Interface crack model using finite fracture mechanics applied to the double pull-push shear test

Depósito de Investigación de la Universidad de Sevilla https://idus.us.es/ This is an Accepted Manuscript of an article published by Elsevier in International Journal of Solids and Structures, Vol. 188-189, on April 2020, available at: https://doi.org/10.1016/j.ijsolstr.2019.09.008 © 2019 Elsevier. En idUS Licencia Creative Commons CC BY-NC-ND Interface crack model using finite fracture mechanics applied to the double pull-push shear test. M. Mu˜noz-Rejaa,∗, P. Cornettib, L. T´avaraa, V. Mantiˇca aGrupo de Elasticidad y Resistencia de Materiales, Escuela T´ecnica Superior de Ingenier´ıa, Universidad de Sevilla, Camino de los Descubrimientos s/n, 41092 Sevilla, Spain bDipartimento di Ingegneria Strutturale, Edile e Geotecnica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy Abstract An analytical procedure predicting a debond (interface crack) onset and growth in an adhesive joint between two beams or plates is developed and applied to a specific configuration often used in reinforcement tests in civil engineering. The procedure is based on Timoshenko beam theory and Linear Elastic-(Perfectly) Brittle Interface Model (LEBIM) combined with the Coupled Criterion of the Finite Fracture Mechanics (CC-FFM) for mixed-mode fracture. First, a sixth order differential equation in the shear stresses along the adhesive layer is deduced and solved, leading to closed form expressions for both shear and normal stresses in the adhesive. Then, the critical value of the applied load necessary to produce debonding is predicted by coupling a stress and an energy condition based on: (i) the stress distribution produced in the interface before the debond onset and (ii) the energy released during the debonding process along the interface. Although the developed procedure can be applied to several types of joints with different geometries, materials and loads (e.g., double lap joint tests including adherents made of steel or composites), herein it is applied to the double pull-push shear test where the debond onset and growth between a Carbon Fibre Reinforced Polymer (CFRP) laminate and a concrete block occurs. Although the debond is produced under predominant fracture mode II, ∗Corresponding author. Tel.: +34 954487299. Email addresses: [email protected] (M. Mu˜noz-Reja), [email protected] (P. Cornetti), [email protected] (L. T´avara), [email protected] (V. Mantiˇc) Preprint submitted to Journal of L A T EX Templates August 27, 2019 it is shown that relevant normal (peeling) stresses associated to mode I may appear as well. A comparison of the present solution with a previous one by the shear-lag model is provided as well. Keywords: LEBIM, FFM, double pull push shear test, composite structure, bond slip 1. Introduction Structural integrity of composite components is in general determined by the durability and quality of their adhesive joints and interfaces among other reasons. The joint quality in composites must be assured at several levels including fibre-matrix interfaces, interfaces between laminae within a laminate, adhesive5 joints between laminates and other structural parts, etc. Thus, an adequate characterisation of interfaces is a primary concern in composite components, and many researchers investigated composite joints at micro, meso and macro scale and developed several methodologies to predict the interface failure. In many of these investigations, the behaviour of the interface/joint is modelled as10 a distribution of linear elastic springs, sometimes also referred to as Winkler’s spring bed, elastic interface or weak interface. Prandtl [1], Volkersen [2] and Goland and Reissner [3] were the first to use this model for adhesive joints. A comprehensive study of the model from the fracture mechanics point of view was presented in [4], where the differences between cracks at perfect and elastic inter-15 faces are described. To enable modeling of crack onset and propagation at such interfaces the constitutive law of initially linear-elastic springs can be completed by a perfectly brittle failure criterion, see [5] for a review. Such a model is usually referred to as Linear Elastic-(Perfectly) Brittle Interface Model (LEBIM). Nowadays many authors use LEBIM for different applications [6, 7, 8, 9, 10, 11].20 The LEBIM has proven to be a very efficient numerical tool when the overall stiffness of the system (including adherents and adhesives) is essentially ruled by the adherent stiffness [12]. Nevertheless, in some problems (due to the geometry and loading configuration) the influence of the interface stiffness on the overall 2 system stiffness is relevant. In such cases an adequate spring stiffness is needed25 in order to correctly model the crack behaviour obtaining a better correlation with experimental results. The failure criterion proposed in [13, 14] and based on the Finite Fracture Mechanics (FFM) [15] approach is used in the present investigation to improve the initial LEBIM formulation. It couples an energy and a stress based criterion,30 each of them being a necessary but not sufficient condition to produce either a debond onset or growth, while their simultaneously fulfillment represents a sufficient condition. This Coupled Criterion of FFM (CC-FFM) was recently used by many authors to solve relevant engineering problems of fracture onset and