Micromechanical modeling of the constitutive response of FRCM composites

Construction and Building Materials 236 (2020) 117539

Contents lists available at ScienceDirect

Construction and Building Materials journal homepage: www.elsevier.com/locate/conbuildmat

Micromechanical modeling of the constitutive response of FRCM composites Francesca Nerilli a,⇑, Sonia Marfia b, Elio Sacco c a

Università degli Studi Niccolò Cusano – Telematica Roma, Via Don C. Gnocchi 13, 00166 Roma, Italy Dipartimento di Ingegneria, Università degli Studi di Roma Tre, Via Vito Volterra 62, 00146 Roma, Italy c Dipartimento di Strutture per l’Ingegneria e l’Architettura, Università di Napoli ‘‘Federico II”, Via Claudio 21, 80125 Napoli, Italy b

h i g h l i g h t s
The constitutive behavior of FRCM composites is numerically investigated.
The proposed micromechanical model accounts for the nonlinear micromechanisms.
The cracking and the slippage of the fibers within the mortar are taken into account.
The shear behavior of FRCM material is investigated, also under cyclic loading.

a r t i c l e

i n f o

Article history: Received 2 July 2019 Received in revised form 12 September 2019 Accepted 7 November 2019

Keywords: FRCM Micromechanics Nonlinear interface

a b s t r a c t In this paper, the tensile and shear constitutive behavior of Fiber Reinforced Cementitious Matrix (FRCM) composites is addressed. A nonlinear finite-element computational approach is developed, by accounting for the micromechanical mechanisms via the introduction of interface elements inside and between the FRCM constituents, i.e. fibers and mortar. The damage, the friction and the unilateral contact are considered for reproducing the mortar cracking and the slippage between the fibers and the mortar. The numerical approach is illustrated and it is validated with an available experimental result, highlighting the effectiveness of such proposed modeling approach for reproducing the constitutive tensile behavior of FRCM materials. Finally, the shear behavior is investigated, also considering cyclic loading patterns under different compressive load levels. Ó 2019 Elsevier Ltd. All rights reserved.

1. Introduction In the field of structural engineering, in the last decades, the use of innovative materials has widely spread. In detail, mainly in the framework of retrofitting of existing structures, the attention of the scientific community and of the manufacturers was focused on materials ecocompatible with the supports, in particular for the structures of historical and monumental interest. At this aim, the FRCMs (Fiber Reinforced Cementitious Matrix) can be considered a good alternative due to their good properties, such as good high fire resistance, permeability, applicability on wet surfaces, reversibility, easiness and reduced costs of installation. They are made by two layers of mortar in which a fabric mesh, with spaced fibers disposed along two orthogonal directions, is embedded. The production procedure, that foresees the laying of the mortar and of ⇑ Corresponding author. E-mail addresses: [email protected] (F. Nerilli), [email protected] (S. Marfia), [email protected] (E. Sacco). https://doi.org/10.1016/j.conbuildmat.2019.117539 0950-0618/Ó 2019 Elsevier Ltd. All rights reserved.

the fiber mesh properly compacted, allows to the mortar to pass through the mesh. This ensures the load transfer between the mortar and the fibers. Different textile materials can be adopted, such as carbon, AR-glass, basalt, steel, PBO, aramid fibers, with or without coating, that, as well as the different cement-based or limebased mortar, confer specific mechanical properties to the composite. In the last years, many experimental works have been carried out in order to investigate both the tensile behavior of FRCM specimens [1–7] and the debonding phenomenon between FRCM from masonry [8–11] or concrete members [12]. In the framework of the tensile constitutive behavior, differently from the most known FRPs (Fiber Reinforced Polymers), FRCMs present a trilinear stress–strain law, defined by three mechanical phenomenon: the cracking of the mortar, the slippage of the fiber within the mortar and the failure of the specimen. This latter can occur due to the fiber failure or to a sudden load loss caused by the fiber-mortar interface failure. Two different tensile test set-ups have been identified: the clamping set-up and the clevis

2

F. Nerilli et al. / Construction and Building Materials 236 (2020) 117539

set-up, differing for the way of gripping method [5,13]. Differently from the clevis test set-up, with the clamping testing procedure the specimen is gripped at the extremities applying an orthogonal compressive pressure that results in the Coulomb friction through which the load in transferred to the specimen. This test set-up, does not allow the slippage of the fibers in the clamped zones [5]. These two different testing procedures result also in a different stress-strain law. Nevertheless, with the clamping method it is possible to evaluate the effective constitutive tensile behavior of the FRCM material, due to the homogeneous distribution of the tensile stresses within the specimen cross-section. In this context, aim of the numerical application is to reproduce the typical tensile behaviour of a representative unit cell of the FRCM material far from the clamped zones. Together with the experimental investigations, some numerical approaches have been proposed. Mobasher et al. [14] suggested an incremental procedure based on the laminate theory that considers the crack evolution and the stiffness degradation. In [15] the tensile stress-strain behavior is simulated by means of a reverse engineering approach: starting from the modeling of an ideal yarn and comparing the results with the experimental evidences, the influence of microscopic deficiency mechanisms on the macroscopic behavior is evaluated. In [16] a reduced two-dimensional numerical model is presented. Leurini et al. [17] proposed a nonlinear numerical procedure to reproduce the global stress-strain law of FRCM specimens. In [18] the authors used commercial codes for reproducing numerical results of FRCM tensile behavior and for identifying the mechanical model of the cementitious matrix. When applied on masonry or concrete structures, the debonding failure between the composite and the support is an important issue to investigate. Experimental evidences show that the FRCMs exhibit a different debonding failure modes with respect to the most common FRP, for which a cohesive debonding that interests the support can occur [19–23]. As a matter of fact, the progressive damage of the mortar, after the achievement of its tensile strength, significantly influences the shear-stress transfer between the constituents and the most frequent failure mode happens due to the failure of the fiber-mortar interfaces with the slippage of the fibers within the mortar [9]. The debonding mechanism between FRCM and masonry panels are analytically and numerically investigated in some recent papers [8,24–31]. In [24] a cohesive interface crack model is presented to simulate the load-displacement curve of an experimentally-tested FRCM-masonry system, taking into account the interlocking between the textile and the matrix and the friction phenomena. The influence of the fiber-mortar interface on the debonding FRCM-masonry behavior is taken into account in [29], where a simple theoretical model is proposed. With the same aim, a meso-scale approach is adopted in [26] and implemented in a 3D finite element model. Less are the analytical and numerical contributions for the study of the debonding phenomenon between FRCM composites and concrete supports [32–35]. The constitutive tensile behavior of FRCM composites and the debonding phenomenon between FRCM and the supports are key topics when the behavior of structural strengthened members want to be evaluated. Nevertheless, some strengthening techniques concern the application of FRCM materials on structural elements subjected to shear load, i.e. masonry walls subjected to in-plane loads. In such a cases, an accurate numerical modeling should also take into account the constitutive shear behavior of the strengthening material. To the best of authors’ knowledge, there is no experimental works that investigate this feature. Thus, the aim of this work is to numerically evaluate the shear behavior of a carbon-based FRCM composite. In this paper a micromechanical nonlinear numerical approach is defined, that takes into account the damage and the friction

phenomena between the FRCM constituents. Firstly, the FRCM tensile behavior of an experimentally-tested specimen is modeled. The numerical model is validated in good agreement with the experimental results and it is used to evaluate the shear behavior of the same FRCM material.

