Investigation on the characterization and modelling of Fracture Process Zone behavior in Concrete Beams subjected to Three-Point Loading Tests

H. S. VishwanathaDepartment of Civil Engineering, M.E.I Polytechnic; Department of Technical Education, Bangalore, Karnataka, India hs_vish@yahoo.co.inS. MuralidharaDepartment of Civil Engineering, B.M.S. College of Engineering; Vishveshwarya Technical University, Bangalore, Karnataka, Indiamurali.civ@bmsce.ac.inB. K. Raghu PrasadDepartment of Civil Engineering, Indian Institute of Science, Bangalore, Karnataka, Indiabkriis@gmail.com

Introduction

The fracture process zone (FPZ) in concrete is a critical region where micro-cracks develop and coalesce, eventually leading to macroscopic crack propagation. The size and characteristics of the FPZ are influenced by the material heterogeneity and the applied loading conditions. In three-phase concrete materials with pre-existing cracks, three distinct crack propagation paths can typically be identified:
✓ ✓ ✓\checkmark✓ Cracks within Cement Mortar: These cracks propagate through the cement mortar phase of the material.
✓ ✓ ✓\checkmark✓ Cracks along the Mortar-Aggregate Interface: In this situation, cracks propagate along the interface between the cement mortar and the aggregate particles.
✓ ✓ ✓\checkmark✓ Cracks Kinking into the Interface: These cracks deviate from their initial path and intersect the mortar-aggregate interface at an oblique angle.
Each of these propagation paths may have its own criterion governing crack growth. Determining the direction of crack initiation or extension involves applying the appropriate crack propagation criterion for each case. This process enables the prediction of crack behaviour and its interaction with the material phases in three-phase concrete systems.
Researchers have extensively studied the FPZ to understand its role in the overall fracture behavior of concrete. Digital image correlation (DIC) and electronic speckle pattern interferometry (ESPI) have enabled detailed observations of the FPZ, revealing intricate crack patterns and stress distributions within this zone [ 1,2 ]. The present study focuses on a detailed analysis of the FPZ and its evaluation for beams ranging from small to very large sizes ( 75 mm to 1000 mm ).

Model generation

The mortar, composed of fine aggregates and cement, typically acts as the composite matrix. Aggregates, the strongest constituents of concrete, account for approximately 75 % 75 % 75%75 \%75% of its volume, with coarse aggregates making up 40 − 50 % 40 − 50 % 40-50%40-50 \%40−50% based on the specific design mix [4]. The coarse aggregates embedded in the mortar matrix are the primary reinforcing elements. They significantly influence the fracture behavior by bridging cracks, altering the FPZ characteristics, and increasing the materials resistance to crack propagation. The interfacial transition zone (ITZ), which generally forms around the aggregates, is often considered the weakest region [3].
The key steps in parameterized modeling [3] for constructing a two-dimensional concrete model involve determining the total area occupied by aggregates in accordance with a specified aggregate gradation. Particles larger than 4.75 mm are categorized as coarse aggregates, which are distributed within the concrete to act as the reinforcing matrix phase. Aggregates of various sizes are created and randomly arranged within the defined area.
In this study, spherical aggregates (appearing circular in 2D models) were randomly distributed. Using the Monte Carlo method, aggregates were generated and randomly placed within the specimen's specified dimensions and aggregate fraction. A Python script was utilized, incorporating a loop with check-and-reject functions to ensure proper placement. Small gaps were intentionally left for the cement matrix during the distribution process, and the number of iterations was predefined. The fig.1(a) shows flow chart adopted for distribution of different sizes of coarse aggregates. An aggregate volume fraction of 40 % 40 % 40%40 \%40% was used in this analysis. Fig. 1(b) shows typical aggregate distribution after executing Python script. During the simulation, the aggregate positions were varied in each iteration. Three iterations for each specimen size were carried out to represent the randomness of aggregate distribution in three trials.
In this study, a Python script was imported into Abaqus for conducting the analysis. The gematrical properties as per Tab. 1 adopted and the materials assigned as per the Tab. 2. The boundary conditions provided to suit simply supported beam with central point load. A displacement load was applied, consisting of 10,000 load steps with a constant load increment. This loading procedure allowed for the crack to penetrate through the entire beam height, providing a detailed understanding of the fracture process in concrete. The flowchart adopted for model generation and analysis is depicted in fig. 3.
Due to the model's inherent randomness in aggregate distribution, achieving a well-structured mesh presents challenges. Notably, issues arise with the interfacial mesh between aggregates and mortar. These issues encompass smaller mesh areas, higher mesh density, and non-uniform mesh sizes compared to other regions. These mesh-related complications hinder the convergence of the realistic aggregate model, largely due to mesh distortion.
In essence, crack propagation and fracturing in concrete numerical simulations are intricately tied to mesh generation. For finite element simulations where the cohesive crack paths are not known in advance, rather fine levels of discretization must be used to reduce the dependence of the cohesive crack paths on the mesh size. However, reducing the element size leads to an increase in computational cost, so a balance between computational cost and accuracy must be established. To this end, a mesh convergence study is performed in this work in terms of the mean stress-strain curve and the final fracture crack paths to find an optimal mesh size.
This study examined five different mesh sizes ( 0.5 mm , 1 mm , 2 mm 0.5 mm , 1 mm , 2 mm 0.5mm,1mm,2mm0.5 \mathrm{~mm}, 1 \mathrm{~mm}, 2 \mathrm{~mm}0.5 mm,1 mm,2 mm, and 4 mm ). A mesoscale concrete model size of 150 mm × 150 mm × 50 mm 150 mm × 150 mm × 50 mm 150mmxx150mmxx50mm150 \mathrm{~mm} \times 150 \mathrm{~mm} \times 50 \mathrm{~mm}150 mm×150 mm×50 mm with coarse aggregate volume fraction of 40 % 40 % 40%40 \%40% was utilized. The Material properties as per Tab. 2 adopted. Following discretization, all specimens underwent with a uniform displacement of 0.15 mm in the x -direction. The mean stress-strain curves for each mesoscale specimen are presented in fig. 2. Mean stress was determined by dividing the total reaction forces at the left edge nodes by the specimen's length at each step frame.
Figure 1(a): Flow chart showing generation of aggregates using Python script.
Figure 1(b): Aggregates distribution.
Figure 2: Mean stress-strain curve for a concrete sample shown in with different levels of discretization.
The stress-strain curves and macrocrack patterns for specimens with mesh sizes of 1 mm and 2 mm showed strong agreement with the experimental results obtained from [12]. However, meshes finer than 1 mm resulted in poorly meshed regions, while coarser meshes caused element distortion and unfavorable mesh angles. Considering that mesh sizes between 1 mm and 2 mm are widely employed in prior studies [5], this research adopted mesh sizes within this range based on a mesh sensitivity analysis.
The convergence study underscored the importance of fracture paths in selecting an appropriate mesh size in addition to stress-strain curves. Although larger meshes may provide similar stress-strain results, they often lead to significantly different fracture paths. Conversely, smaller meshes yielded consistent fracture paths and stress-strain curves, but further reduction drastically increased computational demands.
Figure 3: Flow chart illustrating methodology.

Geometrical properties of beams

Tab. 1 shows the geometrical properties of the beams adopted in the present study. The beams were grouped into five beams. Each type of beam was analysed with three different configurations of coarse aggregate (CA). In the nomenclature employed: B-SB75, the first letter B represent beam series, SB represents small beam and last digit
represents depth or size of beam. The MB stands for medium beam, LB for large beam, VB for very large beam & HB for huge beam.
Beam ID
Depth of beam
(D) mm
Depth of beam (D) mm| Depth of beam | | :--- | | (D) mm |
Breath of beam
(b) mm
Breath of beam (b) mm| Breath of beam | | :--- | | (b) mm |
Length of the beam
(L) mm
Length of the beam (L) mm| Length of the beam | | :--- | | (L) mm |
Span
(S) mm
Span (S) mm| Span | | :--- | | (S) mm |
Central part of beam
( S 0 ) mm S 0 mm (S_(0))mm\left(\mathrm{S}_{0}\right) \mathrm{mm}(S0)mm
Central part of beam (S_(0))mm| Central part of beam | | :--- | | $\left(\mathrm{S}_{0}\right) \mathrm{mm}$ |
Notch to depth ratio
α 0 = a 0 / D α 0 = a 0 / D alpha_(0)=a_(0)//D\alpha_{0}=a_{0} / \mathrm{D}α0=a0/D
Notch to depth ratio alpha_(0)=a_(0)//D| Notch to depth ratio | | :--- | | $\alpha_{0}=a_{0} / \mathrm{D}$ |
Depth of notch
( a 0 ) mm a 0 mm (a_(0))mm\left(a_{0}\right) \mathrm{mm}(a0)mm
Depth of notch (a_(0))mm| Depth of notch | | :--- | | $\left(a_{0}\right) \mathrm{mm}$ |
 S/D
B-SB75 75 150 250 225 50 0.25 18.75 3
B-MB150 150 150 550 450 100 37.5
B-LB250 250 150 1000 750 150 62.5
B-VB500 500 150 2000 1500 300 125
B-HB1000 1000 150 4000 3000 600 250
Beam ID "Depth of beam (D) mm" "Breath of beam (b) mm" "Length of the beam (L) mm" "Span (S) mm" "Central part of beam (S_(0))mm" "Notch to depth ratio alpha_(0)=a_(0)//D" "Depth of notch (a_(0))mm" S/D B-SB75 75 150 250 225 50 0.25 18.75 3 B-MB150 150 150 550 450 100 37.5 B-LB250 250 150 1000 750 150 62.5 B-VB500 500 150 2000 1500 300 125 B-HB1000 1000 150 4000 3000 600 250 | Beam ID | Depth of beam <br> (D) mm | Breath of beam <br> (b) mm | Length of the beam <br> (L) mm | Span <br> (S) mm | Central part of beam <br> $\left(\mathrm{S}_{0}\right) \mathrm{mm}$ | Notch to depth ratio <br> $\alpha_{0}=a_{0} / \mathrm{D}$ | Depth of notch <br> $\left(a_{0}\right) \mathrm{mm}$ | S/D | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | B-SB75 | 75 | 150 | 250 | 225 | 50 | 0.25 | 18.75 | 3 | | B-MB150 | 150 | 150 | 550 | 450 | 100 | | 37.5 | | | B-LB250 | 250 | 150 | 1000 | 750 | 150 | | 62.5 | | | B-VB500 | 500 | 150 | 2000 | 1500 | 300 | | 125 | | | B-HB1000 | 1000 | 150 | 4000 | 3000 | 600 | | 250 | |
Table 1: Geometrical properties of the beams.
To save computational time and memory, the central part of the beam is modeled as a heterogeneous section with a fine mesh, while the side parts are modeled as homogeneous sections. This approach is illustrated in Fig.4,5,6. The heterogeneous section in the central part allows for a detailed representation of the microstructure where stress and strain concentrations are expected to be higher, whereas the homogeneous side sections reduce computational demands without significantly affecting the accuracy of the overall model. This method strikes a balance between accuracy and computational efficiency, ensuring that the key areas of interest are accurately simulated.
Figure 4: Spherical shape aggregates (SSA-Model).
Figure 5: Interfaces in microstructure.
Figure 6: Heterogeneous region for SSA models.

