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________

* Corresponding author. Present address: EGIS Industries, 4 rue Dolors Ibarruri, 93188 - Montreuil France.

Tel.: +33 6 19 71 33 14. Fax: +33 1 73 13 19 00. E-mail: gianluca.ruocci@egis.fr (G. Ruocci)

An improved damage modelling to deal with the variability of

fracture mechanisms in FRP reinforced concrete structures

Gianluca Ruocci1,*

, Pierre Argoul2, Karim Benzarti

1, Francesco Freddi

3

1Universit Paris Est, IFSTTAR, MAST, 14-20 Boulevard Newton, F-77447 Marne la Valle Cedex 2, France

2Universit Paris Est, Laboratoire Navier (UMR 8205), CNRS, ENPC, IFSTTAR, Ecole des Ponts ParisTech, 6&8

Avenue Blaise Pascal, F-77455 Marne la Valle Cedex 2, France

3Universit di Parma, Dept. of Civil-Environmental Engineering and Architecture, Viale Usberti 181/A, I-43100

Parma, Italy

ABSTRACT A new way of modelling is developed and proposed to predict different damage scenarios of

concrete elements strengthened by externally bonded Fibre Reinforced Polymer (FRP) plates. The bonded assembly

is modelled as a three-domain system with concrete, glue and FRP reinforcement assumed as damageable materials

being connected together by two interfaces. Interaction between domain and interface damage is introduced.

Detachment between FRP reinforcement and concrete in a single lap shear test configuration is analysed by

implementing the equations governing the damage model obtained in a finite element code. The damage evolution is

characterised through various indexes, which makes it possible to discriminate the failure mechanism when varying

properties of the glue or interfacial characteristics. Comparison between simulations and experimental tests shows

the accuracy of the damage model prediction and its capability to detect different failure modes; in particular, this

new modelling approach allows distinguishing between an adhesive failure at a glue-substrate interface and a

cohesive failure of the glue layer.

KEYWORDS:epoxy (A), concrete (B), fracture mechanics (C), durability (D), damage modelling.

1. IntroductionThe adhesive bonding of Fibre Reinforced Polymer (FRP) composites is one of the most effective

solutions for the rehabilitation of civil structures. This retrofit technique consists in the strengthening of

concrete structures by means of externally bonded composite plates or carbon fibre sheets. Nowadays,

both the repair of damaged structures and the upgrading of structurally deficient civil infrastructures are

carried out through the gluing of stiff external reinforcements. Despite the popularity of this promising

technology, some issues concerning the variability of the damaging behaviour and the durability of the

adhesive joint are still matters of concern.

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The long-term performances of the glued assembly may be critically affected by several

environmental factors. For instance, large cycles of variation or consistently high values of humidity and

temperature are known to possibly affect the mechanical properties of the glue, and/or weaken the

physico-chemical linkages between the polymer adhesive and the substrates [1-4]. In this particular

context, the effectiveness of the stress transfer and consequently, of the strengthening becomes

undermined. Depending on the degradation mechanisms involved, failure modes are characterised by

cohesive cracking, adhesive debonding or intermediary situations. The bearing capacity and durability of

FRP-strengthened concrete elements can thus be significantly reduced according to the activated failure

mechanism.

Therefore, a reliable retrofit design of concrete structures reinforced by bonded composites plates

requires a model capable of satisfactorily predicting the multiple failure phenomena due to the different

states that the assembly may experience. The purpose of this research is to investigate the failure

mechanisms of FRP-strengthened concrete elements through the damage modelling which considers both

the interfacial debonding and cohesive fracture. The proposed methodology relies on the model proposed

by Frmond et al. [5], which considers damage as the result of microscopic motions whose power is

introduced in the principle of virtual power. On the basis of this theoretical framework, Freddi et al. [6]

developed a model to describe the damage behaviour of bonded assemblies. Both the damage in the

volume of the adherents and the failure of the adhesive interface, as well as their interactions, are taken

into account in order to deal with different failure phenomena.

In addition, Benzarti et al. [7] adopted a simplified version of the former model, whose validity is

restricted to the case of rate independent problems, e.g. quasi-static fracture tests, and which involves a

reduced set of physical parameters and damage coefficients. The simplified formulation proved to

satisfactorily describe the detachment process (fracture propagation in 2D) for a single-lap shear test

carried out on a FRP-concrete assembly. However, only a single condition was analysed, that referred to

an unaltered state of the constitutive materials (in particular the bulk epoxy adhesive) characterised by

means of some standard mechanical tests.

In the present study, different sets of mechanical characteristics and damage coefficients are

considered in order to evaluate the variation in the failure mechanism due to an alteration of the

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constitutive material and/or interfacial properties. Unlike in former studies, the actual thickness of the

glue joint is considered in the damage modelling formulation. Indeed, the previously adopted interfacial

solution was unable to distinguish between the adhesive failure at the interface and the cohesive fracture

within the glue layer. Moreover, a further advantage of this new approach consists in the possibility to

differentiate the characteristics of the interfacial areas between the glue and specific substrates. As clearly

outlined in [810], the glue penetrates porous media with a small thickness acting as a reinforcement of

concrete and locally changing its characteristics.

The paper is organized as follows. In Section 2, the damage modelling is addressed and the basic

equations governing the problem are briefly reviewed. The simplified approach, assuming that the

adhesive bond is a simple contact surface, is first recalled to describe the theoretical framework. The

enhanced modelling of the glue layer is then presented, the computational differences with the former

model are pointed out and the advantages are discussed in details. Next, in Section 3, different indexes are

proposed to characterise the progression of damage in the assembly subjected to mechanical loading. A

numerical example of FRP-reinforced concrete specimen tested in a single shear configuration, as well as

the corresponding experimental tests, are presented in section 4 to investigate how an alteration of the

glue properties affects the fracture process. In addition, the numerical results obtained from the two

damage models are compared and discussed in the light of the experimental evidences. Finally, general

conclusions drawn from this study as regards to damage modelling in bonded assemblies are presented in

Section 5.

