Accepted Manuscript

Delamination process analysis of frp-to-parent material bonded joints with and

without anchorage systems using the distinct element method

Hugo C. Biscaia, Rui Micaelo, João Teixeira, Carlos Chastre

PII: S0263-8223(14)00188-3

DOI: http://dx.doi.org/10.1016/j.compstruct.2014.04.021

Reference: COST 5674

To appear in: Composite Structures

Please cite this article as: Biscaia, H.C., Micaelo, R., Teixeira, J., Chastre, C., Delamination process analysis of frp-

to-parent material bonded joints with and without anchorage systems using the distinct element method, Composite

Structures (2014), doi: http://dx.doi.org/10.1016/j.compstruct.2014.04.021

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DELAMINATION PROCESS ANALYSIS OF FRP-TO-PARENT MATERIAL

BONDED JOINTS WITH AND WITHOUT ANCHORAGE SYSTEMS USING

THE DISTINCT ELEMENT METHOD

Hugo C. Biscaia1, Rui Micaelo2, João Teixeira2, Carlos Chastre3

Abstract

This study looks at the analysis of the interface between Fiber Reinforced Polymer

(FRP)-to-parent material bonded interfaces. The performance of FRP-to-parent material

bonded joints for the Externally Bonded Reinforcement (EBR) technique is numerically

modelled with the PFC2D software which is based on the Distinct Element Method

(DEM). It is believed that this represents the first time the DEM has been used to

simulate the delamination process of FRP-to-parent material bonded joints. In order to

validate the analysis performed with the DEM, a Pull-out test with no slip constrains

was modelled and different linear bond-slip laws were assumed. The numerical results

revealed that the DEM is capable of estimating with good accuracy the exact solutions

of bond stresses, strains or slippages along the bonded length for linear bond-slip laws.

The bi-linear law available in PFC2D was then compared to the numerical results

obtained from other another code developed by the author. The delamination process of

Pull-out tests with slip constrain at one of the free ends of the FRP plate is also

described and analyzed. The results obtained from the DEM revealed that the

delamination process ends with stiffness equal to the axial stiffness of the FRP plate.

1hb@fct.unl.pt Tel: +351-21-2948580 Fax: +351-21-2948398 UNIC/ Faculdade de Ciências e Tecnologia/ Universidade Nova de Lisboa 2829-516 Caparica - Portugal 2 Faculdade de Ciências e Tecnologia/Universidade Nova de Lisboa 2829-516 Caparica - Portugal 3 UNIC/Faculdade de Ciências e Tecnologia/Universidade Nova de Lisboa 2829-516 Caparica - Portugal

- 2 -

This evidence highlights the need to design mechanical anchor devices capable of

preventing premature debonding which is known to occur on EBR systems.

Keywords: FRP; Delamination; Distinct Element Method; Interface; Bond-slip.

1. Introduction

The strengthening of structures with Fiber Reinforced Polymer (FRP) composites has

been a method frequently used by engineers. Different methods have been used: the

Externally Bonded Reinforcement (EBR), the Near Surface Mounted (NSM) using

plates and NSM rods. These methods all have advantages and disadvantages. For

instance, the premature debonding rupture is usually associated to the EBR method,

therefore being a very important drawback.

In the work developed by Carvalho et al. [1, [2], reinforced concrete (RC) beams

strengthened with Carbon Fiber Reinforced Polymers (CFRP) were used and both EBR

and NSM systems were studied. The studies mentioned reported that the EBR analyzed

systems debonded earlier than the NSM systems. Consequently, the RC beams

strengthened with NSM achieved the highest rupture loads. For that reason, the

performance of the EBR systems can be improved if conjugated with a mechanical

anchorage system. However, other issues arise, e.g. where should the mechanical

anchors be located? How does the delamination process develop in such circumstances?

Even with these anchor systems, does it mean that the strengthening of RC beams with

the EBR system is more efficient than the NSM system?

In order to give an answer to these questions, several authors, e.g. [13-[16], have

been developing experimental work. So far the studies, in general, have been focusing

on the modelling of the interface between the FRP plate and the bonded substrate. Of

- 3 -

course, the Finite Element Method (FEM) has been the most used method for the

estimation of the delamination process of FRP-to-parent material interfaces. Many

advantages can be enumerated for the use of FEM but some drawbacks are too

important to be ignored. For instance, in the case of FRP-to-concrete interfaces, the

approach adopted to simulate the cracking behaviour of concrete can affect the bond

stresses along the interface. As the discrete crack model shows a discrete separation

between elements, this seems to be a more realistic way to estimate the debonding

phenomenon. However, there are two important drawbacks with this model: one is the

implication that the mesh should be refined where the main cracks will occur, which is a

difficult task to carry out before the finite element (FE) analysis; the other drawback lies

in the reformulation of the mesh at each step, leading to a slow computational

calculation process. The smeared crack model has no disadvantages of this type.

According to Teng et al. [17] and Lu et al. [18], the smeared crack model can be used

but a fine mesh should be placed in the vicinity of the FRP-to-concrete interface. These

authors recommend that the dimension of the finite elements in the vicinity of the

interface should be similar to the thickness of concrete associated to the detachment

observed in bending tests.

The premature debonding phenomenon is frequently observed at the anchorage

region of RC beams EBR with FRP composites. In order to study this phenomenon,

some studies have been focused on double or simple pull-out tests [19-[27] that have the

purpose of isolate the Mode II (i.e. pure shear) debonding of FRP-to-parent materials

from the other two fracture modes I and III. The Mode II fracture energy is usually

associated to the debonding of FRP composites externally bonded to RC beams.