propagation, see [16] for a review. Especially, Mu˜noz-Reja et al. [17]35 and Cornetti et al. [9] applied the CC-FFM to LEBIM for solving problems of interface crack onset and propagation at micro and macro scales, respectively. One of the aims of the present investigation is to determine the load that originates a debond along a weak interface in the Double Pull Push Shear (DPPS) specimen including a reinforcement adhesively bonded to a concrete40 block. Since Hillerborg et al. [18] proposed the Cohesive Zone Model (CZM) to model fracture in concrete structures, it has been used by numerous authors (e.g. [19, 20, 21, 22]). CZMs often provide accurate predictions of damage propagation in concrete thanks to suitably fitted cohesive softening functions. However, there are still some difficulties in fitting these softening functions to45 the behaviour of the material, specially under mixed mode of fracture to achieve an adequate energy interpretation of the fracture path [23], and also in achieving a satisfactory computational efficiency and to guarantee the convergence of FEM computations using CZMs [24]. For these reasons, the CC-FFM applied to LEBIM could open an alternative way to CZMs for predicting the onset and50 propagation of debonds at interfaces in joints, whereas CZMs remain as a reference model in this field. In fact, recently several comparisons of the CCFFM vs. CZM were published [25, 26, 27, 28, 29, 24]. In [9] the debond onset in DPPS was studied using the CC-FFM but considering only shear stresses along the interface. Thus, in the present investigation,55 3 DPPS is analysed by means of the CC-FFM taking into account both normal (peeling) and shear stresses along the interface, in a similar way as done in [30, 6, 31, 32] for other test configurations. The study is divided in two parts: (i) the development of the analytical solution for the normal and shear stress fields in the DPPS test along the interface between two solids, and (ii)60 the application of the CC-FFM for the prediction of the debond onset. Finally, the obtained results are compared with those proposed in [9], and also some available experimental data and predictions by a CZM. 2. The Double Pull-Push Shear test Adhesive joints between adherents were intensively studied in the last decades.65 Investigations were mainly focused on the used materials (for both adherents and adhesives) and the models used to get the stress field in the joints that was used to apply a failure criterion. The different models for failure of adhesive joints can be divided into two groups: (i) analytical (closed-form), and (ii) numerical models. The selection of the model is not an easy task and depends on70 the final objective (attempting to optimize the used resources), see [33, 34] for reviews of existing models. As the evaluation of shear strength in adhesive joints is an important issue, several types of test were proposed, e.g. Single Lap Joint (SLP) and Double Lap Joint (DLJ) tests. A review of different methods used to characterize the75 bond-slip behaviour can be found in [35, 36]. The DPPS test, see Fig. 1, can be considered as a modified version of the DLJ test including different boundary conditions. The DPPS test is widely used to investigate the adhesion between a Carbon Fibre Reinforced Polymer(CFRP) laminate and a concrete block or clay brick [36].80 Although normal (peeling) stresses are neglected in several models of bonded joints, some investigations including experimental results have shown that normal stresses along the interface may play an important role in the debond onset and propagation [22, 37, 38, 39], and that especially for small overlap lengths 4 z,w x,u hb Reinforcement Adhesive Concrete block P P 2P y,φ t =t l 2 hr br Figure 1: Double Pull-Push Shear (DPPS) test. their contribution may not be neglected. Actually, there are tests where the85 debond is produced at the free end of the reinforcement instead of the loaded end for specific geometry and load configurations [22]. The model proposed in this work is intended to be a general model for any kind of double joint tests. Thus, to illustrate its capabilities a study of peeling in the DPPS is carried out.90 2.1. Analytical model for the DPPS test The model developed in this work (Fig. 2) uses a simplified geometry that includes symmetry boundary conditions along the midplane of the concrete block and it is studied under the following main assumptions: •The adherents are modelled as Timoshenko beams. Due to symmetry95 conditions, the concrete block is subjected to normal stresses and is able to deform in extension, while the outer CFRP laminates are subjected to tension, bending and shear (and are deformable accordingly). •The adhesive is modelled as an elastic interface of negligible thickness, i.e. a continuum spring distribution able to transfer normal and shear stresses.100 These values are assumed to be representative of the stress field in the midthickness plane of the adhesive layer, the remaining stress components in the adhesive are neglected. •Plane strain hypothesis is considered and a linear-elastic behaviour of the adherents, the reinforcement being orthotropic and the block isotropic.105 5 •The width of the adherents and the adhesive is equal tr=tb. h l x σ σ b hr t =tbr x x r b z y Figure 2: Boundary conditions of the DPPS test solved. 