2. Damage ModellingIn the structural response of the FRP reinforcement of concrete elements a key role is played by the

adherence between concrete and composite material determined by the adhesive layer. Particularly, as

evidenced in Figure 1 the separation of the FRP from the support can take place in the concrete (failure

1), among concrete and the adhesive layer (failure 2) inside the glue (failure 3) or at the interface between

the glue and the reinforcement (failure 4).

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Figure 1. Possible failure modes in concrete substrate reinforced by FRP element.

Usually, the debonding process is modeled concentrating and limiting the nonlinear effect in a

fictitious zero thickness cohesive interface between the concrete and the reinforcement (see [11] for an

exhaustive review). This approach revealed to correctly reproduce the macroscopic evidence and the

structural response of cohesive failure mode located in the substrate but, other more complex fracture

processes cannot be exactly reproduced by this simplified approach. In fact, in order to describe the

variety of fracture processes both the materials (substrate, glue, reinforcement) and the interfaces

(substrate-glue, glue-reinforcement) have to be modeled in such a way that rupture may initiate and

develop in different locations.

Starting from these physical evidences we adopted a damage model that is capable to simulate all

these failure modes. Indeed, all involved materials are damageable and cohesion can be lost at the

interfaces. For clarity, a model with two materials and one interface will be presented first in order to

illustrate the main basis and after the formulation will be extended to the case of three materials and two

interfaces.

2.1 Two-domains damage model

The theoretical fundamentals of the damage model adopted here to simulate the detachment between

FRP and concrete are briefly reviewed in this section. For a more detailed summary, the reader can refer

to [57, 12]. The equations of motion result from the principle of virtual power. In this formulation, the

virtual power is assumed to depend on the strain rate, damage velocity and its gradient. For the sake of

simplicity, a quasi-static isothermal problem is considered. As such, acceleration forces can be set to zero

and thermal effects can be neglected.

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Let us consider a system made of two domainsi , i = 1, 2, subjected to mixed boundary conditions

and connected by a cohesive interface or contact surface denoted 21 =s . For the case at hand,

the two domains are the concrete specimen and the FRP reinforcement, and the zero thickness cohesive

interface is defined as the adhesive layer. For each material, the state quantities are the macroscopic

damage quantity i , its gradient igrad and the deformation tensor i . All of these variables depend on

time tand position x. The damage quantity can be interpreted as the volume fraction of active links or

undamaged material; thus its values vary between 0 and 1, where 1 represents the undamaged state and 0

corresponds to completely cracked zones. The gradient of i accounts for local interactions of damage at

a specific point with damage in the surrounding area. Similarly, the deformation i accounts for the local

interaction of displacement at a point with displacement in its neighbourhood.

Concerning the contact surface s , the state quantities are i) variables which describe the evolution of

the surface (or glue layer in the present case), such as the surface damage quantity s and its gradient

sgrad , the latter accounting for the local damage interaction on the surface, and ii) quantities which

describe the macroscopic and microscopic interactions between the domains and the contact surface. The

macroscopic mechanical interactions are described by the gap between the domains ( )21 uu , where iu

represents the displacement field of each domain i .

Combining the principle of virtual power with the constitutive laws leads to three sets of equations of

motion as described in Freddi et al. [6] and in Benzarti et al. [7]. The first one (1) is the classical equation

of motion in classical mechanics, while the others are non-standard differential equations that describe the

damage evolution in the domains (2) and at the interface (3):

0ii =+ fdiv , in i , (1)

( )i i i i ik F = , in i , (2)

( ) ( ) ( )2 1 ,1 1 ,2 2s s s s s s s s sk G k k = u -u , on s (3)

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where div and are the divergence and Laplace operator respectively, i i i i = C is the stress

tensor associated to the elasticity fourth order tensori

C , while if represents the volume exterior force.

The function ( )ii

F is the failure criterion that governs the rupture mechanisms in the domain and whose

expression can be specified for each material. Analogously, the function ( )12 u-usG is the failure

criterion for the interface.

The coefficient ik in eq. (2) is the damage extension parameter, which controls the size of the

transition zone between sound and damaged material. Increasing the value of ik leads to transitions from

concentrated damage (fracture) to widespread damage in the whole domain. The parameters

k of eq. (3)

assumes similar meaning for the bonded interface s . Moreover, s,1k and s,2k are the surface-volume

interaction parameters, which quantify the influence of damage in the domains on damage at the interface.

Setting this parameter to 0 creates a damage barrier at the interface. The physical meaning of the model

parameters is presented in [6] and a sensitivity analysis was presented in [13], where the influence of the

damage coefficients on the bulk and interfacial damaging was highlighted too.

The boundary conditions for the domains read:

iii gN = , in ( )21\ i , (4)

0i

i =

N

ik

, in ( )21\ i , (5)

where iN is the outward normal to the domain i and ig represents the surface exterior force.

The boundary conditions on the contact surface are:

( ) ( )i 1 2 1s sk = N x u - u , for 21 x , (6)

( ) ( )i 2 2 1s sk = N y u - u , for 21 y , (7)

( )11,1

11

=

ss

kkN

, (8)

( )22,2

22

=

ss

kk

N

. (9)

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where sk is the surface rigidity of the bonded interface s . The adhesive source of damage is represented

by ( )12 u-usG .

The initial conditions:

( ) ( )0,0 1i i = =x x , in i , (10)

( ) ( )0,0 1s s = =x x , in 21 , (11)

simply assume the integrity of the volume and interface areas at the initial time t0= 0. Finally, in order to

prevent material healing, the value of i and s are updated to satisfy the irreversibility conditions for

two subsequent loading steps t andt + dt

( ) ( ),i it t dt +x x, , ( ) ( ),s st t dt +x x, .