2. Modeling of FRCM composite material Fiber Reinforced Cementitious Matrix (FRCM) is made of two external layers of mortar and an inner layer of textile mesh. The mesh is composed by fibers laid in two preferential orthogonal directions, i.e. the warp and the weft direction. The final thickness of this composite material, generally of about 10 mm, is small with respect to its in-plane dimensions. Experimental investigations for classical tensile tests [36–40], carried out adopting optical and scanning electron microscopy, have shown that the mechanical response of the material is significantly influenced by the cracking nucleation and evolution in the mortar and at the fiber-mortar interface, followed by the slippage of the fibers within the mortar. Thus, it is very interesting and challenging to derive the response of the FRCM, through a micromechanical approach, i.e. performing a nonlinear homogenization. A sketch of the FRCM material is illustrated in Fig. 1. As it is shown in the figure, the FRCM is characterized by a repetitive micro-structure that allows to define a unit cell (UC), with dimensions d1 and d2 along the two orthogonal directions. A coordinate system x1 ; x2 is introduced in the UC. The FRCM UC can be modeled as a three layer laminate, with the two mortar outer layers denoted as Xm ¼ A t m , and the inner layer Xf ¼ A tf , made of the fibers and of the mortar surrounded by the frame of fibers, with A the in-plane area of the UC. The two mortar outer layers have both thickness tm , while the inner layer is characterized by a thickness equal to size of the fibers, tf , as schematically illustrated in Fig. 2. Actually, the modeling interest concerns only the in-plane behavior of FRCM, neglecting the out-of-plane response of the laminate, developing the analysis in the framework of the twodimensional (2D) small strain regime. In fact, the FRCM laminate is modeled as three superimposed layers in plane stress state, with each layer consisting in a membrane structural element. In order to account in a simple and effective way for the material nonlinearities, on the basis of the experimental evidences, the fracturing process is modeled introducing suitable interfaces, located in the positions where experiments show potential fractures. Thus, the considered FRCM unit cell is characterized by:
the cementitious mortar, modeled as linear elastic isotropic material with elastic modulus Em and Poisson’s coefficient mm ;
the fibers in the warp or in the weft directions, modeled with linear elastic isotropic material with elastic modulus Ef and Poisson’s coefficient mf ;
the mortar-mortar and the fiber-mortar interfaces, denoted as I1 , to simulate the fiber–matrix decohesion in the inner layer of the laminate and the matrix fracturing process, as illustrated in Fig. 3(a);
the fiber-mortar interfaces, denoted as I2 , to simulate the detachment of the fibers from the outer mortar layers, as shown in Fig. 3(b). In particular, fiber-mortar interfaces I1 are introduced in the inner layer along the surfaces separating the fiber from the matrix. Mortar-mortar interfaces are placed in the outer layers along the directions corresponding to the fiber boundaries. The presence of these interfaces allows to reproduce the cracking when the UC is subjected to tensile stresses along the fiber directions. Moreover,

F. Nerilli et al. / Construction and Building Materials 236 (2020) 117539

3

Fig. 1. Sketch of the FRCM material and identification of the repetitive unit cell.

Fig. 2. Sketch of the FRCM as a three layers laminate.

Fig. 3. Definition of the interface type: I1 in-plane interface and I2 between layers interface.

diagonal mortar-mortar interfaces are also introduced into the UC in the three layers. Of course, the length of the diagonal interfaces in the inner layer is lower than the ones in the outer layers because of the presence of the fibers, as shown in Fig. 3(a). The diagonal

interfaces are activated when the UC is subjected to shear stress. It can be remarked that, as each layer is modeled as a membrane, the I1 interface is a link between two lines. Interface I2 joins the layers each others, resulting as a link between two surfaces.

4

F. Nerilli et al. / Construction and Building Materials 236 (2020) 117539

Denoting by E ¼ fE11 E22 C12 gT the overall strain prescribed on the UC in the x1  x2 plane, the micromechanical and homogenization problem consists in determining the overall stress R ¼ fR11 R22 R12 gT representing the mechanical response of the FRCM. In the typical layer of FRCM, the displacement field u ¼ fu1 u2 gT is represented in the form:

" u ð x1 ; x2 Þ ¼

x1

0

0

x2

1 x 2 2 1 x 2 1

# ~ ðx1 ; x2 Þ; Eþu

ð1Þ

~ ¼ fu~1 u~2 g is a perturbation displacement field arising where u because of the heterogeneity of the UC, due to the different constituent materials, i.e. fiber and mortar, and to the presence of interfaces. Furthermore, for the periodicity of the considered composite ~ is the periodic part of the displacement field. material, u Taking into account that the three layers are subjected to the same average strain E, the overall strain can be evaluated considering only one of the three layers as, for instance: T

1 E¼ A

8 > n1 Z < 0 S > : n2

9 0= > n2 u ds; > ; n1

ð2Þ

where S is the boundary of A and n1 and n2 are the components of the outer normal vector to the boundary of the mortar layer. Note that even if the average strain is equal in each layer, the average stress is different in each layer, depending on the different constituents; in fact, it results:

R¼

2 A tm

Z At m

r dV þ

1 A tf

Z At f

r dV;

ð3Þ

Fig. 4. Local interface coordinate system.

tangential to the interface and N is the direction orthogonal to its plane. In the local system, the relative displacement can be defined  T with three components as s ¼ sT 1 sT 2 sN . The traction in the undamaged and damaged part of the representative area is evaluated as:

su ¼ K s   sd ¼ K s  si

2

KT

0

6 with K ¼ 4 0

KT

0

0

0

3

7 0 5; KN

ð4Þ

where si is the inelastic relative displacement arising on DAd . Note that an isotropic behavior in the interface plane is assumed. The stress su is due to the elastic behavior of the undamaged area DAu , while the stress sd is due to unilateral contact as well as friction phenomena, occurring in the damaged part DAd . The T inelastic relative displacement si ¼ fp ^cg accounts for the slip

where r is the value of the local stress in the UC.

p ¼ fp1 p2 gT due to the frictional effects and for the unilateral contact. The latter effect is evaluated setting the contact displacement as:

3. Interface model

^c ¼ HðsN ÞsN ;

