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One of the primary objectives in the design of composite structures is the prevention of premature bond failure. Therefore, the characterization of cohesive behavior is an important field of study in structural engineering. Using fracture mechanics principles, the cohesive behavior of an epoxy bonded coarse silica sand aggregate bond interface is studied in this paper, with a focus on finding a general analytical form of idealizing its behavior when used in a specimen possessing asymmetric and inhomogeneous qualities. Two series of smallscale specimens were experimentally tested under mixedmode bending (MMB) conditions, where it was found that there was negligible influence exerted on the fracture energy of the interface due to changes in the mixedmode ratio or initial crack length. Using finite element analysis (FEA) methods, an appropriate bilinear tractionseparation model was developed to both validate as well as obtain a set of consistent parameters applicable to all tested specimens. Comparison of the Global Method and the Local Method, used to obtain partitioned Mode I and Mode II fracture energy values from MMB specimens, were made, with the conclusion that both methods are adequate in the calculation of the total fracture energy though the Local Method should be used to obtain accurate partitioned Mode I and Mode II fracture energy values. Idealization of the bond interface using the cohesive parameters derived can be accurately achieved by the use of both contact interactions and cohesive elements in twodimensional and threedimensional FE models, though the results obtained using contact interactions would be expected to exhibit greater global stiffness.
In the study of composite structures, the behavior of the bond layer between adjacent material interfaces is of particular interest, since an effective bond is critical to achieve optimum performance of the structure rather than the undesirable premature failure at the bond interface. Fracture mechanics theories have been applied to the study of bond interface behavior, initially through the strengthbased failure criterion, developed by Inglis [
Abundant research has been conducted in previous decades in the effort to characterize bond behavior through a combination of experimental testing and applied analytical techniques, used in conjunction with finite element analysis (FEA) methods. Analysis of pure Mode I action (normal separation at the interface) and pure Mode II action (shear separation along the interface) have primarily been performed on homogenous specimens [
The aim of this research is to determine, with reasonable accuracy, the cohesive parameters of a bonded coarse silica sand aggregate layer intended for enhanced bonding between pultruded glass fiber reinforce polymer (GFRP) plates and ultrahigh performance concrete (UHPC) in fullscale hybrid beams and bridge systems [
This paper includes three main portions. The first section covers an extensive experimental study of smallscale specimens tested under mixedmode bending (MMB) loading conditions with attention towards the relationship between the applied modemixity as well as the crack length initially introduced in the specimen with respect to the bond performance. In the subsequent section, two distinct numerical methods, the Global [
The research performed in this study broadens the current understanding regarding the stateoftheart of bond interface behavior through detailed experimental and numerical analysis of an epoxy bonded coarse silica sand aggregate layer, an improved method of bonding between high performance materials. The findings provide a closedform method for the derivation of consistent partitioned cohesive parameters for MMB specimens. The set of parameters, validated for both 2D and 3D modeling environments in ABAQUS, include considerations for factors such as material inhomogeneity and geometric asymmetry.
For the determination of the cohesive bond properties, MMB specimens were chosen. The advantage of testing a single series of MMB specimens compared with parallel tests of double cantilever beams (DCB) and endnotch flexure (ENF) specimens is the intermixed nature of the Mode I and Mode II behaviors, allowing for interactions between the cohesive laws governing each failure mode [
The tested specimens consisted of a 9.6 mm thick pultruded GFRP plate bonded to a 50 mm thick castinplace layer of UHPC via the epoxy bonded coarse silica sand aggregate layer, with a constant length and width of 460 and 130 mm, respectively, for the entire specimen. The MMB specimens were designed based on the concepts presented in ASTM Standard D667106 [
The GFRP plates used were specified by the manufacturer to have a minimum flexural strength of 400 MPa with a flexural modulus of elasticity equal to or greater than 25 GPa [
Experimental testing of MMB specimens using a test matrix of varying initial crack lengths and mixedmode ratios was conducted. In order to optimize the number of possible test specimens, the test configuration and dimensions allowed for the cohesive interfaces from both ends of the specimen to be tested. The test matrix, with the specimen IDs, is presented in
Test matrix and specimen IDs.