This approach [6, 7] has proved to be sufficient in capturing the main physical phenomena. More

sophisticated constitutive laws, such as those considered in [14, 15], could be adopted; however the

enhancement of the results that could be derived is secondary for the purposes of this study.

The two domains here considered are the concrete element and the FRP reinforcement. In the

following equations, they are distinguished by the subscripts c and p, which refer to concrete and

composite, respectively (i=c,p). The subscript s refers to the adhesive layer, whose thickness is

neglected in order to simulate the joint as a contact surface. In the previous equations (2)-(3), the failure

criterions have to be specified. They are expressed in term of elastic energy and also represent the sources

of damage. In particular, in the case of FRP reinforcement, whose mechanical behaviour law is mainly

symmetric, the source of damage ( )ii

F reported in (2) takes the form:

( ) pp

ppppp

w

F =

C

::2

1

, (12)

while in the case of concrete, which exhibits asymmetric behaviour in tension and compression, damage

is mainly related to extension. Thus, the failure criterion takes only the positive part of the deformation

tensor+c into account:

( )c

cccc

wF

= ++ cc C ::

2

1. (13)

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The coefficienti

w in equations (12) and (13) is the initial damage threshold expressed in terms of

volumetric energy. It derives from the rupture criterion, which is adapted for each material.

Analogously, the damage threshold sw , i.e., Duprs energy that accounts for the glue cohesion, permit to

introduce the source of damage assumed for the interface:

( ) ( )s

scp

scps

wkG

=

2

2

u-uu-u . (14)

2.3 Three-domains damage model

An improvement in the way of modelling glued assemblies affected by damage is adopted here. Its

main characteristic consists in the introduction of a third damageable domain representing the glue layer,

and consequently the presence of two contact surfaces (interfaces) that connect this new domain to the

adjacent adherents. The glue layer is modelled as a volume and its actual thickness is taken into account

in order to capture the phenomena related to the cohesive failure of the adhesive joint. In this context, the

damage behaviour can be fully characterised by considering both the adhesive failure at the interface

between different components and the cohesive fracture within the glue layer.

This variant leads to a new geometry of the system that contains three domains i , i = 1, 2, 3,

corresponding to the concrete element, FRP reinforcement and glue joint, respectively. The interface

311 =s connects the concrete and glue domains, while the interface 232 =s creates

a mechanical link between the adhesive layer to the composite plate. An illustration of the difference

between the two versions of the model is given in Figure 2.

1 s 2 1 1s 3 2s 2

Figure 2: The geometries of the two versions of the bonded assembly modelling: two domains and one interface

(left) and three domains and two interfaces (right).

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( ) ( )1

1211

2

s

scg

scgs

wkG

= u-uu-u , (18)

( ) ( )2

2222

2

s

spg

spgs

wkG

= u-uu-u . (19)

Finally, the surface-volume interaction parameters cs1,k and gs1,k are introduced to respectively

quantify the influence of the damage in concrete on the damage at the first contact surface and that of the

glue on that of the same interface. Similarly, gs2,k accounts for the damage relationship between the glue

and second contact surface, while ps2,k links damage in composite with that on the second interface.

The most important result that we obtain from taking the actual thickness of the joint into account is

the uncoupling of the adhesive and cohesive failure mechanisms. The former is assigned to the two

contact surfaces, while the latter is related to the glue layer. This variant of the damage model also gives

the opportunity to choose the values of the interface characteristics in order to simulate the creation of a

thin transition zone (also called interphase) with characteristics differing from that of the glue and the

substrate, and resulting either from the penetration of the glue into the porosity of the substrate or from

physico-chemical interactions between the glue components and the surface of the adherent [16].

However, it is a hard task to select the characteristics of these transition zones (thickness and mechanical

properties) which are very dependent on the kinetics of diffusion phenomena and requires sophisticated

characterization tools to be properly assessed. For the sake of simplicity, in the following, the interfaces

are imagined to be part of the glue layer and their mechanical properties are selected accordingly. These

additional advantages provided by the new damage model will be evaluated in a forthcoming work.

3. Damage characterisationThe characterisation of damage in bonded assemblies deals with the fundamental issues listed below:

- damage occurrence (failure criterion),

- damage intensity,

- damage location and extension,

- damage evolution in time and space.

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The first item represents the basis of the damage assessment process. A threshold checking approach

based on one or a combination of parameters is commonly used towards this aim. Maximum stress or

strain are the quantities generally adopted, with the latter more effective when the adhesive joint exhibits

significantly nonlinear behaviour. In both cases, a reference value is selected and failure is assumed to

have occurred when the maximum strain or stress reaches its ultimate value. The drawbacks of this

approach are clearly evident: the structural integrity is related to the maximum value of a single feature,

thereby discarding its distribution within the specimen. Moreover, the selection of the threshold value is a

difficult task to accomplish because of its dependency on the mechanical properties of materials; only the

condition of complete failure is considered, whereas the transition through intermediate damage states is

neglected.

The state quantity introduced in the proposed damage model is well-suited for quantifying the

extent of damage and describing its evolution since it is a function of time and position. Its physical

meaning refers to the volume fraction of active links, or undamaged material; thus this variable is directly

related to the local structural integrity of the specimen. Compared to the classical threshold approaches,

the damage parameter provides afuzzydescription of the damage state, which also involves partially-

failed situations. The null value represents open fractures, or complete detachment between two bonded

components, while values in the range 10

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The average damage index provides a quantification of the damage extent, which is a function of the

damage intensity (the value of k ) and the extension in the domain, or along the interface (expressed in

terms of the number of affected nodes dN ). The occurrence of damage in a quasi-static rupture test leads

to an index value lower than 1. As damage increases, the index decreases because of the addition of new

damaged nodes and the lowering of the values of those already affected. Unlike the simple damage

quantity , complete failure does not necessarily correspond to the 0 value of the average damage index.