The FEM has been widely used but alternative and more efficient methods have

also been found. For instance, Ferracuti [24], Martinelli et al. [28] or Biscaia et al. [29,

- 4 -

[30] developed discrete models based on the Finite Difference Method (FDM), in which

the system of nonlinear equations was solved by the Newton-Raphson Method. This

kind of interface modelling has been an important tool in contributing to the knowledge

about the full range delamination process of FRP-to-parent materials. Other authors

[31], have developed other discrete methods where the interface is discretized with

interface elements based on non-linear springs.

As an alternative to all the numerical studies done previously on the

delamination process of FRP-to-parent material interfaces with FEM or FDM, the

Distinct Element Method (DEM) may have potentialities that have been neglected and

have not been well explored. This method demands less computational effort when

fracture and large displacements problems are combined with very fine discretized

systems. Thus, the constituents that compose the materials can be discretized up to the

micro-scale level. To date, there has been a noticeable lack of studies examining the

debonding phenomenon process with DEM. The method consists of the assembly of

small rigid particles which interact at soft contacts. From the contact between particles

the stresses and displacements are then calculated. During the problem simulation

course, the contacts may fail depending on the contact law and new contacts may be

created as well. The continuously increasing number of users and fields of application

demonstrates the potentialities of this numerical method. Some studies [32-[34],

modelled the FRP composites using the DEM model. Due to the advantages of the

DEM model in the microscopic modelling of the damage and fracture in composite

materials, Sheng et al. [33] simulated the transverse cracking initiation and propagation

of composite laminates under transverse tension (perpendicular to the direction of the

fibres). The use of rectangular, hexagonal and random fibre distributions were simulated

as well. The volume of fibres used was another parameter analysed in [33]. According

- 5 -

to those authors, the DEM model was capable of predicting the experimental findings

and allowed the authors to conclude that the distribution and volume fraction of fibres

affect the transverse cracking path and the mechanical behaviour of the composite.

Yang et al. [34] analyzed the fibre-to-matrix interface on composite materials

with the DEM model. The contact between the fibre and the matrix was simulated with

a bi-linear constitutive law and what they found was that the DEM model captured the

initiation and propagation of the interfacial debonding. The experimental results were

therefore estimated by the DEM model. Based on their findings, Yang et al. [34]

claimed that this type of modelling can be used on the simulation of other materials and

structural damages in more complicated reinforced composite systems.

Due to the absence of DEM applications for the modelling of FRP-to-parent

material (such as concrete, steel, glulam, among others) interfaces, this paper takes

advantage of the potentialities of this method to analyse the performance of such kinds

of interfaces applied on EBR strengthening systems, with and without anchorage

systems, i.e. with and without slip constraints, respectively. The Particle Flow Code in

Two Dimensions, PFC2D v3.0 [35], was used and a plain GFRP composite under

tension and two different Pull-out tests were simulated.

2. Numerical modelling

The DEM was initially presented by Cundall in 1971 for the study of blocky rock

systems [36]. During the last 4 decades many researchers have contributed to the

development and spread of the method in different engineering fields, from construction

to the pharmaceutical industry. The method is based on the interaction of a collection of

variable-size rigid discrete bodies, with finite displacements and rotations, which allows

full detachment/separation of bodies, and it is able to recognize new contacts as the

- 6 -

calculation proceeds. While in the field of mechanics it is considered that dimensions of

the body are negligible, and only represent a single point in space, in DEM formulation,

the body (particle) occupies/represents a certain volume of the represented space.

There are two main principles in the DEM formulation: (i) the force-

displacement law, where the contact force acting on two entities in contact is derived

from the relative displacement between the entities; (ii) the law of motion, where the

motion of a rigid particle is determined by the resulting force and moment vectors

acting upon it. Fig. 1 illustrates in a flowchart the general DEM calculation cycle. The

dynamic behaviour is numerically represented by an algorithm with explicit timestep

that uses central differentiation scheme for velocities and accelerations. The timestep

(Δt) adopted in the calculation has to be limited so that during a single timestep the

disturbances can only propagate to its immediate neighbours.

Fig. 1. General DEM calculation cycle. Adapted from [35].

The force-displacement law relates relative displacements and forces at each contact,

either being particle-to-particle or particle-to-wall contacts. As illustrated in Fig. 2, the

contact is described in terms of a point xi[c] lying on a contact plane that is defined by a

unit normal vector ni. The contact force is decomposed into a normal component (Fin)

acting in the direction of the normal vector and, a shear component (Fis) acting in the

contact plane. The magnitude of the normal contact force is calculated according to

���

��� ×= (1)

- 7 -

where, Kn is the normal stiffness and Un is the contact normal displacement (overlap).

Kn is a secant modulus, representing the total displacement and force, while the shear

stiffness (ks) is a tangent modulus, representing incremental displacement (ΔUs) and

force (ΔFs). Therefore, in each step, the shear component force increment is calculated

by

���

����� ×−= (2)

and added to the previous step’s shear contact force (Fis). The shear force is controlled

by the slip model defined as

��

��� ×=���

(3)

where Fsmax is the maximum shear force allowed; μ is the friction coefficient. If Fs >

Fsmax there is a slip and the shear force reduces to the maximum magnitude allowed.

Normal and shear stiffnesses are dependent on the selected contact model.