2.1.1. Governing equations Referring to the free-body diagram of an element of length dxrepresented in Fig. 3, the following equilibrium equations can be written for the CFRP laminate: x d N N r +d N r M r V r M r M r +d V V r +d r b hr z,w x,u y,φσ σ τ τ σ τ N b N b +d N b h b 2 r r Figure 3: Equilibrium of an infinitesimal element of a beam. 6 dNr dx−τ= 0 (1) dVr dx+σ= 0 (2) dMr dx−Vr−hr 2τ= 0 (3) where, Nr,Vrand Mrare the axial force, the shear force and the bending moment per unit width, respectively; τand σare the shear and normal stresses along the adhesive; and hris the thickness of the CFRP laminate.110 Regarding the concrete block, due to the symmetry conditions, the only non-trivial equilibrium equation is dNb dx+ 2τ= 0,(4) Then, the kinematic equations for the CFRP laminate are εr=dur dx(5) χr=dϕr dx(6) γr=dwr dx+ϕr(7) where ur,εr,ϕr,χr,wr, and γrare the longitudinal displacement, longitudinal membrane strain, rotation, curvature, transverse displacement and shear strain, respectively. Due to the symmetry condition, the only significant kinematic equation for the concrete block is εb=dub dx(8) where uband εbare the longitudinal displacement and axial strain, respectively. Assuming an orthotropic CFRP laminate whose material symmetry axis 1 coincides with the reinforcement direction, the material plane 23 becomes a material symmetry plane. Then, constitutive equations can be deduced, in a similar way as in [8], from the strain-stress law for a generalized plane strain state [40, 41]      ε1 ε2 ε6      =     s0 11 s0 12 0 s0 12 s0 22 0 0 0 s0 66     ·     σ1 σ2 σ6      , s0 IJ =sIJ −sI3s3J s33 (9) 7 where s0 IJ are the reduced elastic compliances, in this case material axis 3 is the direction where plane strain conditions apply. Hence, the reduced elastic compliances used in this work are: s0 11 =1−ν13ν31 E1 =1 E0 r and s0 66 =1 G12 =1 Gr (10) where, E0 ris the elastic modulus for plane strain conditions and Gris the inplane shear modulus. Let the material axes 1, 2 and 3 correspond to the (geometric) beam axes x,zand y, respectively. Then, the constitutive equations for the CFRP laminate become εr=Nr E0 rhr (11) χr=12Mr E0 rh3 1 (12) γr=Kz Vr Grhr (13) Kzbeing the shear factor of a Timoshenko beam (6/5, for a rectangular section). Additionally, the only constitutive equation for the concrete block is εb=Nb 2E0 bhb (14) where, E0 b=Eb (1−ν2 b)is the elastic modulus for plane strain conditions for an isotropic material. As the model developed is one-dimensional, the quantities do not vary along115 with the thickness. Therefore, it is assumed the adherent and adhesive thicknesses are enough to achieve plane strain conditions, see (10) and (14). Also, because the thinness of the reinforcement considered in the DPPS test, its shear deformability could often be neglected. However, as stated above, the analytical model is expected to be applied also to thick outer adherents and, therefore,120 (13) is considered for the sake of generality. As the adhesive layer is modelled by a spring distribution, then the normal and shear stresses are directly related to the respective relative displacements, δnand δt, between the adherents. Then, the Timoshenko beam kinematical 8 0.0 0.1 0.2 0.3 0.4 0.5 ξ ξ P P -2.0 0.0 2.0 4.0 6.0 σ (ξ), τ (ξ) ξ=0.485 σ (ξ) τ (ξ) Figure 5: Dimensionless normal and shear stress distributions along the interface in the DPPS test. and shear stress values are obtained at the right end of the interface (loaded end), coinciding with the zone where loads are applied over the concrete block and laminate, see Fig. 4. Regarding the normal stress distribution, it is interesting to notice that a large increase of compressions appear at the right end but also175 tensions appear in a neighbour small zone. Moreover, tensions also occur at the opposite (left) end, free end in Fig. 4. It is evident that the shape of this stress distribution will affect both the debond onset and propagation. The implementation of the presented elastic interface model in a numerical code can be straightforward. Also, if the performed numerical analysis is180 linear-elastic, the analytical values would be in good agreement with the stress distribution along the adhesive mid-line provided by the numerical analysis. Of course, some differences are expected at the very end of the overlap, the analytical model and the numerical analysis providing a non-zero and vanishing shear stress, respectively. However, this differences can be avoided by keeping a185 segment of a free edge in the concrete and in the reinforcement in the numerical analysis. In [31] a comparison between the analytical and numerical solutions for bonded double lap joints is studied. Note that in [31], the analytical solution 15 is very similar to the model presented herein, however, a system of eight linear equations is used instead of six linear equations detailed in this investigation.190 2.2. Shear stress distribution using the shear lag model It is worth recalling the solution provided by the simpler shear lag (SL) model introduced in [9]. Accordingly, the adhesive is modelled as a bed of tangential springs, which are not able to transfer normal stresses (kn= 0); thus, neither peeling stresses nor compressions in the adhesive layer are present in the SL model. Moreover, the adherents are assumed to withstand only axial forces. Regarding the shear stresses at the interface, the SL solution (using the dimensionless quantities defined above) [9] yields ˆτSL =sκ(1 + ρ) η cosh