Indeed, some portions of the specimen could be only partially damaged when rupture occurs, and the

contribution of their nodes in (20) would prevent the index from reaching 0. The unchanged value of the

damage index throughout two or a few more consecutive time steps can be assumed as reliable evidence

of complete failure occurrence. Moreover, the availability of the two damage quantities i and s

allows the damaging process to be followed separately in the domains and at the interface.

3.2 Damage distribution vectors

Although the average damage index provides an effective estimate of the overall damage extent at

each step of the rupture process, no indication is given about the position of the failure within the

specimen and its evolution in space. A new damage feature is introduced to deal with damage

localization. Depending on the dimension (2D or 3D) of the problem being analysed, a different amount

of distinct quantities is defined: one for each coordinated direction. Each quantity is a vector (or a matrix

in the 3D case) representing the distribution of damage along an axis (or over a plane in the 3D case).

Values at nodes are averaged along the direction of each coordinate axis, which gives X representing

the damage distribution over the X axis and Y that over the Y axis. The valleys in the damage

distributions identify the areas where damage is more extended and pronounced. An explanatory example

is shown in Figure 3, where the damage pattern on a two-dimensional domain is depicted in terms of a set

of level curves whose colours assume the coldest tonalities for the lowest values of the damage parameter

(failure). The shape of the two distributions (Figure 3b-c) provides a clear idea of the damage location

within the domain and along the axes, as well as the amount of damage, which was depicted through the

amplitude. The evolution of damage is reflected in the spreading of the distribution along the axes and the

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increase of the amplitude of the distribution. Some examples of the evolution of the damage distributions

will be provided in the next section with reference to the present case study.

Figure 3: (a) level curves of a damage pattern defined on a two-dimensional domain, (b) damage distribution X

along the horizontal axis and (c) damage distributionYalong the vertical axis.

3.3 Failure indexes

A final effort in the characterisation of the damaging process is the fusion of the features concerning

the damage extent and its propagation in time. The aim is to condense into a single scalar value those

characteristics that are most significantly affected by the parameters variation in the modelling of damage

behaviour. The so-called failure index is derived accounting for three different sources: the amount of

energy released throughout the damaging process, the time lapse needed to reach complete rupture and

the overall damage amount at failure. With reference to Figure 4, which shows the evolution of the

average damage index throughout the rupture test of a FRP-strengthened concrete element, the first

quantity is identified by the coloured area

A , the second refers to the interval of time t between the

damage triggering tt and the instant of the rupture occurrence ft and the third is given by the ultimate

reduction of the average damage index, noted . The variables (excluding the last one) are normalized

in order to assume values in the range [0, 1] and then combined according to:

ff t

A

t

t

t

AI

=

= , with [ ]1,0

I . (21)

a)

b)

c)

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Therefore, the failure index

I can be defined as the value of the average damage index that

equals the surface of the two shaded areas split by the horizontal dash-dotted line in Figure 4. Values of

the failure index

I close to 1 represent a detachment process protracted in time and extended in space.

The so-defined failure index is able to discriminate different failure mechanisms. Depending on the

mechanical properties of the components, their aging and the efficacy of the bond, a FRP-strengthened

concrete element shows types of rupture differently localised in the components and damaging processes

spread over a variable lapse of time. For instance, rupture can be abrupt, which signifies a fragile

behaviour, or prolonged in time, as typical of ductile materials. Both rupture time protraction and space

extension are phenomena that indicate a pronounced capacity of the assembly to redistribute stress. The

failure index implicitly takes into account redistribution and dissipative capacity (the amount of released

energy

A ) and quantifies the tightness of the assembly. The highest the index value the highest the

amount of energy which is needed to achieve complete failure.

The formulation presented in (21) applies to all the elements of the assembly depending on the average

damage index that in the present case is selected among c , g , p , 1s and 2s . Thus, the failure

index can be computed separately for each adherent, adhesive and interface, leading toc

I

orp

I

,g

I

,

1sI

or

2sI

, respectively. The three domains can be grouped in a single index

VI

, while the two

interfaces considered as a whole are represented byS

I

. In addition, the combination of failure indexes

related to different components of the assembly provides the means to identify the type of rupture as a

function of a scalar value. In particular, for the improved damage model, the distinction of the glue layer

from the interfaces and substrates allows to compute a global bulk index

I that is expressed as in (22):

sV

c

II

II

+= , with [ [1,0

I , (22)

wherec

I

is the failure index of the concrete substrate. A value close to 1 for the global bulk index

I

refers to a type of rupture that mainly involves the concrete substrate. Conversely, very low values of the

global bulk index identify the case of adhesive failure. More specifically, the occurrence of the interfacial

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failure is pointed out by using a second combination of the interface and volume failure indexes. Thus,

the global interface indexs

I is defined as:

sV

s

sII

I

I

+= , with [ ]1,0 sI . (23)

High values for the global interface indexs

I define the adhesive failure at the interface. For more

detailed studies that adopt the three-domains damage model, the global interface index can be split in two

by distinguishing the failure index for the interface 1s from that for 2s . In the following sections, the

failure indexes and their combinations will be used as criteria for assessing the influence of the

mechanical and bonding characteristics of the epoxy on the fracture behaviour of a glued assembly.

0 10 20 30 40 50 60 70

0.4

0.5

0.6

0.7

0.8

0.9

1

Time [s]

Averagedamageindex

A

t

tt ft

0

f

I

Figure 4: The three components (

A , t and ) of the failure index

I characterizing the damage evolution.