The motion of a single particle is described by a translation and a rotation

determined from the resultant moments and force acting upon it. The translation (x) is

determined with double integration of the Newton’s second law of motion

( )���

���� −×= �� (4)

where, Fi is the resulting force in i (1, 2, 3) co-ordinate directions; m is the particle total

mass; gi is the body force acceleration; �

��� is the particle translational acceleration. The

- 8 -

angular rotations about the principal axes of inertia of the particle are governed by

Euler’ equations of motion, as shown below:

���������������� −+⋅= � (5.a)

���������������� −+⋅= � (5.b)

���������������� −+⋅= � (5.c)

where M1, M2 and M3 are the components of the resulting moment on the principal axes;

��� ,

�

�� ,�

�� are the angular accelerations about the principal axes; I1, I2 and I3 are the

principal moments of inertia of the particle. The time integration scheme calculates the

velocities at Δt/2 while the positions and accelerations are determined at Δt.

Fig. 2. Particle-particle (a) and particle-wall (b) contacts notation (adapted from [35]).

In this study, the DEM modelling was developed with PFC2D that obeys the above-

mentioned conditions

[37]. In a two dimensional system there are only two force components and one moment

component. As PFC deals only with circular/spherical particles, the 2D modelled

assembly can be seen as a collection of variable-radius cylinders or, alternatively, as a

collection of variable-radius spheres whose centroids lie upon the same vertical plan.

The PFC includes two different bonding models, the contact-bond and the

parallel-bond, which can be implemented to model behaviour of contacts in tension.

The contact-bond corresponds to a point bond of infinitesimal size that can only

- 9 -

transmit force and inactivates the slip model while the parallel-bond is a finite-size

piece of cementitious material between particles that can transmit both forces and

moments, and does not preclude the possibility of slip. The parallel-bond can be

regarded as a set of elastic springs with normal and shear stiffness. In this study only

parallel-bonding was implemented. The constitutive behaviour is linear elastic and is

defined by the following five parameters: normal (kn) and shear (ks) stiffness

(stress/displacement); normal (σmax) and shear (τmax) strength (stress); bond radius (r).

Considering that the bond between particles corresponds to an elastic beam, the elastic

moduli in axial conditions is

[ ] [ ]( )���

���� +×= (6)

The PFC can deal with different contact constitutive models and several are already

included, e.g. the linear elastic, the Hertz-Mindlin, some viscoelastic models like the

Burgers model, displacement-softening model. In the present case study, the

delamination process of the FRP-to-parent material has the FRP composite under

tension whereas the interface contacts are subjected to positive shear loadings. The FRP

composite has a linear with fragile rupture constitutive behaviour which was modelled

with the parallel-bond model. The FRP-to-parent material interface was modelled using

the displacement-softening model. Fig. 3 illustrates these constitutive behaviours.

Fig. 3. Constitutive behaviours of the FRP composite and the contact between the FRP composite and

the bonded substrate.

- 10 -

In the case of the softening model, the contact between the FRP and the bonded

substrate is simulated similarly to the Cohesive Zone Model (CZM) in continuum

mechanics. The interfacial delamination is classified as Mode I, Mode II or Mixed

Mode (combination of Modes I and II) according to the stress field at the tip of the

crack. In the particular case of the delamination process of the FRP-to-parent material

under analysis, only Mode II is relevant. The Mode II fracture energy (GFII) released

during the delamination process is defined according to

�

��� �����

�

�

�

⋅= τ (7)

where Uult is the ultimate displacement (or slip) of the interface.

The simulation process is usually carried out with the following steps: (1)

assembly generation; (2) definition of contact models; (3) set-up of boundary

conditions, e.g. restriction of particles movements or assembly stresses; (4) loading; (5)

calculation; (6) analysis. Some of these tasks are not as straightforward as in other

numerical methods, especially the first one. The assembly generation consists of

positioning the collection of particles with various sizes and shapes in a specific

location and creating the contacts. The structure obtained from the particles and contacts

network determines the behaviour of the assembly to loading. It varies with the selected

combination of the particle’s size distribution and the method of bringing the particles

together in an assembly. Several methods for creating assemblies for DEM modelling

can be found in [37]. Some examples are illustrated in Fig. 4.

Fig. 4. Assembly examples used in DEM simulations with: quadrangular (a); linear (b); hexagonal (c);

and random (d) arrangements of the particles.

- 11 -

Loading is considered active when forces are defined to act on the particles’ gravity

centre or movements are imposed on particles (via wall movements or particle

displacements), while it is passive when disturbances are created in the assembly, e.g.

elimination and overlap generation of particles. The calculation phase is usually divided

into several parts with a certain number of calculation cycles and timestep values. Both

variables can either be defined by the user or determined automatically, with the

timestep given being a percentage of the predicted critical timestep that is related to the

maximum eigenvalue of the total system and, the required number of calculation cycles

can be set to stop the simulation when the sum of the out-of-balance forces is less than a

specified value.

Regarding the results analysis task, the contact forces and particle displacements

have a discrete distribution over the modelled medium. The computation of stresses and

strain-rates requires specific procedures as described in [37].

3. Modelling composite flat coupons under tension

The first step of the modelling with the DEM was to predict an FRP composite under

uniaxial tension. The GFRP composites consist of glass fibres impregnated by an epoxy

resin (matrix) but the proportions of each are not known. In order to reproduce its

behaviour in DEM, the final composite behaviour was used. For the assembly, different

arrangements were initially tested, namely the linear, the quadrangular with 2 layers and

the hexagonal, as there were no differences between them, for the uniaxial tension test,

the linear arrangement was adopted because of its simplicity.

The numerical results were compared with the Biscaia’s experiments [38] for

GFRP composites. The mechanical properties of the composite are listed in Table 1.

