Numerical study of the FRP-concrete bond behavior under thermal variations

2.1 Definition of the thermomechanical properties

Polymers and their composites can be used under a varying range of thermal conditions (compared to the glass transition temperature T g ), such that it becomes important to evaluate their performances under temperature variations that may produce changes in physical–mechanical properties. It is well known that the mechanical properties of polymers depend on their molecular weight and temperature. Among them, the elastic normal and shear moduli E ( t , T ) and G ( t , T ) depend on time t and temperature T , where the polymer stiffness can vary due to possible molecular rearrangements. In Figure 1, for example, we plot the modulus vs temperature curve referred to a typical polymer with a secondary relaxation and a clear subdivision in four regions. It is worth noticing the glassy zone, where the elastic modulus can present two secondary transitions (named relaxation γ and β , respectively), and a third one, called glassy transition (or transition α ). For higher temperatures, the glassy transition takes place in a transition zone with a decreasing value of the elastic modulus. Here, T g is identified experimentally as the modulus inflecting point considering the temperature. Graphically, it can be obtained as the intersection between the tangents starting from the extreme points of the drastic modulus loss range. For higher temperatures, the rubbery zone is reached. Here the modulus has lower values and keeps constant, as visible in the third zone in Figure 1, with a plateau representing the elastic zone of the rubbery state. A mixed zone takes place, which features a rapid decrease in the elastic modulus. The polymer chain movement is described with the reptation concept, as introduced by De Gennes [78] for the first time in terms of wormlike movements of the chains. In the final viscous flow, the linear polymer becomes a viscous fluid with an abrupt reduction of the stiffness property. This is not true for crosslinked polymers due to their strong chemical bonds, so their modulus keeps almost constant until degradation is reached.

Figure 1

Elastic modulus of a linear polymer subjected to a temperature variation.

In order to employ polymer matrix composites in a transition zone, it is necessary to describe analytically the polymer property variation (e.g., the stiffness) caused by a temperature change. Mahieux [79,80] studied this problem and proposed a relation valid for every type of polymer (thermosetting, thermoplastic, amorphous, linear, semicrystalline, and crosslinked), for every molecular weight, and for a full range of temperature. In the rubbery zone, the temperature dependence of the stiffness property E can be defined as follows:

where ρ is the density, RT is the room temperature, and M C is the molecular weight. However, this is true only for temperatures higher than the glass transition temperature T g .

Different theories allow us to define the variation of the elastic properties in the glassy state but only for small variations. Among many, the van Krevelen [81] theory suggests the following relations:

where G is the shear modulus; T is the temperature; and the subscripts g , c, sc, m , and r refer to the glassy, crystalline, semi-crystalline, melting, and reference states, respectively. Also, in this case, note that these equations can be applied only to the glassy state (service temperature lower than T _g), and do not refer to the polymer microstructure.

Among different possibilities, in this study, Mahieux’s approach [80] is adopted due to its general nature with respect to other existing theories. Such approach, indeed, describes the polymer behavior on a full-range temperature, including the transitions, and relates the polymer mechanical response to its microstructure. For a consistent definition of the local behavior for such material, during its relaxation time, spring and dashpots (dampers that withstand displacement through viscous friction) are considered in the model, such that the Young’s modulus E ( t ) varies with time t as follows:

where τ i is the relaxation time. Such equation can be written in an integral form as

where H is a distribution function, defined as

In the viscous region, the polymer reptation is usually modeled as follows:

with Q being the activation energy of the process. The Boltzmann distribution denotes a casual process, so that the bonds are broken following an indeterminate sequence, while the Weibull distribution allows us to describe an interactive fracturing process. Note that bond breaking influences the other bonds: for example, the secondary transitions (Figure 1) result from highly localized molecular movements. Therefore, the secondary bonds must be broken to allow movement of a lateral group or a few principal chains. Due to the atomic distance variation, which induces an interaction force distribution, the secondary bonds break in different moments. Compared to the Boltzmann distribution, the Weibull distribution seems to provide a more reliable mathematical representation of the debonding process and accounts for the secondary transition β . The mechanical response is given by the movement of little chain segments (monomers). Given the high number and force of the bonds involved in this relaxation process, it is possible to associate a Weibull coefficient m 1 with transition β , so that Eq. (5) becomes