4. Experimental and numerical tests4.1 FRP-reinforced concrete specimen

The accuracy of the damage model presented in Section 2 is assessed through its comparison with the

results of an experimental campaign carried out jointly by IFSTTAR and the Department Laboratoire

dAutun in France. A large number of FRP reinforced concrete elements were tested under varying

characteristics of the epoxy adhesive. Each specimen consisted of a concrete block of dimensions 210

210 410 mm

3

cast using CEM II 32.5 cement and silico-calcareous aggregates (610 and 1020 mm)

and a water-to-cement ratio of 0.55. A maturation period of 28 days was enforced in order to obtain a

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compressive strength of 32 4MPaand a tensile strength of 3.6 0.1 MPa. A 100 mmwide and 1.5 mm

thick strip of CFRP plate was bonded using a two-component epoxy adhesive on the upper side of the

concrete substrate after a sand-blasting treatment. The adhesive layer measures 200 mmin length and 0.75

mmin thickness and it was positioned 50 mmaway from the front side of the concrete block in order to

prevent edge effects (Figure 5a). This procedure is commonly adopted in experimental campaign of

debonding tests as in [19] and [20]. Former studies [7] have shown the influence of the geometric

configuration of the composite gluing on the observed failure mechanism for a single lap joint shear test.

The configuration chosen here, from those tested in [7], leads to cohesive fracture in the adherent near the

bonded surface. A nearly uniform concrete cover is detached from the block. Conversely, when the glued

surface is prolonged up to the edge of the specimen, a concrete wedge is removed from the blocks

corner. The gap between the edge of the concrete block and the adhesive layer allows a more regular

detachment growth along the specimens length, which is desirable when the aim is to characterize

different failure mechanisms. For more details about the physical properties and mechanical compositions

of the specimen constituents, the reader can refer to [1].

410

210

200 70

50

15

50

90

135

180

410

210

grips

Fstrain gauges

170

X

Y

(a) (b)

(c)

Figure 5: (a) Geometry, (b) test setting and (c) mesh of the FRP-reinforced concrete specimen: units in mm.

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The mechanical characterisation of the bonded assembly was performed through a single lap joint

shear test carried out using a hydraulic jack with a 100 kN maximum force capacity. The boundary

conditions of the test are shown in Figure 5a. The specimen is placed in a steel frame specifically

designed to carry out this kind of test [7]. Thick rigid walls around the concrete prism impede any

translation (Figure 5b). A specific support equipped with adjustable screws acting on the specimens

position was used to ensure a good alignment of the jack with the reinforcement and to prevent any

rotation of the specimen during the test. The clamping of the free end of the reinforcement is made with

conical grips 70 mm away from the glue joint (this distance is more or less arbitrary, but the main thing is

that it remains constant during al the test campaign). The test was carried out at a constant displacement

rate of 6 m/suntil the complete detachment of the FRP overlay.

The shear tests were repeated on several sets of specimens with varying properties of the epoxy

adhesive. The experimental campaign showed a variety of failure mechanisms that can be grouped into 3

main types. The upper and lower faces of fractured surfaces obtained from the shear test for these 3

groups are depicted in Figure 6. The first fractured surface refers to debonding at the interface between

the FRP and adhesive (Figure 6a), the second shows pure concrete cracking (Figure 6b) and the third

involves cohesive failure within the epoxy joint (Figure 6c). In a first step, we will focus on the last two

failure modes, i.e. the cohesive failures within concrete and glue, which were obtained for states of the

epoxy adhesive related to different ageing times. The mechanical properties of the epoxy adhesive were

obtained through tensile-loading tests performed on dumbbell glue samples. Exposure to hydrothermal

ageing at 40C and 95% R.H. for 8 months revealed a decrease by a factor of 4 compared to the initial

state for both the tensile strength (from 251 MPa to 6.60.5 MPa) and the Youngs modulus (from

4.90.2MPato 1.30.2MPa) of the glue. Moreover, the extensive plasticization of the polymer network

by water molecules from the ageing environment enhanced the ductility of the second state of glue

(reference to Figure 9a in [1]). Therefore, in the following, the first state of epoxy will be referred as the

elastic and fragile initial state, whereas the second as the elasto-plastic and ductile state observed after 8

months hydrothermal ageing.

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(a) (b) (c)

Figure 6: Failure surfaces for shear tests carried out on specimens with different adhesive characteristics: (a) FRP-

glue interface debonding for poor adhesive properties of the glue, (b) concrete cracking for elastic and fragile glue

and (c) complete cohesive failure of the adhesive layer for elasto-plastic and ductile glue; dimensions are in mm.

4.2 Numerical simulations

The numerical simulations of the experimental shear test are carried out by implementing the damage

model in a FE code based on the Open Source package Deal.II [17]. Both of the 2D models presented in

Section 2 are considered. In the following, the model where the thickness of the adhesive layer is

neglected will be referred as Model 1 and the other as Model 2. In both models, the concrete and FRP

plate are represented by quadrilateral elements in plane strain condition with polynomial approximation

of degree 1. The introduction of the actual thickness of the adhesive layer in the second model entails the

adoption of a further class of quadrilateral elements for the third domain and two interfaces at the upper

and lower sides of the bond. Interface elements with one infinitesimal dimension are used to split the

domains and to model the interface damage coupling effect. The mesh of the first model, consisting of

4996 nodes and 5222 elements, is illustrated in Figure 5c. The mesh of the second model, consisting of

5398 nodes and 6022 elements, differs from the former only in the elements used to model the adhesive

layer and second interface. Convergence was studied at each time step as difference between the results of

two consecutives iterations and it was considered achieved for values lower than 1e-5. Convergence was

achieved for both models throughout the whole shear test, with slightly higher computation time for

Model 2 due to its larger mesh. The boundary conditions introduced in the simulations are represented in

Figure 5 and reproduce the clamping conditions adopted in the experimental tests.

200

100

200

100

200

100

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dimension of the interface. Smaller values would have provided equivalent results. The damage threshold

parameters w are obtained by adopting the expression of the Duprs energy, which at glue interface and

domain reads respectively:

T

s

eg

sk

w2

12

= , (24)

g

eg

gG

w

2

2

1= , (25)

whereg

G is the shear modulus of the glue andeg corresponds to the stress at the end of the elastic

domain; in the case of the epoxy glue in the state 1 (brittle elastic behaviour), eg is assumed equivalent

to the mean tensile strength, whose value is equal to 25 MPa. As regards the glue in the state 2 (elasto-

plastic behaviour), the elastic limit eg derived from the tensile tests is reduced to 2 MPa, which

corresponds to 1/3 of the measured tensile strength (see Figure 9a in [1]). It is worthy to highlight that the

two damage thresholds as defined in (24) and (25) have the dimensions of energy per unit volume and

area, respectively. To give a physical interpretation, damage thresholds can be considered as the energy

that has to be produced to break the microscopic links in a unit volume (or area) of material.