The interface is defined as the system of two separated surfaces, Sþ and S occupying the same set, so that the interface has zero thickness. At a point of the interface, two different displacements can occur, as they correspond to the displacements of the point belonging at the two different surfaces, i.e. uþ ðx1 ; x2 Þ is the displacement in the point on Sþ and u ðx1 ; x2 Þ is the displacement in the point on S . The kinematics of the interface is defined by the relative displacement sðx1 ; x2 Þ ¼ uþ ðx1 ; x2 Þ  u ðx1 ; x2 Þ of the two surfaces. The work conjugate quantity to the relative displacement is the traction s associated to the surface of the interface. The traction s can be evaluated through the stress r associated with the interface plane with normal unit vector n. The relationship between the relative displacement s and traction s defines the constitutive law of the interface. In the following, a constitutive interface model considering the presence of damage, friction and unilateral effects that occur due to the opening and possible reclosure of micro-cracks, is presented on the basis of the approach proposed in [41,42] in the general framework of the three-dimensional problem. Then, the model will be specialized to the I1 and I2 interface-types of interest. At the typical point of the interface, the damage parameter D is introduced as the ratio between the size of the micro-cracks, DAd , and the representative area DA, i.e. D ¼ DAd =DA, so that 0 6 D 6 1. Consequently, the undamaged area can be determined as the difference between the representative area and its microcracked part, i.e. DAu ¼ DA  DAd . The local coordinate system of the interface is schematically represented in Fig. 4, where T 1 and T 2 are the in-plane directions

ð5Þ

where the symbol HðsN Þ denotes the Heaviside function, which assumes the following values: HðsN Þ ¼ 1 if sN > 0 and HðsN Þ ¼ 0 otherwise. The classical Coulomb yield criterion is considered in order to account for the evolution of the inelastic slip occurring in DAd :

    / sd ¼ l sdN  þ ksdT k;

ð6Þ

where l is the friction coefficient, the symbol h:i denotes the negative part of the number, and sdN is the component of the stress in n oT the normal direction, while sdT ¼ sdT1 sdT2 is the stress vector in the tangential plane of the interface. The evolution of the relative inelastic displacement vector p is governed by the non-associated flow rule:

p_1 ¼ c_

sdT1 ksdT k

p_2 ¼ c_

sdT2 ksdT k

ð7Þ

where c_ is the plastic multiplier and the following Khun-Tucker conditions have to be fulfilled:

 

 

c_  0; / sd  0; c_ / sd ¼ 0:

ð8Þ

The damage parameter D is evaluated considering both mode I and mode II. Indeed, two quantities that depend on the mechanical parameters of the interface in both the tangential and in the normal directions are introduced:

gT ¼

s0T s0T ; 2GcT

gN ¼

s0N s0N ; 2GcN

ð9Þ

5

F. Nerilli et al. / Construction and Building Materials 236 (2020) 117539

where s0T ; s0N are the peak stresses corresponding to the first cracking relative displacement s0T ; s0N and GcT ; GcN are the specific fracture energies in mode II and mode I, respectively. Because of the in-plane isotropic behavior, the parameter gT is equal in all the tangential directions. Then, a parameter g that couples the two fracture modes is introduced:

g¼

1h

a

2

i ksT k2 gT þ s2N gN ;

with a ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ksT k2 þ hsN i2þ ;

ð10Þ

 T where sT ¼ sT 1 sT 2 . Moreover, the following equivalent relative displacement Y is evaluated as:

Y¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Y 2T þ Y 2N :

ð11Þ

with:

YT ¼

ksT k ; s0T

YN ¼

hsN iþ : s0N

ð12Þ

The damage parameter evolution is governed by the following equations:

n n  oo D ¼ max 0; min 1; D history



D¼

Y 1 : Y ð1  gÞ

ð13Þ

Considering the damage parameter and the stress vector in the undamaged and damaged part of the representative area, the overall interface stress vector, denoted by s, is computed as:





s ¼ ð1  DÞsu þ Dsd ¼ K s  Dsi :

ð14Þ

The described model is considered for the implementation of both the I1 and I2 interfaces, as defined in Section 2, for the modeling of the FRCM unit cell. In detail, the interface I1 is characterized by a behavior that takes into account both damage and friction. The I1 interface is defined in the plane of each FRCM layer. In this case the interface joins two lines in the two-dimensional model, as represented in Fig. 3(a). Thus, the local coordinate system is reduced to only one tangential direction and by the normal direction, T and N, respectively, setting, for instance, T=T 1 . The numerical model, is, hence, simplified. The stiffness matrix becomes the 2  2 diagonal matrix K1 = diagðK T ; K N Þ. In the equations above introduced, the norms of the tangential stress and of the inelastic relative displacement coincide with the absolute values of the nontrivial components, i.e. ksdT k ¼ jsdT j and ksT k ¼ jsT j, respectively. The interface type I2 is implemented considering that only fracture mode II can happen, reproducing the sliding between the FRCM layers, as illustrated in Fig. 3(b), as opening (i.e. delamination) is neglected in the actual model. Consequently, only the tangential components of the relative displacement, denoted with the subscript T 1 and T 2 and lying in the x1  x2 plane, are considered, as shown in Fig. 4. The mechanical behavior of the interface is governed only by the damage phenomenon. Thus, the constitutive law, in terms of traction and relative displacement vectors, is given by the equations:

su ¼ K2 s; sd ¼ 0;

ð15Þ

where stiffness matrix is given by K2 = diagðK T ; K T Þ, with s ¼ sT and s ¼ sT . The equivalent relative displacement, introduced in Eq. (11) becomes Y ¼ Y T .

3.1. Numerical procedure The evolutive equations, governing the interface behavior above described, is solved developing a step-by-step time-integration algorithm. In particular, the analysis process is subdivided into a finite number of steps. Once the solution at time tn is determined, the solution at the actual time tnþ1 ¼ tn þ Dt is evaluated adopting a backward-Euler implicit integration procedure. In the following, the quantities relative to the previous time step tn are denoted with the subscript n, whereas the quantities at time t nþ1 are without subscript. The analysis of the progressive damage of the interfaces involves the above mentioned material models, consisting in an elasto-damage problem that is herein numerically implemented via a nonlinear finite element approach. Assuming the relative displacement at the actual time to be known, the damage parameter D is evaluated from Eq. (13). If D ¼ 0, the interface is in elastic regime and DAd ¼ 0 and DA ¼ DAu ; thus, the overall stress is s ¼ su . Otherwise, if D > 0, the stress vector depends on the stress computed in both the undamaged and damaged part. In this last case, the contact displacement ^c is computed by Eq. (5) and, then, a predictorcorrector technique is adopted to evaluate the inelastic slip relative displacement p. A trial state is determined from Eqs. (4) and (6), considering the slip relative displacement obtained at the previous time step, as:



: D E   / sdtrial ¼ l sdN;trial þ ksdT;trial k:

sdtrial ¼ K s 



pn ^c

ð16Þ ð17Þ



  If / sdtrial 6 0, the trial state is the solution of the time step; thus, the inelastic slip p at the current time step is equal to pn evaluated at the previous time step, and sd ¼ sdtrial . Finally, the overall stress in the interface at time t is evaluated by Eq. (14). In the case   / sdtrial > 0, the inelastic slip relative displacement is evaluated as:

p ¼ pn þ Dp;

with Dp ¼ Dc

sdT : ksdT k

ð18Þ

The corrector phase is performed adopting a Newton-Raphson-type iterative procedure; to this end, the equations governing the evolution problem are written in the residual form:

8 sdT > > > Rp ¼ Dp  Dc ksdT k ¼ 0 > > > > > ( )! < p n þ Dp d ¼ s  K s  R ¼ 0: s > > > ^c > > > > > : R/ ¼ lsdN þ ksdT k ¼ 0

ð19Þ

 T  T Defining R ¼ Rp Rs R/ and p ¼ Dp sd Dc , the typical Netwton iteration is:

Rjpkþ1 ¼ Rjpk þ

@R

dp ¼ 0; @ p pk

ð20Þ

where k is the iteration number, dp is the increment of p between two consecutive iterations and

@R h @R ¼ @p @p with:

@R @ sd

@R @c

i

ð21Þ

6

F. Nerilli et al. / Construction and Building Materials 236 (2020) 117539

2

2

1 60 6 6 KT @R 6 ¼6 @p 6 60 6 40 0

0

3

1 7 7 7 0 7 7 KT 7 7 7 0 5 0

0

3

1

2 d 6 Dc B s T 1 C 6 ksd k @ ksd k2  1A 6 T T 6 6 6 6 6 1 d d 6 d 3 sT sT Dc @R 6 6 ksT k 1 2 ¼ @ sd 6 6 6 1 6 6 6 0 6 6 0 6 4 sdT 1

7 07 2 d 3 7 sT 1 7 7 6 ksdT k 7 0 2 1 7 7 6 7 6 sd 7 sd 6 T2 7 C 7 Dc B T 2 6 d 7 @ ksd k2  1A 0 7 7 @R 6 ksT k 7 ksdT k T 7 7 6 7 @ c ¼ 6 0 7: 7 7 6 6 0 7 0 07 7 7 6 7 7 6 4 0 5 1 07 7 7 0 0 17 5 sdT 1 ksdT k3

sdT1 sdT2 Dc

2

ksdT k

ksdT k

l

ð22Þ

Once the problem (20) is solved, i.e. when kRk < tolerance, the interface stress is computed using the Eq. (14). The numerical procedure is schematically described in the flowchart represented in Fig. 5. The evaluation of the consistent tangent stiffness matrix is performed differentiating the traction s with respect to the relative displacement s. From Eq. (14), it results:

Kt ¼

  @s @D @si : D ¼ K I  si  @s @s @s

ð23Þ

From the Eq. (9)–(13) the partial derivative @D=@s is:

8 > > <0

@D ¼ @s > > : @D

@Y @Y @s



if þ @D @g

@g @s

if

D 6 Dn

! D ¼ Dn

DP1

!D¼1



Dn < D < 1

@Y YN ¼ HðsN Þ; @sN Ys0N

@g 2s2 ðg  g Þ ¼ N T 4 N sT 1 ; @sT 1 a

ð27Þ

@Y 2sT ¼  2 ; @sT 2 Y s0 2 T

@g 2s2 ðg  g Þ ¼ N T 4 N sT 2 ; @sT 2 a

ð28Þ

The partial derivative of the vector si in Eq. (23) is evaluated computing the derivative of ^c and of p, resulting:

8 0 if > > >8 9 0 > @ ^c < > ¼ < = @s > > > 0 > if > :: ; 1

sN 6 0 sN > 0

:

ð29Þ

The partial derivative of p with respect to s is more complicated to derive, because the dependency of the p on the inelastic relative displacements in T 1 and T 2 directions and in the N direction. Thus, a procedure used for its evaluation is described in the following. Considering the residual defined in (19), also function of the relative displacement s, its differential results:

dR ¼

@R @R @R @R dp þ d dsd þ ds ¼ 0; dc þ @p @s @c @s

ð30Þ

that allows to evaluate the derivative of p with respect to s as:

ð24Þ

! D ¼ D

8 > > <

@p @s

9 > > =

 1 @p @R @R d ¼ @@ss ¼ > @ @s @s > p > : @c > ;

ð31Þ

@s

where:

@D 1 ¼ 2 ; @Y Y ð g  1Þ

@Y 2sT ¼  1 ; @sT 1 Y s0 2 T

@D Y 1 ¼ ; @ g Y ðg  1Þ2

with @R=@ p defined by formula (22) and

ð25Þ

@ g s2T 1 ðgN  gT Þ þ s2T 2 ðgN  gT Þ ¼ 2HðsN ÞsN ; @sN a4 ð26Þ

2

0

6 0 6 6 K T @R 6 ¼6 @s 6 6 0 6 4 0 0

0 0 0 K T 0 0

0

3

0 7 7 7 0 7 7: 0 7 7 7 K N 5 0

Fig. 5. Flowchart of the numerical procedure implemented for the interface model.

ð32Þ

7

F. Nerilli et al. / Construction and Building Materials 236 (2020) 117539

From Eq. (31) the first two lines of the solution represent the 2  3 matrix @p=@s. These latter are the partial derivatives of the inelastic relative displacement p with respect to the components of the displacement s. This numerical procedure is implemented in the finite element code FEAP [43] as 2 + 2 nodes and 4 + 4 nodes finite element with zero thickness for I1 and I2 interfaces, respectively. Linear shape functions are adopted in the finite element modeling. It is worth noting that, considering the I1 interface, such numerical procedure is simplified, because only one tangential direction is activated, reducing the vector and matrices dimensions. For the interface I2 the procedure is easier. In this case, the plasticity laws are not considered, because of sd ¼ 0 due to the occurrence only of the damaging phenomenon, as indicated in Eq. (15). In such a case the consistent stiffness matrix becomes:

Kt ¼

@s @D : ¼ ð1  DÞK  Ks  @s @s

ð33Þ

The partial derivative @D=@s is directly computed as:

@D @D @Y T ¼ ; @s @Y T @s

ð34Þ

where:

@D 1 ¼ 2 ; @Y T Y T ðgT  1Þ

8 2s 9 8 @Y 9 T1 > > pffiffiffiffiffiffiffiffiffiffiffiffi T > > 2 2 < = < sT þsT = @sT @Y T 1 1 1 2 1 ¼  2 @Y ¼  2 : 2sT > @s 2 > pffiffiffiffiffiffiffiffiffiffiffiffi > s0T : @sTT ; s0T > : ; 2 s2 þs2 T1