- 12 -

The radius of the particles was selected in each case such that the thickness of the GFRP

composite, 2.54 mm, is truly represented in the model (e.g. 2 layers, being 1.57 mm

each). The particles were bonded with the parallel-bond model. The simulation was

carried out in deformation control mode by defining a constant velocity for the particles

at the ends of the assembly. Given the high values of the contacts stiffness and the small

mass of the particles, the timestep adopted for the numerical calculations has to be tiny.

Timestep values in the range of 1e-9 to 1e-11 were tested, from which the value of 1e-

10 was adopted considering the balance between time of simulation and numerical

results accuracy.

Since displacement loading was always used in the modelling, the influence of loading

velocities on the solutions is analyzed by comparing the stress-strain curves from a

parallel-bond model submitted to different tensile loading velocities. From Fig. 5, the

range of loading velocities tested showed no significant changes on the global

constitutive behaviour of the GFRP composite. However, looking closer, the models

with lower loading velocities have more regular behaviour, as illustrated in the detailed

graph in Fig. 5. Even so, compared to the experimental data, the relative errors found

are less than 1.0%. The irregularities found in the cases with higher velocity loadings

can be attributed to the fact that the stresses are not uniformly distributed between

particles along the FRP length. Therefore, in the following simulations the adopted

deformation velocity was 0.05 to 0.10 m/s.

Fig. 5. Numerical and experimental results for the uniaxial tension tests.

4. Validation of the numerical modelling

In order to check the possibility of modelling the delamination process of FRP-to-parent

material interface with the Distinct Element Method (DEM) with PFC2D [35], the Pull-

- 13 -

out test shown in Fig 6 was modelled. The DEM model consists of a linear arrangement

of the FRP plate and the substrate in a two-layer system, with bonded length Lb, where

the parallel bond and the displacement softening models were adopted for interaction of

the two different particles. For the purpose of model validation the interface

performance was modelled using different contact model properties and the numerical

results were compared to exact solutions. To prevent undesired ruptures and/or stress

disturbances in the assembly, the loads were applied to the FRP plate with the minimum

velocity achieved in Section 3 (v=0.05 m/s). For the sake of simplicity, the constitutive

behaviour of the contacts between the FRP and the substrate elements were defined with

the linear and bi-linear laws.

Fig. 6. Pull-out test with no slip constrains at the opposite side of the loaded plate end with details of the

DEM used.

The governing equation of FRP-to-parent material interface for EBR systems is [24, [25,

[29-[31, [39[43]:

( ) ( )�

�

�

=⋅

−��

��

�

��

��� τ (8)

where Ef is Young's modulus of the FRP composite; tf is the thickness of the FRP

composite; s is the slippage and τ(s) is the shear stress defined by the local bond-slip

law; x is the horizontal distance to the FRP pull point. Linear bond-slip laws such as the

rigid-plastic, linear with a fragile rupture and rigid-softening linear laws were chosen to

validate the numerical modelling because, in such cases, Eq. (8) has an exact solution

- 14 -

and the numerical results can be compared to the exact solution. Independently of the

bond-slip law chosen, the boundary conditions are always the same, e.g. [29, [30]:

��

��

��

�

��

��

⋅−==

=�ε (9.a)

and

�===

���

�

��

�� ε (9.b)

where F is the load applied at the pull end of the FRP plate; Af is the cross section area

of the FRP composite; Lb is the effective bond length.

The full range delamination process of the FRP-to-parent material in a Pull-out

test is described in [24, [29, [44] with no slip constraints. The same behaviour is also

described in [41, [43] but in those cases the full range delamination process is based on a

Pull-out test that simulates two consecutive cracks, i.e. both ends of the FRP plate are

loaded.

Moreover, the snap-back effect after maximum load is reached, described in [24,

[44, [45], is a theoretical concept and, until now, has never been experimentally

observed. Mainly for that reason, the present work does not aim to represent such full

range delamination behaviour. Instead, as mentioned above, the aim is to understand the

capabilities of DEM in the estimation of the interface behaviour until maximum load

(Fmax) is reached. Up to this point, the delamination process can be represented by a

load-slip at x=0 mm graph. Depending on the bond-slip law used, the load increases

with the slippage of the interface. If the bonded length is equal to or longer than the

- 15 -

effective bond length (Leff) then when maximum load is reached a plateau is visible. If

the bonded length is shorter than the effective bond length then, the maximum load

transmitted to the FRP plate is lower than that obtained in the previous case and, at the

same time, no plateau can be seen in the load-slippage at x=0 mm graph. According to

equilibrium principle shown in Fig. 7, the load transmitted to the FRP plate is

determined according to:

( )∫ ⋅= ��

������

�

τ (10)

where bf is the width of the FRP plate and the slippage is determined by one of these

methods: (a) from the readings of a displacement transducer at x=0 mm; or (b) from the

measurements taken from strain gauges bonded to the FRP plate along the bonded

length. In this last case, the slippage at x=0 mm is obtained according to:

( ) ( ) ( )( )

( ) ( )��

��

��

������

�

��

��

�−⋅+

−⋅

−−

−= +++

+

++ ��

�

�

�

�

�

�

εεε (11)

where x corresponds to the axis parallel to the bonded length (see Fig. 7); (εi+1 - εi) and

(xi+1 - xi) are, respectively, the strain and the distance between two consecutive strain

gauges.

In the following sub-sections, the loads were monitored in two distinct ways: (a)

from the contact force between the first two particles of the FRP plate; and (b) from the

adhesion loads along the interface. Thus, the two terms of the first member of equation:

( )∫ Δ=⋅−⋅⋅ ��

����������

�

τε (12)

- 16 -

are constantly controlled throughout the delamination modelling. The second member in

Eq. (12) is the error obtained from the numerical measurements and, therefore, is the

error obtained in the modelling process. Ideally, Δ should be equal to zero (resulting Eq.