Similarly, damage thresholds in the adherents (concrete and CFRP plate) are respectively given by:

c

ecc

Ew

2

2

1= and

p

ep

PE

w

2

2

1= (26)

As both adherents exhibit brittle elastic behaviours, elastic limits can be replaced by mean tensile

strengths, which take values of 2.5 and 2800MPafor concrete and composite, respectively.

The damage extension parameters kare set according to previous studies [6, 7] and kept unchanged

for both the cases. In Model 2, the damage extension coefficient gk referring to the volume of the glue is

obtained by multiplying the interfacial parameter sk of Model 1 by the actual thickness of the adhesive

layer. In order to simplify the comparison between the models, both the interfaces of Model 2 are

assumed to be part of the glue layer and their mechanical properties and damage features are selected

accordingly. Consequently, the surface-volume interaction parameters gs1,k and gs2,k for Model 2 are

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equal to cs1,k , which corresponds to cs,k for Model 1. This value can be numerically estimated by

considering the bond strength of the glue on the concrete, as was done in [6]. In the case of the second

state of adhesive, the value of cs1,k is lowered in order to consider the reduction of the interaction between

the interface and the volume of the adhesive due to the weakening of the glue bonding. Finally, since the

interaction between the glue and FRP plate is negligible, the interaction parameter ps,k for Model 1 and

ps2,k for Model 2 are set to be practically zero for both states of adhesive.

4.3 Results and discussion

The damage patterns obtained at the end of the shear test for the two damage models and the two

states of epoxy are collected in Figure 7. For the sake of clarity in the representation, only the volume

damage i is portrayed in Figure 7, being the one that affects more the actual failure mechanism for both

the states of epoxy. The nodes where i is equal to 0 represent the complete material failure that leads to

opening of macro-cracks. The portions of the specimen characterised by values of i smaller that 1 are

partially damaged, e. g., occurrence of micro-cracking. The fracture within concrete produced by the stiff

adhesive (glue state 1) is accurately depicted by both of the two damage models (Figure 7a and Figure

7c). Conversely, only the improved damage model is capable of predicting the cohesive failure within the

adhesive layer when the weak glue is involved (Figure 7d). The simplified damage model provides a

misleading prediction that entails the crack of concrete immediately below the interface with the glue

(Figure 7b). The limits of Model 1 in predicting the fracture mode can be ascribed to the lack in the

description of the three-phase assembly. For this reason, in the following sections, the characterisation of

the damage behaviour for both adhesive states is carried out only by means of the second damage model.

The position of the fracture at the end of the shear test can be determined also by the displacement

field. A sudden change of displacement over a small area (large gradient) identifies a detachment within

the material, i.e., a fracture. The horizontal displacements illustrated in Figure 8 are in perfect agreement

with the volume damage patterns and show clearly the position of the fracture line in different domains

according to model and state that are considered.

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extent a few seconds after the damage triggering in concrete. The adhesive failure at the interfaces does

not occur at all for the first state of glue ( ( ) ( ) ttt ss == 121 in Figure 9b).

Figure 8: Horizontal displacements for (a) Model 1 and epoxy state 1, (b) Model 1 and epoxy state 2, (c) Model 2

and epoxy state 1, (d) Model 2 and epoxy state 2 (displacements in mm).

The weak adhesive shows a failure mechanism that is diametrically opposed to the former one. The

dashed line of g in Figure 9a decreases smoothly from the very first stages of the test. A small rise of

the index occurs towards the end, when a new large portion of the glue layer is affected by damage but to

a minor extent. In fact, the average in equation 20 is shifted towards higher values because a large number

of new nodes affected by minor damage (value of k close to 1) is taken into account. This behaviour

can be physically interpreted by micro-cracking, i.e. formation of microscopic cracks whose coalescence

leads to a macroscopic crack. It is authors belief that the loss of monotonic decrease with loading of the

average damage index is not misleading because it can be referred to a unique and well-defined

phenomenon, i.e., minor damage propagation.

The trend of thec index proves that concrete is practically unaffected by the cohesive failure within

the glue layer. Unlike, 1s and 2s indexes portrayed in Figure 9b show the same smooth evolution

of g . This is the result of the enhanced surface-volume interaction that is gained for the weak epoxy.

(a)

(b)

(c)

(d)

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24

The circles depicted in Figure 9 identify the time instants that exhibit significant changes in the

damage evolution, such as damage triggering, turning points in the trend and final failure. They find

correspondence in the evolution of the damage patterns as a function of time, portrayed in Figures 10-11,

which clarify the process of the failure mechanisms observed in the experimental tests for the strong and

weak adhesive, respectively. In the first case (Figure 10), the crack in the concrete block originates a few

millimetres under and away from the front side of the interface (t = 27s). The fracture proceeds

backwards along the adhesive layer until it reaches the unglued concrete surface when complete failure

occurs (t = 58s). When the weak glue is involved (Figure 11), the fracture initiates within the adhesive

layer at the front side of the interface (t = 52s) and proceeds backwards with a constant increase rate.

When the crack entails about 80% of the length of the adhesive layer (t = 59s), damage rapidly propagates

in the rest of the joint and a sudden rupture occurs (t = 60s).

