bond–slip curve: the non-linear interface law is calibrated on the basis of this local result. By the second method the non-linear interface law is chosen a priori and the material parameters are calibrated in order to obtain the best fitting curve for the load–displacement diagram. Both methods present some drawbacks. The direct derivation of the bond–slip curve from the experimental axial strain of the bonded plate is strongly influenced by the onset of discrete cracks in the quasi-brittle substrate material which, generally, produces sudden variations of the measured strain. By the indirect method, the choice of the local bond–slip curve is absolutely arbitrary and it is obvious that rather different interface laws may lead to similar load–displacement curves. Therefore, the principal objective of the present study is to contrive a simple procedure which permits to derive the mechanical quantities characterizing any interface constitutive law from the results of the experimental pull tests. The objective has been attained in the ambit of the indirect method making use of a simple schematization of the structural problem already applied for the same [2] or similar studies [9,10]. In particular, assuming a specific non-linear softening response of the FRP–substrate material and analyzing the experimental data coming out from the laboratory tests, in a first step of the procedure the local value of the adhesion strength and the interface softening modulus are identified on the basis of the size

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1959

effect laws formulated along the analytical treatment of the structural problem. In the same step, it is possible to evaluate the bond length which classically allows to distinguish large specimens from the small specimens and it is proved that this length coincides with the effective bond length of the assembly [2,3]. In the second step, considering the elastic branch of the specimen’s response, the elastic stiffness of the interface is calibrated and consequently other important mechanical quantities can be calculated such as the interface fracture energy. The application to some cases whose results are reported in the literature shows the effectiveness of the procedure. The paper is organized as follows. In the next Section the general assumptions adopted for the study are presented and in Section 3 the elastic and post-elastic solutions of the problem are illustrated. In Section 4 the concept of effective bond length is discussed in detail and in Section 5 the size effect laws, discovered for the FRP–substrate material assembly, are described. In Section 6 the theoretical model is applied for the simulation of single shear pulling tests whose experimental results are reported in literature. Finally, in Section 7 the principal conclusions of the study are reported. 2. General assumptions With reference to the FRP–substrate material bonded joints, in Fig. 1 a possible physical schematization of the contact layer is illustrated. Two sub-layers can be distinguished. The first one is constituted by the thin film of adhesive which connects the lamina to the substrate. The latter corresponds to a thickness of consolidated material of the substrate whose depth depends on the porosity of the material and on the fluidity of the adhesive. As the tensile and shear strength of the adhesive and of the consolidate material are usually higher than the tensile and shear strength of the substrate, failure will normally occur in the substrate. In this case, a thin layer of the substrate remains on the FRP lamina. Bond failure at the interfaces between the consolidated material and the adhesive or the adhesive and the FRP lamina occurs if the bond surface is not properly prepared for FRP application process. Interlaminar failure is a secondary failure mode, that occurs after the bond fracture has started in the support, and, hence, does not usually determine the maximum transferable load. This mechanical problem can be studied making use of the zero-thickness interface concept for which two adherents, i.e. the composite lamina and the substrate material, come into contact through an interaction surface. In particular, for this mechanical problem, an elasto-plastic interface model is used to represent the adhesive layer and the consolidated substrate material assembly, Fig. 1. The following classical hypotheses are considered:  the strain state is uniform through the contact layer thickness derived from the relative displacements of the two adherents;  the continuity of contact tractions occurs at the interface;  the additive decomposition in the elastic and inelastic parts is assumed for the total relative displacement. The constitutive interface laws relate the contact tractions to the displacement discontinuities and the non-linear context requires that these relations must be expressed in rate form. Furthermore, in order to describe the evolution of interface mechanical state, some internal variables must be involved in the formulation. Since the objective of the paper is the study of the decohesion process in presence of a pure shear stress state, normal stresses and related friction effects are neglected in the formulation of the constitutive laws. Therefore, in the case of delamination in pure mode II, the following general form of the Helmholtz free energy density can be assumed:

wðue ; nÞ ¼ we ðue Þ þ wi ðnÞ;

ð1Þ

where we is the elastic component of the free energy (i.e. the elastic strain energy) and wi is the internal one stored at the interface and related to the changes of the interface internal properties. ue and n represent the elastic relative displacement in the tangential direction and the kinematic internal variable, respectively.

Fig. 1. Physical schematization of the contact layer.

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The two energy components in Eq. (1) have the following expressions:

1 EI u2e ; 2 1 wi ðnÞ ¼ h n2 ; 2 we ðue Þ ¼

ð2aÞ ð2bÞ

where EI is the interface tangential elastic stiffness and h the hardening/softening parameter (negative for softening behaviour). The second principle of thermodynamics, taking into account the balance equation (first principle), can be written as the Clausius–Duhem inequality, which, for an isothermal purely mechanical evolutive process, reads

D :¼ s u_  w_ P 0;

ð3Þ

where D is the interface dissipation density or net entropy production and s is the tangential traction. The inequality (3) represents the thermodynamic admissibility condition for the interface constitutive laws and in view of the definition (1) of the Helmholtz potential it can be expressed in the following form:

D¼



s

 @w @wi _ n P 0: u_ e þ s u_ p  @ue @n

ð4Þ

Since Eq. (4) has to hold for every incremental deformation process, including the purely elastic ones, from the position (2a) it follows that

s :¼

@we ¼ EI ue : @ue

ð5Þ

The other partial derivative appearing in Eq. (3) is used to define the proper static internal variable energetically conjugate to n, namely

v :¼

@wi ¼ hn: @n

ð6Þ

Eqs. (5) and (6) define the interface state equations: in particular the first one is directly related to the elastic interface behaviour, whereas the second one is the hardening/softening elasto-plastic state equation. With the positions (5) and (6), the intrinsic interface dissipation density assumes the form