T2

ð35Þ

4. Numerical applications Some numerical applications are developed in order to assess the efficiency of the proposed FRCM model and the related numerical procedure in describing the tensile and shear mechanical response of the material. In the framework of 2D plane stress regime, initially the tensile mechanical behavior of FRCM is studied and the numerical results are compared with experimental ones available in literature. In particular, the carbon-based FRCM (CFRCM) specimen experimentally-tested in [4,17] is considered. The rectangular 400  40  10 mm coupon made with a balanced carbon fiber mesh (equal in both the grid directions) is tested with the clamping test set-up. Then, in the second numerical application the shear response of the FRCM is investigated. The geometry of the FRCM unit cell is illustrated in Fig. 2 and the geometrical and mechanical parameters are reported in Table 1, with clear meaning of the symbols and setting f tm the tensile strength of the mortar and dm1 and dm2 the side length of the mortar area of the inner layer along x1 and x2 direction, respectively. It is worth noting that the fiber thickness t f , indicated in Table 1, is the actual thickness of one roving, computed as the ratio between the roving cross-section area and its width, different from the equivalent fiber thickness, as defined in [4]. Considering the manufacturing process of the FRCM material, the discontinuity along the thickness direction, i.e. along the x3 direction, is due to the presence of the fibers. As a matter of fact,

the mortar that passes through the textile grid can be considered as a single continuum material. This fact is taken into account in the model, tieing the three layers along the out-of-plane direction only in correspondence of the central mortar area, that is represented by the grey hatched area in Fig. 6. In the model, the interfaces I1 -type are the in-plane mortarmortar and fiber-mortar interfaces, while I2 -type consider only the out-of-plane fiber-mortar interfaces. Thus, in the following two mechanical constitutive characterizations of the different interfaces, i.e. mortar-mortar and fiber-mortar, with different mechanical properties, are considered and reported in Table 2. Assuming that the thickness of the interface is equal to 1 mm, the normal and tangential stiffnesses of the mortar-mortar interface are set out by the elastic and shear modulus of the mortar, Em and Gm , respectively. The mortar-mortar maximum traction s0N and s0T , in the normal and tangential direction, are set with values close to the mortar tensile strength. The fracture energy GN , equal along the normal and tangential directions, is determined on the basis of data available in literature, typical of the analysed material. As also reported in [4] the mortar used for the FRCM manufacturing is a grout system based on cement and a low dosage of dry organic polymers (less than 5% by weight). In the graph in Fig.1 in [44] the specific fracture energy of cementbased materials are related to the maximum aggregate sizes. The value of the fracture energy is about 0.009–0.1 N/mm. The fibermortar interface, joining different kind of materials, are considered with different properties than that of the mortar-mortar interface. In correspondence of this interface, a material discontinuity is present. Although an experimental mechanical characterization of the fiber-mortar interface is not directly available, some experimental data can be found in [1,2,4]. On the basis of the results reported in these works the mechanical parameters of the fiber-mortar interface are estimated. Thus, the stiffness of such interface is set to be neither weaker nor stiffer than the mortar-mortar interface, and the same K N and K T values are considered. Due to the discontinuity, that represents a weakness, both the tangential and normal strength, s0T and s0N , respectively, are evaluated to be half of the strengths of the mortar-mortar interface, as also can be detected in [2] from Fig. 6-a (mortar-mortar) and 6-c (fiber-mortar). Finally, the energy fractures GN and GT are computed in order to have the half maximum slip of that of the mortar-mortar interface. 4.1. Tensile constitutive behavior The unit cell subjected to a tensile state is studied and the numerical results are compared with the experimental ones reported in [4]. Two slightly different finite element models are considered:
Model 1. The outer layers are separated from the inner layer in correspondence of the fiber area, and are tied in correspondence of the central mortar area. In this case only the I1 -type interfaces are introduced, in each layer, as represented in Fig. 7.
Model 2. In this model both the interfaces I1 and I2 are adopted. The interfaces I1 -type are adopted as in Model 1, while the I2 type interfaces are introduced between the outer and inner layers, as illustrated in Fig. 7.

Table 1 FRCM geometrical and mechanical properties. Geometry

CFRCM [4]

Mechanical properties

d1 [mm]

d2 [mm]

dm1 [mm]

dm2 [mm]

tm [mm]

t f [mm]

Em [MPa]

mm [–]

f tm [MPa]

Ef [MPa]

mf [–]

10

10

6

6

4.9475

0.105

2870

0.15

2.02

203000

0.1

8

F. Nerilli et al. / Construction and Building Materials 236 (2020) 117539

Fig. 6. Geometry of the constituents within the layers.

Table 2 Mechanical parameters of the interfaces.

Mortar-mortar interface Fiber-mortar interface

K T [N/mm3]

s0T [N/mm2]

GcT [N/mm]

K N [N/mm3]

s0N [N/mm2]

GcN [N/mm]

l

1248 1248

1.5 0.75

0.012 0.003

2870 2870

1.5 0.75

0.012 0.003

0.5 0.5

Fig. 7. Definition of the two models for reproducing the tensile behavior of the FRCM specimen.

The FE two-dimensional mesh for the three layers is made by 3072 4-nodes quadrilateral finite elements, 336 2 + 2 nodes I1 interface elements for both Model 1 and Model 2 and 1247 4 + 4 nodes I2 interface elements for Model 2. The mesh size of the quadrilateral elements is about of d = 0.3 mm. The positive average strain E11 is considered and the periodic boundary conditions for the perturbation displacement field are prescribed on the UC.

The numerical results, in terms of overall stress–strain relationship R11  E11 , obtained from the Model 1 and Model 2, are compared with the experimental curve, as represented in Fig. 8. It should be underlined that the experimental trilinear law is based on the evaluation of the fiber tensile stress R11;f , which is the ratio between the tensile load and the fiber grid area in the warp direction. Thus, the numerical curve is evaluated computing the    R11;f ¼ R11 d2 2tm þ t f =Af , as the ratio between the total force

F. Nerilli et al. / Construction and Building Materials 236 (2020) 117539

9

Fig. 8. Stress-strain law. Black line: experimental curve; blue dotted line: numerical result. Point A: cracking; point B: slippage; point C: failure.

in x1 direction and the textile fiber area Af . It can be observed that the numerical results obtained from the micromechanical modeling, well reproduce the tensile behavior of the experimentallytested FRCM specimen. The difference of the tensile curve resulted from the two models is undetectable. In order to deeper investigate the failure mechanisms and the damaging process of the material, the damage state occurring at

the interfaces in correspondence of the three characteristic points, denoted with the letters A; B and C in Fig. 8, are evaluated. Then, considering these load steps, both the deformed mesh of the FRCM cell and the stress state of the interfaces are plotted in Fig. 9–11, respectively, with the contour plot of the displacement component u1 . As a notation rule, the circle items on the s-s curves refer to the state of the interfaces lying along the x2 direction, while the trian-

Fig. 9. Results at the load step A: cracking. On the top: deformed mesh of the layers. Below: interfaces state.

10

F. Nerilli et al. / Construction and Building Materials 236 (2020) 117539

Fig. 10. Results at the load step B: slippage. On the top: deformed mesh of the layers. Below: interfaces state.

Fig. 11. Results at the load step C: ultimate state. On the top: deformed mesh of the layers. Below: interfaces state.

