Fracture simulation of concrete beams to assess softening behavior by varying different fractions of aggregates

Simulating the concrete fracture unlike other elastic and brittle materials quite different due to its quasibrittleness. The present research focussed on assess softening behavior by varying different fractions of aggregates  and cement matrix in micro details. Extended Finite Element Method (XFEM) for crack modeling implemented for simulating and visualizing crack propagation through Cement matrix, Interfacial Transition Zone (ITZ) and Aggregates . This approach permits the initializing crack by from enrichment zone and propagation of crack through element by traction separation law .The crack formation initiates when the maximum principal tensile stress reaches the tensile strength. The work involves creating python script for iterative process of random distribution of aggregates with in the matrix using Monte Carlo method and creating Cohesive zone element for zero thickness ITZ. introduces a finite element modeling technique for investigating multiscale fracture characteristics. This approach encompasses multiple levels of analysis, including the generation of aggregate particles using a Monte Carlo method implemented via a Python script. Additionally, we replicate the Interfacial Transition Zone (ITZ) between aggregate and mortar in the model. The load-deflection curves can be used to assess the softening behavior of concrete and suggest the realistic fraction of coarse aggregate in mix proportion to impart more ductility to beams.


INTRODUCTION
oncrete undergoes various adverse loading conditions.During seismic loads, ductility plays an important factor [40].Most of the concrete structures were designed considering bending moments and shear forces.Concrete, with its heterogeneous mesostructure, demonstrates complex mechanical responses when subjected to high-rate loads [29].
A comprehensive description of failure mechanisms is crucial to ensuring the safety of concrete structures [27].During the design, importance is not emphasized on concrete behavior during fracture, which is significantly important during the postpeak softening process [6].Several computational techniques have been utilized to simulate concrete materials.Many of these models primarily focused on homogenized macroscopic material behavior, often neglecting the intricate micro-or meso-structure of concrete materials [3,15,17].Concrete comprises three phases: the mortar, the coarse aggregates, and the interfacial transition zone (ITZ) situated between the aggregate and mortar [9,10,28].

C
Simulating the post-peak softening process using mesostructures of concrete consisting of a three-phase system is a challenging job [33].The latest studies show that computed tomography scans can be used to generate the mesostructures of concrete, which involves complexity and high cost [19,24].To achieve a realistic simulation within a reasonable computation time, finite element analysis yields favorable results [26].The cracking behavior in numerical simulations can be modeled using the softening (failure) behavior of concrete [5,8,18,7].The surface-based cohesive zone model between the aggregate and cement matrix was simulated using the Traction Separation Law (TSL) [41].The fundamental concept of the cohesive zone model, initially proposed by Barenblat [1], is to investigate the behavior of the material within a region directly ahead of a traction-free crack tip.Hillerborg introduced the initial cohesive crack model to simulate discrete cracks in the fracture process zone (FPZ) of concrete [2].The cohesive zone model is a straightforward approach to modeling the fracture of concrete specimens, demonstrating strong agreement with experimental results.It can effectively predict the behavior of intact structures [11].Crack propagation can be simulated using the XFEM method, in which material properties can be provided as input under damage for TSL initiation [12,20,30].Both the relative proportion of aggregates and the ITZ assume pivotal roles in determining the behavior of concrete grades during the fracture process.The present study consists of creating finite element models of different fractions of aggregate and cement matrix along with ITZ and studying the post-peak softening behavior of concrete models.The models were validated using experimental data.
The study focuses on analyzing various aggregate fractions and investigating the influence of the Interfacial Transition Zone (ITZ) during the fracture process.The final conclusion is drawn from the analysis of load-deflection curves obtained from simulations.

NUMERICAL SIMULATIONS
he present work involves the numerical modeling of fracture tests of geometrically similar three-point bending (TPB) beams with a constant length-to-depth ratio.The simulation of the TPB specimen performed by the finite element analysis incorporating the concrete softening behavior predicts the load-displacement curves.The 2D beam models were developed using a Python script to study the effects of different proportions of coarse aggregates (uniform size and distributed size) and cement matrix, both with and without an Interfacial Transition Zone (ITZ).The boundary conditions were applied to simulate simply supported conditions.The loading was applied in a displacement-controlled manner by means of an analytically rigid body acting on the upper surface of the beam in its mid-span.The loading allows for crack penetration through the entire beam height.Adequate contact properties between supports and beams are applied.
In essence, crack propagation and fracture in concrete numerical simulations are intricately tied to mesh generation.
Recognizing the influence of mesh sizes on numerical simulations, the present study investigated five distinct mesh sizes (0.5mm, 0.75mm, 1mm, 2mm, and 4mm) [36].Notably, mesh sizes less than 1mm in the current study's model result in the development of inadequately meshed regions for fine meshes and element distortion with unfavorable mesh angles for coarser meshes.Since a 1mm mesh size was used in many studies [25,29,27], the different mesh sizes were set around 1 mm in this study.A concrete beam is modeled by solid, deformable, finite elements meshed with quad elements.XFEM parameters are provided using the Abaqus-Cae software.