Material properties

In the FE model, the solid elements that represent the mortar and aggregates are assumed to behave in a linear elastic manner. Most of the deformation occurring between two solid elements is absorbed by the cohesive elements. The solid elements only deform when the cohesive elements are within their linear elastic range. Due to the very high initial stiffness of the cohesive elements, they experience negligible deformation within this range, resulting in minimal deformation of the solid elements.
Elements Parameter Density ( kg / m 3 kg / m 3 kg//m^(3)\mathrm{kg} / \mathrm{m}^{3}kg/m3 ) Young's Modulus, E (MPa) Poisson's ratio, v v vvv Traction-Separation Law
Elastic Stiffness (MPa) Cohesive strength (MPa) (Damage initiation) Fracture energy ( N / mm N / mm N//mm\mathrm{N} / \mathrm{mm}N/mm )
Aggregate 2800 47200 0.2 - - -
Cement matrix 2400 29200 0.2 - - -
Bulk Homogeneous beam part 2400 36100 0.2 - - -
Cohesive CIEs 2400 - - 10 6 10 6 10^(6)10^{6}106 3.5 0.168
ITZs 2300 - - 10 6 10 6 10^(6)10^{6}106 2.4 0.115
Elements Parameter Density ( kg//m^(3) ) Young's Modulus, E (MPa) Poisson's ratio, v Traction-Separation Law Elastic Stiffness (MPa) Cohesive strength (MPa) (Damage initiation) Fracture energy ( N//mm ) Aggregate 2800 47200 0.2 - - - Cement matrix 2400 29200 0.2 - - - Bulk Homogeneous beam part 2400 36100 0.2 - - - Cohesive CIEs 2400 - - 10^(6) 3.5 0.168 ITZs 2300 - - 10^(6) 2.4 0.115| Elements | Parameter | Density ( $\mathrm{kg} / \mathrm{m}^{3}$ ) | Young's Modulus, E (MPa) | Poisson's ratio, $v$ | Traction-Separation Law | | | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | | | | | | Elastic Stiffness (MPa) | Cohesive strength (MPa) (Damage initiation) | Fracture energy ( $\mathrm{N} / \mathrm{mm}$ ) | | | Aggregate | 2800 | 47200 | 0.2 | - | - | - | | | Cement matrix | 2400 | 29200 | 0.2 | - | - | - | | Bulk | Homogeneous beam part | 2400 | 36100 | 0.2 | - | - | - | | Cohesive | CIEs | 2400 | - | - | $10^{6}$ | 3.5 | 0.168 | | | ITZs | 2300 | - | - | $10^{6}$ | 2.4 | 0.115 |
Table 2: Material properties adopted in FE analysis for concrete beam [6, 7].

Validation of the model

The effect of shapes of aggregate particles investigated with beams of same geometry and material parameters with spherical shape. Three point bending test conducted and load-deflection curves plotted. Fig. 7 shows the results of 10 beams with spherical aggregates takes average peak load 7.10 kN . The maximum discrepancy of the peak load between the experimental and numerical values is 6.5 % 6.5 % 6.5%6.5 \%6.5% for the Spherical shape aggregates.
From the final crack distribution and crack development process, as shown in Fig. 8, it is evident that the macroscopic crack initiates near the notch tip, where stress concentration is highest due to the applied boundary conditions. The crack propagates upward, interacting with the mortar and aggregates, ultimately pointing toward the loading point. The figure illustrates the progression of the crack through the heterogeneous section, influenced by the relative positions of aggregates and the stress field distribution.
The crack of the spherical aggregate model begins developing in small aggregates and mortar and reaches aggregate 'a' as shown in Fig.8, where the development of cracks is blocked and continues to develop upward along the boundary of aggregate 'a'.
Figure 7: Load-deflection curves for B-SB75.
The same phenomenon, known as the aggregate blocking effect, is seen for aggregates b-c-d-e-f in Fig.8. Therefore, it can be concluded that the aggregate geometry has the effect of blocking and guiding the crack, while the blocking effect tends to occur in the spherical aggregate model. The results indicated that the geometric properties of aggregates have both blocking and guiding effects on crack development. The present study focuses on medium-strength concrete, where the crack propagation predominantly occurs along the weak interfacial transition zone (ITZ) due to the relatively lower strength of the matrix compared to the aggregate. However, in high-strength concrete, the ITZ is significantly strengthened due to the reduced porosity and improved bonding between the matrix and the aggregate. As a result, the crack is more likely to propagate through the aggregate itself, destroying it and altering its trajectory.
Figure 8: Crack Propagation under TPB test at heterogeneous section (Central part of beam).

Identification of loading steps

To investigate crack propagation in the fracture procedure, ten steps were selected, as shown in Fig. 9. Based on the results of analysis, the load deflection curve of the TPB with the notch-to-depth ratio of 0.25 plotted, and the results are shown in Fig.9. In Fig. 9, P 1 P 1 P_(1)\mathrm{P}_{1}P1 point selected after initial crack load. At P 1 P 1 P_(1)\mathrm{P}_{1}P1, cracking starts at the notch tip. As cracking progresses, the load in P 2 P 2 P_(2)\mathrm{P}_{2}P2 approaches the maximum load with a less steep slope than in P 1 P 1 P_(1)\mathrm{P}_{1}P1. The load at P 2 P 2 P_(2)\mathrm{P}_{2}P2 reaches
its peak value and from this point onwards leads to unstable fracture, which is reflected in areas P 3 P 3 P_(3)\mathrm{P}_{3}P3 to P 10 P 10 P_(10)\mathrm{P}_{10}P10 of the loaddisplacement curve.
Figure 9: Typical Load- Displacement curve showing different loading steps to evaluate FPZ for B-MB150

ANALYSIS OF LOAD-CMOD AND LOAD-CTOD PLOTS

Fig. 10 depicts the load-crack mouth opening displacement (P-CMOD) and load-crack tip opening displacement (PCTOD) curves for five concrete beam sizes (B-SB-75, B-MB-150, B-LB-250, B-VB-500, and B-HB-1000) as determined from the analysis. The P-CMOD curve exhibits an approximately linear trend up to the peak load, transitions to a nonlinear phase near the peak, and then demonstrates distinct softening behavior beyond the peak load. Furthermore, the results indicate that the peak load increases with the size of the beams. In Fig.10, "PC" denotes the loading step at which the fracture process zone reaches its maximum length.
To highlight the differences between CMOD and CTOD, both values from the same beam are shown in Fig.10. It is evident that the shape of the P-CTOD curve closely resembles that of the P-CMOD curve, though the P-CTOD values are consistently smaller than the corresponding CMOD values under identical conditions.
(a) B − SB 75 B − SB 75 B-SB75\mathrm{B}-\mathrm{SB} 75B−SB75
(b) B-MB150
Figure 10: Load-CMOD and Load-CTOD plots.
As shown in Fig.11, no microcracks are observed in the concrete material when the load is below 40 % 40 % 40%40 \%40% of the peak load. During this phase, the P-CMOD curve remains nearly linear, signifying a linear elastic stage with only a slight difference between CMOD and CTOD. Once the load exceeds the proportionality limit, the difference between CMOD and CTOD progressively increases with the applied load. As crack propagation occurs, the rotation angle between the crack mouth and crack tip increases, further amplifying the variation in COD values.
Figure 11: CTOD versus Percentage of Peak loading during pre peak and post peak.