By substitution of time τ 1 with the characteristic instantaneous temperature and by introducing a conversion constant, we have

where β 1 is a characteristic temperature (the transition temperature beta) and H 1 0 beta transition (i.e., the relaxation entity). The same approach can be applied to the other transitions for an improved number of segments involved in the relaxation stage. A novel Weibull coefficient can be associated with each stage, which refers to the number of intermolecular bonds involved. For this reason, a more general law can be applied to all transformation stages by adding all their components

Commercial polymers typically feature from one up to three transitions ( 1 ≤ N ≤ 3 ), whose coefficients H i (magnitude of phase transition) can be defined with different methods. Hereafter, we assume the following thermal dependence of the elastic modulus [79], as plotted in Figure 2.

1. For materials that are not subjected to any transition before becoming rubbery/viscous

   (12) E = E 3 e − ( T / T 3 ) m 3 .

2. For materials featuring two transitions

   (13) E = ( E 2 − E 3 ) e − ( T / T 2 ) m 2 + E 3 e − ( T / T 3 ) m 3 .

3. For materials featuring three transitions

Figure 2

Definition of the input parameters for the E − T relation.

In these three relations, temperatures ( T i ) are expressed in Kelvin (K) and refer to each transition phase depicted in Figure 2. The thermal values can be defined considering the maximum points of the curve resulting from a dynamic mechanic analysis or considering the inflection points of a differential scanning calorimetry curve. At the same time, E i corresponds to the instantaneous stiffness modulus assumed by the material at the beginning of a transition phase: E 1 is the instantaneous modulus of the polymer for low temperatures, E 2 is the instantaneous modulus immediately after the beta transition, and E 3 is the instantaneous modulus at the beginning of the rubbery state (Figure 2). Moreover, the parameters m i stand for the Weibull coefficients related to the stress distribution during a debonding phenomenon. The value of m is small if there is a wide distribution of the bonding forces, and it becomes high for homogeneous materials with a limited distribution of bonding forces, or amorphous materials. Moreover, the m value depends on the impediment degree of a molecular motion (i.e., crosslinking, molecular weight, crystallinity): if the motion is limited, m assumes a low value; if the motion is strictly limited, m becomes much lower, getting a similar behavior to the Boltzmann distribution. In the viscous flow and for cross-links, the gradient slope decreases for an increased cross-linking degree. For highly cross-linked materials, the viscous flow can disappear.

2.2 Interfacial constitutive law and finite-element algorithm

For a numerical description of the mixed mode debonding process, an uncoupled cohesive law is here assumed to model the adhesive interface within the FRP-to-concrete joint both in the normal and tangential directions. An uncoupled cohesive zone model is here selected, as typically used when the interface separation is enforced to occur in a single predefined direction, such that either Mode I and II separations occur. This means that the normal and tangential stresses, p N and p T , depend on the relatively normal and tangential displacement, g N , and g T , separately.