F. Nerilli et al. / Construction and Building Materials 236 (2020) 117539

gle items refer to the state of the interfaces along the x1 direction. Furthermore, the different colors distinguish the interface typology (mortar-mortar: blue color; fiber-mortar: red color). From the numerical results it emerged that the first branch of the curve (from zero to point A) is characterized by a linear elastic behavior up to the first cracking of the mortar. Considering the load step very close and just behind the point A, the results are summarized in Fig. 9. At this point, the x2 interfaces reach their normal tensile strength s0N , as showed in Fig. 9b) and Fig. 9(d). Then, a crack occurs and the x2 interfaces, both in the outer and inner layers, start to open. The x1 mortar-mortar interface elements -blue triangles in Fig. 9(d)- in the outer layers are not active. In the inner layer the x1 interface elements – red triangles in Fig. 9(d) – start to slip along the load direction (sT -sT curve). At the point B, the vertical interfaces continue to open but do not carry further load. This evidence is well depicted in Figs. 10c) and in Fig. 10(d). The behavior of the fiber-mortar x1 interfaces, illustrated in Fig. 10(b), shows that a relative slippage occurs with a mixed fracture mode. The results obtained at the point C are represented in Fig. 11. The vertical interfaces are completely opened and the slippage between the fibers and the mortar in the load direction occurred. It is worth underlining, that the I1 interfaces in the Model 2 exhibit the same damaging pattern occurred in the Model 1 and represented in Figs. 9–11. Thus, for sake of brevity, they are not again represented. In Fig. 12 the stress state of the 4 + 4 nodes I2 interfaces in correspondence of the load step A, B and C, as defined in Fig. 8, is

11

reported. The red cross items refer to the I2 elements that lie along the x1 direction, while the green cross items to that along the x2 direction. Considering the x1 interfaces, between the cracking point A and the stage B, the damage starts along the x1 direction, as showed in sT1 -s1 graphs. Finally, at point C these interfaces are completely damaged along the load direction. In the other direction sT2 -s2 graphs  they remain elastic. The x2 interfaces exhibit a linear elastic behavior until the ultimate point C, both in x1 and x2 direction. At the point C a little level of damaging can be detected for the these interfaces along the x1 direction. The check of the interface damage state allows to understand the failure mode of the FRCM. From the experimental results [4], for the considered carbon-based FRCM specimen, the most frequent detected failure mode depends on the slippage of the fibers within the mortar, evident in the load-displacement curves in [4]. In this context, the predicted failure mode seems to be consistent with the experimental one. Nevertheless, the experimental evidence has shown, in some specimens, also a damage of the fiber close to the main cracks. This phenomenon can not be detected by the model, in which the fiber is modeled as linear elastic material, but it will be considered as a future improvement. 4.2. Shear behavior The above outlined modeling approach is used in order to evaluate the behavior of the same FRCM cell subjected to shear in the 2D regime. In order to consider the damaging phenomenon of

Fig. 12. Definition of the 4 + 4 nodes interface elements between layers (green area). In the graphs: s-s behavior of the I2 interfaces in the x1 and x2 directions. The letter A, B and C refer to the critical load steps.

12

F. Nerilli et al. / Construction and Building Materials 236 (2020) 117539

Fig. 13. Sketch of the mesh and definition of the interfaces at the outer and inner layers. The different colors refer to the interfaces, differently named, where D denotes the diagonal interfaces, H the horizontal ones and V the vertical ones.

inclined in-plane interfaces, a different mesh of the FRCM unit cell is constructed, that considers the possibility to include diagonal interfaces, as represented in Fig. 13. It is worth observing that the mesh represented in Fig. 13 is more versatile: if a mixed load condition, at instance, want to be investigated for other applications, this discretization can be adopted, activating all the vertical, horizontal and diagonal interfaces. In this model only the in-plane interfaces, i.e. I1 type, are taken into account. The central mortar area is always tied between the three layers. The FE two-dimensional mesh is made by 4032 4-nodes quadrilateral finite elements and 512 2 + 2 nodes interface elements. The mesh size of the quadrilateral elements is about of d = 0.3 mm. The mesh accuracy is validated through the modeling of the tensile behavior of the FRCM, resulted in the same curves, as represented in Fig. 8 and in Figs. 9–11, respectively. In Fig. 13 the FRCM mesh is represented and the I1 interface elements are depicted with different coloured lines. In detail, the diagonal interfaces are mortar-mortar interfaces and in the outer

layers are named D1 and D2 -red and orange colors-, while in correspondence of the inner layer are called D3 and D4 and are represented with the pink and violet colors in Fig. 13. In correspondence of the inner layer also the ‘‘physical” fiber-mortar interfaces are taken into account, called H5 and V6, along the x1 -direction and x2 -direction, respectively. The mechanical properties of such interfaces are those reported in Table 2. The shear load condition consists in applying the average strain field E12 . The lability of the model is avoided fixing the displacements, in both x1 and x2 directions, of two symmetric points and the periodic conditions are considered. Fig. 14 shows the results in terms of shear stress–strain law. It should be underlined that, in this case, the stress R12 it is not related to the fiber area. The nonlinearity of such curve depends on the damaging process of the interfaces. Two characteristic points have been identified: point I refer to the first cracking of one of the two diagonal interface; point II is related to the state at which a first point of the interfaces reaches the complete damage. The deformed meshes, illustrated in Fig. 14, define the failure

Fig. 14. Shear stress-strain law and deformed mesh.

13

F. Nerilli et al. / Construction and Building Materials 236 (2020) 117539

Fig. 15. Point I: stress state of the interfaces, where

N

denote the behavior in the normal direction and

T

in the transversal direction.

Fig. 16. Point II: stress state of the interfaces, where

N

denote the behavior in the normal direction and

T

in the transversal direction.

mechanisms for which a diagonal interfaces opens in the outer and inner layers, where the mortar is present. The opposite diagonal interfaces, indeed, are subjected to compression. In Figs. 15 and 16, the stress state of the interfaces are depicted, in correspondence of the points I and II, respectively. The colors refer to the different interfaces, as defined above in Fig. 13. In correspondence of the point I, in Fig. 15, the mortar-mortar normal strength s0N is achieved for the D1 and D3 interfaces, while the mortar-mortar D2 and D4 interfaces are subjected to compres-

sion. Furthermore, the H5 (blue color) and V6 (green color) fibermortar interfaces are active, exhibiting a mixed-mode state. In Fig. 16, at the point II, the interfaces continue to damage, but one element of both the interfaces D1 and D3 is completely damaged. At this state, the fiber-mortar interfaces are highly compressed, and reach in one point, the compressive strength, about 20 MPa, as defined in [4]. In detail, the complete damage of the open interfaces and the achievement of the compressive strength is assumed to be the ultimate state.

14

F. Nerilli et al. / Construction and Building Materials 236 (2020) 117539

Fig. 17. Cyclic shear behavior of the FRCM UC.

is achieved in one element. The results show that the compression improves the shear strength and the ultimate strain value. Furthermore, the larger E22 , the more the unloading curves follow the loading curves, approaching an elastic behavior. 5. Conclusions

Fig. 18. Cyclic shear behavior of the FRCM UC, under different compression states.