D ¼ su_ p  vn_ P 0:

ð7Þ

The inelastic dissipative mechanism is driven by a simple linear activation function, defined in the space of the static variables as follows:

Up ðs; vÞ ¼ jsj  ða0 þ vÞ 6 0;

ð8Þ

where a0 is the initial adhesion or the adhesion of the virgin joint. Assuming that the model belongs to the class of the generalized standard materials, the complete set of evolutive constitutive relations can be derived by a specific maximum dissipation theorem, i.e. by maximization of the functional (7) under the admissibility condition (8). Adopting the Lagrange multiplier method, this maximization problem is equivalent to the following unconstrained stationarity problem:

min max Lðs; vÞ ¼ s u_ p  v n_  k_ p Up k_ p

s;v

ð9Þ

where k_ p is the plastic multiplier. The Kuhn–Tucker conditions of problem (11) provide the elasto-plastic evolutive laws

@L ¼0 ) @s @L ¼0 ) @v

u_ p ¼ k_ p sgn s;

ð10aÞ

n_ ¼ k_ p ;

ð10bÞ

with the following loading–unloading conditions:

Up 6 0;

k_ p P 0;

Up k_ p ¼ 0:

ð11Þ

The constitutive Eqs. (5), (6), (10) and (11) describe the interface response in pure mode II depicted in Fig. 2. For a monotonic displacement history (u P 0) the bilinear response can be expressed as

s ¼ EI u for 0 6 u 6 uE ; s ¼ sI ðuF  uÞ for uE 6 u 6 uF ; s ¼ 0 for u > uF ;

ð12Þ ð13Þ ð14Þ

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Fig. 2. Bilinear bond–slip one-dimensional model.

where sI is the interface softening modulus defined as

EI h : EI þ h

sI ¼ 

ð15Þ

Let ^ x be an interface point actually traversed by the decohesion and crack fronts during the loading process. The decohexÞ ! uðtd Þ ¼ uE and exhausts at tc ð^ xÞ ! uðtc Þ ¼ uF . The energy density G dissipated along the decosion process starts at td ð^ hesion process is

G¼

Z

t c ð^xÞ

_ _ nÞdt Dðu; ¼

t d ð^xÞ

Z

tc ð^xÞ

_ ðs u_  wÞdt:

ð16Þ

t d ð^xÞ

The above equation can be easily rewritten in the following form:

G¼

Z

uF

sdu þ we ðuE Þ þ wi ðuF Þ:

ð17Þ

uE

where the first and second term of the right-hand side of the equality (17) represent the specific surface fracture energy Gc, or the area under the su response illustrated in Fig. 2. Following Polizzotto [8], the third term can thought of to represent a surface energy Gin, locked in the microstructure, which is made available at the beginning of the decohesion process. Therefore,

G ¼ Gc þ Gin ;

ð18Þ

where

Gc ¼

Z

uF

sdu ¼ 

0

Gin ¼ wi ðuF Þ ¼ 

a20 ; h

a20 : h

ð19aÞ ð19bÞ

3. FRP lamina–substrate interaction In this section, the stress transfer mechanism between the FRP lamina and the substrate is analyzed for the case of the single shear pulling test. In particular, the solution of the mechanical problem is pursued by application of the bilinear interface constitutive model above presented. The theoretical study is conducted making use of a simple mechanical schematization in which the FRP lamina is considered as a beam or a plate on a deformable foundation. This schematization has been adopted in other works such as the study of the elastic response of the steel and CFRP plates bonded to concrete [2], the solution of the fibre pull-out problems considering the elasto-plastic cohesive law here used [9] and the mechanical description of the delamination phenomenon in composite materials accounting for frictional effects [10]. The beam has length L, coinciding with the bond length, and rectangular cross section (b  t) where b is the width and t is the thickness. The FRP material is indefinitely elastic and the support is modelled as a Winkler non-linear foundation whose constitutive laws are those previously described for the interface. Under these assumptions, the considered failure mode consists in the debonding of the lamina from the substrate material whereas the rupture of the reinforcement lamina is not admitted. The mechanical scheme adopted is represented in Fig. 3.

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Fig. 3. Mechanical scheme adopted for the simulation of the single shear pulling test.

Following [9], the elastic governing equations for the structural system in the range x 2 [0,L] studied are:  Beam indefinite equilibrium equation:

N0 ðxÞ  tðxÞ ¼ 0

ð20Þ

 Beam kinematic equation:

eðxÞ ¼ u0 ðxÞ

ð21Þ

 Beam constitutive equation:

NðxÞ ¼ K B eðxÞ

ð22Þ

 Interface constitutive equation:

tðxÞ ¼ K I ½uðxÞ  up ðxÞ

ð23Þ

where t(x) = s(x) b is the interaction force per unit length, N(x) the beam axial force, KB the beam axial stiffness and KI is the interface stiffness evaluated as KI = EI b.