ξqκ(1+ρ) η sinh λqκ(1+ρ) η(37) Note that the SL model is not a particular case of the present one, i.e. it cannot be achieved by setting kn= 0 into the present developed model (for instance, the rotational equilibrium given by Eq. (3) would never be satisfied). Nevertheless, the shear stress field provided by the SL model, Eq. (37), agrees195 very well with the one provided by Eq. (28). The differences obtained between both models will be analysed in detail in the following section for several characteristic parameters of the DPPS test. 2.3. Analysis of the stress distributions along the interface for different geometrical and mechanical parameters200 As mentioned above, the present model is intended to be a general model for any double joint test. For this reason, both the normal and shear stresses must be considered equally in the model rather than just shear stresses. For example, the present procedure can be applied to FRP reinforcements subjected to inclined loads.205 In fact, a parametric study of the different parameters involved in the DPPS test presented in this section is motivated by works of Carrara et al. [39, 22]. 16 Theses works highlighted that for several geometries the normal stresses on the free edge can influence significantly in the debond onset and propagation. Specifically, this section presents the influence of the dimensionless parame-210 ters defined in Table 1 on the stress distributions are analysed using Eqs. (28) and (29). Additionally, the SL solution given by Eq. (37) will also be depicted for comparison purposes. In the following analyses, the values presented in Table 2 will be used except for the parameter to be analysed in each case, which will vary in the range defined in each figure.215 The first analysed parameter is λ, representing the length of the bonded joint (overlap length), Figs. 6 and 7, show shear and normal stress distributions for several overlap lengths λ. Small differences are shown for ˆτvalues obtained using the present model and using the SL model in [9] for all analysed values of λ. It is noticeable that the highest differences occur at the right (loaded) end220 of the interface, which is the zone where the debond onset is expected to take place. 0.0 0.5 1.0 1.5 2.0 0 1 2 3 4 5 6 7 ξ τ(ξ) λ = 2.00 λ = 1.00 λ = 0.50 λ = 0.25 continuous line line with markers τ(ξ) τ (ξ) SL 0.20 0.21 0.22 0.23 0.24 0.25 5.0 5.5 6.0 6.5 7.0 τ(ξ) ξ τ(ξ) τ (ξ) SL for λ = 0.25 for λ = 0.25 (a) (b) Figure 6: (a) Dimensionless shear stress distribution along the interface for different λvalues, with η= 0.00728, ρ= 0.142, κ= 0.25 and ζ= 29.67 (b) Zoom at the right (loaded) end of the interface for λ=0.25. It is interesting to recall the importance of the normal stress distribution, 17 0.0 0.5 1.0 1.5 2.0 -3 -2 -1 0 1 ξ σ(ξ) ξ (a) (b) 0.02 0.03 0.04 0.05 0.06 0.07 0.08 -0.04 -0.02 0.00 0.02 0.04 σ(ξ) λ = 2.00 λ = 1.00 λ = 0.50 λ = 0.25 λ = 2.00 λ = 1.00 λ = 0.50 λ = 0.25 continuous line σ(ξ) Figure 7: (a) Dimensionless normal stress distribution along the interface for different λvalues, with η= 0.00728, ρ= 0.142, κ= 0.25 and ζ= 29.67. (b) Zoom at the left (free) end of the interface. see Fig. 7, especially for small overlap lengths, where large tension values appear in the left end with traction free boundary conditions. Interestingly, the size of225 the zone under traction is independent of the overlap lenght, as highlighted in Fig. 7(b). These results are in accordance with previous investigations [38, 39, 22, 37]. The second parameter to be analysed is η=hr lch which is directly related to the laminate (reinforcement) thickness. Also here shear stress distributions230 obtained by the present model and by the SL model present small differences, see Fig. 8. The larger differences are obtained at the loaded end, where these differences slightly increase for larger values of the laminate thickness. Fig. 9 presents the influence of ηon the normal stress distribution. Results show that the zone of tensions and the maximum tension value at the left235 (free) end increase with increasing values of laminate thickness, whereas, at the opposite (loaded) end, a decrease of the maximum compression value occurs for larger laminate thickness although the zone of compressions is larger. The behaviour associated with ηshows that, for larger laminate thickness, shear 18 0.0 0.1 0.2 0.3 0.4 0.5 0 5 10 15 η = 0.02 η = 0.01 η = 0.005 η = 0.001 continuous line line with markers τ(ξ) τ (ξ) SL ξ τ(ξ) (a) 0.40 0.42 0.44 0.46 0.48 0.50 3 4 5 6 7 8 ξ (b) τ(ξ) ξ=0.485 ξ=0.485 Figure 8: (a) Dimensionless shear stress distribution along the interface for different ηvalues, with λ= 0.485, ρ= 0.142, κ= 0.25 and ζ= 29.67. (b) Zoom at the loaded end. 0.0 0.1 0.2 0.3 0.4 0.5 -4 -3 -2 -1 0 1 0.476 0.480 0.484 -4 -3 -2 -1 0 0.00 0.04 0.08 0.12 0.0 0.2 0.4 0.6 0.8 ξ=0.485 (a) ξ (b) ξ (c) ξ σ(ξ) σ(ξ) σ(ξ) η = 0.02 η = 0.01 η = 0.005 η = 0.001 continuous line σ(ξ) Figure 9: (a) Dimensionless normal stress distribution along the interface for different ηvalues, with λ= 0.485, ρ= 0.142, κ= 0.25 and ζ= 29.67. (b) Zoom at the loaded end. (c) Zoom at the traction free end. 19 stresses are more uniformly distributed, while compressions at the loaded end240 are balanced (at least partially, in view of another intermediate zone of moderate tensions) by tensions at the free end. This fact may cause, for some geometries, debond onset at the free end. It should be noted that although several ηvalues are used in Figs. 8 and 9. The comparison is made at constant relative axial stiffness between block and245 reinforcement, i.e. at constant ρ=E0 rhr E0 bhb. Then, the influence of the parameter ρis analysed in Figs. 10 and 11. Results show that a decrease of the block axial stiffness raises the shear stress peak in the loaded end as depicted in Fig. 10. Additionally, compressions in the loaded end increase while tensions decrease250 in the traction free end, as shown in Fig. 11. It is noteworthy that this behaviour is the opposite as the one observed when ηincreases. 