EVALUATION OF FRACTURE PROCESS LENGTH

As shown in Fig. 12, the FPZ consists of a main crack and numerous secondary cracks. The main crack begins above the notch due to stress concentration at the notch tip and propagates upward due to bending. The secondary cracks primarily initiate and propagate through the weak interfacial transition zones (ITZs) along the boundaries of the strong aggregate particles. The main crack extends by bridging the interfacial micro-cracks that form around adjacent aggregates.
Figure 12: Development of cracks at interfaces.
During the growth of the main crack, many new cracks appear while some earlier-formed cracks cease to grow. The evolution path of the FPZ is highly irregular due to the random spatial distribution of aggregate particles with weak ITZs. The FPZ forms before the load reaches its maximum value and can expand significantly around the peak load. After the peak load, the FPZ continues to extend upward with the appearance of many new cracks, while the main crack forms and propagates. Since the extension of the FPZ is generally measured from the stress-free crack mouth ( w 0 w 0 w_(0)w_{0}w0 ) to the crack tip, the length of the FPZ may gradually decrease during the post-peak stage. This process is illustrated in Fig. 13. The findings of this study regarding the development mechanism of cracks in concrete align well with experimental observations using DIC, X-ray, CT, and AE techniques [9-13].
(i) At P 1 P 1 P_(1)\mathrm{P}_{1}P1
(vii) At P C P C P_(C)\mathrm{P}_{\mathrm{C}}PC
(viii) At P 7 P 7 P_(7)\mathrm{P}_{7}P7
(ix) At P 8 P 8 P_(8)\mathrm{P}_{8}P8
Figure 13: Evaluation of Fracture Process Zone for B-SB75.