3.1. Elastic behaviour The elastic response of the structural system is governed by the following second order differential equation, carried out on the basis of equilibrium, kinematic and elastic constitutive relations for the beam and for the interface

K B u00 ðxÞ  K I uðxÞ ¼ 0:

ð24Þ

Integration of Eq. (24) gives

uðxÞ ¼ C 1 eax þ C 2 eax ;

ð25Þ

where a has been posed equal to

sffiffiffiffiffiffi KI : a¼ KB

ð26Þ

The two constants C1 and C2 are evaluated requiring that N(0) = 0 and N(L) = F. Therefore, the beam displacement field is obtained

coshðaxÞ uðxÞ ¼ F pffiffiffiffiffiffiffiffiffiffi K I K B sinhðaLÞ

0 6 x 6 L:

ð27Þ

The structural system behaves elastically up to u(L) = uE, corresponding to the applied force

FE ¼

A0

a

tanhðaLÞ;

ð28Þ

where A0 ¼ a0 b. 3.2. Elasto-plastic behaviour Increasing the F value beyond the elastic limit FE, the decohesion process takes place propagating from the boundary of the beam where the load is applied. As shown in Fig. 4a, at a generic time t, the interface can be divided in two parts: the first one of length q where decohesion occurs and the remaining part whose behaviour is still elastic.

A. Cottone, G. Giambanco / Engineering Fracture Mechanics 76 (2009) 1957–1976

1963

(a)

(b) Fig. 4. Geometrical description of the elastic and elasto-plastic zones at the interface (a) and of the beam-joint crack propagation (b).

The governing relation for the decohesion zone is a second order differential equation having the following form:

u00 ðxÞ K B  SI ðuF  uðxÞÞ ¼ 0 L  q 6 x 6 L

ð29Þ

with SI = sI b. Integrating Eq. (29), the displacement field for the decohesion process zone is derived

uðxÞ ¼ C 3 eia^ x þ C 4 eia^ x þ uF

Lq6x6L

ð30Þ

where

sffiffiffiffiffiffi SI : a^ ¼ KB

ð31Þ

C3 and C4 are two constants that can be evaluated imposing the continuity of the displacements (25) and (30) in x = L  q, and using the boundary condition N(L) = F. Therefore, the final expressions of the displacement field for the two zones are

coshðaxÞ 0 6 x 6 L  q; cosh½aðL  qÞ ^ ðL  xÞ ^ ðL  x  qÞ A0 cos ½a F sin ½a  þ uF uðxÞ ¼  ^ q ^ q SI a^ K B cos½a cos½a uðxÞ ¼ uE

ð32Þ L  q 6 x 6 L:

ð33Þ

In order to derive the complete solution of the problem, the extension of the decohesion zone can be identified by imposing the continuity of the beam axial strain e(x) in x = L  q. This condition leads to a non-linear equation expressed in a nonalgebraic way. Alternatively, the extension of the decohesion zone q can be assigned and the continuity of the beam axial strain provides the corresponding value of the applied force

F¼

  ^ q A0 cos½a a^ ^ q þ tanh ½aðL  qÞ : tan½a a a^

ð34Þ

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Eqs. (32)–(34) give the complete solution of the lamina–substrate interaction for an assigned value of q during the decohesion process and before fracture initiation (uE < u(L) < uF). The fracture initiation in the joint occurs in correspondence of the beam end at tc(L) when u (L) = uF. Using this condition, from Eq. (33) the value of the applied force at the beam end at tc(L) is obtained

F ðt C ðLÞÞ ¼

A0

a^ sinða^ qÞ

ð35Þ

:

The value of the decohesion length q at tc (L) must satisfy the following equality, obtained substituting Eq. (35) in (34):

a ^ q: cot½a a^

tanh ½aðL  qÞ ¼

ð36Þ

As it is illustrated in Appendix A, the q value at tc (L) can be determined numerically from Eq. (36). Since the left-hand side of the above equation is always positive, the range of variation for the decohesion length is univocally defined

0 6 q 6 qi ¼

p ^ 2a

ð37Þ

:

Eq. (37) shows clearly that the maximum extension of the decohesion zone qi is independent on the beam length. Moreover, it can be considered as an internal length (or characteristic length) of the joint-beam assembly, determined by the elastic stiffness of the reinforcement and by the interface softening modulus. Along the elasto-plastic stage, firstly, the applied force increases from the limit elastic value FE up to the maximum value Fmax. Subsequently, a non-linear global softening response takes place from the Fmax value to F(tC(L)) value of Eq. (35). In correspondence of the Fmax the decohesion length must satisfy the following non-algebraic equation

tanh ½aðL  qÞ 

a^ ^ q ¼ 0 tan½a a

ð38Þ

Also in this case, the q value can be derived numerically as described in detail in Appendix A. 3.3. Crack propagation The above reported condition (36) relates the decohesion process zone extension to the time tc(L) in which the crack zone starts at the end of the beam and propagates modifying its extension q*. At the generic time t > tc(L) the interface presents three zones as illustrated in Fig. 4b. For the cracking zone the beam displacement field can be easily obtained considering that in this zone the strain state must be uniform, thus

uðxÞ ¼ C 5 x þ C 6

L  q 6 x 6 L:

ð39Þ

The displacement along the beam, therefore, can be obtained from Eqs. (25), (30) and (39) imposing the static boundary conditions N(0) = 0 and N(L) = F and the kinematic conditions at the transition points between the elastic and the elasto-plastic zones (u(L  q  q*) = uE) and the elasto-plastic and the cracking zones (u(L  q*) = uF). Once evaluated C5 and C6 constants, the displacement functions assume the following forms:

coshða xÞ 0 6 x 6 L  q  q ; cosh ½aðL  q  q Þ ^ ðL  q  xÞ A0 sin ½a L  q  q 6 x 6 L  q ; uðxÞ ¼ uF  ^ qÞ SI sin ða F uðxÞ ¼ uF þ ðx  ðL  q ÞÞ L  q 6 x 6 L: KB

uðxÞ ¼ uE

ð40Þ ð41Þ ð42Þ

In order to identify the extension of the three zones the continuity condition of the beam strain e(x) must be imposed for x = L  q  q* and for x = L  q*. Therefore, making use of Eqs. (40)–(42) for the beam strain evaluation, the lengths q and q* are obtained

1



A



q ¼ ^ Arcsin ^ 0 ; a aF

ð43Þ 0

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 1 a F A2 q ¼ L  q  Arctanh@ 1  0 2 A: A0 a a^ 2 F

ð44Þ

Eqs. (40)–(44) provide the complete solution for the beam-joint assembly for t > tc(L), when the crack propagates towards the interface.