0.0 0.1 0.2 0.3 0.4 0.5 0 2 4 6 8 10 12 14 ξ (a) ξ (b) 0.40 0.42 0.44 0.46 0.48 0.50 6 8 10 12 14 ρ = 5.00 ρ = 1.00 ρ = 0.50 ρ = 0.10 ξ=0.485 ξ=0.485 τ(ξ) τ(ξ) continuous line line with markers τ(ξ) τ (ξ) SL Figure 10: (a) Dimensionless shear stress distribution along the interface for different ρvalues, with λ= 0.485, η= 0.00728, κ= 0.25 and ζ= 29.67. (b) Zoom at the loaded end. Finally, Figs. 12 and 13 show stress distribution for different κ=kt knvalues. It is interesting to notice, as indicated in [5, 43], that κ < 0.50 for isotropic adhesive layers while κ > 0.50 values may be associated to orthotropic adhesive255 20 0.0 0.1 0.2 0.3 0.4 0.5 5 4 3 2 1 0 1 0.40 0.44 0.48 5 4 3 2 1 0 1 0.00 0.04 0.08 0.00 0.10 0.20 0.30 (a) (b) (c) ξ σ(ξ) σ(ξ) σ(ξ) ξ ξ ρ = 5.00 ρ = 1.00 ρ = 0.50 ρ = 0.10 ξ=0.485 continuous line σ(ξ) Figure 11: Dimensionless normal stress distribution along the interface for different ρvalues, with λ= 0.485, η= 0.00728, κ= 0.25 and ζ= 29.67. (b) Zoom at the loaded end. (c) Zoom at the traction free end. layers. In the DPPS test, stresses along the interface are mainly dominated by shear stresses. Thus, for smaller κvalues, the stress distribution along the interface is more uniform. As can be expected, this effect is more pronounced for shear stresses. Nevertheless, it is noticeable that compressions almost disappear in260 the traction free end for larger κvalues. This behaviour is very similar to the one obtained for η. 3. The Coupled Criterion of Finite Fracture Mechanics (CC-FFM) applied to linear-elastic interfaces in the DPPS test In this section, the CC-FFM [13, 14, 44] is used to predict the critical load265 which generates the crack onset in the interface in the DPPS test. The CC-FFM uses both the stress and incremental energy criteria, predicting the onset of a crack with a finite (dimensionless) length ∆λwhen both criteria are simultaneously fulfilled. 21 0.0 0.1 0.2 0.3 0.4 0.5 0 5 10 15 0.40 0.42 0.44 0.46 0.48 0.50 6 8 10 12 14 κ = 1.50 κ = 0.50 κ = 0.25 κ = 0.10 ξ=0.485 ξ=0.485 ξ (a) ξ (b) τ(ξ) τ(ξ) continuous line line with markers τ(ξ) τ (ξ) SL Figure 12: (a) Dimensionless shear stress distribution along the interface for different κvalues, with λ= 0.485, η= 0.00728, ρ= 0.142 and ζ= 29.67. (b) Zoom at the loaded end. 0.0 0.1 0.2 0.3 0.4 0.5 -6 -5 -4 -3 -2 -1 0 1 (a) (b) (c) ξ 0.40 0.44 0.48 -5 -4 -3 -2 -1 0 1 0.00 0.04 0.08 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ξ=0.485 κ = 1.50 κ = 0.50 κ = 0.25 κ = 0.10 σ(ξ) σ(ξ) σ(ξ) ξ ξ continuous line σ(ξ) Figure 13: (a) Dimensionless normal stress distribution along the interface for different κ values, with λ= 0.485, η= 0.00728, ρ= 0.142 and ζ= 29.67. (b) Zoom at the loaded end. (c) Zoom at the traction free end. 22 As explained in the previous section, the linear elastic behaviour of an undamaged spring, located at an interface point x, is given by relations σ(x) = knδn(x) and τ(x) = ktδt(x), see Eqs. (15) and (16). Following [4, 7, 5], the Energy Release Rate (ERR) due to an interface fracture is actually given by the energy stored in the spring (per unit area) before fracture. Thus, the ERR at an interface point can be expressed by the interface tension and shear stresses acting there G(x) = GI(x) + GII (x) = hσ(x)i+hδn(x)i+ 2+τ(x)δt(x) 2=hσ(x)i2 + 2kn +τ2(x) 2kt (38) where h·i+=(·)+|·| 2stands for the positive part of a real number, also known270 as Macaulay brackets. Consequently, the ERR at an interface part under compression is associated to pure mode II, i.e. G=GII. If the dimensionless variables and parameters defined in Table 1 are used, also a suitable dimensionless ERR can be defined in terms of the dimensionless stresses, ˆτ(ξ) and ˆσ(ξ), obtained in the previous section, cf. [44, 45], ˆ G(ξ) = 2ktl2 ch P2G(ξ) = ˆ GI(ξ) + ˆ GII (ξ) = κhˆσ(ξ)i2 ++ ˆτ2(ξ) (39) Then, the definition of the energy based fracture-mode-mixity angle ψ(ξ), originally given in terms of the ERR components as tan2ψ=ˆ GII (ξ) ˆ GI(ξ)for a positive GI(ξ)>0, can be extended to cover the whole angle range as follows [5] tan ψ(ξ) = √κ−1ˆτ(ξ) ˆσ(ξ),for −π≤ψ(ξ)≤π(40) The plot of the fracture-mode-mixity angle ψ(ξ) at undamaged points ξ, for the DPPS test model with the parameters given in Table 2, is shown in Fig. 14. As can be observed, significant compressions appear in the loaded end while275 significant tensions occur in the free end. In the following, the CC-FFM will be applied to the DPPS test considering the parameter values defined in Table 2. 