Variation of FPZ length

The precise measurement of the fracture process zone (FPZ) is crucial for accurately predicting the fracture behavior of concrete. In this study, the FPZ length ( l FPZ l FPZ l_(FPZ)l_{\mathrm{FPZ}}lFPZ )was observed to investigate its changes during crack growth. The critical crack tip opening displacement or Stress-free Crack opening displacement ( w 0 w 0 w_(0)w_{0}w0 ) is a measure of the displacement at the crack tip when the material reaches a critical state before rapid crack propagation.
Figure 14: Development of fracture process zone length
At the initial loading stage (Refer Fig.14(i)), the crack tip opening displacement (CTOD) at the tip of the macrocrack is smaller than the w 0 ( w 0 w_(0)(:}w_{0}\left(\right.w0( CTOD ≤ w 0 ) ≤ w 0 {: <= w_(0))\left.\leq w_{0}\right)≤w0). With the process of loading, the CTOD gradually increases. When the it reaches critical value, maximum length of FPZ developed near notch area (Refer Fig.14(ii)). During post peak loading stage crack tip opens with visible crack as shown in Fig. 14(iii). The stages of complete fracture process depicted in Fig. 14.
The ligament length ( D − a 0 D − a 0 D-a_(0)\mathrm{D}-a_{0}D−a0 ) is the portion of the uncracked cross-section between the notch tip and the opposite edge of the beam or specimen. The ligament length determines the region available for the formation and development of the FPZ, where micro-cracking, aggregate bridging, and energy dissipation occur. A longer ligament length provides more material to resist crack propagation, leading to a larger FPZ. Ligament length directly affects the amount of fracture energy available for crack propagation. A shorter ligament reduces the capacity for energy dissipation in the FPZ, often leading to more brittle fracture behavior, whereas a longer ligament supports more stable crack growth.
For B-SB75, fracture process under various loading steps depicted in Tab. 3. At loading step P 1 P 1 P_(1)\mathrm{P}_{1}P1, a FPZ is observed forming ahead of the notch tip. The FPZ length measures 4.75 mm , which is 0.085 times the ligament length ( D − a 0 = 75 − 19 = 56 mm D − a 0 = 75 − 19 = 56 mm D-a_(0)=75-19=56mm\mathrm{D}-a_{0}=75-19=56 \mathrm{~mm}D−a0=75−19=56 mm ). The crack tip opening displacement ( w w www ) is measured at 4.88 μ m 4.88 μ m 4.88 mum4.88 \mu \mathrm{~m}4.88μ m.
As loading progresses to P 2 P 2 P_(2)\mathrm{P}_{2}P2 (the peak load), the FPZ length ( l FP Z l FP Z l_(FPZ)l_{\mathrm{FP} Z}lFPZ ) extends to 0.135 times the ligament length, with a w w www of 7.56 μ m 7.56 μ m 7.56 mum7.56 \mu \mathrm{~m}7.56μ m. By loading step P 5 P 5 P_(5)\mathrm{P}_{5}P5, the l FPZ l FPZ l_(FPZ)l_{\mathrm{FPZ}}lFPZ increases to 0.698 times the ligament length, with a w w www of 49.65 μ m 49.65 μ m 49.65 mum49.65 \mu \mathrm{~m}49.65μ m, resulting in the FPZ appearing as an irregular band. At loading point P 6 P 6 P_(6)\mathrm{P}_{6}P6, with a w w www of 51.25 μ m 51.25 μ m 51.25 mum51.25 \mu \mathrm{~m}51.25μ m, the l FPZ l FPZ l_(FPZ)l_{\mathrm{FPZ}}lFPZ equal to 45.83 mm , 0.85 45.83 mm , 0.85 45.83mm,0.8545.83 \mathrm{~mm}, 0.8545.83 mm,0.85 times the ligament length. When loading step P 7 P 7 P_(7)\mathrm{P}_{7}P7 reached, the l FPZ l FPZ l_(FPZ)l_{\mathrm{FPZ}}lFPZ becomes 35 mm . Therefore it is observed that maximum l FPZ l FPZ l_(FPZ)l_{\mathrm{FPZ}}lFPZ developed between P 6 P 6 P_(6)\mathrm{P}_{6}P6 and P 7 P 7 P_(7)\mathrm{P}_{7}P7 loading steps.
From the visualization technique available in Abaqus CAE, the exact point P C P C P_(C)\mathrm{P}_{\mathrm{C}}PC at which the crack tip opens up which is called Stress-free Crack opening displacement ( w 0 w 0 w_(0)w_{0}w0 ) could be noted. At that load the value of l F P Z = 47.5 mm l F P Z = 47.5 mm l_(FPZ)=47.5mml_{F P Z}=47.5 \mathrm{~mm}lFPZ=47.5 mm and w 0 = 52.67 μ m . P C w 0 = 52.67 μ m . P C w_(0)=52.67 mum.P_(C)w_{0}=52.67 \mu \mathrm{m} . \mathrm{P}_{\mathrm{C}}w0=52.67μm.PC is observed to be 26 % 26 % 26%26 \%26% of P max P max  P_("max ")\mathrm{P}_{\text {max }}Pmax  (Tab. 3). Further examination of the FPZ's variation, depicted in Fig.13, reveals that at Loading steps P 8 , P 9 P 8 , P 9 P_(8),P_(9)\mathrm{P}_{8}, \mathrm{P}_{9}P8,P9, and P 10 P 10 P_(10)\mathrm{P}_{10}P10, the w w www exceeds w 0 w 0 w_(0)w_{0}w0, leading to the formation of a new traction-free crack ahead of the notch tip. Interestingly, despite the crack tip advancing by 113.1 μ m 113.1 μ m 113.1 mum113.1 \mu \mathrm{~m}113.1μ m and 150.7 μ m 150.7 μ m 150.7 mum150.7 \mu \mathrm{~m}150.7μ m at loading steps P 8 P 8 P_(8)\mathrm{P}_{8}P8 and P 9 P 9 P_(9)\mathrm{P}_{9}P9, respectively, the FPZ length diminishes by 22.9 mm and 17.5 mm , respectively. Fig. 15 shows fracture process zone at heterogenous section at P 3 P 3 P_(3)\mathrm{P}_{3}P3, P C P C P_(C)\mathrm{P}_{\mathrm{C}}PC and P 7 P 7 P_(7)\mathrm{P}_{7}P7 for B-SB75 extracted from analysis results.
Figure 15: Fracture process zone for B-SB75.
The final macrocrack trajectory in B-SB75, shown in Fig.16(iii), aligns well with the propagation path of the FPZ determined by the DIC method [9] and AE method [11] in Fig. 15 (i,ii).
Figure 16: Propagation path of the FPZ
Loading steps
PrePeak
Peak
Post Peak
PrePeak Peak Post Peak| PrePeak | | :--- | | Peak | | Post Peak |
P Max P Max  P_("Max ")\mathrm{P}_{\text {Max }}PMax  ) B-SB75
4.85 7.4 20.5 34.5 49.0 50.5 85.0 111.0 148.0 194.0 51
w w www 4.90 7.6 20.9 35.5 50.0 51.5 87.0 115.0 152.0 198.0 53
( μ m μ m mum\mu \mathrm{m}μm ) 4.89 7.68 20.7 35.18 49.95 51.75 87.1 113.3 152.2 198.2 54
Avg. 4.88 7.56 20.70 35.06 49.65 51.25 86.37 113.10 150.73 196.73 52.67
CoV 0.44 1.56 0.79 1.19 0.93 1.05 1.12 1.45 1.28 0.98 2.37
l F P Z l F P Z l_(FPZ)l_{F P Z}lFPZ 4.7 7.5 21.0 26.0 38.5 46 34.5 22.5 17.2 12.5 47.0
4.8 7.7 21.2 26.8 39.5 45 35.5 23.3 17.8 12.9 48.0
( μ m μ m mum\mu \mathrm{m}μm ) 4.75 7.6 21.1 26.5 39.3 46.5 35.0 22.9 17.5 12.7 47.5
Avg. 4.75 7.60 21.10 26.43 39.10 45.83 35.00 22.90 17.50 12.70 47.50
CoV 0.86 1.07 0.39 1.25 1.10 1.36 1.17 1.43 1.40 1.29 0.86
B-MB150
7.5 13.5 24.3 40.32 51 60.5 92.0 116.0 150.2 220.2 69
w w www 7.6 13.8 26.5 41.5 54 61.8 94.5 120.0 154.9 225.4 74
( μ m μ m mum\mu \mathrm{m}μm ) 7.55 13.6 27 40.04 53.5 61.13 93.34 118.1 160.2 230.5 76
Avg. 7.55 13.63 25.93 40.62 52.83 61.14 93.28 118.03 155.10 225.37 73
CoV 0.54 0.91 4.52 1.56 2.48 0.87 1.10 1.38 2.63 1.87 4.03
9.3 29.8 50.1 70 87.0 91 77 61.5 52.5 37.5 90
l F P Z l F P Z l_(FPZ)l_{F P Z}lFPZ 9.7 31.2 51.3 71.4 88.5 90 70.6 66.5 56.2 38 92
( μ m μ m mum\mu \mathrm{m}μm ) 9.5 30.5 50.7 69 87.5 89 75 58.8 44.3 38.34 95
Avg. 9.50 30.50 50.70 70.13 87.67 90.00 74.20 62.27 51.00 38 92.33
B-LB250
w w www 9.5 15 35.2 65.0 80.2 90.0 128.0 187.0 243.0 338.0 95
9.3 16.4 38.2 69.0 78.3 96.0 135.0 195.0 256.0 355.0 100
( μ m μ m mum\mu \mathrm{m}μm ) 9.2 18.2 40.3 67.1 83.4 92.2 132.9 192.1 250.1 346.6 98
Avg. 9.33 16.53 37.90 67.03 80.63 92.73 131.97 191.37 249.70 346.53 97.67
CoV 1.34 7.92 5.52 2.44 2.61 2.67 2.22 1.73 2.13 2.00 2.10
24.0 27.0 39.0 75 90 110 72.0 56.0 40.0 30.0 120
26.0 30.0 43.0 84 95 125 79.0 64.0 47.0 35.0 135
l F P Z l F P Z l_(FPZ)l_{F P Z}lFPZ 25.1 28.5 41.0 67.6 85 110 74.0 60.0 44.0 33.4 126
Avg. 25.03 28.50 41.00 75.53 90.00 115.00 75.00 60.00 43.67 32.80 127
CoV 3.27 4.30 3.98 8.88 4.54 6.15 3.93 5.44 6.57 6.36 4.85
B-VB-500
8.7 16.5 38.0 80.0 105.0 120.0 164.0 210.0 296.0 530.0 128