Specimen ID  Initial crack length ( 
Moment arm ( 

Sa80c250  80  250 
Sa100c250  100  250 
Sa120c100  120  100 
Sa120c175  120  175 
Sa120c250  120  250 
Sa120c325  120  325 
Sa120c400  120  400 
Sa140c250  140  250 
Sa160c250  160  250 
The values for the crack lengths and moment arms were chosen to maximize the diversity of configurations tested, with crack lengths ranging incrementally from 0 to 0.5
Testing was conducted under displacement control mode, with a vertical loading rate of 1 mm/min at the load actuator. Three linear variable differential transformers (LVDTs) were positioned at quarterpositions along the top of the GFRP plate. Since both sides of all specimens were to be tested, the first tested side (Side A) was loaded up to and past the peak load, in a manner that would provide an adequate depiction of its behavior. It was then unloaded prior to the onset of unstable crack propagation along the bond interface, once the crack growth approached the midspan of the specimen. The second tested side (Side B) was, however, allowed to undergo complete failure.
Specimen dimensions and test setup (all dimensions are in mm).
The loaddeflection curves, describing the relationship between the applied actuator load and the downward displacement at the point of load application, from the nine tested specimens (with two sets of data obtained for each specimen) are presented in
Experimental loaddeflection comparison for specimens tested at
Experimental loaddeflection comparison for specimens tested at
Complete debonding, characterized by separation of the GFRP plate from the UHPC layer, occurred in all of the tested specimens except for Sa120c100, Sa120c250 and Sa80c250; in these specimens, tensile cracking in the UHPC layer within the flexural region was observed without separation of the GFRP plate from the UHPC layer. Tensile cracking in the UHPC may have been due to higher stress field concentrations caused by a reduction in the UHPC thickness at the ends of the specimens; the additional space was used to accommodate the steel builtup anchor, as shown previously in
All of the tested specimens exhibited near linearelastic behavior during the first phase of loading up to the peak load. Once the load applied approached the peak load, the global stiffness of all the specimens was observed to decrease, where some of the loaddeflection curves peaked at the maximum load and others demonstrated a smooth transition through a zero slope plateau. Past the peak load, crack propagation commenced and it was seen that the specimens did continue to provide resistance to the applied load, though typically at a level lower than the peak load. The decrease in strength was particular dramatic for specimens with crack lengths (
Representative photographs illustrating typical failure mode: (
Typically, in bond behavior studies where at least one of the adherends is a cementitious material such as reinforced concrete, two types of cohesive failure mechanism are considered. The first is failure through the interior of the adhesive layer and the second is interface cracking within the concrete adherend. However, the unique combination of high performance materials used in this study, namely GFRP and UHPC, allows for the elimination of the latter failure mechanism. This is due to the enhanced overall strength of the UHPC confirmed through previous studies [
The typical cohesive parameters used to characterize the behavior of a bond interface in the CZM are: the critical fracture energy, effective nominal stress, the damage initiation ratio and the initial material stiffness per unit area. For simple test configurations in elastic homogeneous specimens, the cohesive parameters can be easily derived from calculations based on the experimental loaddeflection curve. Nevertheless, with increasing complexities, such as the introduction of mixedmode loading, asymmetry in the crosssectional dimensions as well as an inhomogeneous composition, more detailed derivations will be required to determine the cohesive parameters. Various research studies have been conducted over the years with the objective of obtaining accurate methods to numerically determine values for fracture energy under different configurations. These studies include considerations for loading configurations [
Williams [
In this following section, an extension to the work originally completed by Williams, derived by the authors, will be presented, resulting in a set of equations for the partitioned critical fracture energy for a MMB asymmetric bilayer specimen with an inhomogeneous crosssection. A diagram illustrating the components of the crack tip contour rotation is provided in
Diagram illustrating crack tip contour rotations used in the Global Method [
Given a height factor, ξ, equal to:
It follows that the expression for the moment of inertia are:
Expanding on the equations derived by Williams [
The total fracture energy,
In order to obtain the fracture energy associated specifically with Mode I and II action, the applied moments can be described by the following linear combinations:
To determine the value of the coefficients
Under pure Mode II loading,
Under pure Mode I loading,
Substituting the expressions for
The total fracture energy would be equal to the linear sum between Equations (12) and (13), with the following relationship, where:
The equivalent flexural stiffness,
Hutchinson and Suo [
Diagram illustrating crack tip stress field parameters used in the Local Method.
Dundurs’ elastic mismatch parameters, where α measures the mismatch in the inplane tensile modulus across the interface and β measures the mismatch in the inplane bulk modulus, are given as:
The complex interface equivalent modulus of elasticity is given as:
The neutral axis of the composite beam ahead of the crack can be found at a distance of Δ
The dimensionless crosssection,
The complex stress intensity factor,
The complex stress intensity factor can be partitioned into its Mode I (real) and Mode II (imaginary) components, where
The total energy release rate for crack advance in the interface,
Firstly, it should be noted that the effect of the apparatus weight was not considered in the derivations presented in
Possible nonlinearity on the experimental results produced by the configuration of the loading apparatus used, specifically in regards to the horizontal force induced in the system at higher actuator displacements, should also be considered. It was found through previous research that the application of load above the lever arm may induce nonlinearity in the results, sometimes up to 30% [
The investigation conducted above shows that the nonlinearity effect due to the horizontal force induced in the lever arm is within a range between 1% and 4%. The difference is thereby demonstrated to be marginal and in line with the expected error for MMB testing [
Nonlinearity imposed due to induced horizontal force on lever arm.