23 ψ ( ξ )º 0.0 0.1 0.2 0.3 0.4 0.5 75 80 85 90 95 100 105 110 ψ (λ =0.00) HS a ψ (λ =0.12) HS a ψ (λ =0.20) HS a ψ ( ξ )º ξ ξ = 0.485 Figure 14: Fracture-mode-mixity angle ψ(ξ) in degrees computed along the undamaged interface, for the parameters shown in Table 2. Additionally, ψadefined in (44) is indicated for several values of λHS. 3.1. Energy based criterion Similarly as in [44, 45], the incremental energy criterion under mixed mode is satisfied when Zλ λ−∆λ G(λ0) dλ0≥Zλ λ−∆λ Gc(ψ(ξ)) dξ, (41) where, according to Fig. 15, ∆λis a finite crack advance and G(λ0) is the ERR280 at the tip of a virtual interface crack of length ∆λ0computed by (38). Gc(ψ(ξ)) is the fracture energy associated to an undamaged interface point ξbefore the debonding propagation. It is given as a function of the fracture-mode-mixity angle ψ(ξ) defined in Eq. (40), which is coherent with the main hypothesis of FFM assuming an abrupt onset of a crack of a finite size, see [46] for a discussion.285 Fig. 15 shows the way how λ0is associated to a virtual-interface-crack tip, and ξto a point in an initially undamaged segment ∆λof the interface. Figs. 16 and 17, respectively, show the plots of dimensionless stresses ˆτ(λ0) and ˆσ(λ0) at the tip of the virtual interface crack of length ∆λ0and overlap 24 Notice that both the modulus of the critical traction vector tc(ψ(ξ)) as well as the previously defined fracture energy Gc(ψ(ξ)) depend on the same fracture-325 mode-mixity angle ψ(ξ) evaluated at a considered undamaged point ξbefore the debond onset, see Fig. 14. If the Hutchinson-Suo empirical law (44) is assumed for ˆ Gc(ψ), then, similarly as in the energy criterion, interface fracture at a point ξis allowed only if |ψ(ξ)|<¯ ψa(λHS). 0.0 0.1 0.2 0.3 0.4 0.5 0 1 2 3 4 5 6 7 0.45 0.47 5.0 6.0 7.0 0.00 0.04 0.08 0.55 0.65 0.75 0.46 0.48 t (ξ) [8] t (ξ) SL t (ξ) ξ ξ ξ t (ξ)t (ξ) (a) (b) (c) ξ = 0.485 Figure 22: Dimensionless modulus of tractions ˆ t(ξ), for the parameter values given in Table 2. Fig. 22 shows distributions of ˆ t(ξ) for the present model and the SL model.330 Noteworthy, the behavior of ˆ t(ξ) is very similar for both models except for the end zones of the interface, as could be expected from the plots of tractions shown in Section 2.3. Distributions of ˆ tc(ξ) for several values of λHS are presented in Fig. 23. As could be expected, smaller values of λHS lead to higher differences in the very335 end zones of the interface. Then, the stress criterion in Eq. (47) for a finite crack advance ∆λfrom the loaded end of the interface can be expressed in terms of the dimensionless function s(ξ) = ˆ tc(ψ(ξ)) ˆ t(ξ)(53) 31 0.0 0.1 0.2 0.3 0.4 0.5 1.0 1.5 2.0 2.5 ξ t (ξ) c 0.40 0.44 0.48 1.0 1.2 1.4 1.6 1.8 2.0 2.2 0.00 0.04 0.08 0.7 0.8 0.9 1.0 t (ξ) c t (ξ) c ξ ξ (a) (b) (c) t (λ =0.50) HS t (λ =0.30) HS t (λ =0.20) HS ξ = 0.485 c c c t (SL) c Figure 23: Dimensionless critical modulus of traction ˆ tc(ξ), for the parameter values given in Table 2. as σr x ¯τc≥s(ξ) ηfor all ξ, λ −∆λ≤ξ≤λ(54) or as σr x ¯τc≥¯s(λ−∆λ) η,where ¯s(λ−∆λ) = max λ−∆λ≤ξ≤λs(ξ) (55) is a monotonous function. Function s(ξ) is shown in Fig. 24 for several λHS values. This function characterizes the load factor (LF) required to produce the debond onset according to the stress criterion. Noteworthy, a local minimum of the function s(ξ) in the340 neighbourhood of the loaded end indicates a highly stressed region there. This observation agrees with the experimental data in [37], where small cracks are observed in this zone prior to the failure. 3.3. The coupled criterion A way to characterize the predictions of the coupled criterion of FFM+LEBIM, applied in the present study, is to use a suitably defined dimensionless characteristic parameter [9, 17] defined, using parameters for pure fracture mode II, 32 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.5 1.0 1.5 0.36 0.38 0.40 0.42 0.44 0.46 0.48 0.0 0.1 0.2 0.3 0.4 s (λ =0.50) HS s (λ =0.30) HS s (λ =0.20) HS s ξ = 0.485 s ( ξ ) (a) (b) s ( ξ ) SL ξξ Figure 24: Dimensionless function of the stress criterion s(ξ), for the parameter values shown in Table 2. as µ=2kt¯ GIIc ¯τ2 c =τ2 max ¯τ2 c ,(56) where τmax and ¯τcare the maximum and critical shear stresses associated to345 the energy and stress criteria, respectively. Thus, for µ= 1 the present model reverts to the original LEBIM. For increasing value of µand for a fixed ¯ GIIc and ¯τc, the interface becomes stiffer, and for µ→ ∞ a perfect (rigid) interface is obtained. As can be seen from the previous sections, the fracture energy and the strength of the interface are independent data in the present FFM+LEBIM ap-350 proach. Recall that in the original LEBIM these variables were directly related by an equality including the interface stiffness [5, 43]. To combine both criteria in Eq. (46) and Eq. (55), the energy criterion is rewritten in a form similar to that of the stress criterion σr x ¯τc≥√µ ηpg(λ−∆λ).(57) Then, by combining Eqs. (55) and (57), the coupled criterion of FFM+LEBIM 33 can be expressed in the present case as σr x ¯τc≥σr c ¯τc = min ∆λmax ¯s(λ−∆λ) η,√µ ηpg(λ−∆λ),(58) where the minimum is achieved at ∆λ= ∆λc. √µ g (λ-∆λ), s (λ-∆λ) λ-∆λ (a) (b) 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.2 0.4 0.6 0.8 1.0 g with µ=8 g with µ=8 SL g with µ=4 g with µ=4 SL g with µ=1 g with µ=1 SL s s SL λ’ = 0.485 0.25 0.30 0.35 0.40 0.45 0.50 0.0 0.1 0.2 0.3 0.4 0.5 0.6 √µ g (λ-∆λ), s (λ-∆λ) λ-∆λ λ’ = 0.485 Figure 25: Intersection of the curves pµg(λ−∆λ) and s(λ−∆λ) for the present model and the SL model, for several values of µand for the parameter values shown in Table 2. Fig. 25 shows failure curves for the energy and stress criteria obtained with the present model and the SL model, using the parameters given in Table 2355 and λHS = 0.5. The figure includes the solutions for several values of µ. It can be observed that both criteria coincide for µ= 1, predicting an infinitesimal crack initiation with ∆λc= 0, but for increasing µvalues, ∆λcand the failure load increase as well. For the chosen value λHS the differences between the developed model and the SL model are quite small, however, for smaller values360 of λHS they can become quite large. Finally, debond growth (crack propagation) is presented in Fig. 26 for µ= 8 and µ= 4. The intersection of both (energy and stress based) criteria allows to obtain the load factor (LF) required to produce a debond onset or growth with a finite length ∆λc. Then, for each debond a “new” overlap length, λ0=λ−∆λc,365 34 √µ g (λ-∆λ), s (λ-∆λ) λ-∆λ s √µ g (λ-∆λ) virtual LF actual LF ∆λ 2 ∆λ 3 λ-∆λ (a) (b) ∆λ 4 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ∆λ∆λ 13 ∆λ 2 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 0.6 √µ g (λ-∆λ), s (λ-∆λ) ∆λ 1 Figure 26: Prediction of the stepwise debond growth (propagation) in the DPPS test, for the parameter values shown in Table 2, with λHS = 0.5 and for (a) µ= 8 and (b) µ= 4. is used in the coupled criterion. To be consistent with the non-local method used, horizontal lines are represented for each LF with its corresponding ∆λc, thus a stepped graph representing a series of finite crack advances is obtained. From a physical point of view, these results show that fracture is actually unstable under load-control and the first LF will produce a complete debond of370 the interface. Hence, the jumps included in the stepped graph are associated to a kind of hypothetical LFs. It should be mentioned that for small overlap lengths also another debond might occur in the opposite (free) end. This alternative failure mechanism is not studied in the present investigation, assuming that the unstable character375 of the crack growth will produce a one-way debond advancing. 35 4. Comparison between experimental results and theoretical predictions This section aims to compare the predictions obtained by the presented procedure and some experimental data available in literature, where numerous380 types of shear test can be found, see [35, 50] for a review. In every case, the results obtained depend on the laboratory test setup, the mechanical properties of materials and the type of control during the tests. Thus, for an adequate comparison, the mechanical and geometrical characteristics of the concrete block and reinforcement are necessary. Moreover, the mechanical behaviour of the385 interface depends not only on the adhesive but also on the concrete block characteristics and the bonding process. This is due to the experimentally observed fact that the debond surface is typically produced in a thin layer of concrete close to the reinforcement laminate. This is the reason why it is difficult to determine the parameters kt,kn,¯ GIIc, ¯τcand λHS. Therefore an inverse analysis390 is an attractive option to obtain these parameters, cf. [51, 52], however a large number of test would be required. As the objective of this investigation is not a complete interface characterization, a simplified inverse analysis, providing only the fracture toughness and the critical shear stress, is applied. The experimental results by Carrara et al. [39], studying a shear test for395 different reinforcement lengths shorter than the effective anchorage length, are used to compare with the present theoretical predictions. Note that for the reinforcement lengths longer than the effective anchorage length, the values of ∆λcand σc x(the critical values of σr x) are always the same. For this reason, these tests are a good option for an inverse analysis, since the only parameter400 which significantly varies is the length of the joint. Four different tests in [39] are used for the comparison, namely 30B, 60C, 90B and 120B. In Table 4, the mechanical and geometrical characteristics of the concrete block and reinforcement used in [39] are indicated. As no further information was found, the reinforcements are considered homogeneous405 and isotropic. 36 Table 3: Geometrical and elastic properties of the different tests obtained from [39]. lb(mm) hb(mm) tb(mm) Eb(GPa) νb Concrete block 300 90 150 28.7 0.20 Reinforcements lr(mm) hr(mm) tr(mm) E1(GPa) ν13 30B 31 1.32 30.5 168.5 0.248 60C 61 1.33 30.3 168.5 0.248 90B 91 1.31 29.3 168.5 0.248 120B 121 1.22 29.8 168.5 0.248 In Table 4, the interface properties used in the present procedure are shown. Additionally, λHS = 0.5 [53] is assumed due to the shear test was carried out in a predominant fracture mode II. The ktvalue is calibrated in order to fit the slopes of the load-displacement curves for every test. The relation κ=kt kn= 0.25410 is taken assuming an isotropic behaviour of the interface. Finally, to determine ¯ GIIc and ¯τa simplified inverse analysis with the three tests is performed. Table 4: Interface properties of the different tests used in the presented procedure. Specimens kt(MPa/mm) κ=kt kn ¯ GIIc (N/mm) ¯τc(MPa) λHS 30B 280 0.25 0.42 5.8 0.5 60C 90 0.25 0.42 5.8 0.5 90B 340 0.25 0.42 5.8 0.5 120B 65 0.25 0.42 5.8 0.5 In Fig. 27, the load-displacement predictions are presented. Pis the load applied on the reinforcement and the ”slip” is the relative tangential displacement (δt) produced between the loaded edge of the reinforcement and the concrete415 block. The thick lines represent