w w www 9.5 18.5 41.0 90.0 115.0 135.0 185.0 245.0 318.0 570.0 145
( μ m μ m mum\mu \mathrm{m}μm ) 9.2 17.7 39.8 85.0 111.0 127.0 172.9 229.2 307.4 550.0 138
Avg. 9.13 17.57 39.60 85.00 110.33 127.33 173.97 228.07 307.13 550.00 137
CoV 3.61 4.68 3.11 4.80 3.72 4.81 4.95 6.27 2.92 2.97 5.09
65.0 70.0 82.0 119.0 130.0 160.0 145.0 120.0 74.0 67.0 170
l F P Z l F P Z l_(FPZ)l_{F P Z}lFPZ 73.0 80.0 89.0 125.0 140.0 180.0 160.0 145.0 92.0 75.0 172
( μ m μ m mum\mu \mathrm{m}μm ) 69.0 75.0 85.5 123.2 140.2 170.2 153.9 133.3 84.2 71.0 174
Avg. 69.00 75.00 85.50 122.40 136.73 170.07 152.97 132.77 83.40 71.00 172
CoV 4.73 5.44 3.34 2.05 3.48 4.80 4.03 7.69 8.84 4.60 0.95
B-HB-1000
15 30 70 105 115 139 195 263 370 650 171
w w www 17 34 80 120 130 154 210 270 400 662.5 175
( μ m μ m mum\mu \mathrm{m}μm ) 17.6 32.91 74.06 111.5 122 143 200.4 280 410 670.5 165
Avg. 16.53 32.30 74.69 112.17 122.33 145.33 201.80 271.00 393.33 661.00 170
CoV 6.72 5.23 5.50 5.48 5.01 4.36 3.07 2.57 4.32 1.28 2.41
34.0 70.0 130.0 170 201.5 210 251.2 233.9 200 180 270
l F P Z l F P Z l_(FPZ)l_{F P Z}lFPZ 37.0 80.0 140.0 160.4 205 214.1 264.9 220 205 190.2 285
( μ m μ m mum\mu \mathrm{m}μm ) 35.71 74.07 133.3 181.4 180 236 244.5 225 207.3 189.7 240
Avg. 35.57 74.69 134.43 170.60 195.50 220.03 253.53 226.30 204.10 186.63 265
CoV 3.45 5.50 3.09 5.03 5.65 5.19 3.35 2.54 1.49 2.52 7.06
Loading steps "PrePeak Peak Post Peak" P_("Max ") ) B-SB75 4.85 7.4 20.5 34.5 49.0 50.5 85.0 111.0 148.0 194.0 51 w 4.90 7.6 20.9 35.5 50.0 51.5 87.0 115.0 152.0 198.0 53 ( mum ) 4.89 7.68 20.7 35.18 49.95 51.75 87.1 113.3 152.2 198.2 54 Avg. 4.88 7.56 20.70 35.06 49.65 51.25 86.37 113.10 150.73 196.73 52.67 CoV 0.44 1.56 0.79 1.19 0.93 1.05 1.12 1.45 1.28 0.98 2.37 l_(FPZ) 4.7 7.5 21.0 26.0 38.5 46 34.5 22.5 17.2 12.5 47.0 4.8 7.7 21.2 26.8 39.5 45 35.5 23.3 17.8 12.9 48.0 ( mum ) 4.75 7.6 21.1 26.5 39.3 46.5 35.0 22.9 17.5 12.7 47.5 Avg. 4.75 7.60 21.10 26.43 39.10 45.83 35.00 22.90 17.50 12.70 47.50 CoV 0.86 1.07 0.39 1.25 1.10 1.36 1.17 1.43 1.40 1.29 0.86 B-MB150 7.5 13.5 24.3 40.32 51 60.5 92.0 116.0 150.2 220.2 69 w 7.6 13.8 26.5 41.5 54 61.8 94.5 120.0 154.9 225.4 74 ( mum ) 7.55 13.6 27 40.04 53.5 61.13 93.34 118.1 160.2 230.5 76 Avg. 7.55 13.63 25.93 40.62 52.83 61.14 93.28 118.03 155.10 225.37 73 CoV 0.54 0.91 4.52 1.56 2.48 0.87 1.10 1.38 2.63 1.87 4.03 9.3 29.8 50.1 70 87.0 91 77 61.5 52.5 37.5 90 l_(FPZ) 9.7 31.2 51.3 71.4 88.5 90 70.6 66.5 56.2 38 92 ( mum ) 9.5 30.5 50.7 69 87.5 89 75 58.8 44.3 38.34 95 Avg. 9.50 30.50 50.70 70.13 87.67 90.00 74.20 62.27 51.00 38 92.33 B-LB250 w 9.5 15 35.2 65.0 80.2 90.0 128.0 187.0 243.0 338.0 95 9.3 16.4 38.2 69.0 78.3 96.0 135.0 195.0 256.0 355.0 100 ( mum ) 9.2 18.2 40.3 67.1 83.4 92.2 132.9 192.1 250.1 346.6 98 Avg. 9.33 16.53 37.90 67.03 80.63 92.73 131.97 191.37 249.70 346.53 97.67 CoV 1.34 7.92 5.52 2.44 2.61 2.67 2.22 1.73 2.13 2.00 2.10 24.0 27.0 39.0 75 90 110 72.0 56.0 40.0 30.0 120 26.0 30.0 43.0 84 95 125 79.0 64.0 47.0 35.0 135 l_(FPZ) 25.1 28.5 41.0 67.6 85 110 74.0 60.0 44.0 33.4 126 Avg. 25.03 28.50 41.00 75.53 90.00 115.00 75.00 60.00 43.67 32.80 127 CoV 3.27 4.30 3.98 8.88 4.54 6.15 3.93 5.44 6.57 6.36 4.85 B-VB-500 8.7 16.5 38.0 80.0 105.0 120.0 164.0 210.0 296.0 530.0 128 w 9.5 18.5 41.0 90.0 115.0 135.0 185.0 245.0 318.0 570.0 145 ( mum ) 9.2 17.7 39.8 85.0 111.0 127.0 172.9 229.2 307.4 550.0 138 Avg. 9.13 17.57 39.60 85.00 110.33 127.33 173.97 228.07 307.13 550.00 137 CoV 3.61 4.68 3.11 4.80 3.72 4.81 4.95 6.27 2.92 2.97 5.09 65.0 70.0 82.0 119.0 130.0 160.0 145.0 120.0 74.0 67.0 170 l_(FPZ) 73.0 80.0 89.0 125.0 140.0 180.0 160.0 145.0 92.0 75.0 172 ( mum ) 69.0 75.0 85.5 123.2 140.2 170.2 153.9 133.3 84.2 71.0 174 Avg. 69.00 75.00 85.50 122.40 136.73 170.07 152.97 132.77 83.40 71.00 172 CoV 4.73 5.44 3.34 2.05 3.48 4.80 4.03 7.69 8.84 4.60 0.95 B-HB-1000 15 30 70 105 115 139 195 263 370 650 171 w 17 34 80 120 130 154 210 270 400 662.5 175 ( mum ) 17.6 32.91 74.06 111.5 122 143 200.4 280 410 670.5 165 Avg. 16.53 32.30 74.69 112.17 122.33 145.33 201.80 271.00 393.33 661.00 170 CoV 6.72 5.23 5.50 5.48 5.01 4.36 3.07 2.57 4.32 1.28 2.41 34.0 70.0 130.0 170 201.5 210 251.2 233.9 200 180 270 l_(FPZ) 37.0 80.0 140.0 160.4 205 214.1 264.9 220 205 190.2 285 ( mum ) 35.71 74.07 133.3 181.4 180 236 244.5 225 207.3 189.7 240 Avg. 35.57 74.69 134.43 170.60 195.50 220.03 253.53 226.30 204.10 186.63 265 CoV 3.45 5.50 3.09 5.03 5.65 5.19 3.35 2.54 1.49 2.52 7.06| Loading steps | PrePeak <br> Peak <br> Post Peak | | | | | | | | | | | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | | $\mathrm{P}_{\text {Max }}$ ) | B-SB75 | | | | | | | | | | | | 4.85 | 7.4 | 20.5 | 34.5 | 49.0 | 50.5 | 85.0 | 111.0 | 148.0 | 194.0 | 51 | | $w$ | 4.90 | 7.6 | 20.9 | 35.5 | 50.0 | 51.5 | 87.0 | 115.0 | 152.0 | 198.0 | 53 | | ( $\mu \mathrm{m}$ ) | 4.89 | 7.68 | 20.7 | 35.18 | 49.95 | 51.75 | 87.1 | 113.3 | 152.2 | 198.2 | 54 | | Avg. | 4.88 | 7.56 | 20.70 | 35.06 | 49.65 | 51.25 | 86.37 | 113.10 | 150.73 | 196.73 | 52.67 | | CoV | 0.44 | 1.56 | 0.79 | 1.19 | 0.93 | 1.05 | 1.12 | 1.45 | 1.28 | 0.98 | 2.37 | | $l_{F P Z}$ | 4.7 | 7.5 | 21.0 | 26.0 | 38.5 | 46 | 34.5 | 22.5 | 17.2 | 12.5 | 47.0 | | | 4.8 | 7.7 | 21.2 | 26.8 | 39.5 | 45 | 35.5 | 23.3 | 17.8 | 12.9 | 48.0 | | ( $\mu \mathrm{m}$ ) | 4.75 | 7.6 | 21.1 | 26.5 | 39.3 | 46.5 | 35.0 | 22.9 | 17.5 | 12.7 | 47.5 | | Avg. | 4.75 | 7.60 | 21.10 | 26.43 | 39.10 | 45.83 | 35.00 | 22.90 | 17.50 | 12.70 | 47.50 | | CoV | 0.86 | 1.07 | 0.39 | 1.25 | 1.10 | 1.36 | 1.17 | 1.43 | 1.40 | 1.29 | 0.86 | | B-MB150 | | | | | | | | | | | | | | 7.5 | 13.5 | 24.3 | 40.32 | 51 | 60.5 | 92.0 | 116.0 | 150.2 | 220.2 | 69 | | $w$ | 7.6 | 13.8 | 26.5 | 41.5 | 54 | 61.8 | 94.5 | 120.0 | 154.9 | 225.4 | 74 | | ( $\mu \mathrm{m}$ ) | 7.55 | 13.6 | 27 | 40.04 | 53.5 | 61.13 | 93.34 | 118.1 | 160.2 | 230.5 | 76 | | Avg. | 7.55 | 13.63 | 25.93 | 40.62 | 52.83 | 61.14 | 93.28 | 118.03 | 155.10 | 225.37 | 73 | | CoV | 0.54 | 0.91 | 4.52 | 1.56 | 2.48 | 0.87 | 1.10 | 1.38 | 2.63 | 1.87 | 4.03 | | | 9.3 | 29.8 | 50.1 | 70 | 87.0 | 91 | 77 | 61.5 | 52.5 | 37.5 | 90 | | $l_{F P Z}$ | 9.7 | 31.2 | 51.3 | 71.4 | 88.5 | 90 | 70.6 | 66.5 | 56.2 | 38 | 92 | | ( $\mu \mathrm{m}$ ) | 9.5 | 30.5 | 50.7 | 69 | 87.5 | 89 | 75 | 58.8 | 44.3 | 38.34 | 95 | | Avg. | 9.50 | 30.50 | 50.70 | 70.13 | 87.67 | 90.00 | 74.20 | 62.27 | 51.00 | 38 | 92.33 | | B-LB250 | | | | | | | | | | | | | | | | | | | | | | | | | | $w$ | 9.5 | 15 | 35.2 | 65.0 | 80.2 | 90.0 | 128.0 | 187.0 | 243.0 | 338.0 | 95 | | | 9.3 | 16.4 | 38.2 | 69.0 | 78.3 | 96.0 | 135.0 | 195.0 | 256.0 | 355.0 | 100 | | ( $\mu \mathrm{m}$ ) | 9.2 | 18.2 | 40.3 | 67.1 | 83.4 | 92.2 | 132.9 | 192.1 | 250.1 | 346.6 | 98 | | Avg. | 9.33 | 16.53 | 37.90 | 67.03 | 80.63 | 92.73 | 131.97 | 191.37 | 249.70 | 346.53 | 97.67 | | CoV | 1.34 | 7.92 | 5.52 | 2.44 | 2.61 | 2.67 | 2.22 | 1.73 | 2.13 | 2.00 | 2.10 | | | 24.0 | 27.0 | 39.0 | 75 | 90 | 110 | 72.0 | 56.0 | 40.0 | 30.0 | 120 | | | 26.0 | 30.0 | 43.0 | 84 | 95 | 125 | 79.0 | 64.0 | 47.0 | 35.0 | 135 | | $l_{F P Z}$ | 25.1 | 28.5 | 41.0 | 67.6 | 85 | 110 | 74.0 | 60.0 | 44.0 | 33.4 | 126 | | Avg. | 25.03 | 28.50 | 41.00 | 75.53 | 90.00 | 115.00 | 75.00 | 60.00 | 43.67 | 32.80 | 127 | | CoV | 3.27 | 4.30 | 3.98 | 8.88 | 4.54 | 6.15 | 3.93 | 5.44 | 6.57 | 6.36 | 4.85 | | B-VB-500 | | | | | | | | | | | | | | 8.7 | 16.5 | 38.0 | 80.0 | 105.0 | 120.0 | 164.0 | 210.0 | 296.0 | 530.0 | 128 | | $w$ | 9.5 | 18.5 | 41.0 | 90.0 | 115.0 | 135.0 | 185.0 | 245.0 | 318.0 | 570.0 | 145 | | ( $\mu \mathrm{m}$ ) | 9.2 | 17.7 | 39.8 | 85.0 | 111.0 | 127.0 | 172.9 | 229.2 | 307.4 | 550.0 | 138 | | Avg. | 9.13 | 17.57 | 39.60 | 85.00 | 110.33 | 127.33 | 173.97 | 228.07 | 307.13 | 550.00 | 137 | | CoV | 3.61 | 4.68 | 3.11 | 4.80 | 3.72 | 4.81 | 4.95 | 6.27 | 2.92 | 2.97 | 5.09 | | | 65.0 | 70.0 | 82.0 | 119.0 | 130.0 | 160.0 | 145.0 | 120.0 | 74.0 | 67.0 | 170 | | $l_{F P Z}$ | 73.0 | 80.0 | 89.0 | 125.0 | 140.0 | 180.0 | 160.0 | 145.0 | 92.0 | 75.0 | 172 | | ( $\mu \mathrm{m}$ ) | 69.0 | 75.0 | 85.5 | 123.2 | 140.2 | 170.2 | 153.9 | 133.3 | 84.2 | 71.0 | 174 | | Avg. | 69.00 | 75.00 | 85.50 | 122.40 | 136.73 | 170.07 | 152.97 | 132.77 | 83.40 | 71.00 | 172 | | CoV | 4.73 | 5.44 | 3.34 | 2.05 | 3.48 | 4.80 | 4.03 | 7.69 | 8.84 | 4.60 | 0.95 | | B-HB-1000 | | | | | | | | | | | | | | 15 | 30 | 70 | 105 | 115 | 139 | 195 | 263 | 370 | 650 | 171 | | $w$ | 17 | 34 | 80 | 120 | 130 | 154 | 210 | 270 | 400 | 662.5 | 175 | | ( $\mu \mathrm{m}$ ) | 17.6 | 32.91 | 74.06 | 111.5 | 122 | 143 | 200.4 | 280 | 410 | 670.5 | 165 | | Avg. | 16.53 | 32.30 | 74.69 | 112.17 | 122.33 | 145.33 | 201.80 | 271.00 | 393.33 | 661.00 | 170 | | CoV | 6.72 | 5.23 | 5.50 | 5.48 | 5.01 | 4.36 | 3.07 | 2.57 | 4.32 | 1.28 | 2.41 | | | 34.0 | 70.0 | 130.0 | 170 | 201.5 | 210 | 251.2 | 233.9 | 200 | 180 | 270 | | $l_{F P Z}$ | 37.0 | 80.0 | 140.0 | 160.4 | 205 | 214.1 | 264.9 | 220 | 205 | 190.2 | 285 | | ( $\mu \mathrm{m}$ ) | 35.71 | 74.07 | 133.3 | 181.4 | 180 | 236 | 244.5 | 225 | 207.3 | 189.7 | 240 | | Avg. | 35.57 | 74.69 | 134.43 | 170.60 | 195.50 | 220.03 | 253.53 | 226.30 | 204.10 | 186.63 | 265 | | CoV | 3.45 | 5.50 | 3.09 | 5.03 | 5.65 | 5.19 | 3.35 | 2.54 | 1.49 | 2.52 | 7.06 |
Table 3: w ( μ m ) w ( μ m ) w(mum)w(\mu \mathrm{~m})w(μ m) and l F P Z ( mm ) l F P Z ( mm ) l_(FPZ)(mm)l_{F P Z}(\mathrm{~mm})lFPZ( mm) at P 1 P 1 P_(1)\mathrm{P}_{1}P1 to P 10 P 10 P_(10)\mathrm{P}_{10}P10.
Tab. 4 shows the maximum FPZ length developed in all five beams under consideration. It is observed the maximum FPZ length developed at 20 % − 26 % 20 % − 26 % 20%-26%20 \%-26 \%20%−26% of post peak load. It is observed that, when small and large beams are made from the same concrete, the fracture process zone (FPZ) significantly influences the energy dissipation characteristics in each beam.
In small beams, the FPZ is relatively large compared to the size of the uncracked ligament. This extensive FPZ leads to higher energy dissipation per unit area of crack growth due to more pronounced microcracking, aggregate interlocking, and frictional effects. Consequently, the fracture process in small beams exhibits a more ductile-like behavior, characterized by a gradual post-peak decline and a pronounced tail in the load-displacement curve.
In large beams, the FPZ is proportionally smaller relative to the beam size. The energy dissipation per unit area of crack growth is lower because the fracture behavior shifts toward a more brittle regime. Large beams typically show a steeper drop in the load-displacement curve after the peak load, reflecting reduced energy dissipation and a quicker transition to failure.
Beam ID Iterations P C P C P_(C)\mathrm{P}_{\mathrm{C}}PC l FPZ l FPZ l_(FPZ)l_{\mathrm{FPZ}}lFPZ (mm)
1 28 % 28 % 28%28 \%28% of P Max P Max  P_("Max ")\mathrm{P}_{\text {Max }}PMax  47.0
2 25 % 25 % 25%25 \%25% of P Max P Max  P_("Max ")\mathrm{P}_{\text {Max }}PMax  48.0
B-SB-75 3 24 % 24 % 24%24 \%24% of P Max P Max  P_("Max ")\mathrm{P}_{\text {Max }}PMax  47.5
Avg. 26 % 26 % 26%26 \%26% of P Max P Max  P_("Max ")\mathrm{P}_{\text {Max }}PMax  47.50
CoV 6.62 0.86
1 20 % 20 % 20%20 \%20% of P Max P Max  P_("Max ")\mathrm{P}_{\text {Max }}PMax  90
2 21 % 21 % 21%21 \%21% of P Max P Max  P_("Max ")\mathrm{P}_{\text {Max }}PMax  92
B-LB-150 3 24 % 24 % 24%24 \%24% of P Max P Max  P_("Max ")\mathrm{P}_{\text {Max }}PMax  95
Avg. 22 % 22 % 22%22 \%22% of P Max P Max  P_("Max ")\mathrm{P}_{\text {Max }}PMax  92.33
CoV 7.84 2.23
1 21 % 21 % 21%21 \%21% of P Max P Max  P_("Max ")\mathrm{P}_{\text {Max }}PMax  120
2 19% of P Max P Max  P_("Max ")\mathrm{P}_{\text {Max }}PMax  135
B-MB-250 3 20 % 20 % 20%20 \%20% of P Max P Max  P_("Max ")\mathrm{P}_{\text {Max }}PMax  126
Avg. 20 % 20 % 20%20 \%20% of P Max P Max  P_("Max ")\mathrm{P}_{\text {Max }}PMax  127
CoV 4.08 4.85
1 18 % 18 % 18%18 \%18% of P Max P Max  P_("Max ")\mathrm{P}_{\text {Max }}PMax  170
2 19% of P Max P Max  P_("Max ")\mathrm{P}_{\text {Max }}PMax  172
B-VB-500 3 22 % 22 % 22%22 \%22% of P Max P Max  P_("Max ")\mathrm{P}_{\text {Max }}PMax  174
Avg. 20 % 20 % 20%20 \%20% of P Max P Max  P_("Max ")\mathrm{P}_{\text {Max }}PMax  172
CoV 8.64 0.95
1 23 % 23 % 23%23 \%23% of P Max P Max  P_("Max ")\mathrm{P}_{\text {Max }}PMax  270
2 19 % 19 % 19%19 \%19% of P Max P Max  P_("Max ")\mathrm{P}_{\text {Max }}PMax  285
B-HB-1000 3 21 % 21 % 21%21 \%21% of P Max P Max  P_("Max ")\mathrm{P}_{\text {Max }}PMax  240
Avg. 21 % 21 % 21%21 \%21% of P Max P Max  P_("Max ")\mathrm{P}_{\text {Max }}PMax  265
CoV 7.78 7.06
Beam ID Iterations P_(C) l_(FPZ) (mm) 1 28% of P_("Max ") 47.0 2 25% of P_("Max ") 48.0 B-SB-75 3 24% of P_("Max ") 47.5 Avg. 26% of P_("Max ") 47.50 CoV 6.62 0.86 1 20% of P_("Max ") 90 2 21% of P_("Max ") 92 B-LB-150 3 24% of P_("Max ") 95 Avg. 22% of P_("Max ") 92.33 CoV 7.84 2.23 1 21% of P_("Max ") 120 2 19% of P_("Max ") 135 B-MB-250 3 20% of P_("Max ") 126 Avg. 20% of P_("Max ") 127 CoV 4.08 4.85 1 18% of P_("Max ") 170 2 19% of P_("Max ") 172 B-VB-500 3 22% of P_("Max ") 174 Avg. 20% of P_("Max ") 172 CoV 8.64 0.95 1 23% of P_("Max ") 270 2 19% of P_("Max ") 285 B-HB-1000 3 21% of P_("Max ") 240 Avg. 21% of P_("Max ") 265 CoV 7.78 7.06| Beam ID | Iterations | $\mathrm{P}_{\mathrm{C}}$ | $l_{\mathrm{FPZ}}$ (mm) | | :--- | :--- | :--- | :--- | | | 1 | $28 \%$ of $\mathrm{P}_{\text {Max }}$ | 47.0 | | | 2 | $25 \%$ of $\mathrm{P}_{\text {Max }}$ | 48.0 | | B-SB-75 | 3 | $24 \%$ of $\mathrm{P}_{\text {Max }}$ | 47.5 | | | Avg. | $26 \%$ of $\mathrm{P}_{\text {Max }}$ | 47.50 | | | CoV | 6.62 | 0.86 | | | 1 | $20 \%$ of $\mathrm{P}_{\text {Max }}$ | 90 | | | 2 | $21 \%$ of $\mathrm{P}_{\text {Max }}$ | 92 | | B-LB-150 | 3 | $24 \%$ of $\mathrm{P}_{\text {Max }}$ | 95 | | | Avg. | $22 \%$ of $\mathrm{P}_{\text {Max }}$ | 92.33 | | | CoV | 7.84 | 2.23 | | | 1 | $21 \%$ of $\mathrm{P}_{\text {Max }}$ | 120 | | | 2 | 19% of $\mathrm{P}_{\text {Max }}$ | 135 | | B-MB-250 | 3 | $20 \%$ of $\mathrm{P}_{\text {Max }}$ | 126 | | | Avg. | $20 \%$ of $\mathrm{P}_{\text {Max }}$ | 127 | | | CoV | 4.08 | 4.85 | | | 1 | $18 \%$ of $\mathrm{P}_{\text {Max }}$ | 170 | | | 2 | 19% of $\mathrm{P}_{\text {Max }}$ | 172 | | B-VB-500 | 3 | $22 \%$ of $\mathrm{P}_{\text {Max }}$ | 174 | | | Avg. | $20 \%$ of $\mathrm{P}_{\text {Max }}$ | 172 | | | CoV | 8.64 | 0.95 | | | 1 | $23 \%$ of $\mathrm{P}_{\text {Max }}$ | 270 | | | 2 | $19 \%$ of $\mathrm{P}_{\text {Max }}$ | 285 | | B-HB-1000 | 3 | $21 \%$ of $\mathrm{P}_{\text {Max }}$ | 240 | | | Avg. | $21 \%$ of $\mathrm{P}_{\text {Max }}$ | 265 | | | CoV | 7.78 | 7.06 |
Table 4: Maximum fracture process zone length.
Thus, small beams tend to display more ductile behavior, while large beams exhibit more brittle characteristics due to the relative influence of the FPZ on energy dissipation.