Specimen ID  Angle of rotation (degrees)  Horizontal Force (kN)  Moment (kNm)  Nonlinearity due to horizontal force (%)  

Horizontal Force  Vertical Force  
Sa80c250A  0.66  0.018  0.003  0.394  1 
Sa80c250B  0.64  0.015  0.002  0.336  1 
Sa100c250A  0.77  0.017  0.003  0.317  1 
Sa100c250B  0.65  0.016  0.002  0.351  1 
Sa120c100A  1.56  0.097  0.015  0.362  4 
Sa120c100B  1.69  0.10  0.015  0.336  4 
Sa120c175A  1.50  0.047  0.007  0.314  2 
Sa120c175B  1.10  0.042  0.006  0.379  2 
Sa120c250A  0.93  0.020  0.003  0.301  1 
Sa120c250B  0.78  0.018  0.003  0.325  1 
Sa120c325A  1.0  0.017  0.003  0.315  1 
Sa120c325B  0.76  0.011  0.002  0.261  1 
Sa120c400A  1.03  0.014  0.002  0.300  1 
Sa120c400B  1.09  0.014  0.002  0.291  1 
Sa140c250A  0.93  0.014  0.002  0.218  1 
Sa140c250B  0.97  0.017  0.003  0.250  1 
Sa160c250A  1.17  0.018  0.003  0.222  1 
Sa160c250B  1.54  0.026  0.004  0.240  2 
The two analytical techniques, in the form presented in the
Firstly, a comparison between the values of total fracture energy obtained through the two different techniques showed that identical results (or near identical results in the case of the plane strain formulation using the Local Method) were obtained, which agrees with similar previous studies [
Comparison of derived fracture energy.
Specimen ID  Global Method [ 
Local Method (plane stress) [ 
Local Method (plane strain) [ 















Sa80c250  0.443  0.443  0.0002  2138  0.443  0.196  0.247  0.792  0.421  0.186  0.235  0.792 
Sa100c250  0.335  0.335  0.0002  2138  0.335  0.148  0.187  0.792  0.319  0.141  0.178  0.792 
Sa120c100  0.966  0.965  0.0013  721  0.966  0.412  0.554  0.743  0.919  0.392  0.527  0.743 
Sa120c175  0.769  0.769  0.0005  1470  0.769  0.336  0.433  0.777  0.732  0.320  0.412  0.777 
Sa120c250  0.662  0.662  0.0003  2138  0.662  0.292  0.370  0.792  0.370  0.278  0.352  0.792 
Sa120c325  0.572  0.572  0.0002  2704  0.572  0.254  0.318  0.799  0.544  0.242  0.302  0.799 
Sa120c400  0.597  0.596  0.0002  3179  0.597  0.266  0.331  0.804  0.568  0.252  0.315  0.804 
Sa140c250  0.491  0.491  0.0002  2138  0.491  0.217  0.274  0.792  0.467  0.206  0.261  0.792 
Sa160c250  0.668  0.668  0.0003  2138  0.668  0.295  0.373  0.792  0.636  0.281  0.355  0.792 
Note: fracture energy values provided in kJ/m^{2}.
A brief discussion pertaining to the relationship between changes in test configurations and the resulting fracture parameters will be presented here. For ease of analysis, the plane stress partitioned fracture energy values calculated using the Local Method (as given in
In regards to the effects of changing the initial crack length of the specimen, it is difficult to determine a reliable relationship due to the large scattered nature of the data points; the small number of duplicate tests for each test configuration is also a contributing factor. The
Relationship between partitioned fracture energy and initial crack length.
Relationship between partitioned fracture energy and applied modemixity ratio.