the solutions obtained by the present procedure and the thin lines are the experimental results in [39] for the different configurations. Recall that in the present model, only the debond onset is predicted 37 P (KN) slip (mm) Thick lines: Presented P. Thin lines: Tests [27] 0 5 10 15 20 0.00 0.05 0.10 0.15 0.20 0.25 0.30 60C at F.E. 90B at L.E. 120B at L.E. 30B at F.E. Figure 27: Experimental load vs. displacement plots obtained from [39] compared with the failure load predictions by the present procedure (Present P.). F.E. and L.E. mean the free end and loaded end correspondingly. since the boundary conditions include a load control. In this sense, the arrow size is associated to the first finite debond-size produced in the debond onset,420 whereas the subsequent unstable debond propagation is indicated by the dashed line. However, the tests are driven under displacement control and are able to capture even a snap-back behaviour, which was one of the aims in [39]. Except for this difference, an excellent agreement between the experimental response and the model predictions is observed.425 Noteworthy, the debonds are not always produced at the loaded edge of the reinforcement. For the shortest joints, the debonds are produced at the free edge. Specifically, in this case, the failure occurs by peeling for specimens 30B and 60C, similarly as described in [39]. The procedure presented is also able to predict this detail in the specimen failure, since the load necessary to430 produce the debond at the free end for specimens 30C and 60B is smaller than load necessary to originate the failure at the loaded edge. For the prediction of 38 the load at the free edge one can change the reference system for finite crack advance description or switch the boundary conditions between the free and loaded edges. Although the difference between these two loads is not large in435 specimen 60C, this difference is sufficient for the present procedure to predict the debond at the free edge using the interface properties in Table 4. Table 5: Comparison of the maximum load for every specimen. Specimens Tests [39] Present Procedure CZM [22] 30B 5.5kN at F.E. 4.5kN at F.E. 5.8kN at F.E. 60C 10.0kN at F.E. 10.1kN at F.E. 10.6kN at F.E. 90B 12.6kN at L.E. 12.9kN at L.E. 13.8kN at L.E. 120B 12.8kN at L.E. 11.8kN at L.E. 15.3kN at L.E. it should be noted that only a few CZMs are able to model adequately the tests by Carrara et al. [39]. This is due to a fracture mixed mode model, similar to that proposed by Carrara et al. [22], is necessary to predict the phenomenon440 of the free edge failure. In Table 4, the results in [22] are compared with the predictions obtained with the present procedure. This table shows an excellent agreement between the maximum load achieved in the tests and the predictions by the present procedure and by [22]. Note that in [22] the stiffnesses of the interface (ktand kn) are taken from the Italian standards. A discussion about445 the very wide range of values of the parameters can be found in that work. kt values are fitted in the present work similarly as in [22]. 5. Conclusions One of the goals of this study was the development of an analytical model able to characterize the crack onset and propagation along the interface between450 two adherents of any thickness/stiffness in double joints. Therefore, the present model is based on the Timoshenko beam theory and the linear elastic interface model. 39 As an application, the crack onset and propagation along the interface in the Double Pull-Push Shear (DPPS) test is studied using this original analytical455 solution. The peeling stresses in the the shear tests keep the discussion about the fracture energy still open [22, 35, 38]. The fracture energy in Mode I in the interface is significantly lower than the fracture energy in Mode II for these tests, therefore, even a small increase in the peeling stress could reduce the460 bearing capacity of the interface [35]. For this reason, the expressions for the normal and shear stresses in the adhesive layer are deduced for the DPPS test in a suitable dimensionless form, pointing out the role of the main characteristic variables and parameters of this test. Several parametric studies are carried out to analyse the influence of the geometry and material (adherents and adhesive)465 characteristics on the stress distributions. The obtained results show that: •Normal stresses at the loaded end are important for reinforcement stiffness larger than the block stiffness (i.e., large values of ρ), relatively small values of the reinforcement thickness (i.e., small values of η), and for adhesives whose shear stiffness is larger than the normal stiffness (i.e., for large470 values of κ). •The normal stresses become more important for small overlap lengths because these stresses increase their presence throughout the interface. •Tensions at the free end can increase achieving relevant values although compressions at loaded end decrease, e.g. when the laminate (reinforce-475 ment) thickness increases (i.e., for large values of ηor for small overlap lengths). •When tensions become relevant at the free end, the shear stresses present a quasi-uniform distribution [22] (i.e., for large values of ηand small values of ρ).480 In general, these results are in agreement with some experimental data in the literature [22, 37, 38]. 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