(10)) but in the validation of the numerical process the value of Δ is not exactly equal to

zero but is approximately zero.

The relative displacements between the FRP plate end and the substrate at x=0

mm were measured by the displacement monitoring of the first particle of the FRP

plate.

Fig. 7. Static equilibrium of the forces along the interface.

In the numerical examples presented in the following sub-sections, reference parameters

have been assumed unless otherwise stated, i.e. the thickness of the FRP plate is tf =

2.54 mm, the width is bf = 80 mm, the bonded length is Lb= 250 mm and Young’s

modulus is Ef = 23.35 GPa. For the definition of the bond-slip laws the reference

parameters are τmax = 4.24 MPa, smax = 0.101 mm and sult = 0.720 mm.

4.1. Rigid-plastic bond-slip law

In the first attempt to validate the model, the rigid-plastic bond-slip law was

implemented in the contacts between the particles of the FRP plate and the substrate.

The delamination process in such a case is described in [30]. The solution of Eq. (8) is:

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛ ⋅⋅−⋅+⋅−⋅

⋅⋅⋅

=��

�

�

����

��� ��

���

��

�����

�

���

��

ττ (13)

- 17 -

where a is the length where bond stresses are not zero and it can be determined by

considering that:

∫ ⋅=�

�����

�

������τ � � � � � � � � (14)

and, thereby [30]:

��

�

�

⋅=

���

τ (15)

The equations that define the bond stresses and the strains along the interface are

reported in [30].

Fig. 8 shows the load-slippage curve, as well as the slippage, the strains in the

FRP composite and the bond stresses along the bonded length. The numerical model

estimated accurately the exact solution derived from de solution of Eq. (8), which

proves that the performance of an interface with a local behaviour described by a rigid-

plastic bond-slip law can be modelled with DEM. Particularly the Fmax/FmaxT vs. Leff/Leff

T

graph, which compares the ratio between the maximum load transmitted to the FRP

plate obtained from the numerical analysis (Fmax) with the theoretical maximum load

transmitted to the FRP plate FmaxT, as:

��

��

��

�

����� ⋅⋅⋅= ����

(16)

and the effective bond length (Leff) determined from the numerical analysis and the

theoretical effective bond length (LeffT) defined according to [29-[30]:

- 18 -

��

�

��

���

�

��

�

��

��

⋅⋅=

�

(17)

where GFII is the fracture energy of the interface defined as:

( )∫= ����

��

�����

�

τ (18)

In this graph the numerical results show that the normalized maximum load (Fmax/FmaxT)

is always lower than that proposed by Neubauer and Rostásy [19] and Teng et al. [46],

whose recommendations are, respectively:

⎟⎟⎠

⎞⎜⎜⎝

⎛−⋅==

���

�

���

�

��

�

�

�

�

�

�

�

���

���β (19)

and

⎟⎟⎠

⎞⎜⎜⎝

⎛⋅==

���

�

��

�

�

�

�

�

���

���

���

πβ (20)

The numerical solution showed that both proposals [19, [46] overestimate the values of

βL defined in Eqs. (19) and (20). Moreover, the effective bond length defined in Eq. (17)

was accurately predicted.

- 19 -

Fig. 8: Performance of the FRP-to-parent material when a rigid-plastic bond-slip curve is used.

4.2. Linear with fragile rupture bond-slip law

The contacts between the particles of the FRP plate and the substrate were then

modelled with the linear with fragile rupture bond-slip law. The delamination process in

such cases is fully described in [30]. The solution of Eq. (8) is now:

( ) ( )( )

�

�

����

��

���

�

��

⋅⋅−⋅

⋅⋅⋅⋅

=�

��

�

����

����

ααα

α (21)

where

����� ⋅⋅

=���

����

�

τα (22)

Fig. 9: Performance of the FRP-to-parent material when a linear with fragile rupture bond-slip curve is

used.

In this particular case, the Fmax/FmaxT vs. Lb/Leff

T graph shows that the results of the

numerical model are slightly higher than the proposals made by Neubauer and Rostásy

[19] and Teng et al. [46]. This might be due to the assumption made in [30] where it

was assumed that tanh(4)≈1, which may lead to conservative effective bond lengths. For

instance, assuming that tanh(2)≈1, the effective bond length would be one half of the

effective bond length obtained with Eq. (23) and the numerical results in the Fmax/FmaxT

vs. Lb/LeffT graph would be closer to the proposals of Neubauer and Rostásy [19] and

Teng et al. [46]:

- 20 -

��

�

��

���

�

��

�

��

��

�

�

⋅⋅= (23)

4.3. Rigid-softening linear bond-slip law

In this case, the contacts between the particles of the FRP plate and the substrate were

modelled with the rigid-softening linear bond-slip law. The delamination process in

such cases is described in [30] as well. Introducing this bond-slip law into Eq. (8) the

solution is:

( ) ( )( ) ���

���

�

�

��

���

�

�� +⋅

⋅−⋅⋅

⋅⋅⋅−=

�

��

�

���

���

ααα

α (24)

where

�������� ⋅⋅

= ����

�

τα (25)

and a is the length corresponding to bond stresses different from zero defined by the

following expression:

⎟⎟⎠

⎞⎜⎜⎝

⎛

⋅⋅⋅⋅⋅=

����������

�

�

��

������

�

αα (26)

From Fig. 10, the numerical results show, once more, that DEM results are in agreement

with the exact solution. In this case, the results shown in the Fmax/FmaxT vs. Lb/Leff

T graph

- 21 -

are quite similar to the proposal made by Teng et al. [46]. Note that the maximum load

transmitted to the FRP plate (FmaxT) is defined according to Eq. (16) and the theoretical

effective bond length (LeffT) is determined by [30]:

��

�

��

���

�

��

�

��

��

��

⋅⋅⋅= π

(27)

Fig. 10: Performance of the FRP-to-parent material when a rigid-softening bond-slip curve is assumed.