This choice derives from the possibility of using different values of interfacial fracture energies for both Mode I and Mode II, in line with the experimental evidence. For a compressive state in the normal direction (i.e., for g N < 0 and p N < 0 ), we enforce a non-penetration condition by using a Penalty method, within a generalized contact algorithm, based on the minimization of a modified potential W ¯ = W + W C , which involves both the elastic part W and contact part, W C = ε N g N , with ε N being the Penalty parameter. We select a bilinear cohesive traction-separation law for the normal and shear directions (Figure 3) because of its capability to simply define the main three interfacial properties, namely, the fracture energies, the cohesive strengths p N max and p T max , and the elastic stiffness [82]. For such simple reasons, the bilinear model is largely used, also in design guidelines, to model the FRP-to-brittle substrate interfaces [83] in lieu of more complicated nonlinear models. Thus, the energy release rates for Modes I and II correspond to the areas under the respective curves, integrated up to the current values of g N and g T [84], such that a mixed mode fracture criterion must be properly introduced to govern a failure process. In the present work, we selected one of the simplest possible criteria to handle the fracturing process, i.e.,

where G IC and G IIC stand for the fracture energies in pure Modes I and II, respectively.

Figure 3

Relation at the interface between the traction and the relative displacement: (a) normal direction and (b) tangential direction.

A generalized NTS contact element is adopted to handle the interfacial problem for both the normal and tangential forces within a unified formulation, as also described in the study of Dimitri et al. [24]. Based on the contact status, an automatic switching procedure is applied to select a cohesive or contact model. Each element contribution, for both cohesive and contact forces, is automatically added to the virtual global work as

where F N = p N A and F T = p T A stand for the normal and tangential interfacial forces, and A is the contact area associated with each contact element.

Figure 4 depicts the geometrical scheme of the selected FRP-to-concrete pull-out test. From a numerical standpoint, the adherend is discretized with 2D linearly elastic beam elements, while the concrete substrate is modeled with plane-stress four-node isoparametric elastic elements.

Figure 4

Setup of the single-lap shear test.

The nonlinear problem is solved with a Newton–Raphson iterative scheme, where the global tangent stiffness matrix is consistently obtained from a linearization of all the terms in Eq. (16). Such linearization yields the following relation:

where δ and Δ refer to the virtual variation and linearization, respectively. For more details about the geometrical quantities δ g N , δ g T , Δ g N , Δ g T , Δδ g N , and Δδ g T , the reader is referred to Ref. [85]. The partial derivatives of the normal and shear forces with respect to the normal and shear relative displacements depend on the cohesive law parameters. According to the selected traction separation law (Figure 3), it is

3.1 Effect of bond length

The numerical test is repeated for an increased value of the bond length l from 50 up to 400 mm in steps of 50 mm, keeping the other parameters constant, as summarized in Table 3 and Figure 5, together with the corresponding peak load P max . It is worth observing that an increased bond length of the FRP on the substrate up to 200 mm yields larger peak loads, beyond which it remains almost constant, in line with findings by Takeo et al. [86] and Ueda et al. [87]. Similar comments can be repeated by looking at the force–displacement curves in Figure 6, which clearly show an increased initial stiffness (in the ascending branch together with an increased value of the peak load and an increased displacement capacity of the specimen, because of an improved transfer of local interfacial stresses among FRP and substrate in the bonded portion, especially in the shear direction. It is expected, indeed, that an increased bond length increases the Mode II stress components, in lieu of decreased values of Mode I stresses, and an overall decrease of the interfacial mixed-mode stress distribution within the specimen.

Figure 5

Variation of the maximum load with the bond length.

Figure 6

Load–displacement curves for different bond lengths.

3.2 Effect of the reinforcement elastic modulus

The analysis has been extended by considering three different types of composite reinforcement, namely, a glass FRP with an elastic modulus E f = 100 GPa (labeled as GFRP100), an aramid FRP with E f = 150 GPa (labeled as AFRP150), and a carbon-based reinforcement (CFRP) with a different elastic modulus E f of 190, 210, 230, 250, 390 and 450 GPa, while keeping all the other parameters constant.