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In particular, also in this stage, if the extension of the decohesion length is assigned, Eqs. (43) and (44) provide the value of the applied force F and the crack length q*, respectively. The full response of the beam and of the interface is given by Eqs. (40)–(42) for the identified three regions. Finally, a particular condition occurs when the decohesion length reaches the limit value qi. In this case, the applied force and the crack length assume the following values

F¼

A0 a^

q þ qi ¼ L:

ð45Þ

The last condition indicates that the elastic zone disappears and in the interface only two regions can be distinguished: the elasto-plastic part qi long and the cracked part having length q* = L  qi.

4. Effective bond length and shear pulling test response This section concerns with the study of the shear pulling test response for the three cases in which the length of the bonded strip is less, equal or greater than the internal length qi. 4.1. Case I: L < qi The typical response of the beam-interface assembly when the beam length is less than the internal length qi is depicted in Fig. 5a. The response is linear elastic until the pulling force reaches the elastic limit FE, expressed by Eq. (28) and corresponding to the displacement uE of the beam end. Beyond the elastic limit value (point E), three different stages can be distinguished. In the first stage (branch E–M) the response is non-linear and regulated by Eqs. (32) and (33); the pulling force increases together with the extension q of the decohesion zone. The maximum value of the applied force is reached when condition (38) is satisfied. The second stage (branch M–U) is the non-linear softening response which terminates when the elasto-plastic zone covers the entire bond length (q = L). At the point U the applied force and the beam end displacement can be evaluated from Eqs. (33) and (34)

^ LÞ A0 sinða ; ^ a A0 ^ LÞ: cosða uðLÞU ¼ uF  SI

FU ¼

ð46Þ ð47Þ

Finally, in the third stage the behaviour of the assembly is fully elasto-plastic; the differential governing equation is the (29) with the integration constants evaluated on the basis of the boundary conditions u0 ð0Þ ¼ 0; u0 ðLÞ ¼ F=K B , thus

uðxÞ ¼ uF 

F ^ xÞ: cosða ^ sin ða ^ LÞ KBa

ð48Þ

By the above equation it is easy to verify that in this branch the overall softening response of the assembly is linear and the limit value of the displacement is equal to uF. Two interesting observations can be made for this case. The first one is that the mechanical response of the system, after the maximum force value is reached, shows an unstable behaviour under a driving force process, whereas the behaviour is stable in presence of a displacement controlled test. The second observation regards the branch U-F which does not correspond to a crack propagation phase along the interface, since softening evolves with different velocities leading to the simultaneous debonding condition at all interface points: u(x) = uF, "x 2 [0, L] at tc(L). 4.2. Case II: L = qi The typical beam-interface response for this case is illustrated in Fig. 5b. As in the previous case, the bonded joint behaves elastically until the FE value of the applied force is reached. The propagation of the decohesion zone, at the beginning, coincides with the non-linear increasing of the applied force (branch E–M); afterwards, a non-linear softening response takes place terminating at the point U, where the length of the decohesion zone equals the beam length, q = qi = L at t = tc(L). Obviously, at the inner free end of the beam, the discontinuous displacement takes the elastic limit value, u(0) = uE; making use of this condition and imposing that u0 (0) = 0, from Eq. (30) the beam displacements are derived:

uðxÞ ¼ uF 

A0 ^ xÞ: cosða SI

ð49Þ

The value of the applied force at point U is:

F U ¼ K B u0 ðLÞ ¼

A0 : a^

ð50Þ

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(a) FU

(b) U

(c) U

Fig. 5. Beam-interface response for different bond length.

At the generic time t > tc(L), the fracture propagation along the joint occurs so that the whole joint is divided into two parts: a decohesion zone of length q < qi and a cracked zone q* long. The displacement fields for the two zones are

A. Cottone, G. Giambanco / Engineering Fracture Mechanics 76 (2009) 1957–1976

1967

(a)

(b)

Fig. 6. Beam displacement field for different values of the applied force in the range [FU, 0]: (a) L = qi and (b) L > qi.

expressed by Eqs. (30) and (39). With reference to the decohesion zone, the unknown constants can be evaluated considering 0 that the beam strain is zero at the inner free end, i.e. u (0) = 0, and that at the crack tip the displacement value is * u(qi  q ) = uF. The above boundary conditions lead to the relevant result that the cracking zone can not propagate and the beam external end is fixed:

q ¼ 0; uðqi Þ ¼ uF for t > tc ðLÞ:

ð51Þ

The displacement field depends only by the applied force and is given by the following equation:

uðxÞ ¼ uF 

F

a^ K B

^ xÞ cosða

ð52Þ

with F varying in the range [FU, 0]. The force–displacement response is represented by the vertical line U–F in Fig. 5b. In Fig. 6a the beam displacement field for different values of the applied force in the range [FU, 0] is illustrated.