4.4. In the choice between bilinear or nonlinear bond-slip laws

Several authors, e.g. [19, [40-[42, [47], have analysed the performance of FRP-to-parent

materials based on bi-linear bond-slip laws. However, other authors, e.g. [24, [29, [25,

[48], use nonlinear bond-slip laws based essentially on Popovics' formula [49] or the

exponential bond-slip laws [50, [51] for the particular analysis of FRP-to-concrete

interfaces. Nevertheless, the implementation of bi-linear bond-slip laws in some

situations where the bonded substrate is concrete has provided adequate results, e.g. [24,

[52-[55]. These curves are all graphically represented in Fig. 11 and are mathematically

defined as follows:

(i) Bi-linear:

( )

⎪⎪

⎩

⎪⎪

⎨

⎧

≤≤−

+⋅−

≤≤

=���������

���

���

���

���

���

�

�

���

�

�

�

��

�

�

�

�

�

��

�

�

�

���

���

���

���

ττ

(28)

- 22 -

(ii) Popovics' formula [49]:

( )

( )�

�

�

�

�

�

��

⎟⎟⎠

⎞⎜⎜⎝

⎛+−

⋅=

���

������

�

ττ

(29)

where n is a constant and, in a preliminary analysis, Nakaba et al. [48] recommend

n=3.0.

(iii) Mentioned in Dimande's work [50]:

( )���

�

������

�

�

�

�

��−

⋅=ττ

(30)

(iv) Based on the proposal of Dai et al. [51]:

( ) ( ) ( )

⎟⎟⎠

⎞⎜⎜⎝

⎛−⋅=

⋅−⋅−������

�������

���

�

�

�

�

�

��

�

ττ

(31)

where maximum bond stress and maximum slip are determined, respectively, according

to:

�

�

���

���

����� ⋅⋅⋅

=ε

τ (32)

- 23 -

where εmax is the maximum strain in the FRP composite experimentally obtained from

the Pull-out test and

( )�

�

���

���

= (33)

where B is a parameter defined from the experimental strain vs. slip curve.

Fig. 11. Comparison between different nonlinear bond-slip laws and the bi-linear law.

At this point, it is important to mention that the contact models available in PFC2D [35]

are based on linear and bi-linear laws. Therefore, in order to validate the approximation

made with bi-linear bond-slip laws, the numerical results obtained with the DEM were

compared to the numerical strategy adopted in [29, [30]. Fig. 12 and Table 2 illustrate

this comparison. It should be noted that, in general, it makes no significant difference

which of the bond-slip laws is adopted. The greatest differences found are on the

maximum load transmitted to the FRP plate and on the bond stress distribution along

the interface corresponding to the softening stage of the interface. The main reason for

this can be explained by the same point: the softening part of the bond-slip law is much

softer than in the other nonlinear bond-slip laws. The area beneath the bi-linear bond-

slip law (see Eq. (18)) leads to the highest values of fracture energy of the interface,

which increases the maximum load transmitted to the FRP plate (Eq. (16)). At the same

time, during the softening stage of the interface, and for the same slippage value, the

bond stresses are highest when the bi-linear bond-slip law is used.

- 24 -

The relative errors found on the maximum load transmitted to the FRP plate and on the

effective bond length are not so high (less than 16% in the case of Fmax and less than

13% in the case of Leff as listed in Table 2) and therefore, the bi-linear bond-slip law

may be a good and easier alternative to the nonlinear bond-slip laws. For all the reasons

explained above, the modelling made in the following section was based on a bi-linear

bond-slip law.

Fig. 12: Performance of the FRP-to-parent material when linear and nonlinear bond-slip curves are

assumed.

5. Delamination process of pull-out tests using a mechanical anchor device

Some publications, e.g. [24, [56], analysed the interface performance when a mechanical

fixing device is placed on the opposite side of the loaded plate end, as schematically

shown in Fig. 13. The use of a mechanical fixing device allows the constraining of the

relative displacements (or slips) between the FRP and the bonded substrate, thus

changing the delamination process discussed previously in section 4. Still, the process

that leads to the complete delamination of the FRP-to-parent material bonded joints is

dependent on the bonded length. Therefore, two different cases lead to two different

delamination processes of the interface performances: (1) the bonded length is the same

as or longer than the effective bond length (Lb ≥Leff) or (2) the bonded length is shorter

than the effective bond length (Lb < Leff). The resulting delamination processes are

described next, where the bilinear bond-slip law was always assumed.

Fig. 13. Scheme of the FRP anchorage region related to the pull-out test with a mechanical anchor device

placed at the end of the specimen.

- 25 -

5.1. Delamination process 1: with Lb ≥Leff

In this sub-section, the delamination process is based on the Pull-out test represented in

Fig. 14. Particular, the case of Lb ≥Leff is analysed. Two forces are now applied to the

FRP plate: one is the load applied to the FRP plate and the other one is the reaction at

x=Lb due to the slip constraints at that point. This reaction is insignificant (~0) for the

initial stages of the delamination process because the FRP-to-parent material interface

behaves similarly to the situations where no slip constraints are imposed on the opposite

side of the loaded plate end. This discussion is already widely reported in literature, e.g.

[29-[31, [38-[40, [44, [48].