GENERATION OF RANDOM AGGREGATES
he mortar, which is made of fine aggregates and cement, usually serves as the composite matrix.The aggregates, being the strongest component, make up about 75% of the concrete volume, with 40-50% of them being coarse aggregates depending on different design mixes.The ITZ, typically forming around the aggregates, is generally T T regarded as the weakest part [17,28,39].For the present study, covering a range from low-strength concrete (M20) to highstrength concrete (up to M45), aggregate fractions of 38%, 40%, 50%, and 54% were selected [32].Spherical aggregates were randomly distributed and used in this study.
The aggregates are generated with a random distribution within the specified size of the model and the aggregate fraction by adopting the Monte Carlo method [16].Aggregates are placed without overlapping each other.To ensure this condition is met, a loop to check and reject functions has been coded in the Python script.During the distribution process, small gaps are specifically created for the cement matrix.The number of iterations should be defined before running the script.
The Python code was developed to generate concrete mesostructures with aggregate volume fractions of up to 54%.But when dealing with instances where the aggregate volume fraction exceeds 54%, the computational demands notably increase.This increase leads to a greater demand for conducting intersection checks among aggregate particles that are already positioned, along with a decreased probability of identifying empty spaces within the mortar matrix.The Python script is designed for creating models, positioning aggregates, defining and assigning material properties, creating steps, loading and assigning boundary conditions, assembling and developing interactions with the model, and meshing complete, which can be directly submitted for job-run analysis.The aggregate positions are changed by each iteration in the Python script.For the present study, three iterations are considered for each case (Model-1, Model-2, and Model-3).

GENERATION OF RANDOM AGGREGATES COHESIVE ELEMENT MODEL OF THE INTERFACIAL TRANSITION ZONE (ITZ)
he thickness of the ITZ lies between 10 and 50 µm; achieving this level of precision using the Finite Element Method (FEM) proves challenging.As a solution, the present work employs the Cohesive Element Model (CEM) to simulate the ITZ [41].In this context, the ITZ is treated as having a thickness of zero [38], which retains the relevant mechanical properties of the actual ITZ to achieve the accuracy of the simulation, and all ITZs in concrete can be represented by a zero-thickness element.This can be achieved by the following methods: i) Cohesive elements share nodes with other elements.ii) Contact interactions between cohesive elements and other features.iii) Contact interaction with the cohesive zone and one part.The first approach adopted in the present study is to create a zero-thickness cohesive zone on the contact surface of the aggregate and mortar.ITZ properties are assigned to the cohesive zone using a Python script in the finite element software [34].
The deterioration of the Cohesive Element (CE) is categorized into four distinct phases: the initial linear elastic phase, the phase of damage initiation, the phase of damage progression, and the ultimate complete damage phase.The phase that pertains to the CE's ability to sustain damage while still maintaining its functionality is referred to as the online resilience phase of the CE's damage response.
In the present study, the cohesive zone model is adopted to simulate fracture in a particular zone.However, it does not yield any damage fracture since we are adopting the XFEM method.
T FINITE ELEMENT MODELS he proposed finite element models for the present study are shown in Fig. 2. The material properties considered for analysis [30]:  fter adjusting the input parameters of the TSL based on calibration, we carried out mesoscale concrete simulations [41].These simulations were then validated against experimental results obtained from TBD fracture tests in reference [39].To the validate model, three different FE models of length = 400 mm, breadth = 100 mm, and depth = 100 mm with notch depth = 25 mm were developed using a Python script, each with varying aggregate distributions, covering a volume fraction range of 38% to 54%.The model's reliability was checked by the normal distribution and the Gaussian probability density function.The XFEM method is adopted for simulations.The load-deflection curves were plotted by averaging the results from each of these simulations.Fig. 3 shows the load-deflection curves plotted from the TBD simulations, along with the experimental data.
. Comparing the numerical simulations with the experimental results yielded remarkable consistency, affirming the capability of the proposed approach to accurately predict crack initiation and propagation in composite materials like concrete.The average peak load values recorded during the experiments were 6.54 kN, while the result obtained from the simulation was 5.95 kN.This slight deviation of approximately 10% lower results in the simulations compared to the experimental study can be attributed to the simplifications made in the analysis, such as the use of circular aggregates and the adoption of a two-dimensional plane stress model.These simplifications were necessary to manage computational time and memory consumption, as a full 3-D analysis would have been significantly more resource-intensive.Furthermore, when examining the post-peak softening slope (represented by tan θ), the experimental study yielded an average value of 51, whereas the result from the simulation yielded 41.67.This range in the simulation results closely aligns with the observed real-world behavior, suggesting that our FE model analysis produced a realistic curve reflecting the concrete's response during and after the peak load