4.3. Cyclic shear behavior When applied on existing structures, the strengthening material has the aim to retrofit the structural members also against seismic load, characterized by cyclic behavior. The model illustrated in the previous paragraph is used to evaluate the cyclic behavior of the FRCM unit cell, subjected to shear. In Fig. 17 the R12 -E12 law is represented, where one loading–unloading cycle is considered, and the deformed meshes at E12 =0, after the first unload, and at the failure point E12 ¼ 0:5% are also represented. As it was expected, the diagonal interfaces open one at a time, considering opposite loading pattern. During the unloading the opened interfaces reclose. In Fig. 18, the cyclic shear stress–strain law is reported for different level of transversal compression, along the x2 -direction. This latter is introduced as a constant average strain field E22 , during the shear loading pattern. The ultimate state for each curve is set to be the stage for which at least one point of one diagonal interface is completely damaged and the mortar-mortar compressive strength

In this paper, the mechanical behavior of an experimentallytested FRCM specimen is investigated via a computational micromechanical approach. In the framework of the numerical modeling at the micro-scale and because of the periodical geometry of the textile mesh, a single unit cell (UC) was taken into account. The FRCM UC was modeled as a three layer laminate, considering the outer mortar layers and the inner fiber-mortar layer. The numerical model was faced via an incremental strain formulation in a 2D plane stress regime. The possible occurrence of nonlinear mechanical phenomena at the micro-scale were detected via the introduction of interface elements accounting for: i) the mortar cracking occurrence in the layers plane, ii) the different constituents, fiber and mortar, within the inner layer, iii) the slippage between the layers. The modeling approach of the first two interfaces (i-ii) takes into account the combined effect of the damage and the friction, while for the interlayer interfaces (iii) only the damage is taken into account. The theoretical formulation and the main aspects of the numerical implementation of the interface models are presented and discussed. In the framework of the FRCM constitutive tensile behavior, numerical results computed via the proposed approach were successfully compared with available experimental data in terms of the tensile stress–strain law, also furnishing a physically consistent description of progressive damage patterns. As the numerical results showed, the main damage mechanisms concern the mortar cracking and the slippage of the fibers within the mortar at the inner layer, in good agreement with available benchmarking evidence. Nevertheless, the numerically-obtained tensile constitutive behavior seems to strongly depend to the in-plane interfaces. In fact, the introduction of the interlayer interfaces did not give a significant variation in the tensile stress–strain curve. Furthermore, the same modeling approach was used for evaluating the shear behavior of the same FRCM element. At this end, a new accurate mesh was defined and validated. The shear stress– strain law was evaluated and the micromechanical mechanisms

F. Nerilli et al. / Construction and Building Materials 236 (2020) 117539

were investigated. In detail, the damaging process is assessed by the opening of a crack in correspondence of a diagonal mortarmortar interface, both in the inner and in the external layers. As an assumption, the shear failure of the FRCM was assumed to occur when the diagonal crack was completely open. With the same modeling approach, the cyclic behavior in shear of FRCM unit cell was evaluated, also considering different compressive levels. As a matter of fact, in the proposed model, any limitation is enforced to the compressive stress in the mortar, assumed to behave as linear elastic. In order to deeply investigate the actual shear failure mode of the unit cell, increasing the shear strain level, the limited compressive strength of the mortar has to be considered. The proposed computational micromechanical strategy can be considered as an effective tool to predict the mechanical behavior of FRCM composites under tensile loads, also ensuring the detection of the damage initiation as well as an effective description of damage patterns during the loading process. This simplified approach allows also to give a first characterization of the shear behavior of FRCM composites where no suitable experimental results are provided. Nevertheless, because of the unavailability of experimental evidence, the numerical modeling assumptions have to be confirmed. Thus, an experimental mechanical characterization of the shear behavior of FRCM is highly desirable. Indeed, the final goal of this study is to understand and to simulate the tensile and shear behavior of the FRCM material in order to implement it in a FE code, entailing an easy simulation of the mechanical behaviour of structural members strengthened with FRCM material. Furthermore, this approach can be used in the framework of multi-scale modeling of full-scale structural applications. Finally, the proposed modeling approach can be also adapted to the modeling of the out-of-plane behaviour of masonry structural members. As perspective work, in order to take into account the proper non-linearity of the mortar material, a nonlocal damage model will be considered and introduced in the model. Moreover, in such a modeling approach a limitation of the compressive stress of the non-linear interfaces will be introduced. Finally, the out-of-plane micromechanical response of the composite will be addressed. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgement and compliance with ethical standards Funding: This study was funded by ReLUIS (Italian Department of Civil Protection) e by the Excellent Center ‘‘Distretto tecnologico per le nuove tecnologie applicate ai beni e alle attività culturali”. References [1] G. De Felice, M. Aiello, C. Caggegi, F. Ceroni, S. De Santis, E. Garbin, N. Gattesco, Ł. Hojdys, P. Krajewski, A. Kwiecien´, M. Leone, G. Lignola, C. Mazzotti, D. Oliveira, C. Papanicolaou, C. Poggi, T. Triantafillou, M. Valluzzi, A. Viskovic, Recommendation of RILEM technical committee 250-CSM: test method for textile reinforced mortar to substrate bond characterization, Mater. Struct. 51 (4) (2018) 95. [2] A. Bellini, M. Bovo, C. Mazzotti, Experimental and numerical evaluation of fiber-matrix interface behaviour of different FRCM systems, Compos. Part B: Eng. 161 (2019) 411–426. [3] S. De Santis, H.A. Hadad, F. De Caso y Basalo, G. De Felice, A. Nanni, Acceptance criteria for tensile characterization of fabric-reinforced cementitious matrix systems for concrete and masonry repair, J. Compos. Constr. 22 (6) (2018) 04018048. [4] F. Carozzi, C. Poggi, Mechanical properties and debonding strength of Fabric Reinforced Cementitious Matrix (FRCM) systems for masonry strengthening, Compos. Part B: Eng. 70 (2015) 215–230.