0 10 20 30 40 50 60 70 80 90 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time steps [s]

Averag

edamageindex

cglue state 1

gglue state 1

cglue state 2

gglue state 2

Average

damageindex

Time steps [s]

c glue state 1

g glue state 1

c glue state 2

g glue state 2

0 10 20 30 40 50 60 70 80 90 100

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time steps [s]

Averagedamageindex

s1

glue state 1

s2

glue state 1

s1

glue state 2

s2

glue state 2

Time steps [s]

Average

damageindex

s1 glue state 1

s2 glue state 1

s1 glue state 2

s2 glue state 2

(a) (b)

Figure 9: Evolution of throughout the shear test: (a) concrete and glue volume indexes for strong (solid lines) and

weak (dashed lines) epoxy; (b) interface indexes for strong (solid lines) and weak (dashed lines) epoxy.

The propagation of damage in space is analysed by means of the vectorial damage indexes related to

the distribution of damage along the coordinated axes (see 3.2). The distributions presented in Figure 12

refer to the three significant stages of the damaging process already pointed out in the case of the stiff

glue (Figure 10). The propagation of the crack along the adhesive layer can be clearly recognized from

the spreading of the damage distribution X along the X axis (in Figure 12a, the range of X coordinates

whose X values is less than 1 widen with the test). Differently, the extension along the vertical

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25

direction does not take place (in Figure 12b, Y coordinates affected by damage remain unchanged

throughout the test), while an increase in the extent of Y is observed.

Figure 10: Evolution of the volume damage patterni

sampled at the most significant stages (see circles of Fig.8a)

of the shear test carried out on the strongly bonded assembly (glue state 1).

Figure 11: Evolution of the volume damage pattern i sampled at the most significant stages (see circles of Fig.8b)

of the shear test carried out on the weakly bonded assembly (glue state 2).

The distributions depicted in Figure 13 describe the spatial evolution of the damaging process

observed for the weakly bonded specimen (Figure 11). Analogously to Figure 12, the longitudinal

propagation of fracture is shown by the spreading of the damage distribution X along X (Figure 13a)

and the increasing extent of Y for the other direction (Figure 13b). The discrepancy between the final

values attained by the damage distributions X and Y proves that the failure propagation takes place

t = 58 s

t = 27 s

t = 54 s

t = 60 s

t = 52 s

t = 59 s

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mainly along the horizontal axis. Indeed, in Figure 13a the portion of failed material along Y is so limited

that averaging over the whole height of the specimen leads to values of X very close to 1. Therefore,

one can deduce that damage is spread over a large length along X, but not in the other direction.

Moreover, in Figure 13b, small values of Y and its sharp distribution along Y suggest that the damaged

zone is limited to the thickness of the adhesive layer for all of the three time steps. In conclusion,

combining the analysis of the two distributions provides a clear outlook of the area affected by damage.

0 50 100 150 200 250 300 350 400 4500.92

0.94

0.96

0.98

1

Abscissas [mm]

Damagedistribution

t = 27 s

t = 54 s

t = 58 s

X coordinates [mm]

Dam

agedistribution

(a)

0 50 100 150 200 2500.5

0.6

0.7

0.8

0.9

1

Ordinates [mm]

Damagedistribution

t = 27 s

t = 54 s

t = 58 s

Y coordinates [mm]

Damagedistribution

(b)

Figure 12: Evolution of the damage distribution Xalong the X axis (a) and Yalong the Y axis (b) for the strongly

bonded assembly: comparison between the time steps t = 26 s, t = 52 sand t = 56 s. Origin of coordinates located at

the bottom-left corner of the concrete prism (Figure 5a).

0 50 100 150 200 250 300 350 400 4500.9975

0.998

0.9985

0.999

0.9995

1

Abscissas [mm]

Damagedistribution

t = 52 s

t = 59 s

t = 60 s

X coordinates [mm]

Damagedistribution

(a)

195 200 205 210 215

0.6

0.7

0.8

0.9

1

Ordinates [mm]

Damagedistribution

t = 52 s

t = 59 s

t = 60 s

Y coordinates [mm]

Damagedistribution

(b)

Figure 13: Evolution of the damage distribution Xalong the X axis (a) and Yalong the Y axis (b) for the weakly

bonded assembly: comparison between the time steps t = 52 s, t = 59 sand t = 60 s. Origin of coordinates located at

the bottom-left corner of the concrete prism (Figure 5a).

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Finally, the failure index is computed for the concrete substratec

I

, adhesive layerg

I

and interface

sI

. These indexes quantify to what extent each component of the glued assembly is affected both in time

and space. Moreover, the combination of the failure indexes related to different components according to

(22) and (23) provides the global bulk

I and interfaces

I indexes, respectively.

The values obtained for the two states of epoxy are summarised in Table 2. These results quantify the

change in the failure mechanism depending on the mechanical and bonding properties of the adhesive. In

particular, the stiff and resistant epoxy leads to concrete fracture (high values forc

I

and

I ), whereas

the weak and ductile glue moves the failure towards an adhesive decohesion that involves also slight

interface separation (high values forg

I

,s

I

ands

I ). The comparison between the numerical and

experimental results in terms of failure mode and maximum shear load is presented in Table 3. A good

agreement is observed for both the types of glue state.

Table 2. Failure indexes and their combinations for the two adhesive states

cI

gI

sI

I

sI

Glue state 1 0.277 0.004 0 0.986 0

Glue state 2 4.67 x 10-5 0.505 0.245 6.2 x 10-5 0.333

Table 3. Comparison between numerical and experimental results for the two adhesive states

Numerical Experimental

Failure mode Max shear load [kN] Failure mode Max shear load [kN]

Glue state 1 concrete fracture 51 concrete fracture 54

Glue state 2cohesive crack

within the glue layer47

cohesive crack

within the glue layer46

The deformation of the composite plate throughout the shear test was measured by means of five strain

gauges placed in the locations that are shown in Figure 5a. In Figure 14, the experimental measurements

are compared with the results of the numerical simulations that are carried out for both types of epoxy. A

good agreement between the outcomes of the damage model and the experimental evidence is observed in

both cases. The shape of the strain curves changes from a piecewise linear elastic-pseudo plastic profile

for the stiff epoxy (Figure14a) to a more rounded trend for the ductile epoxy (Figure 14b). This result

confirms the evolution of the load transfer mechanism outlined in [1] as a function of the stiffness of the

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by decreasing the values of damage parameters related to the cohesion of the interfaces

w and the

interaction between interface and adherent/adhesive is,k . The damage threshold and damage interaction

parameters of the upper and lower interfaces are changed in turn in order to obtain the adhesive failure at

FRP/epoxy and epoxy/concrete interface, respectively. In this way, the former assumption that considered

both the interfaces as part of the glue layer is released to imagine them as different entities separately

affected by different phenomena. Obviously, the impossibility to distinguish between the two interfaces

prevents Model 1 to discriminate between the two interfacial failures.