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4.3. Case III: L > qi The beam-interface response is elastic until the applied force reaches the elastic limit value FE. Afterwards, the decohesion zone propagates from the outer border of the interface and the elasto-plastic response of the assembly is represented by the E–M–P branches depicted in Fig. 5c. In this stage, assigned the extension of the decohesion zone q, from Eqs. (33) and (34) the displacement of the beam and the applied force are derived, respectively. At point M the maximum value of the applied force is reached and Eq. (38) is verified. The elasto-plastic stage terminates at point P, where u(L) = uF, and the cracking zone starts to develop. The branch P–U, corresponding to crack propagation along the interface, can be easily calculated imposing the extension of the decohesion zone q and deriving the applied force F and the extension of the cracking zone q* from Eqs. (43) and (44), respectively. The displacement field of the beam is given by Eqs. (40)–(42). At point U, the extension of the decohesion zone reaches the limit value qi and, as it has been demonstrated in Section 3.3, the elastic zone disappears, so that the interface remains divided into two parts: the elasto-plastic one and the cracked one (q* + qi = L). It can be noticed that the condition reached at point U is identical to the situation described in the previous subsection (case II) when t = tc(L). Therefore, at this point the crack can not propagate and the beam cross section posed at the boundary between the elasto-plastic zone and the cracking zone remain fixed. In the elasto-plastic zone, the loss of cohesion results in a decreasing of the tangential stresses resultant and, for the equilibrium condition along the axial beam direction, the applied force F decreases too. Consequently, for that part of beam with the cracked joint, linear elastic unloading occurs which, at the assembly level, corresponds to the snap-back branch U–F depicted in Fig. 5c. In Fig. 6b the beam displacement field for different values of the applied force in the range [FU, 0] is illustrated. At point U the maximum value of the displacement u(L) is reached and the value can be evaluated from Eq. (42), assuming for F the value FU

uU ¼ uF þ

A0

a^ K B

ðL  qi Þ:

ð53Þ

Therefore, under the same mechanical properties of the beam and of the interface, the maximum displacement for the assembly depends only on the difference between the increment of the beam length and the characteristic length qi. 4.4. Remarks The internal length of the FRP–substrate assembly plays a crucial role in the mechanical response of the single shear pulling test. In particular, if the FRP lamina has a length less than the internal length of the assembly, the whole mechanical response in terms of the force applied versus the displacement of the beam end can be easily obtained with a displacement controlled experiment. If the FRP lamina has length equal or greater than the internal length, a displacement controlled test provides the mechanical response up to a residual value of the applied force FU. In fact, for L = qi at point U the applied force instantaneously drops to zero, showing a global pure brittle response. For L > qi at point U a snap-back path occurs describing a global unstable response. It should be noted that the unstable response is not related to an unstable crack, where for unstable crack it is intended a crack which propagates spontaneously under constant load and imposed boundary displacement [8]. On the contrary, the unstable behaviour of the FRP lamina corresponds to a fixed position of the crack tip and to the elastic unloading of the part of the lamina located between the crack tip and the external beam boundary where the force is applied. In all these three cases studied, the debonding of the lamina from the substrate is due to the progressive decohesion in the plasticized interface where all points, with different velocities, attain the limit discontinuous displacement uF simultaneously. With reference to the maximum value of the applied displacement, the results obtained above show that only for FRP lamina length greater than qi the limit value of u(L) = uF can be exceeded and, in this case, the maximum displacement depends on the difference between the length of the beam and the internal length. Another important aspect of the pull tests on FRP–substrate bonded joints is the existence of a sort of effective bond length qef, beyond which the increment of the bond length does not produce an increment of the ultimate load. On the other hand, an increment of the bond length provides an increment of the ductility of the system, in the sense that an increasing displacement is measured in correspondence of the ultimate load. With reference to the theoretical study presented above, if for ultimate load is intended the load reached at point U of the load–displacement curve, it can be asserted that the effective bond length qef corresponds to the internal length qi. In fact, the ultimate load is

^ LÞ A0 sinða for L < qi ; a^ A0 for L P qi : FU ¼ a^

FU ¼

ð54Þ ð55Þ

A. Cottone, G. Giambanco / Engineering Fracture Mechanics 76 (2009) 1957–1976

1969

5. Size effect In presence of strain-softening materials, the structure strength, defined as the value of a certain nominal stress at the ultimate load, shows a dependence on the size of the structural element [11,12]. Geometrically similar structures of different sizes, subjected to the same kind of experimental test, provide different values of the structural strength, which is also different from the local strength of the quasi-brittle material. In the case of the single shear pulling test of the FRP plate bonded to a substrate material, the nominal tangential traction at the interface can be defined as the applied load divided by the bonded area:

sn ¼

F bL

ð56Þ

The bond strength expression is derived from Eq. (56) substituting to F the ultimate load reported in Eqs. (46) and (50), thus