15

[5] B. Ferracuti, F. Nerilli, On tensile behavior of FRCM materials: an overview, in: Proceedings of the 8th International Conference on Fibre-Reinforced Polymer (FRP) Composites in Civil Engineering, CICE 2016, 2016, pp. 450–455. [6] E. Monaldo, F. Nerilli, G. Vairo, Basalt-based fiber-reinforced materials and structural applications in civil engineering, Compos. Struct. 214 (2019) 246–263. [7] C. de Carvalho Bello, I. Boem, A. Cecchi, N. Gattesco, D. Oliveira, Experimental tests for the characterization of sisal fiber reinforced cementitious matrix for strengthening masonry structures, Constr. Build. Mater. 219 (2019) 44–55. [8] F.G. Carozzi, G. Milani, C. Poggi, Mechanical properties and numerical modeling of Fabric Reinforced Cementitious Matrix (FRCM) systems for strengthening of masonry structures, Compos. Struct. 107 (2014) 711–725. [9] F. Nerilli, B. Ferracuti, Investigation on the FRCM-masonry bond behaviour, in: Proceedings of the 9th International Conference on Fibre-Reinforced Polymer (FRP) Composites in Civil Engineering, CICE 2018 (2018) 17–19.. [10] J. Donnini, S. Spagnuolo, V. Corinaldesi, A comparison between the use of FRP, FRCM and HPM for concrete confinement, Compos. Part B: Eng. 160 (2019) 586–594. [11] P. Askouni, C. Papanicolaou, Textile Reinforced Mortar-to-masonry bond: experimental investigation of bond-critical parameters, Constr. Build. Mater. 207 (2019) 535–547. [12] L. Ombres, Analysis of the bond between fabric reinforced cementitious mortar (FRCM) strengthening systems and concrete, Compos. Part B: Eng. 69 (2015) 418–426. [13] S. De Santis, F. Carozzi, G. De Felice, C. Poggi, Test methods for Textile Reinforced Mortar systems, Compos. Part B: Eng. 127 (2017) 121–132. [14] B. Mobasher, J. Pahilajani, A. Peled, Analytical simulation of tensile response of fabric reinforced cement based composites, Cem. Concr. Compos. 28 (1) (2006) 77–89. [15] U. Haubler-Combe, J. Hartig, Bond and failure mechanisms of textile reinforced concrete (TRC) under uniaxial tensile loading, Cem. Concr. Compos. 29 (4) (2007) 279–289. [16] J. Hartig, U. Häußler-Combe, K. Schicktanz, Influence of bond properties on the tensile behaviour of Textile Reinforced Concrete, Cem. Concr. Compos. 30 (10) (2008) 898–906. [17] P. Bernardi, D. Ferretti, F. Leurini, E. Michelini, A non-linear constitutive relation for the analysis of FRCM elements, Procedia Struct. Integrity 2 (2016) 2674–2681. [18] E. Bertolesi, F. Carozzi, G. Milani, C. Poggi, Numerical modeling of Fabric Reinforce Cementitious Matrix composites (FRCM) in tension, Constr. Build. Mater. 70 (2014) 531–548. [19] F. Nerilli, L. Tarquini, M. Marino, G. Vairo, Numerical modeling of failure modes in bolted composite laminates, AIP Conf. Proc., vol. 1648, AIP Publishing, 2015, p. 570019. [20] F. Nerilli, G. Vairo, Strengthening of reinforced concrete beams with basaltbased FRP sheets: an analytical assessment, AIP Conference Proceedings, vol. 1738, AIP Publishing, 2016, p. 270016. [21] F. Nerilli, G. Vairo, Experimental investigation on the debonding failure mode of basalt-based FRP sheets from concrete, Compos. Part B: Eng. 153 (2018) 205–216. [22] M. Malena, G. De Felice, Debonding of composites on a curved masonry substrate: experimental results and analytical formulation, Compos. Struct. 112 (2014) 194–206. [23] A. D’Altri, C. Carloni, S. De Miranda, G. Castellazzi, Numerical modeling of FRP strips bonded to a masonry substrate, Compos. Struct. 200 (2018) 420–433. [24] F. Carozzi, P. Colombi, G. Fava, C. Poggi, A cohesive interface crack model for the matrix-textile debonding in FRCM composites, Compos. Struct. 143 (2016) 230–241. [25] E. Grande, M. Imbimbo, E. Sacco, Local bond behavior of FRCM strengthening systems: some considerations about modeling and response, Key Eng. Mater. 747 (2017) 101–107. [26] G. Mazzucco, T. D’Antino, C. Pellegrino, V. Salomoni, Three-dimensional finite element modeling of inorganic-matrix composite materials using a mesoscale approach, Compos. Part B: Eng. 143 (2018) 75–85. [27] A. Cascardi, M. Aiello, T. Triantafillou, Analysis-oriented model for concrete and masonry confined with fiber reinforced mortar, Mater. Struct. 50 (4) (2017) 202. [28] M. Malena, Closed-form solution to the debonding of mortar based composites on curved substrates, Compos. Part B: Eng. 139 (2018) 249–258. [29] E. Grande, M. Imbimbo, E. Sacco, Numerical investigation on the bond behavior of FRCM strengthening systems, Compos. Part B: Eng. 145 (2018) 240–251. [30] E. Grande, G. Milani, Interface modeling approach for the study of the bond behavior of FRCM strengthening systems, Compos. Part B: Eng. 141 (2018) 221–233. [31] E. Grande, M. Imbimbo, S. Marfia, E. Sacco, Numerical simulation of the debonding phenomenon of FRCM strengthening systems, Frattura ed Integrita Strutturale 13 (47) (2019) 321–333. [32] C. Carloni, T. D’Antino, L.H. Sneed, C. Pellegrino, Three-dimensional numerical modeling of single-lap direct shear tests of FRCM-concrete joints using a cohesive damaged contact approach, J. Compos. Constr. 22 (1) (2017) 04017048. [33] F. Focacci, T. D’Antino, C. Carloni, L. Sneed, C. Pellegrino, An indirect method to calibrate the interfacial cohesive material law for FRCM-concrete joints, Mater. Des. 128 (2017) 206–217. [34] T. D’Antino, P. Colombi, C. Carloni, L.H. Sneed, Estimation of a matrix-fiber interface cohesive material law in FRCM-concrete joints, Compos. Struct. 193 (2018) 103–112.

16

F. Nerilli et al. / Construction and Building Materials 236 (2020) 117539

[35] P. Colombi, T. D’Antino, Analytical assessment of the stress-transfer mechanism in FRCM composites, Compos. Struct. 220 (2019) 961–970. [36] M. Brockmann, J. Raupach, Durability Investigations On Textile Reinforced Concrete, in: Proceedings of the 9th international conference on durability of building materials and components (CSIRO2002), Brisbane, Australia, paper no. 111, 1–9, 2002.. [37] B. Mobasher, A. Peled, J. Pahilajani, Distributed cracking and stiffness degradation in fabric-cement composites, Mater. Struct. 39 (3) (2006) 317–331. [38] M. Butler, V. Mechtcherine, S. Hempel, Durability of textile reinforced concrete made with AR glass fibre: effect of the matrix composition, Mater. Struct. 43 (2010) 1351–1368. [39] D. Dvorkin, A. Peled, Effect of reinforcement with carbon fabrics impregnated with nanoparticles on the tensile behavior of cement-based composites, Cem. Concr. Res. 85 (2016) 28–38.

[40] J. Donnini, G. Lancioni, V. Corinaldesi, Failure modes in FRCM systems with dry and pre-impregnated carbon yarns: experiments and modeling, Compos. Part B: Eng. 140 (2018) 57–67. [41] G. Alfano, E. Sacco, Combining interface damage and friction in a cohesivezone model, Int. J. Numer. Meth. Eng. 68 (5) (2006) 542–582. [42] E. Sacco, J. Toti, Interface elements for the analysis of masonry structures, Int. J. Comput. Methods Eng. Sci. Mech. 11 (6) (2010) 354–373. [43] R. Taylor, FEAP - A finite element analysis program, 2014.. [44] F. Wittmann, Crack formation and fracture energy of normal and high strength concrete, Sadhana 27 (4) (2002) 413–423.