Table 4 shows the values of the damage parameters that were selected to achieve the two interfacial

failure modes (modes 2 and 4 in Figure 1). The values used for the cohesive cracking mode (failure mode

1 of Figure 1) are given as well for comparison. Large variations of the parameters are necessary to

achieve failure at the interfaces. One possible explanation is that the whole set of material parameters as it

was derived from available experimental measurements is much more sensitive to cohesive cracking than

to interfacial failures. Probably, smaller changes would have been obtained for a set of material

parameters less prone to cracking in the adherent. More rigorous approaches that aim at identifying

parameters by taking their interaction into account are envisaged and will be pursued in future works.

Table 4. Damage parameters used in Model 2 to simulate interfacial failures.

Parameter Description

Failure mode 1

Concrete craking

Failure mode 2

Concrete/epoxy

interface

Failure mode 4

FRP/epoxy

interface

ws1[MPa mm] damage threshold (concrete/epoxy interface) 0.3 3.0 x 10-4 0.3

ws2[MPa mm] damage threshold (FRP/epoxy interface) 0.3 0.3 3.0 x 10-5

ks1,c[MPa mm] damage interaction interface 1 - concrete 0.2 1.0 x 10-3 0.2

ks1,g[MPa mm] damage interaction interface 1 - epoxy 0.2 1.0 x 10-3 0.2

ks2,g[MPa mm] damage interaction interface 2 - epoxy 0.2 0.2 2.0 x 10

-6

The volume and interfacial damage patterns that are obtained for the two adhesive failures are

depicted in Figure 15. In both cases, the improved damage model manages to accurately reproduce the

separation between adherents and adhesive through continuously failed interface. The interfacial fracture

is clearly represented by the straight line along the joint where the surface damage quantity s vanishes

(Figure 15b and zoom). Damage in the volume is limited to a small propagation of the fracture in concrete

at the end of the glue joint when the lower interface is affected (Figure 15a). The FRP/epoxy separation

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(Figure 15d and zoom) that is coupled with concrete cracking at the front side of the joint (Figure 15c) is

comparable to the interface debonding observed in Figure 6a.

Figure 15: Volume and interface damage patterns for the two adhesive failures: (a) volume damage i at the

concrete/epoxy interface, (b) surface damage s at the concrete/epoxy interface, (c) volume damage i at the

FRP/epoxy interface, (d) surface damagesat the FRP/epoxy interface.

The results obtained for the computation of the failure index for concretec

I

, adhesive layerg

I

and

interfaces1s

I

and2s

I

, together with those of the global bulk

I and interfaces1

s

I

and2

s

I

indexes

are summarised in Table 5. As expected, the adhesive failure modes lead to a diametrically opposed result

compared to the cohesive fracture modes of Table 2. The global bulk index

I is drastically reduced,

while the split of the global interface index between the surfaces 1s and 2s provides a clear distinction

of the detached interface.

Table 5. Failure indexes and their combinations for the two adhesive failures

cI

gI

1sI

2sI

I

1

sI

2

sI

Failure mode 2

Concrete/epoxy

interface0.007 0.046 0.448 0 0.015 0.858 0

Failure mode 4FRP/epoxy

interface

0.047 0.003 0 0.132 0.257 0 0.739

(a)

(d)

(b)

(c)

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5. CONCLUSIONSAn improved damage modelling approach model is proposed to deal with different failure modes

depending on the properties of the adhesive. The model involves a three-domain representation of FRP

reinforced concrete structures and accounts for the interaction between domain and interface damage to

predict different failure mechanisms for a single lap joint shear test.

The damage model proves to be capable of accurately reproducing the failure mechanism and fracture

line, outperforming a former two-domain version of the damage model over different debonding modes.

The modelling of the thickness of the adhesive layer allows detecting the occurrence of cohesive failure

within the glue when its mechanical and bonding properties weaken. The sensitivity of the damage

models response to the variation of the epoxy parameters is highlighted. Indeed, the damage behaviour

significantly changes when the weakening of the glue is considered. Adhesive failure modes are

investigated too. The proposed damage model is capable to discriminate between the concrete/epoxy and

FRP/epoxy separation. The application of this new model, once confirmed by more exhaustive studies,

would offer promising prospects to investigate failure modes related to poor adherence of the epoxy on

the substrates.

Original damage and failure indexes are introduced to locate damage in time and space, quantify the

evolution of the detachment behaviour and distinguish different types of failure. Moreover, the distinction

of the glue layer from the interfaces and substrates in the improved damage model allows the computation

of the bulk and interface global indexes that proved to be suited to quantify the contribution of each type

of damage mechanism to the global failure. In future works, the proposed failure indexes will be used as

criteria to guide the selection of the damage model parameters within an optimisation framework aimed at

the definition of parameters sets related to the different failure modes that may occur.

By way of conclusion, the proposed damage model is an effective and robust tool for providing an

accurate description of the fracture behaviour of FRP-reinforced structures when they undergo states

which are capable to activate different damage mechanisms, such as glue weakening and lack of

adherence. Further studies are currently carried out focusing on the relationship between the damage

model parameters and their interactions and the evolution of different failure modes.

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