Similarly, when examining the correlation between the fracture energy and the applied mixedmode ratio, it was found that as the mixedmode ratio increased (by the application of the force through a longer moment arm and thereby increasing the relative magnitude of the Mode I mechanism with respect to the Mode II mechanism), the total fracture energy would decrease in a near linear fashion. Since it has been shown that faster loading rates during experimentation would increase the perceived fracture energy of the interface [
From the theorem of minimum energy, the equilibrium of a conservative system deformed by surface forces must be achieved in such a way that the total potential energy of the system would reach a minimum state [
In order for the derived fracture energy values to be applicable for use through the tractionseparation model (which will be discussed in further detail in
It is important to realize that the calculated values for the partitioned fracture energy (
It can, therefore, be concluded that the critical fracture energy under pure Mode I loading can be taken as the value obtained from the experimental testing of the specimens whereas the critical fracture energy under pure Mode II loading would be a value greater than the calculated value of each individual specimens. It is for this reason that the derived expressions for fracture energy and stress intensity factors provided in
From the analytical derivations performed in
Cohesive parameters of a typical bilinear tractionseparation model [
The relationships between each of the parameters is given as:
The above four parameters are specific to the corresponding Mode I and Mode II critical fracture energy values and do not necessarily need to be the same for both failure modes.
The FE model was developed using ABAQUS 6.9. In the case of the 2D FE model, plane stress quadratic quadrilateral elements were used for both the GFRP and UHPC materials. Three layers of elements were used to represent the GFRP layer with a mesh size in the longitudinal direction equal to 2 mm. The same longitudinal mesh size was used for the UHPC block, with a total of 17 layers through its depth. The layer of cohesive elements, when used in the model, was tied at the top and bottom to the adjacent GFRP and UHPC material parts and had a thickness equal to 1 mm, with a longitudinal mesh size of 0.5 mm; this allowed for a 4:1 mesh ratio between the cohesive elements and the adjacent materials. Similarly, for the threedimensional FE model, linear 3D stress brick elements used. For the GFRP material part, two layers of elements were used with an average mesh size equal to 8 mm in the longitudinal and transverse directions. The UHPC material part had a mesh size equal to approximately 8 mm in all three dimensions. Lastly, the 1 mm thick cohesive layer had a mesh size of approximately 2 mm in order to provide the same 4:1 mesh ratio that was used in the 2D FE model. Representative diagrams showing the fully developed 2D and 3D models are shown in
The cohesive properties were modeled using the power law form of the energy criterion, with a power exponent equal to unity. The failure criterion of the cohesive interface can be expressed by the following equation [
Diagram of developed 2D FE model.
Diagram of developed 3D FE model.
For the initial 2D parametric study, the bond interface was idealized using a contact interaction, with the input parameters for the tractionseparation model equal to those obtained from experimental tests. The advantage of using contact interactions rather than a layer of cohesive element for modeling the bond interface is that it allows for an additional simplification of the tractionseparation model as a twoparameter model for each mode by assuming a default contact penalty constraint to the interaction. This imposes cohesive constraints in the normal and shear direction without the need to provide the values for the initial material stiffness. The assumption effectively removes all influences from the shape of the tractionseparation model, which eliminates the dependence on the initial material stiffness as well as the damage initiation ratio.
Using this initial simplification, a parametric study was then performed by using effective nominal stress values for Mode I equal to 1, 1.5, 2, and 3 MPa in order to determine the cohesive strength of the bond system; this method for determining the cohesive strength is in line with techniques used by other researchers [
Parametric study matrix for Specimen Sa120c100.
Model ID  Fracture energy ( 
Effective nominal stress ( 


Per unit width (mm)  Per 130 mm width  Per unit width (mm)  Per 130 mm width  




( 
( 
( 
( 

Sa120c1001MPav 2D  0.412  0.554  53.55  72.05  1.0  1.35  130  174.90 
Sa120c1001.5MPav 2D  0.412  0.554  53.55  72.05  1.5  2.02  195  262.36 
Sa120c1002MPav 2D  0.412  0.554  53.55  72.05  2.0  2.69  260  349.81 
Sa120c1003MPav 2D  0.412  0.554  53.55  72.05  3.0  4.04  390  524.71 
Load deflection comparison for finite element analysis (FEA) of Sa120c100 with parametric study of Mode I effective nominal stress.
By visual comparison, it was observed that the Model Sa120c1002MPav most closely coincided with experimental results. In order to confirm this hypothesis, the defining properties from Model Sa120c1002MPav were applied to the other eight MMB configurations tested, using the parameters given in
Model validation parameters for mixedmode bending (MMB) specimens.



