Evaluation of CRITICAL CRACK TIP OPENING DISPLACEMENT ( W 0 W 0 W_(0)\mathrm{W}_{0}W0 )

Fracture energy and tensile strength are essential parameters in the application of the bilinear softening model. This study validates the softening model parameters for the FPZ across various beam sizes. As per the cohesive crack model, the stress distribution within the FPZ of concrete is governed by the COD ( σ = f ( w ) σ = f ( w ) sigma=f(w)\sigma=\mathrm{f}(w)σ=f(w) ), with the critical crack opening displacement, w 0 w 0 w_(0)w_{0}w0, being a key parameter of this relationship. Peterson's research [17] indicates that the bilinear
model is an effective representation of the stress-COD relationship. The parameters of this model include the turning point stress σ 1 , w 0 , w 1 σ 1 , w 0 , w 1 sigma_(1),w_(0),w_(1)\sigma_{1}, w_{0}, w_{1}σ1,w0,w1 and f t f t  f_("t ")f_{\text {t }}ft .
Figure 17: The bilinear softening constitutive model of concrete.
Beam ID Iterations Average G f ― G f _ G_(f)_\underline{G_{f}}Gf― Pc P 7 P 7 P_(7)\mathrm{P}_{7}P7 w 0 P 8 w 0 P 8 w_(0)P_(8)w_{0} \mathrm{P}_{8}w0P8 P 9 P 9 P_(9)\mathrm{P}_{9}P9 P 10 P 10 P_(10)\mathrm{P}_{10}P10 Average w 0 w 0 w_(0)w_{0}w0 ( μ m μ m mum\mu \mathrm{m}μm ) α n α n alpha_(n)\alpha_{n}αn
G f ( N / m ) G f ( N / m ) G_(f)(N//m)\mathrm{G}_{\mathrm{f}}(\mathrm{N} / \mathrm{m})Gf(N/m) f t ― f t ¯ bar(f_(t))\overline{f_{t}}ft―
1 50.35 13.99 51 58.2 52.1 54 45
2 47.00 13.05 53 59.8 52.15 51 49
B-SB75 3 53.15 14.76 54 59.8 48 47 48 52 3.73
Avg. 50.17 13.93 52.67 59.27 50.75 50.67 47.33
CoV 5.01 5.02 2.37 1.27 3.83 5.66 3.59
1 83.34 23.15 69 66 68 74 80
2 90.12 25.03 74 74 77 73 78
B-MB150 3 88.51 24.59 76 70 70 71 72 72.8 3.00
Avg. 87.32 24.26 73 70 71.67 72.67 76.67
CoV 3.31 3.31 4.03 4.67 5.38 1.72 4.43
1 92.33 25.64 95 95 98 100 99.8
2 88.02 24.45 100 97 94 95 88
B-LB250 3 90.67 25.19 98 85 90.6 85.4 80 93.30 3.72
Avg. 90.34 25.09 97.67 92.33 94.20 93.47 89.27
CoV 1.96 1.96 2.10 5.69 3.21 6.48 9.11
1 146 40.56 128 136 137 101 129
2 122.15 33.93 145 131 132 98 125.8
B-VB500 3 115.62 32.11 138 120.6 113.6 96.1 124 123.68 3.48
Avg. 127.92 35.53 137 129.20 127.53 98.37 126.27
CoV 10.21 10.22 5.09 4.96 7.89 2.05 1.64
1 149 41.39 171 187.8 195 222 240
2 125.32 34.81 175 185.5 188.9 212 235
B- 3 128.11 35.59 165 176 193.5 208 220 198.30 5.32
HB1000 Avg. 134.14 37.26 170.3 183.10 192.47 214.00 231.67
CoV 7.88 7.88 2.41 2.79 1.35 2.75 3.67
Beam ID Iterations Average G_(f)_ Pc P_(7) w_(0)P_(8) P_(9) P_(10) Average w_(0) ( mum ) alpha_(n) G_(f)(N//m) bar(f_(t)) 1 50.35 13.99 51 58.2 52.1 54 45 2 47.00 13.05 53 59.8 52.15 51 49 B-SB75 3 53.15 14.76 54 59.8 48 47 48 52 3.73 Avg. 50.17 13.93 52.67 59.27 50.75 50.67 47.33 CoV 5.01 5.02 2.37 1.27 3.83 5.66 3.59 1 83.34 23.15 69 66 68 74 80 2 90.12 25.03 74 74 77 73 78 B-MB150 3 88.51 24.59 76 70 70 71 72 72.8 3.00 Avg. 87.32 24.26 73 70 71.67 72.67 76.67 CoV 3.31 3.31 4.03 4.67 5.38 1.72 4.43 1 92.33 25.64 95 95 98 100 99.8 2 88.02 24.45 100 97 94 95 88 B-LB250 3 90.67 25.19 98 85 90.6 85.4 80 93.30 3.72 Avg. 90.34 25.09 97.67 92.33 94.20 93.47 89.27 CoV 1.96 1.96 2.10 5.69 3.21 6.48 9.11 1 146 40.56 128 136 137 101 129 2 122.15 33.93 145 131 132 98 125.8 B-VB500 3 115.62 32.11 138 120.6 113.6 96.1 124 123.68 3.48 Avg. 127.92 35.53 137 129.20 127.53 98.37 126.27 CoV 10.21 10.22 5.09 4.96 7.89 2.05 1.64 1 149 41.39 171 187.8 195 222 240 2 125.32 34.81 175 185.5 188.9 212 235 B- 3 128.11 35.59 165 176 193.5 208 220 198.30 5.32 HB1000 Avg. 134.14 37.26 170.3 183.10 192.47 214.00 231.67 CoV 7.88 7.88 2.41 2.79 1.35 2.75 3.67 | Beam ID | Iterations | Average | $\underline{G_{f}}$ | Pc | $\mathrm{P}_{7}$ | $w_{0} \mathrm{P}_{8}$ | $\mathrm{P}_{9}$ | $\mathrm{P}_{10}$ | Average $w_{0}$ ( $\mu \mathrm{m}$ ) | $\alpha_{n}$ | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | | | $\mathrm{G}_{\mathrm{f}}(\mathrm{N} / \mathrm{m})$ | $\overline{f_{t}}$ | | | | | | | | | | 1 | 50.35 | 13.99 | 51 | 58.2 | 52.1 | 54 | 45 | | | | | 2 | 47.00 | 13.05 | 53 | 59.8 | 52.15 | 51 | 49 | | | | B-SB75 | 3 | 53.15 | 14.76 | 54 | 59.8 | 48 | 47 | 48 | 52 | 3.73 | | | Avg. | 50.17 | 13.93 | 52.67 | 59.27 | 50.75 | 50.67 | 47.33 | | | | | CoV | 5.01 | 5.02 | 2.37 | 1.27 | 3.83 | 5.66 | 3.59 | | | | | 1 | 83.34 | 23.15 | 69 | 66 | 68 | 74 | 80 | | | | | 2 | 90.12 | 25.03 | 74 | 74 | 77 | 73 | 78 | | | | B-MB150 | 3 | 88.51 | 24.59 | 76 | 70 | 70 | 71 | 72 | 72.8 | 3.00 | | | Avg. | 87.32 | 24.26 | 73 | 70 | 71.67 | 72.67 | 76.67 | | | | | CoV | 3.31 | 3.31 | 4.03 | 4.67 | 5.38 | 1.72 | 4.43 | | | | | 1 | 92.33 | 25.64 | 95 | 95 | 98 | 100 | 99.8 | | | | | 2 | 88.02 | 24.45 | 100 | 97 | 94 | 95 | 88 | | | | B-LB250 | 3 | 90.67 | 25.19 | 98 | 85 | 90.6 | 85.4 | 80 | 93.30 | 3.72 | | | Avg. | 90.34 | 25.09 | 97.67 | 92.33 | 94.20 | 93.47 | 89.27 | | | | | CoV | 1.96 | 1.96 | 2.10 | 5.69 | 3.21 | 6.48 | 9.11 | | | | | 1 | 146 | 40.56 | 128 | 136 | 137 | 101 | 129 | | | | | 2 | 122.15 | 33.93 | 145 | 131 | 132 | 98 | 125.8 | | | | B-VB500 | 3 | 115.62 | 32.11 | 138 | 120.6 | 113.6 | 96.1 | 124 | 123.68 | 3.48 | | | Avg. | 127.92 | 35.53 | 137 | 129.20 | 127.53 | 98.37 | 126.27 | | | | | CoV | 10.21 | 10.22 | 5.09 | 4.96 | 7.89 | 2.05 | 1.64 | | | | | 1 | 149 | 41.39 | 171 | 187.8 | 195 | 222 | 240 | | | | | 2 | 125.32 | 34.81 | 175 | 185.5 | 188.9 | 212 | 235 | | | | B- | 3 | 128.11 | 35.59 | 165 | 176 | 193.5 | 208 | 220 | 198.30 | 5.32 | | HB1000 | Avg. | 134.14 | 37.26 | 170.3 | 183.10 | 192.47 | 214.00 | 231.67 | | | | | CoV | 7.88 | 7.88 | 2.41 | 2.79 | 1.35 | 2.75 | 3.67 | | |
Table 5: Evaluation of α n α n alpha_(n)\alpha_{n}αn.
The bilinear model is highly effective in simulating the softening properties of the concrete FPZ. Wittmann et al. [14] proposed a constitutive model for concrete softening based on this approach, observing that the ratio of σ 1 / f t σ 1 / f t sigma_(1)//f_(t)\sigma_{1} / f_{t}σ1/ft for plain concrete is typically below 0.25 . Asmaro [15], through inverse analysis and mode I fracture tests on plain concrete, determined that σ 1 / f t σ 1 / f t sigma_(1)//f_(t)\sigma_{1} / f_{t}σ1/ft falls within the range of 0.4 to 0.53 . Using the cohesive crack model, Huang [16] developed a numerical solution for the softening curve, concluding that σ 1 / f t σ 1 / f t sigma_(1)//f_(t)\sigma_{1} / f_{t}σ1/ft is 0.25 and w 1 / w 0 w 1 / w 0 w_(1)//w_(0)w_{1} / w_{0}w1/w0 is 0.4 .
Petersson [17] suggested that the stress-free crack opening displacement ( w 0 w 0 w_(0)w_{0}w0 ) can be obtained by defining the cohesive stress and crack opening displacement at the kink point as f t / 3 f t / 3 f_(t)//3f_{t} / 3ft/3 and 0.8 G f / f t 0.8 G f / f t 0.8G_(f)//f_(t)0.8 G_{f} / f_{t}0.8Gf/ft, respectively, where f t f t f_(t)f_{t}ft represents tensile strength and G f G f G_(f)G_{f}Gf denotes the fracture energy of concrete. The value of G f G f G_(f)G_{f}Gf is calculated as the area under the load-displacement curve
obtained during three-point bending (TPB) tests. By equating the area under the softening curve to the fracture energy of concrete, wo can be determined.
w 0 = α n G f f t w 0 = α n G f f t w_(0)=alpha_(n)(G_(f))/(f_(t))w_{0}=\alpha_{n} \frac{G_{f}}{f_{t}}w0=αnGfft
The parameter α n α n alpha_(n)\alpha_{\mathrm{n}}αn is determined differently across various models. In Hillerborg's model [25], α n α n alpha_(n)\alpha_{\mathrm{n}}αn is assigned a value of 3.5. According to the CEB-FIP model [18], α n α n alpha_(n)\alpha_{\mathrm{n}}αn depends on the maximum aggregate size, taking a value of 7 for a maximum aggregate size of 16 mm and 5 for a size of 32 mm . However, an alternative version of the CEB-FIP model [19] simplifies the calculation by consistently setting α n α n alpha_(n)\alpha_{\mathrm{n}}αn to 5 , regardless of the maximum aggregate size.
Based on the results of the present study, the α n α n alpha_(n)\alpha_{\mathrm{n}}αn values presented in Tab. 5.
Figure 18: Relationship between the w o w o w_(o)w_{o}wo and G f / f o G f / f o G_(f)//f_(o)G_{f} / f_{o}Gf/fo.
Fig. 18 shows the relationship between the w 0 w 0 w_(0)w_{0}w0 and G f / f f G f / f f G_(f)//f_(f)G_{f} / f_{f}Gf/ff. It can be seen from the Fig. 18 that, the w 0 w 0 w_(0)w_{0}w0 increases with the increasement of G f / f t G f / f t G_(f)//f_(t)G_{f} / f_{t}Gf/ft, which can be well approximated by a power function. It can be written as
w 0 = 3.646 ( G f f t ) 0.983 w 0 = 3.646 G f f t 0.983 w_(0)=3.646((G_(f))/(f_(t)))^(0.983)w_{0}=3.646\left(\frac{G_{f}}{f_{t}}\right)^{0.983}w0=3.646(Gfft)0.983
Figure 19: COD profiles of B-SB75.
Figure 20: COD profiles of B-MB150.
Figure 21: COD profiles of B-LB250.
Figure 22: COD profiles of B-VB500.
Figure 23: COD profiles of B-HB1000.
Fig.19-23 present the COD profiles of all size beams along with their corresponding FPZ lengths. It was observed that after the FPZ was fully developed, the l FPZ l FPZ l_(FPZ)l_{\mathrm{FPZ}}lFPZ decreased as the crack progressed. This phenomenon can be attributed to the boundary effect of the specimen, where the crack tip approached the upper boundary of the beam, leaving an insufficient remaining ligament for the further development of the FPZ.