F U a0 ^ LÞ for 0 6 L 6 qef ; sinða ¼ ^L bL a F U a0 ¼ ¼ for L > qef : ^L bL a

su ¼

ð57Þ

su

ð58Þ

The previous equations can be regarded as the size effect laws for the structure constituted by the FRP lamina and the interface assembly. Following the standard practice, in Fig. 7 the logarithm of the interface strength as a function of the logarithm of the bond length is shown. The size effect behaviour is described by a transitional curve, expressed by Eq. (57), and by the straight line of Eq. (58) having slope 1/1. For small specimens (0 6 L 6 qef), the interface strength varies in the range a0 6 su 6 2 a0/p. Therefore, the upper bound of the interface strength is represented by the interface adhesion provided by the plasticity theory adopted. For large specimens (L > qef), the interface strength varies linearly as the inverse of the length L. The derivation of the size effect laws has interesting implications in the identification of some mechanical properties of the FRP plate–substrate material assembly from the experimental data carried out during single shear pulling test performed on specimens exhibiting different bond lengths. In particular, the structural strengths calculated by Eq. (56) assuming F = FU (where FU are experimental values referred to specimens with different bond lengths), are reported in ^ can be calibrated in order to fit the bilogarithmic plane (Fig. 7) and the sub-optimal values of the parameters a0 and a the experimental points by the size effect laws (57) and (58). The effective bond length can be obtained from the sub-opti^ making use of Eq. (37). Finally, the identification of the interface stiffness can be done either fitting the mal value of a experimental elastic branch of the load–displacement response or, alternatively, by the axial strain of the FRP plate at a load level less than elastic limit value FE. The great advantage of the parameters identification procedure proposed on the basis of the size effect laws is that the interface adhesion and the effective bond length, two very important quantities for practical engineering purposes, can be directly derived from the ultimate values of the pull force, registered during displacement controlled single shear pulling tests performed on specimens with different bond lengths. This procedure is applied in the following section and the analytical approach presented is validated by comparison of numerical and experimental data.

Fig. 7. Size effect laws represented in the bilogarithmic plane Log (su)–Log (L).

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6. Application examples The FRP–substrate bonded joint can be experimentally tested making use of different set-ups which can be referred to as pulling tests, because the composite lamina used in these tests is always subjected to a pulling force. The stress state at the interface can vary with the test set-up used; for instance, the contact tractions can be characterized not only by the tangential stresses but also by normal stresses, which if positive (i.e. tensile stresses) take part to the debonding process, if negative (i.e. compressive stresses) contribute to the residual strength of the interface due to frictional effects. Therefore, different test set-ups can lead to significantly different experimental results. The near end supported single-shear test seems to be the most effective test method to obtain results on the FRP–substrate interaction for its simple applicability and easy interpretation. Furthermore, making use of an appropriate test apparatus, capable of minimizing the load offset, it is possible to derive the joint response in pure mode II in the sense of fracture mechanics. The applications herein presented refer to this kind of test, whose a huge number of results is reported in literature. Numerical calculations have been carried out on two case-studies. The first one regards CFRP strips bonded to concrete blocks and the experimental data and results are reported in the paper of Yao et al. [13]. The latter case-study regards a graphite/epoxy composite material plate adhered to concrete whose results are reported in Chajes et al. [14]. In both cases the experimental results regard different bond lengths and the identification of material parameters is pursued by the approach proposed above. 6.1. Example 1 An extensive experimental program, regarding FRP strips bonded to concrete blocks, has been carried out by Yao et al. [13]. They prepared 72 specimens for the application of the single-shear test and the results, coming from the test on the Series VII of these specimens, are used here to perform the numerical analyses and validate the parameter identification procedure. The specimens of Series VII present different bond lengths, the load offset is zero and the failure mode obtained is homogeneous, represented by cracking of concrete few millimeters from the adhesive-concrete separation surface.

Fig. 8. Yao et al. [13], geometrical scheme of the single shear pulling test.

Table 1 Yao et al. [13], geometrical and elastic properties of the CFRP plates bonded to the prism. Type

t (mm)

b (mm)

EB (GPa)

CFRP

0.165

25

256

Table 2 Yao et al. [13], constitutive parameters values obtained by the proposed identification procedure. a0 (N/mm2)

a^ (1/mm)

sI (N/mm3)

qef (mm)

4.2

0.015

9.50

104.72

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Fig. 9. Yao et al. [13], experimental tests-size effect laws.

Table 3 Yao et al. [13], constitutive parameters adopted for the simulations of the single shear pulling tests. Series

L (mm)

EI (N/mm3)

h (N/mm3)

uE (mm)

uF (mm)

Gc (N/mm)

VII-2 VII-3 VII-6 VII-7

95 145 190 240

48 36 160 40

7.93 7.51 8.96 7.67

0.0875 0.116 0.026 0.105

0.53 0.56 0.47 0.55

1.11 1.17 0.98 1.15

The concrete block has cross section of dimensions 150  150 mm and length of 350 mm, Fig. 8. The CFRP plates were bonded to the upper surface of the prism with anchor lengths varying from 95 mm to 240 mm. The geometrical and elastic properties of the CFRP plates, adopted for the numerical application, are illustrated in Table 1. In the first step of the interface parameters identification, the experimental structure strengths, calculated from the ultimate value of the applied load for the bond lengths equal to 95, 145, 190 and 204 mm, have been placed into the bilogarith^ have been calibrated in order to fit the experimental points by using the size effect mic plane. The two parameters a0 and a laws (57), (58). The sub-optimal parameters value and the effective bond length calculated through expression (37) are reported in Table 2. In Fig. 9 the size effect laws are depicted together with the experimental data. In the second step the interface elastic stiffness has been evaluated fitting the elastic branches of the load–displacement diagrams. The results are reported in Table 3. Three specimens show similar values of the interface stiffness and the specimen VII-6 has a value approximately four times greater. This is probably related to a different procedure adopted for the hand lay-up. In the same Table 3 other interface mechanical quantities are listed such as the softening modulus, the limit displacements and the fracture energy calculated through Eqs. (15), (12), (13) and (19a), respectively. In Fig. 10 the load–displacement curves for the different values of the bond length, obtained by the analytical model, are illustrated and compared with the experimental data. A general good agreement between the experimental response and model predictions can be noticed in terms of elastic behaviour, maximum transferable load and ultimate force. ^ leads to the evaluation of the sub-optimal It should be observed that the identification of the constitutive parameter a value for the effective bond length of the assembly, posed equal to 104.72 mm. Therefore, the experimental test, in which the bond length is 95 mm, is relative to a short plate (L < qef) and the numerical response beyond the ultimate load is of softening type. The other tests regard long bonded plates (L > qef) and the numerical responses exhibit snap-back branches beyond the ultimate load. 6.2. Example 2 Chajes et al. [14] conducted single shear-pulling tests on FRP plates bonded to concrete blocks with the specific intent to investigate the effects of the surface preparation, type of adhesive and concrete strength on average bond strength.