In Table 4 and Figure 7, the variation of the maximum load for a varying elastic stiffness of the FRP strip is summarized. As also expected experimentally in the study of Wu et al. [88], the maximum load significantly increases for an increased elastic modulus E f , reaching the highest values for a CFRP with E f = 390 ÷ 450 GPa . In these last cases, however, the numerical values of P max are more conservative than expected experimentally by Wu et al. [88], because of the adhesive crisis estimations in lieu of their cohesive experimental results found in [88]. This can be addressed to the local behavior of the concrete which may exhibit higher properties with respect to the theoretical prediction. The overall load–displacement response of the specimen is, thus, plotted in Figure 8 for different FRPs. Based on these plots, note that as the elastic modulus E f of the FRP increases, the slope of the curves in the ascending branch increases, together with an increased peak strength P max and interfacial brittleness in the descending branch of the curves.

Figure 7

Variation of the maximum load with the elastic modulus of the reinforcement.

Figure 8

Load–displacement curves for different fiber elastic moduli.

3.3 Effect of the reinforcement thickness

We now assume six different FRP thicknesses t f , i.e., 0.167, 0.222, 0.247, 0.337, 0.55, and 1.4 mm, as commonly used in structural reinforcements. All the other parameters are kept constant. The results are plotted in Figures 9 and 10 and could be compared to the experimental predictions by Ueda et al. [87] and Tan [89]. Even in this case, an increased value of the fiber thickness corresponds to an increased value of the peak load of 7 ÷ 8.5 % , see also the results in Table 5. As also plotted in Figure 10, the global load–displacement curves increase their initial slope in the ascending branch for an increased reinforcement thickness. At the same time, the maximum strength of the specimen tends to increase, and the specimen tends to assume an elastic-brittle behavior up to the final collapse, for an increased reinforcement thickness.

Figure 9

Variation of the maximum load with the fiber thickness.

Figure 10

Load–displacement curves for different fiber thicknesses.

3.4 Effect of the FRP width

The reinforcement width represents another key parameter that can significantly affect the global mechanical behavior of the adhesive joint. A larger dimension of the reinforcing FRP, indeed, can improve the interfacial distribution of stresses, and the related maximum debonding load of the specimen, whose sensitivity is studied here, by selecting the following combinations:

1. b c = 150 mm and b f 50, 60, 80 and 100 mm;

2. b c = 200 mm and b f 120 and 150 mm; and

3. b c = 400 mm and b f 200 and 300 mm,

while keeping all the other parameters constant.

As summarized in Table 6 and plotted in Figure 11, the maximum load P max increases monotonically for higher values of b f , with increased structural stiffness and decreased ductility, as visible in the global load–displacement response in Figure 12. An increased interfacial stiffness, indeed, is expected to yield a more localized distribution of the cohesive stresses in the bond length, with an increased overall fragility of the specimen. Even in this case, the numerical insights are comparable with the experimental findings by Ueda et al. [87] and Tan [89].

Figure 11

Variation of the maximum load with the reinforcement width.

Figure 12

Load–displacement curves for different reinforcement widths.

3.5 Effect of temperature

We now analyze the possible effect of temperature on the mechanical bond properties of concrete members externally strengthened with FRPs while selecting an epoxy adhesive with a glass transition temperature T g ranging between 50 and 80°C . Epoxy adhesives are noteworthy for their susceptibility to mechanical degradation when exposed to high temperatures. Since adhesives are responsible for the bonding behavior among FRPs and concrete, for bond-critical conditions, the structural efficiency of the strengthening system could be highly influenced during a high-temperature exposure [90]. Based on the experimental observations from literature on shear tests, when the temperature in the FRP exceeds 200°C , the reinforcement does not contribute anymore to the load-bearing capacity, because of a softening bonding adhesion, with a general loss of the adhesive transfer between materials and a premature detachment of the FRP from the adherend. Due to the low glass transition temperature of adhesives, their elastic modulus decays drastically at temperatures slightly higher than the ambient ones. In such a context, it is important to define a theoretical correlation between temperature and the global response of the reinforced specimen. Two adhesives are considered, labeled as “Adhesive 1” and “Adhesive 2,” with elastic moduli of 4,157 MPa and 2,595 MPa, respectively. Both adhesives feature two different transition phases, moving from the glassy zone to the rubbery state, such that they lose any reserve of resistance for temperatures higher than 150 – 200°C . In what follows, we refer to the relation proposed by Mahieux [79] to define the variation of the elastic stiffness E, with temperature T