Size effect on the FPZ

As previously mentioned, the size of the FPZ ahead of the notch tip before reaching the peak load can serve as a fundamental explanation for the size effect phenomenon. Therefore, analyzing the characteristics of the FPZ across models of varying sizes holds significant importance. The current study reveals that FPZ widths are relatively consistent across all models, ranging from 3 to 5 mm . This observation aligns with findings by Skarzynski et al. [20] using the DIC technique, as well as previous numerical analyses conducted by Grassl et al. [21]. These results suggest that the FPZ width may be considered a material property.
In crack band theory, the FPZ width is generally assumed to remain constant, with Bažant [22] estimating it to be three times the maximum aggregate size. However, the FPZ widths observed in this numerical study are smaller than the assumed value. Conversely, the absolute length of the FPZ at the peak load shows a strong correlation with specimen size. Specifically, as specimen size increases, the localized damage zone lengthens, growing from approximately 47.5 mm for the smallest beam ( D = 75 mm D = 75 mm D=75mm\mathrm{D}=75 \mathrm{~mm}D=75 mm ) to 265 mm for the largest beam ( D = 1000 mm D = 1000 mm D=1000mm\mathrm{D}=1000 \mathrm{~mm}D=1000 mm ). This pronounced size dependence of FPZ length is attributed to the reduction in stress gradient with increasing beam size.
However, when considering the relative or normalized fracture process zone length, i.e., the ratio of FPZ length to the ligament length above the notch, l F P Z ( D − a 0 ) l F P Z D − a 0 (l_(FPZ))/((D-a_(0)))\frac{l_{F P Z}}{\left(D-a_{0}\right)}lFPZ(D−a0), an opposite trend is observed. This ratio decreases as specimen size increases.
The present findings align with experimental observations [23, 9, 24], suggesting that the normalized FPZ length is not a material parameter but depends on model size. This size-dependent normalized FPZ length provides an intrinsic explanation for the size effect.
From Tab. 4, it is evident that l F P Z l F P Z l_(FPZ)l_{F P Z}lFPZ for the small beam ( D = 75 mm ) ( D = 75 mm ) (D=75mm)(\mathrm{D}=75 \mathrm{~mm})(D=75 mm) is 47.5 mm , while for the large beam ( D = 1000 mm ) ( D = 1000 mm ) (D=1000mm)(\mathrm{D}=1000 \mathrm{~mm})(D=1000 mm), it is 265 mm .
l F P Z − small ( D − a 0 ) = 47.5 ( 75 − 19 ) = 0.84 l F P Z − large ( D − a 0 ) = 265 ( 1000 − 250 ) = 0.35 l F P Z − small  D − a 0 = 47.5 ( 75 − 19 ) = 0.84 l F P Z − large  D − a 0 = 265 ( 1000 − 250 ) = 0.35 {:[(l_(FPZ_(-)"small "))/((D-a_(0)))=(47.5)/((75-19))=0.84],[(l_(FPZ_(-)"large "))/((D-a_(0)))=(265)/((1000-250))=0.35]:}\begin{aligned} & \frac{l_{F P Z_{-} \text {small }}}{\left(D-a_{0}\right)}=\frac{47.5}{(75-19)}=0.84 \\ & \frac{l_{F P Z_{-} \text {large }}}{\left(D-a_{0}\right)}=\frac{265}{(1000-250)}=0.35 \end{aligned}lFPZ−small (D−a0)=47.5(75−19)=0.84lFPZ−large (D−a0)=265(1000−250)=0.35
Thus the condition of l F P Z − large < l F P Z − small l F P Z − large  < l F P Z − small  l_(FPZ_(-)"large ") < l_(FPZ_(-)"small ")l_{F P Z_{-} \text {large }}<l_{F P Z_{-} \text {small }}lFPZ−large <lFPZ−small  and l F P Z − small ( D − a 0 ) > l F P Z − small ( D − a 0 ) l F P Z − small  D − a 0 > l F P Z − small  D − a 0 (l_(FPZ_(-)"small "))/((D-a_(0))) > (l_(FPZ_(-)"small "))/((D-a_(0)))\frac{l_{F P Z_{-} \text {small }}}{\left(D-a_{0}\right)}>\frac{l_{F P Z_{-} \text {small }}}{\left(D-a_{0}\right)}lFPZ−small (D−a0)>lFPZ−small (D−a0) are reaffirmed.
From the crack patterns shown in Fig. 13, it is evident that the FPZ deviates due to the presence of aggregates and the weak interfacial transition zones (ITZs) around them. This suggests that the mesostructure of concrete significantly influences the formation and propagation of the FPZ.

Conclusions

  1. The findings of this study align with experimental results [ 4 , 9 , 24 ] [ 4 , 9 , 24 ] [4,9,24][4,9,24][4,9,24], indicating that the normalized FPZ size is not a material property but is instead influenced by the model size. This size-dependent normalized FPZ length offers a fundamental explanation for the size effect.
  2. Observations from the crack patterns reveal that the FPZ deviates due to the presence of aggregates and the associated weak ITZs, highlighting the role of concrete's mesostructure in the evolution of the FPZ.
  3. The FPZ length increases initially and then decreases as the load changes. The cracks within the FPZ are narrow and regular, with the complete FPZ typically forming during the post-peak load stage.
  4. The stress-free crack opening displacement ( w 0 w 0 w_(0)w_{0}w0 ) can be calculated using the expression α n G f f t α n G f f t alpha_(n)(G_(f))/(f_(t))\alpha_{n} \frac{G_{f}}{f_{t}}αnGfft. In this study, α n α n alpha_(n)\alpha_{n}αn ranges from 3 to 3.73 for beam sizes between 75 mm and 500 mm , consistent with Hillerborg's model [31], which assigns a value of 3.5. However, for a beam size of 1000 mm , α n 1000 mm , α n 1000mm,alpha_(n)1000 \mathrm{~mm}, \alpha_{n}1000 mm,αn increases to 5.32.
  5. The FPZ length along the ligament is dependent on specimen size. Larger specimens exhibit longer regions of dissipated energy, but the increase in FPZ length is non-proportional to size.
  6. In this study, l F P Z − small ( D − a 0 ) l F P Z − small  D − a 0 (l_(FPZ_(-)"small "))/((D-a_(0)))\frac{l_{F P Z_{-} \text {small }}}{\left(D-a_{0}\right)}lFPZ−small (D−a0) is 0.84 , while l F P Z − large ( D − a 0 ) l F P Z − large  D − a 0 (l_(FPZ_(-)"large "))/((D-a_(0)))\frac{l_{F P Z_{-} \text {large }}}{\left(D-a_{0}\right)}lFPZ−large (D−a0) is 0.36 , reaffirming that l F P Z − large < l F P Z − small l F P Z − large  < l F P Z − small  l_(FPZ_(-)"large ") < l_(FPZ_(-)"small ")l_{F P Z_{-} \text {large }}<l_{F P Z_{-} \text {small }}lFPZ−large <lFPZ−small  and l F P Z − small ( D − a 0 ) > l F P Z − large ( D − a 0 ) l F P Z − small  D − a 0 > l F P Z − large  D − a 0 (l_(FPZ_(-)"small "))/((D-a_(0))) > (l_(FPZ_(-)"large "))/((D-a_(0)))\frac{l_{F P Z_{-} \text {small }}}{\left(D-a_{0}\right)}>\frac{l_{F P Z_{-} \text {large }}}{\left(D-a_{0}\right)}lFPZ−small (D−a0)>lFPZ−large (D−a0)
  7. The results provide new insights into the evolution mechanisms of the FPZ from the perspective of micro-crack initiation and propagation.

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