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Fig. 10. Load vs. displacement experimental and numerical plots for the Yao et al. [13], series.

Fig. 11. Chajes et al. [14], geometrical scheme of the single lap shear tests.

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The specimens consist of a 25.4 mm wide FRP (graphite/epoxy) plates bonded to a concrete block (152 mm wide  152.4 mm high  228.6 mm long), with different anchor lengths varying from 50.8 mm (2 in.) to 203.2 mm (8 in.), Fig. 11. The geometrical and elastic properties of the graphite/epoxy plates, adopted for the numerical application are showed in Table 4. As in the previous example, the structure strengths, calculated from the ultimate value of the applied load for different bond lengths (50.8, 76.2, 101.6, 152.4 and 203.2 mm) have been placed into the bilogarithmic plane (Fig. 12) and the ^ of Table 5. size effect curves, which better fit the experimental points, have been obtained with the value of a0 and a The computed effective bond length is 103.4 mm (4.07 in.) which is very close to the value estimated by Chajes et al. (3.75 in.). A single value of the interface stiffness has been calibrated fitting the experimental elastic branches of the microstrain-distance along the bonded plate diagrams. The interface elastic stiffness and the others mechanical quantities, which characterize the bonded joint, are reported in Table 6. The numerical microstrains developed in the bonded FRP plate versus the distance along the plate, for various percentages (20%, 40%, 60%, 80% and 100%) of the ultimate load and for two values of the bond length, 152.4 and 203.2 mm, are illustrated and compared with the experimental responses in Fig. 13.

Table 4 Chajes et al. [14], geometrical and elastic properties of the graphite/epoxy plate bonded to the concrete block. Type

t (mm)

b (mm)

EB (GPa)

graphite/epoxy

1.016

25.4

108.4

Fig. 12. Chajes et al. [14], experimental tests-size effect laws.

Table 5 Chajes et al. [14], constitutive parameters values obtained by the proposed identification procedure. a0 (N/mm2)

a^ (1/mm)

sI (N/mm3)

qef (mm)

7.1

0.01518

25.4

103.4

Table 6 Chajes et al. [14], constitutive parameters adopted for the simulations of the single lap shear tests. EI (N/mm3)

h (N/mm3)

uE (mm)

uF (mm)

Gc (N/mm)

492.13

24.15

0.0144

0.294

1.04

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A. Cottone, G. Giambanco / Engineering Fracture Mechanics 76 (2009) 1957–1976

Fig. 13. Microstrain developed in the plate vs distance along the bonded plate experimental and numerical plots.

7. Conclusions The strengthening technique of structural members making use of external bonding of composite FRP materials is quite usual in civil engineering and a number of research works, which investigate the load transfer mechanism between the reinforcement and the substrate material, are reported in literature. These works treat the mechanical problem with an experimental and/or theoretical point of view. The present paper belongs to the class of the theoretical treatments of the reinforcement–substrate material interaction making use of the interface concept. The attention is pointed out to the debonding of the reinforcement from the substrate material in presence of a pure tangential stress state. This failure mode is often experienced by concrete and masonry elements externally reinforced by composite laminae and can be easily reproduced in laboratory, making use of the single shear pulling test, in which the lamina is bonded to a prism of the material under observation and subjected to a tensile force. The delamination in pure mode II has been studied considering the composite reinforcement as a beam connected to a rigid substrate through an interface, whose constitutive laws are derived from elasto-plasticity theory with softening. The theoretical treatment considers the pure elastic response of the FRP–substrate assembly, the evolution of the elasto-plastic zone and the fracture initiation and propagation at the interface. Some interesting considerations come out from the theoretical study.

