Experimental and Numerical Study of the Tensile Behavior of Dam Concrete

Abstract

Tensile behavior governs the seismic safety of high concrete dams. This study integrates testing with mesoscale simulation to elucidate the tensile-failure mechanisms of dam fully graded concrete. Uniaxial tension, splitting tension, and flexural tests were performed on 450 mm-scale specimens using a 15 MN servo-hydraulic system. A two-dimensional random-aggregate model was then developed with globally inserted cohesive interfaces, and parameters were calibrated against the tests. Across ten random aggregate mesoscale models per loading case, simulations reproduced the measured responses. Predicted failure patterns matched observations, with cracks initiating along interfacial transition zones (ITZs), linking through mortar, and forming through-cracks. Quantified damage evolution revealed three stages—elastic response, ITZ crack initiation and extension, and mortar penetration—with >80% of cumulative damage localized in ITZs. One-at-a-time sensitivity analyses showed that (i) mortar tensile strength primarily controls peak strength but increases brittleness; (ii) ITZ tensile strength governs crack-initiation stress, ITZ shear strength shapes splitting-failure mode, and fracture energies mainly delay post-peak softening; and (iii) aggregate parameters exert comparatively weak influence on macroscopic behavior. The combined experimental–mesoscale framework provides mechanism-based guidance for selecting material parameters in seismic analyses, supporting performance-informed design and assessment of high dams.

1. Introduction

High dams already built in China are widely distributed in the country’s western regions, where concrete dams predominate. These regions are characterized by intense tectonic activity, highly unstable geological conditions, frequent and strong earthquakes, and hydropower-rich basins that commonly coincide with zones of high seismic intensity. Under earthquake excitation, the dynamic response and damage failure processes of concrete high dams constitute a highly complex problem involving pronounced nonlinearity and multiscale coupling. To ensure the seismic safety of such critical infrastructure, the rationality and reliability of dam seismic design must be verified from three complementary perspectives [1]: (i) appropriate selection of input ground motions and accurate simulation of input boundary conditions; (ii) accuracy and applicability of structural dynamic analysis methods; and (iii) the actual mechanical behavior of dam concrete under dynamic loading. Because earthquake-induced failure in concrete dams is predominantly tension-controlled, investigating the tensile behavior of dam concrete is of fundamental importance for seismic design and safety assessment [2].

According to the hydraulic-concrete testing code, concrete tensile properties can be obtained by three laboratory methods—uniaxial tension, splitting tension (Brazilian), and flexure—under which concrete exhibits distinct mechanical responses. Conventional mechanical tests characterize material behavior through macroscopic indicators such as strength, strain, and elastic modulus, implicitly treating concrete as a homogeneous continuum. From a mesoscale perspective, however, concrete is a multiphase composite comprising coarse aggregates, mortar, and the interfacial transition zone (ITZ); its macroscopic response emerges from the phase-level behaviors and their interactions [3]. To uncover the mechanisms underlying macroscopic behavior, mesoscale investigation is indispensable.

Mesoscale simulation provides an effective avenue for such inquiry [4], enabling explicit reconstruction of the random aggregate distribution and faithful representation of the material’s heterogeneous internal architecture. Numerical approaches commonly used to study concrete mesoscale structure include: (1) lattice models [5,6,7,8,9]; (2) rigid-body spring models [10,11,12,13]; (3) discrete element methods [14,15,16,17]; and (4) finite element methods [18,19]. In addition, the cohesive zone model has, in recent years, become a key tool for analyzing fracture in brittle and quasi-brittle materials, directly capturing interface damage evolution at the mesoscale and offering a unified, flexible, and implementation-friendly framework for problems such as concrete and rock microcrack growth and interface failure [4,20,21,22,23].

This study performs uniaxial tension, Brazilian splitting, and four-point bending tests on dam concrete to characterize its tensile performance. At the mesoscale, we employ a cohesive–interface modeling framework to represent the mortar, aggregates, and ITZ, and to simulate crack initiation and propagation. The simulations are used in a comparative analysis of failure mechanisms under direct tension, splitting, and flexure, providing mechanistic insight complementary to the experiments.