Sa80c2502MPav 2D  0.196  0.247  25.44  32  2  2.53  260  328 
Sa100c2502MPav 2D  0.148  0.187  19.25  24  2  2.53  260  328 
Sa120c1002MPav 2D  0.412  0.554  53.55  72  2  2.69  260  350 
Sa120c1752MPav 2D  0.336  0.433  43.73  56  2  2.57  260  334 
Sa120c2502MPav 2D  0.292  0.370  38.02  48  2  2.53  260  328 
Sa120c3252MPav 2D  0.254  0.318  33.01  41  2  2.50  260  325 
Sa120c4002MPav 2D  0.266  0.331  34.56  43  2  2.49  260  323 
Sa140c2502MPav 2D  0.217  0.274  28.21  36  2  2.53  260  328 
Sa160c2502MPav 2D  0.295  0.373  38.38  48  2  2.53  260  328 
For more indepth and precise examination, the initial simplification used in the validation step (
Consistent parameters for MMB specimens.
Parameter  Mode I  Mode II 

Damage initiation ratio (δ_{ratio})  5  
Critical partitioned fracture energy ratio ( 
0.33  
Fracture energy ( 
0.3  0.9 
Effective nominal stress ( 
2  6 
Initial material stiffness ( 
33.33  100 
These parameters were tested in the second series of validation tests in FEA. The results, presented in detail in the subsequent subsection, showed good correlation with the experimental data for all specimens tested.
The last step was the validation of consistent parameters in a 3D model. Cohesive elements, tied at the top and bottom to the GFRP layer and UHPC layer, respectively, were used in this final stage in order to allow for proper mapping and visualization of the crack propagation. It was assumed for the 3D model that the Mode III parameters were identical to those of Mode II [
A detailed comparison of the FEA results obtained from the three series of tests is graphically shown for all of the nine MMB specimens in
Validation of FEA with experimental results using Mode I effective nominal stress = 2 MPa for specimens: (
Firstly, the use of a layer of cohesive elements rather than contact interactions to model the bond layer was found to result in decreased stiffness of the interface. This finding is consistent with previous studies performed under pure Mode II loading [
The results presented in this paper on the behavior and characterization through cohesive parameters of the epoxy bonded coarse silica sand aggregate layer provide a practical set of consistent cohesive parameters that can be applied to FE modeling techniques in both 2D and 3D environments to reasonably approximate the behavior of the bond interface. Comparisons made between the Global and Local Methods for deriving the critical fracture energy of asymmetric, inhomogeneous specimens tested under mixedmode configurations demonstrate the suitability of both methods in the determination of the total critical fracture energy. Nevertheless, partitioned critical fracture energy values calculated through the Global Method heavily overestimates the contribution of Mode I action and should only be recommended for use in simpler configurations consisting of pure mode, symmetric and homogeneous specimens. The derived set of consistent parameters can be applied when modeling with both contact interactions and contact elements, though lower stiffness behavior is to be expected in the latter case. It is recommended for future studies to examine more closely the results obtained from one specific test specimen, which demonstrated an increase in stiffness and strength after crack propagation had initialized, a phenomenon that was later replicated through FEA.
Dimensionless crosssection
Crack length
Section width
Geometric factors
Complex interface equivalent modulus of elasticity
Modulus of elasticity of upper arm, and lower arm
Modulus of elasticity in upper arm, and lower arm, accounting for plane stress and plane strain effects
Equivalent flexural stiffness of uncracked section
Ratio of the mean square between the groups to the mean square within the group as defined in the Single Factor ANOVA test
Mode I, and Mode II fracture energy
Critical fracture energy
Height of upper arm, and lower arm
Dimensionless moment of inertia
Moment of inertia of equivalent section with one material
Moment of inertia of upper arm, and lower arm
Initial material stiffness
Complex stress intensity factor associated with Mode I, and Mode II
Length of MMB specimen
Linear combination of edge moments
Bending moment contribution from Mode I, and Mode II
Bending moment applied to upper arm, and lower arm
Linear combination of edge loads
Effective nominal stress
External work performed
Potential strain energy
Location neutral axis in uncracked section
Neutral axis parameters
Dundurs’ elastic mismatch parameters
Separation at damage initiation, and failure
Damage initiation ratio
Oscillation index
Inplane bulk modulus in upper arm, and lower arm
Partitioning coefficient associated with Mode I
Poisson’s ratio in upper arm, and lower arm
Height factor
Curvature in upper arm, and lower arm
Partitioning coefficient associated with Mode II
Angle shift due to elastic mismatch
The authors would like to thank the following companies for their generous donations of materials used: Pultrall Inc., Lafarge Canada, and Sika Canada Inc. We would also like to acknowledge the University of Calgary and the Natural Sciences and Engineering Research Council of Canada (NSERC) for their financial support towards this research project.
The authors declare no conflict of interest.