Fig. 14. Pull-out test with no displacements at the opposite side of the loaded plate with details of the

DEM used.

When the maximum load transmitted to the FRP plate is achieved, the bond stress field

starts to migrate from the loaded end to the opposite plate end. During this process, the

maximum load transmitted to the FRP plate does not increase because the bond stress

field is the same and equilibrium equation defined in Eq. (10) should be respected. The

novelty of the delamination process comes when the bond stress field arrives to the slip

constraint placed at x=Lb. From this stage forward the resulting delamination failure

process is briefly described.

Fig. 15 highlights the equilibrium of the forces along the interface and the static

equilibrium equation is derived:

- 26 -

( ) ������

��

�+⋅= ∫

�

τ (34)

where R is the reaction force in the simple rolled support placed at x=Lb. Of course, Eq.

(34) is independent of the local bond-slip law and this equation is valid for any bonded

situation like that represented before. Moreover, Eq. (34) is always the same throughout

the delamination process and, in reality, it describes the problem more completely than

the one defined mathematically through Eq. (10).

Fig. 15. Static equilibrium of the forces along the interface.

The delamination failure process is now characterized by an increase of the load applied

to the FRP plate whereas the quantity obtained from the integral (I) defined in Eq. (34)

starts to decrease. The length where the bond stresses were different from zero becomes

shorter than Leff. The remaining bonded length becomes equal to the softening bonded

length (corresponding to the softening bond-slip law) and during the delamination

process, it approximates to zero. At the same time, the reaction force R increases in the

same proportion to the decrease of the integral in Eq. (32). The results are schematically

illustrated in Fig. 16. Note also that the stiffness (K) in this figure is:

�

��

�

��

�

⋅= (35)

which is the axial stiffness of the FRP plate under tension. Therefore, if the bonded

length increases then the represented plateau for maximum load transmitted to the plate

in Fig. 16 increases and K decreases. On the other hand, decreasing the bonded length

- 27 -

until Lb=Leff, only two stages are observed because the constant plateau found before

will no longer appear in this case.

Fig. 16. Performance of the interface at different loading stages for Lb ≥Leff.

5.2. Delamination process 2: with Lb < Leff

The delamination process 2 (bonded length is shorter than the effective bond length)

starts in the same way as the debonding failure process 1, i.e. for low loads, the reaction

force R is zero and during this phase the delamination process is equal to that which

occurred in a Pull-out test with no slip constraints. This subject is also discussed in

literature and therefore, the reader is advised to read e.g. [28-[30, [41-[44]. From the

instant that the bond stresses become different from zero in the vicinity of the slip

constraint, the delamination process becomes different from that discussed in [28-[30,

[44]. Here, two different cases can occur: the bonded length is in elastic and softening

stages (see Fig. 17 - case (a.1)); or the bond length is in an elastic stage (see Fig. 17 -

case (a.2)). From the moment either one of these cases occurs, the maximum load

transmitted to the FRP plate increases with the increase of the reaction load. However, if

at this stage the maximum load has not yet been reached, then the integral in Eq. (32)

continues increasing; otherwise this integral decreases. Either situation will depend on

the bonded length of the interface.

When the ultimate slip (sult) is reached at x=0 mm, the debonding of the interface

begins and increasing the load transmitted to the plate will increase the unbonded length

(Ldb) of the interface (see Fig. 17 - case (b)) where part of the interface begins to

debond. Of course, the length of the interface with bond stresses different from zero is

now shorter than the initial bonded length (Lb,i). This process will evolve until the

- 28 -

maximum bond stress is attained at the point x=Lb (see Fig. 17 - stage (c)). Here the

remaining bonded length is equal to corresponding bond of the softening bond-slip law

(Lb,sof). From this stage on, this bonded length decreases and, the load and the reaction

force will tend to the same value meaning that the FRP plate is fully debonded from the

substrate (see Fig. 17 - stage (d)). Regarding the load-slip at x=0 mm represented in Fig.

17, the ultimate stage of the delamination process is characterized by the tendency of

the load transmitted to the FRP plate (F) and the reaction force (R) becoming equal to a

stiffness equal to the axial stiffness of the FRP plate (K) earlier defined in Eq. (33).

Fig. 17. Performance of the interface at different loading stages for Lb < Leff.

Despite the reduced bonded length of the interface (less than the effective bond length),

two aspects are relevant to highlight in this delamination process: (i) at point x=Lb, the

slip is always zero and consequently the bond stress is zero due to the adopted bond-slip

law for FRP-to-parent material interface; and (ii) the FRP composite may collapse

before attaining the upper stage in the vicinity of stage (c) (see Fig. 17).

5.3. Examples

In order to illustrate the delamination processes discussed in the previous section, the

FRP-to-parent material interfaces were modelled with various bonded lengths (Lb),

namely, 50, 100, 150, 200 and 250 mm. Fig. 18 shows the load vs. slip at x=0 mm for

all bonded lengths obtained from the DEM modelling. The numerical results confirmed

what was described before, i.e. for short bonded lengths the stiffness is higher than that

obtained for long bonded lengths. Thus, if an ultimate slip (sult) of the interface is fixed,

i.e. if the initiation of the detachment of the FRP plate from the substrate is defined by

the ultimate slip, the debonding load (corresponding to the initiation of the detachment

- 29 -

of the FRP plate from the substrate) is much higher on short bonded lengths than on

long bonded lengths. Nevertheless, the strength of the FRP composite should be taken

into account because whilst premature debonding is avoided, rupture of the FRP

composite may occur.

Fig. 18: Delamination process of FRP-to-parent material for several bonded lengths.