RESULTS AND DISCUSSION
s expected, crack initiation occurred at the pre-existing notch located at the midpoint of the beam and propagated towards the loading point [7].This cracking progression traversed through the Interfacial Transition Zone (ITZ) and the cement paste phases, both of which are weaker materials while attempting to bypass the aggregates, as depicted in Fig. 5.b.This observed crack development is reasonable, as aggregates are inherently stronger than the other two phases.Furthermore, it is noteworthy that this crack pattern closely resembles the cracks observed in Kozicki's model [13], which serves to validate our current model.Post-peak softening slope as per Tabs.9 and 10 is shallow for Case 2(b).When θ is smaller and softens more and more, ductility increases more and more.From Tabs. 8, 9, and 10, it is observed that the value of tan θ is less for Case 2(b).The inclusion of ITZ leads to a flat slope, which indicates that ITZ enhances the ductility of concrete.The Load-CMOD (Crack Mouth Opening Displacement) plots have also been obtained, revealing a triphasic pattern of behavior.Initially, during the first stage, deflection shows a linear increase alongside incremental loading.At this point, crack initiation transpires without immediate propagation.Subsequently, in the second stage, nonlinear tendencies emerge, causing the plot's slope to decline until it reaches its zenith.Within this phase, the formation of a fracture zone becomes evident due to the existence of microcracks, and crack propagation occurs at a subdued pace.The third stage is recognized as the strain softening zone, characterized by accelerated crack propagation owing to heightened stress concentration.This concentration of stress is particularly pronounced in the narrow region between the notch tip and the loading point.The stress concentration is higher as the load-carrying capacity decreases, leading to the failure of the specimen.From Tab.11, it is observed that the value of tan θ is less for Case 2(b), indicating the ductile behavior of concrete with respect to CMOD under load.Fig. 14 shows the comparative study of the inclusion of 38% CA for all cases which reveals similar results as that of Load displacement curves.

CONCLUSIONS
n the case of a model without ITZ, the peak load observed is greater than the experimental value; this could be due to the resistance offered by the aggregate well before compared to when the crack was hard to pass through the ITZ.
When the interface is introduced, whose tensile strength is lower than that of the matrix, cracks will progress through the ITZ but pass around the aggregate.In that case, the peak load from the simulation closely matches the experimental result.Particularly, post-peak softening behavior, which is indicated by the gradual decrease in slope, indicates that concrete behavior is relatively ductile in nature.It shows an improvement in ductility due to the introduction of ITZ, which represents a realistic concrete model.The softening slope is the most precious parameter, which tells us what type of mix can give a more softening effect, and it is better to have a shallow softening slope.The Extended Finite Element Method (XFEM), along with Cohesive Element, effectively used for multiscale modeling of concrete to assess fracture behavior, appears to be more appropriate.Modeling the cohesive zone model along with the effect of the texture of aggregates on the simulation of fractures in concrete beams would be the focus of a future study.

Figure 2 :
Figure 2: Finite-element grid model of concrete three-point bending beam with and without ITZ for 38% fraction of CA

Figure 3 :
Figure 3: Comparison between experimental and numerical load-displacement curves.
beam under load b.Deflected beam under peak load

Figure 10 :
Figure 10: Case2(a) Load -Deflection curve for CA of mean size 12.5mm with ITZ.

Figure 12 :
Figure 12: Case2(b) Load -Deflection curve for CA of mean size between 4.75mm to 20mm with ITZ.
(a) and (b) Peak load as per Tabs. 3 and 4: Case 1 (a) ranges from 2.8 kN to 4.0 kN, Case 1 (b) ranges from 4 kN to 5.3 kN, Case 2 (a) ranges from 2.5 kN to 3.5 kN, and Case 2 (b) ranges from 2.2 to 3.6 kN.In the case of uniform aggregate size, the peak load for notched beams shows less compared to non-uniform aggregates due to the resistance to fracture caused by the interlocking of aggregates, and the aggregates are also quite hard.Whereas in the case of a model with ITZ, the peak load reduces due to the ITZ zone.The numerical prediction for the maximum load on the S-B1 beam falls within the range of 3.5 to 3.6 kN, which closely aligns with the experimentally observed values ranging from 3.5 to 3.8 kN[22].Case 2(b) gives a realistic peak load that can be comparable with the experimental data.Fracture energy (GF) as per Tabs.7 and 8 Case 1 (a) ranges from 225 to 296 N/m, Case 1 (b) ranges from 220 to 275 N/m, Case 2 (a) ranges from 120 to 207 N/m, and Case 2 (b) ranges from 120 to 145 N/m.In the case of uniform aggregate size, the peak load for notched beams shows more GF compared to non-uniform aggregates due to the resistance to fracture created by the interlocking of aggregates.Whereas in the ITZ model, GF reduces due to the ITZ zone.The numerical prediction for the GF on the S-B1 beam falls within the range of 120 to 154 kN, which closely aligns with the experimentally observed values ranging from 146N/m to 154N/m kN[22].Case 2(b) gives realistic GF, which can be comparable with the experimental data.

Table 2 :
Material Properties

Table 6 :
Displacement (mm) at maximum load for Case 2 (a) and (b).

Table 11 :
Load vs Deflection-Slope of the curve (tan θ) for all Cases of 38%CA.