Boundary effect on the softening curve of concrete

The apparent fracture energy of concrete experimentally determined on the basis of the work of fracture in bending or wedge splitting tests becomes larger with increasing specimen dimensions. This experimental observation may be attributed to the varying local fracture energy along the crack path. When the crack tip approaches the specimen boundary, the size of the fracture process zone will be reduced and, consequently, only a portion of the fracture energy is activated; i.e., the local fracture energy is getting smaller. The influence of this boundary effect diminishes with increasing specimen size resulting in the size dependence of the apparent fracture energy determined by the work-of-fracture method as an average value in the ligament. With varying local fracture energy, the local softening curve will also show variations. The latter are subject of the present study. Wedge splitting tests with different specimen sizes as well as inverse analyses of these experiments were carried out. For the inverse analyses, the cohesive crack model was adopted and an evolutionary optimization algorithm has been used. The boundary effect on the local fracture properties was taken into account and, as a result, the variation of the softening curve along the crack path could be determined. It was found that the tail of the softening curve is shortened and lowered due to the boundary effect whereas the initial slope of this curve appears to be not affected.     

A bilinear cohesive zone model tailored for fracture of asphalt concrete considering viscoelastic bulk material

A bilinear cohesive zone model (CZM) is employed in conjunction with a viscoelastic bulk (background) material to investigate fracture behavior of asphalt concrete. An attractive feature of the bilinear CZM is a potential reduction of artificial compliance inherent in the intrinsic CZM. In this study, finite material strength and cohesive fracture energy, which are cohesive parameters, are obtained from laboratory experiments. Finite element implementation of the CZM is accomplished by means of a user-subroutine which is employed in a commercial finite element code (e.g., UEL in ABAQUS). The cohesive parameters are calibrated by simulation of mode I disk-shaped compact tension results. The ability to simulate mixed-mode fracture is demonstrated. The single-edge notched beam test is simulated where cohesive elements are inserted over an area to allow cracks to propagate in any general direction. The predicted mixed-mode crack trajectory is found to be in close agreement with experimental results. Furthermore, various aspects of CZMs and fracture behavior in asphalt concrete are discussed including: compliance, convergence, and energy balance.     

Determination of concrete fracture parameters based on two-parameter and size effect models using split-tension cubes

The notched beam specimens have been commonly used in concrete fracture. In this study, the splitting-cube specimens, which have some advantages - compactness and lightness - compared to the beams, were analyzed for the effective crack models: two-parameter model and size effect model. The linear elastic fracture mechanics formulas of the cube specimens namely the stress intensity factor, the crack mouth opening displacement, and the crack opening displacement profile were first determined for different load-distributed widths using the finite element method. Subsequently, four series of experimental studies on cubic, cylindrical, and beam specimens were performed. The statistical investigations indicated that the results of the split-cube tests look viable and very promising.     

Constitutive models for cohesive zones in mixed-mode fracture of plain concrete

Discrete mixed-mode fracture (modes I and II) of plain concrete is investigated using a coupled and an uncoupled cohesive zone constitutive model in a finite element context. Fracture surfaces are confined to inter-element boundaries that are not necessarily coincident with the actual fracture surfaces. For this reason, traction components on the cohesive zone do not correspond to actual values either. In this work is demonstrated that only the coupled model is able to cope with these spurious traction components, that must decrease with crack opening. It is shown also that, in this regard, the key variable is the plastic potential adopted in the integration of tractions. Three mixed-mode fracture examples were tested in this work: a three-point single-edge notched beam, double-edge notched plates under variable lateral and normal deformation and four-point double-edge notched beams. A good fitting with experiments was obtained only for the coupled model. Mode II parameters can change in a large range without noticeable change in results, at least in the tested examples.     

Cohesive crack modelling of a simple concrete: Experimental and numerical results

Fracture tests were performed on six types of simple concrete made with two types of mortar matrix w/c = 0.32 and w/c = 0.42, two types of spherical aggregates (strong aggregates that debonded during concrete fracture, and weak aggregates, able to break), and two kinds of matrix-aggregate interface (weak and strong).The tensile strength, fracture energy and elasticity modulus of the six types of concrete were measured. These results are intended to serve as an experimental benchmark for checking numerical models of concrete fracture and for providing certain hints to better understand the mechanical behaviour of concrete.A bilinear softening function was used to model the fracture of concrete. The influence of the type of matrix, aggregate, and interface strength on the parameters of the softening curve are discussed: particularly, the fracture energy, the cohesive strength and the critical crack opening.     