Fig. 19 illustrates the delamination processes 1 (Lb= 250 mm) and 2 (Lb= 100 mm) for

stages a to d showing the bond stresses and the stress in the FRP composite. The

numerical examples show that with the increase of the transmitted load to the FRP

composite, the length of bond stresses tends to decrease while the reaction force tends to

increase.

Fig. 19: Bond and FRP composite stresses in the interface at different stages of the delamination process

of the FRP-to-parent material interface.

Another important aspect is that when the bonded length is short, the reaction force has

a fast increase for low loads transmitted to the FRP plate without promoting the

debonding of the FRP composite from the substrate. For designers, this aspect is

important and the design load is determined according to:

⎪⎩

⎪⎨⎧

≥⋅⋅⋅⋅

<⋅⋅⋅⋅⋅=

������

��

����

������

��

�����

��

������

������

�

���

���

β

ββ (36)

where Fdb is the load corresponding to the initiation of the detachment of the FRP plate

from the substrate; βL can be determined from Eqs. (19) or (20); and βdb is a coefficient

- 30 -

that takes into account the effect of the mechanical anchor on the load transmitted to de

FRP plate. The βdb coefficient was based on the trend line of the post-debonding

behaviour of the FRP-to-parent material interface with different bonded lengths

depicted in Fig. 20. In this figure, the bonded lengths with bond stresses different from

zero (Lbτ) were monitored throughout the delamination process of the interface and the

results show that Eq. (36) is able to predict the debonding initiation for all interfaces.

Thus, the βdb can be determined according to the following expression:

⎪⎩

⎪⎨

⎧

≥

<⎥⎥⎦

⎤

⎢⎢⎣

⎡+⎟⎟⎠

⎞⎜⎜⎝

⎛×⋅=

����

����

�

���

��

��

��

�

�

���

������

�

���

τβ (37)

where Lbτ is the bonded length with bond stresses different from zero. For short bonded

lengths (Lb < Leff), Lbτ is equal to the bonded length (Lb) because, in such cases, when the

debonding initiation occurs, all the bonded length is being mobilized.

Fig. 20: F/FmaxT vs Lb

τ/Leff relation along the delamination process of the FRP-to-parent material interfaces

for different bonded lengths of 50, 100, 150, 200, and 250 mm.

6. Conclusions

The numerical study herein presented is based on DEM and reports preliminary

numerical results about the delamination behaviour of FRP-to-parent material

interfaces. The study is based on a 2D analysis that allows us to come to the conclusions

summarized below:

- 31 -

• The uniaxial tensile test simulation results of the FRP composite showed that

different regular arrangements for the assembly can be used and, the loading

velocity affects the force-displacement path along the FRP length; and adequate

combination of loading velocity and timestep values is important for the numerical

model stability and the results accuracy;

• Numerical modelling with DEM was validated by comparing the numerical results

with the analytical expressions derived from the linear bond-slip models. The Pull-

out test was assumed and no slip constraints were considered. Maximum loads

transmitted to the FRP plate, strains, bond stresses or the slippage along the bonded

length were well estimated by the model. Moreover, the effective bond length was

another parameter that was also well predicted with the DEM;

• A mechanical anchor system was modelled by constraining the slippage at the end

of the specimen. In such circumstances, the results showed an increase in the

maximum load transmitted to the FRP plate in the delamination process.

Theoretically, the bonded length tends to zero whilst the load increases. However, it

can be concluded that when this length is too short, the stiffness of the system is

quite similar to the FRP axial stiffness under tension;

• The DEM proved it is capable of modelling the delamination process of FRP-to-

parent material in both cases, i.e., Lb < Leff and Lb ≥ Leff;

• Based on the numerical findings, a proposal for the design of bonded joints between

FRP composites and the substrate was made. The maximum load transmitted to the

FRP plate is limited by the initiation of the detachment of the composite from the

substrate, preventing the premature debonding phenomenon. Of course, the

debonding force should be limited by the FRP tensile strength.

- 32 -

The results obtained in this study provide an incentive to move forward with the

application of DEM to model the FRP-to-parent material debonding phenomenon. It is

intention of the authors to model this phenomenon using a 3D DEM formulation in the

future.

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- 36 -

[49] Popovics, S. A numerical approach to the complete concrete stress-strain relation for concrete.

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- 37 -

Table 1. Mechanical properties of the GFRP composites [38]. GFRP flat

couponεfu(1)

(%)

εf m(2)

(%)

ff(3)

(MPa)ffm

(4)

(MPa)Ef

(5)

(MPa)Ef m

(6)

(MPa)

No. 1 2.25 2.20 487.2 513.9 22.14 23.49 No. 2 2.20 507.5 23.09 No. 3 2.14 551.1 25.78No. 4 2.22 509.9 22.96

(1)εf – rupture strain; (2)

εf – average rupture strain; (3) ff – tensile strength; (4) ff – average tensile strength; (5) Ef – Young’s modulus; Ef m – average Young’s modulus.

Table 2: Numerical results obtained from the nonlinear bond-slip laws defined in Fig. 12. Bond-slip law Maximum

load,Fmax (kN)

Relativeerror(%)

Fractureenergy,

GF (N/mm)

Relativeerror(%)

Effectivebond length,

Leff (mm)

Relativeerror(%)

Bi-linear 33.72 - 1.526 - 240 - Popovics' formula [49] 28.39 -15.81 1.231 -19.33 230 -4.17 Mentioned by Dimande [50] 29.32 -13.05 1.170 -23.33 225 -6.25 Proposal of Dai et al. [51] 30.16 -10.56 1.236 -19.00 210 -12.50

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