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Analyzing the elasto-plastic behaviour of the assembly, for any FRP bond length dimension, it is possible to affirm the existence of a maximum extension of the elasto-plastic zone at the interface, which is independent on the geometrical features of the structural system and is only function of the reinforcement elastic parameter and of the interface softening modulus. This maximum extension can be considered as a sort of internal length of the reinforcement-interface system. It has been proved that this internal length coincides with the effective bond length beyond which the increment of the bond length does not produce any further increment of the ultimate load. Moreover, the internal length of the assembly plays a crucial role in the mechanical response of the single shear pulling test. If the actual bond length is less than the internal length, a displacement controlled test can capture the entire load–displacement response characterized by an overall softening branch beyond the ultimate load. For the case of an actual length greater or equal than the internal length, after the ultimate load attainment, the load–displacement diagram shows a pure brittle response (L = qi) or a snap-back branch (L > qi). In both cases, the instability phenomena are caused by the circumstance that the cracking zone can not propagate. At this time the elasto-plastic zone reaches the maximum extension and meanwhile the elastic zone disappears. Afterwards, with the crack tip fixed, the decohesion phenomenon evolves at different velocities along the points of the elasto-plastic zone up to the complete simultaneous debonding of the reinforcement from the substrate. Another important aspect of this study is the size effect evidenced by the analytical treatment of the problem. By the definition of the structural strength for the FRP–substrate material assembly, it has been possible to derive the size effect laws which are useful in the identification of the local interface constitutive parameters and other engineering quantities. The identification parameters procedure has been applied to some experimental tests whose results are reported in literature. With the sub-optimal values of the parameters, the proposed model has been used to simulate the experimental test responses. The good agreement between numerical results and experimental ones validates the effectiveness both of the proposed approach to the parameters identification and of the analytical model. Obviously, the same study can be followed in presence of the different forms of the bond–slip interface constitutive law and the obtained results compared to define assets and drawbacks. Furthermore, the mechanical model can be extended to the case of coexistence of normal and tangential tractions at the interface in order to investigate the effect of the peel stresses on the decohesion process or frictional effects due to compressive normal tractions. The above developments constitute the objects of future papers.

Acknowledgements The authors acknowledge the financial support given by the Italian Ministry of Education University and Research (MIUR) under the PRIN07 Project 2007YZ3B24, ‘‘Multi-scale problems with complex interactions in Structural Engineering”.

Appendix A. Approximate solution of non-algebraic equations The approximate solution of Eq. (36) is carried out rewriting it in the following form:





ha i 1 ^ qÞ 1 þ aa^ cotg ða aðl  qÞ ¼ Arctgh ^ cotg ða^ qÞ ¼ Log ; ^ qÞ 2 a 1  aa^ cotg ða

ð59Þ

from which

"

#

a 1 þ e2aðlqÞ : a^ q ¼ Arctg ^ að1 þ e2aðlqÞ Þ

ð60Þ

Expanding in series the right-hand side of Eq. (60) about the value 1a^ Arctgðaa^ Þ, it is possible to obtain a second order equation in q

Aq2 þ Bq þ C ¼ 0;

ð61Þ

whose coefficients are:



^ ða ^ 2 þ a2 Þ sinh 2a l  a1^ Arctan aa^ 2a3 a A¼ 

2 ; ^ 2 Þ þ ða ^ 2 þ a2 Þ cosh 2a l  a1^ Arctan aa^ ða2  a ^ B¼a

ð62Þ





^ 2 þ a2 ÞArctan aa^ sinh 2a l  1a^ Arctan aa^ ^ 4a3 ða 2a2 a  1 a

þ  

2 ; ^ 2 Þ þ ða ^ 2 þ a2 Þ cosh 2a l  a^ Arctan a^ ða2  a ^ 2 Þ þ ða ^ 2 þ a2 Þ cosh 2a l  1^ Arctan a^ ða2  a a

a

ð63Þ

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A. Cottone, G. Giambanco / Engineering Fracture Mechanics 76 (2009) 1957–1976



ha

ai 2a2 Arctan aa^ a  1

þ 2 C ¼ Arctan coth al  Arctan ^ 2 Þ þ ða ^ 2 þ a2 Þ cosh 2a l  a^ Arctan aa^ a^ a^ a^ ða  a



^ 2 þ a2 ÞArctan aa^ 2 sinh 2a l  a1^ Arctan aa^ 2a3 ða    1 a

2 : a^ ða2  a^ 2 Þ þ ða^ 2 þ a2 Þ cosh 2a l  a^ Arctan a^

ð64Þ

In a similar way, Eq. (38) can rewritten in the following form:

" #   ^ qÞ 1 þ aa^ tgða a^ ^ 1 ; aðl  qÞ ¼ Arctgh tgðaqÞ ¼ Log ^ qÞ a 2 1  aa^ tgða

ð65Þ

from which

"

# a^ 1 þ e2aðlqÞ : a^ q ¼ Arctg að1 þ e2aðlqÞ Þ Expanding in series the right-hand side of the Eq. (63) about the value obtained

ð66Þ

, the second order equation in q is

1 a^ a^ Arctg a

Dq2 þ Eq þ F ¼ 0;

ð67Þ

whose coefficients are

D¼



^ ða ^ 2 þ a2 Þ sinh 2a l  1a Arccot aa^ 2a3 a 

2 ; ^ 2 þ a2 Þ cosh 2a l  a1 Arccot aa^ ^ 2  a2 Þ þ ða ða





^ ða ^ 2 þ a2 ÞArccot aa^ sinh 2a l  1a Arccot aa^ ^ 4a2 a 2a2 a  1 a^

  

2 ; ^ 2  a2 Þ þ ða ^ 2 þ a2 Þ cosh 2a l  a Arccot a ða ^ 2 þ a2 Þ cosh 2a l  a1 Arccot aa^ ^ 2  a2 Þ þ ða ða

    ^ Arccot aa^ 2aa a^ a^  1

F ¼ Arccot coth al  Arccot  2 ^ 2 þ a2 Þ cosh 2a l  a Arccot aa^ ^  a2 Þ þ ða a a ða 2 

^ ða ^ 2 þ a2 ÞArccot aa^ sinh 2a l  a1 Arccot aa^ 2aa þ 

2 : ^ 2 þ a2 Þ cosh 2a l  1a Arccot aa^ ^ 2  a 2 Þ þ ða ða ^þ E¼a

ð68Þ ð69Þ

ð70Þ

The q value is carried out from Eq. (67). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

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