Effect of aggregate shape on the mechanical properties of a simple concrete

The influence of aggregate shape on the fracture energy, tensile strength and elasticity modulus in concrete is considered. For this purpose, eight simple cement-based composites were designed, manufactured and tested, with two purposes: to provide experimental data that can throw some light on this involved problem and help in the design of future cement-based composites, and supply information that can be used as a benchmark for checking numerical models of concrete failure, as this simple composite is amenable to being modelled quite easily. Thirty-six notched beams were tested and values of the fracture energy and elasticity modulus were recorded. The tensile strength was measured from indirect standard tensile tests. Comparison with available experimental data is also included and discussed. Fracture was modelled using a cohesive crack with a bilinear softening function; data of the softening function inferred from the experimental measurements are also provided and discussed.     

Prediction of fracture parameters of concrete by Artificial Neural Networks

Modelling of material behaviour generally involves the development of a mathematical model derived from observations and experimental data. An alternative way discussed in this paper is Artificial Neural Network (ANN)-based modelling which is a subfield of artificial intelligence. The main benefit in using an ANN approach is that the network is built directly from experimental data using the self-organising capabilities of the ANN. In this paper the Two-Parameter Model (TPM) in the fracture of cementitious materials is modelled with a back-propagation ANN. The results of an ANN-based TPM look viable and very promising.     

Micromechanical fracture modeling of asphalt concrete using a single-edge notched beam test

Cracks in asphalt pavements create irreversible structural and functional deficiencies that increase maintenance costs and decrease lifespan. Therefore, it is important to understand the fracture behavior of asphalt mixtures, which consist of irregularly shaped and randomly oriented aggregate particles and mastic. A two-dimensional clustered discrete element modeling (DEM) approach is implemented to simulate the complex crack behavior observed during asphalt concrete fracture tests. A cohesive softening model (CSM) is adapted as an intrinsic constitutive law governing material separation in asphalt concrete. A homogenous model is employed to investigate the mode I fracture behavior of asphalt concrete using a single-edge notched beam (SE(B)) test. Heterogeneous morphological features are added to numerical SE(B) specimens to investigate complex fracture mechanisms in the process zone. Energy decomposition analyses are performed to gain insight towards the forms of energy dissipation present in fracture testing of asphalt concrete. Finally, a heterogeneous model is used to simulate mixed-mode crack propagation.     

Determination of the effective mode-I toughness of a sinusoidal interface between two elastic solids

A finite element model of crack propagation along a sinusoidal interface with amplitude A and wavelength λ between identical elastic materials is presented. Interface decohesion is modeled with the Xu and Needleman (J Mech Phys Solid 42(9):1397, 1994) cohesive traction–separation law. Ancillary calculations using linear elastic fracture mechanics theory were used to explain some aspects of stable and unstable crack growth that could not be directly attained from the cohesive model. For small aspect ratios of the sinusoidal interface (A/λ ≤ 0.25), we have used the analytical Cotterell–Rice (Intl J Fract 16:155–169, 1980) approximation leading to a closed-form expression of the effective toughness, K _Ic , given by where is the work of separation, E is Young’s modulus, and ν is Poisson’s ratio. For A/λ > 0.25, both the cohesive zone model and numerical J-integral estimates of crack tip stress intensity factors suggest the following linear relationship: Parametric studies show that the length of the cohesive zone does not significantly influence K _Ic , although it strongly influences the behavior of the crack between the initiation of stable crack growth and the onset of unstable fracture. An erratum to this article can be found at     

Lifetime evaluation of concrete structures under sustained post-peak loading

Experimental tests on crack propagation in concrete under constant post-peak loading are simulated using the finite element method and the cohesive crack model, in both Mode I and Mixed-mode conditions. The time-dependent behaviour of concrete in the process zone is due to the interaction and growth of microcracks, a phenomenon which, for high constant load levels, turns out to be predominant over linear viscoelastic creep in the bulk material. In mechanical systems based on this type of material behaviour (creep and strain-softening taking place simultaneously), the initial value problem is non-parabolic, i.e., the error at one time level is affected by the accumulation of errors introduced at earlier time levels. Despite these difficulties, the scatter in numerical failure lifetime vs. load level turns out to be negligible in Mode I conditions and practically acceptable in Mixed-mode conditions. Therefore the time-dependent behaviour of the process zone can be inferred solely from the results of direct tensile tests.     