where T i refers to temperatures at each transition, E i is the instantaneous stiffness of the adhesive at the beginning of each plateau region, and m i are the Weibull coefficients, which account for the microstructure of a polymer and the statistics of the bond breakage. More specifically, E 2 and E 3 stand for the instantaneous moduli right after the beta transition and at the beginning of the rubbery plateau, which occur at temperatures T 2 and T 3 , respectively. Based on the experimental data in previous studies [80,91,92], we set high values of m i for amorphous adhesives with homogeneous properties (see Adhesive 1) and narrow distribution of bond strengths, while setting lower values of m i for crosslinked adhesives with a reduced slope of the drop in the viscous flow region (see Adhesive 2). All the selected parameters for Eq. (21) are reported in Table 7, whereas Figures 13 and 14 show the theoretical and experimental variation of the stiffness with temperature, with a good matching among results. Based on a comparative evaluation of the plots in Figure 13a and b, it is worth observing the different behavior in the glass transition zone, with a rapid modulus drop for “Adhesive 1” in lieu of a gradual reduction with temperature for “Adhesive 2,” thus leading to a different behavior of the two adhesives in the rubbery state. Starting with the theoretical variation of the stiffness properties in Eq. (21) for the adhesive layer, we now analyze the sensitivity of the debonding response for the selected pull-out tests, both in terms of maximum load and global time histories. The systematic investigation accounts for the normal and shear elastic stiffness at different temperatures as summarized in Table 8 for both adhesives, whose properties are responsible for the slopes in the ascending branch of the interfacial model in Modes I and II. The numerical tests are performed in displacement control mode.

Figure 13

Variation of the elastic modulus of resins with temperature: (a) Adhesive 1 and (b) Adhesive 2.

Figure 14

Load–displacement response for different temperatures: (a) Adhesive 1 and (b) Adhesive 2.

Figure 14 shows the load–displacement curves for Adhesive 1 (Figure 14a), and Adhesive 2 (Figure 14b), with a noteworthy variation of the global response for an increased temperature, both in the ascending and softening branches. An increased temperature, indeed, drastically reduces the elastic stiffness and the maximum load of all the specimens, while extending the elastic stage for all the specimens, especially for temperatures higher than T g , when adhesives reach the rubbery state, due to an increased molecular motion and overall deformability. At the same time, we summarize the variation of P max with temperature in Table 9, whose results are also plotted in Figure 15 for both adhesives, with a maximum reduction of about 78% for T = 125 ° and T = 160 ° for Adhesives 1 and 2, respectively. Note that P max remains almost constant for T < T g and decreases monotonically when the glass transition takes place, as also expected from experimental evidences. Specimens based on the second adhesive feature a more gradual decay of P max compared to the first adhesive, thus reflecting in a global sense, the local decay of the interfacial properties. Moreover, for “Adhesive 2”, P max starts decreasing for a higher temperature compared to that one based on “Adhesive 1.” For affinity reasons among the global response of P max vs T plotted in Figure 15 and the local properties of adhesives in Figure 13, we propose the following relation among the debonding load P max and temperature T , which could serve as a useful theoretical tool for design purposes, namely

where α i and β i are parameters that depend on the transition states to which the adhesive is subjected, F i is a force parameter for a specific transition state occurring at temperature T i , and T ref , i is a reference temperature. This formulation can be considered as a starting point for future theoretical and experimental developments on the topic, in order to assess and improve the durability of glued joints, and to improve the load-bearing capacities of strengthened structural elements.

Figure 15

Variation of the maximum load with temperature.