Significance of K-dominance zone size and nonsingular stress field in brittle fracture

Stress intensity factor has been used to characterize the fracture toughness of a brittle material. This practice is apparently based on the assumption that the singular stress alone at the crack tip is responsible for fracture and that the nonsingular part of the near tip stress has no effect on fracture. In this study, mode I fracture experiments were conducted on a brittle material (PMMA) with four different specimen configurations. The result indicated that fracture toughness cannot be described by stress intensity alone and that a second parameter representing the influence of the nonsingular stress is needed. A two-parameter fracture model was proposed and validated with the experimental result. This two-parameter model was shown to be able to account for various effects created by specimen configurations, crack sizes, and loading conditions, on the fracture behavior of brittle materials.     

Discrete cohesive zone model for mixed-mode fracture using finite element analysis

The discrete cohesive zone model (DCZM) is implemented using the finite element (FE) method to simulate fracture initiation and subsequent growth when material non-linear effects are significant. Different from the widely used continuum cohesive zone model (CCZM) where the cohesive zone model is implemented within continuum type elements and the cohesive law is applied at each integral point, DCZM uses rod type elements and applies the cohesive law as the rod internal force vs. nodal separation (or rod elongation). These rod elements have the provision of being represented as spring type elements and this is what is considered in the present paper. A series of 1D interface elements was placed between node pairs along the intended fracture path to simulate fracture initiation and growth. Dummy nodes were introduced within the interface element to extract information regarding the mesh size and the crack path orientation. To illustrate the DCZM, three popular fracture test configurations were examined. For pure mode I, the double cantilever beam configuration, using both uniform and biased meshes were analyzed and the results show that the DCZM is not sensitive to the mesh size. Results also show that DCZM is not sensitive to the loading increment, either. Next, the end notched flexure for pure mode II and, the mixed-mode bending were studied to further investigate the approach. No convergence difficulty was encountered during the crack growth analyses. Therefore, the proposed DCZM approach is a simple but promising tool in analyzing very general two-dimensional crack growth problems. This approach has been implemented in the commercial FEA software ABAQUS^® using a user defined subroutine and should be very useful in performing structural integrity analysis of cracked structures by engineers using ABAQUS^®.     

Determination of fracture energy and tensile cohesive strength in Mode I delamination of angle-ply laminated composites

The energy release rate in delamination of angle-ply laminated double cantilever composite beam specimens was calculated using the compliance equation, and interlaminar cohesive strengths were obtained. Instead of the traditional approach of a beam on an elastic foundation, a second-order shear-thickness deformation beam theory (SSTDBT) was considered. The equilibrium equations were obtained using the principle of minimum total potential energy and the system of ordinary differential equations were solved analytically. The problem was solved for [0°]₆ , [±30°]₅, and [±45°]₅ laminates with mid-plane delaminations and the results were verified using experimental evidence available in the literature.     

Cohesive zone model for the prediction of interfacial shear stresses in a composite-plate RC beam with an intermediate flexural crack

In this paper, an analytical method is developed to predict the distribution of interfacial shear stresses in concrete beams strengthened by composite plates. Accurate predictions of such stresses are necessary when designing to prevent debonding induced by a central flexural crack in a FRP-plated reinforced concrete (RC) beam. In the present analysis, a new theoretical model based on the bi-linear cohesive zone model for intermediate crack-induced debonding is established, with the unique feature of unifying debonding initiation and growth. Adherent shear deformations have been included in the present theoretical analyses by assuming a parabolic shear stress through the thickness of the adherents, verifying the cubic variation of the longitudinal displacement function, whereas all existing solutions neglect this effect. The results obtained for interfacial shear stress distribution near the crack are compared to the Jialai Wang analytical model and the numerical solutions are based on finite element analysis. Parametric studies are carried out to demonstrate the effect of the mechanical properties and thickness variations of FRP, concrete and adhesive on interface debonding. Indeed, the softening zone size is considerably larger than that obtained by other models which neglect adherent shear deformations. However, loads at the limit of the softening and debonding stages are larger than those calculated without the thickness effect. Consequently, debonding at the interface becomes less apparent and the lifespan of our structure is greater.     

Size effect in the strength of concrete structures

This paper reports on the range of applicability of the various size effect formulae available in the literature. In particular, the failure loads of three point bend (TPB) beams are analysed according to the size effect formulae of Bažant and of Karihaloo for notched beams and according to those of Bažant and of Carpinteri for unnotched beams, and the results of this analysis presented. Improvements to Karihaloo’s size effect formula are also proposed.     

A two-dimensional in situ fatigue cohesive zone model for crack propagation in composites under cyclic loading

In this paper a two-dimensional fatigue cohesive zone model (CZM) for crack propagation in composites under cyclic loading has been formulated and validated through successful predictions of fatigue crack growth under pure and mixed mode conditions for several different composites. The proposed fatigue CZM assumes simple power-law functions for fatigue damage accumulation of which the damage parameters can be calibrated from simple fatigue tests under pure mode I and mode II conditions. The model relies solely on the in situ cohesive responses for fatigue damage rate calculation, enabling the differentiation of the local elemental load history from the global load history. An effective cycle jump strategy for high-cycle fatigue has also been proposed. It has been demonstrated that once calibrated, the fatigue CZM can predict the Paris laws for the pure modes. Furthermore, it can predict the Paris laws of any mixed-mode conditions without the need of additional empirical parameters. This is of significant practical importance because it leads to greatly reduced experimental needs for mixed mode crack propagation widely observed in composites under cyclic loads.     

A nonlinear viscoelastic fracture analysis of concrete/FRP delamination in aggressive environments

In this paper, the durability of the bondline between concrete and FRP reinforcement was characterized at various temperature and humidity levels. The linear and nonlinear viscoelastic constitutive behavior of the epoxy bondline was characterized and used for a nonlinear viscoelastic fracture analysis of delamination. A hygrothermal nonlinear viscoelastic pseudo-stress model was developed and calibrated in order to compute a generalized J integral. Driven wedge tests were conducted for examining the fracture behavior of the interface. A finite element analysis was developed for determining the cohesive zone size and the generalized J integral at various temperature and humidity levels. The fracture energy obtained from these parameters greatly depended upon crack growth rate, temperature and humidity.     

Cohesive zone modeling of interface fracture in elastic bi-materials

Some basic issues regarding the cohesive zone modeling of interface fracture between two dissimilar elastic materials are studied. The dependence of the cohesive energy density on the phase angle is first discussed under small scale cohesive zone conditions. It is then shown that in general the stress singularities in tension and shear cannot be simultaneously removed at the cohesive zone tip if a single cohesive zone length is adopted for both tensile and shear fracture modes. Finally, the energy dissipation at the tip of a prescribed cohesive zone is examined using a bilinear cohesive zone model under the uncoupled tension/shear conditions.     

Determination of fracture parameters for crack propagation in concrete using an energy approach

Double-G fracture model, a new analytical model describing fracture behaviour on cracked concrete, was recently proposed by Xu and Zhao based on the conception of energy release rate. This model couples the Griffith brittle fracture theory with the bridging softening property of concrete, and it is an extension of double-K fracture model proposed by Xu and Reinhardt. In this model, two fracture parameters, i.e., the initiation fracture energy release and the unstable fracture energy release , are termed to distinguish the different crack propagation stages undergoing during the whole fracture process in concrete. The difference between the two parameters, written as , is assumed to come from the contribution by aggregate bridging interlock, which results in the presence of fracture process zone. In our present work, firstly, the new model is elaborately introduced. Then, fracture tests are conducted, where besides three-point bending beams, a new testing method, called wedge-splitting on compact tension is adopted. In the experiments, electrical strain gauges are used to measure initial cracking load. Based on recorded load-crack mouth opening displacement curve (P-CMOD) or load-displacement curve (P-δ) and load-strain curve (P-ε), double-G fracture parameters are experimentally determined. Further more, based on the assumed three parameters relationship among , and , unstable fracture energy release are calculated. A comparable result between the measured and the calculated confirms this assumption. In order to verify the feasibility of this new model, the effective double-K fracture parameters converted by double-G fracture parameters using are compared with the double-K fracture parameters calculated by double-K fracture model. It is found that there is a good agreement. Another two series of different initial crack-depth ratios three-point bending beams carried out by Refai and Swartz are also collected to provide more experimental verification. It shows that the results obtained from the double-G fracture model agree well with those of double-K fracture model.     

Cohesive-bridging zone model of FRP-concrete interface debonding

External bonding of FRP plates or sheets has emerged as a popular method for strengthening reinforced concrete structures. Debonding along the FPR-concrete interface can lead to premature failure of the structures. In this study, a combined cohesive/bridging zone model is presented to simulate the debonding procedure between the FRP and concrete interface. In this model, the crack processing zone of the interface is modeled by a cohesive zone model and the particle interlocking zone of the interface is modeled by a bridging zone model. Two different linearly softening bond stress-slip laws are used to describe these two different zones. Closed-form solutions of interfacial stress, FRP stress and ultimate load are obtained for a typical single-lap specimen and verified with experimental results. The pulling force applied to the FRP plate is found to be proportional to the square root of the energy release rate at the debonding tip for this model. Such a relationship is then extended to any general shapes of bond stress-slip law through J-integral method. A new approach to experimentally determine the bond stress-slip law is also proposed.