2. Experimental Methods

2.1. Materials and Specimens

All materials used to cast the laboratory concrete specimens, with the exception of water and cement, were sourced directly from the construction site. The cement was identical to that used in the project and procured directly from the cement plant: a 42.5-grade moderate-heat Portland cement, with 28-day compressive and flexural strengths of 51.9 MPa and 9.0 MPa, respectively. Two types of admixtures were employed. The mineral admixture was Class I fly ash (water-demand ratio 95%, fineness 6.15%). The chemical admixtures comprised a retarding high-range water reducer and an air-entraining agent to enhance workability. In dam concrete, water-reducing admixtures are incorporated to lower the water–binder ratio and cement content while maintaining adequate workability. This helps to reduce the heat of hydration and improve both strength and durability. Air-entraining agents are added to introduce uniformly distributed microscopic air bubbles, thereby enhancing workability. The combined use of these two admixtures allows for an optimized balance between strength, temperature control, and durability, and is therefore fully justified from an engineering standpoint in dam concrete applications. To be consistent with actual engineering practice, the same two chemical admixtures were also incorporated in the concrete mixtures prepared in the laboratory. Aggregates were manufactured from on-site granodiorite: the coarse aggregate consisted of four size fractions—5–20, 20–40, 40–80, and 80–150 mm—blended at a mass ratio of 2:2:3:3; the fine aggregate was manufactured sand with a fineness modulus of 2.96 and a stone-powder content of 12.8%. The mixture proportions of the test specimens are listed in

Table 1. Proportion of mixture of testing concrete.

Strength Grade	Gradation	Mass per Unit Volume of Concrete (kg/m3)
		Water	Cement	Fly Ash	Sand	Coarse Aggregate				Water-Reducing Agent (%)	Air-Entraining Agent (%)
						5 mm–20 mm	20 mm–40 mm	40 mm–80 mm	80 mm–150 mm
C18030	4	85	123	66	481	342	342	513	513	0.8	0.058

. The dosage of the water reducer was kept constant, while the air-entraining agent was slightly adjusted to ensure the measured slump and air content met the design specifications.

The size of the fully graded compressive tests specimens was 450 × 450 × 450 mm, while the size of the corresponding wet-screened specimens was 150 × 150 × 150 mm. The size of the fully graded flexural tensile test specimens was 450 × 450 × 1700 mm, and the size of the corresponding wet-screened test specimens was 150 × 150 × 550 mm. The fully graded specimens were placed and vibrated in steel molds in two layers, and the wet-screened specimens were placed using a vibrating table. The molds were removed 4 to 6 days after the specimens were formed and then transferred to a standard curing room to cure for 180 days.

The preparation of all fully graded concrete specimens strictly adhered to the Test code for hydraulic concrete (SL/T 352-2020) [24]. For the uniaxial tensile test, cylindrical specimens with a diameter of 450 mm and a length of 1350 mm were used. The two end zones (375 mm at each end) were cast with a higher strength to anchor the reinforcing bars connected to the testing machine, while the central 600 mm consisted of the test concrete. For the splitting tensile test, 450 mm × 450 mm × 450 mm cubes were prepared. For the flexural test, prismatic specimens measuring 450 mm × 450 mm × 1700 mm were employed.

2.2. Experiment Device

An MTS 15 MN large-scale dynamic materials testing system (MTS Systems Corporation, Eden Prairie, MN, USA), housed at the China Institute of Water Resources and Hydropower Research in Beijing, China, was employed, as shown in Figure 1. The machine provides capacities of 15 MN in compression and 8 MN in tension, with an overall frame stiffness of 6 MN/mm. The actuator’s maximum speed is 50 mm/s, meeting the requirements for dynamic concrete testing at seismic loading rates. The control architecture is fully digital: any measured channel can be selected as the feedback signal, and multi-channel measurements are processed in real time via programmable control to govern loading. The data acquisition frequency was set to 6144 Hz to ensure complete capture of the dynamic response.

For the uniaxial tension tests, the specimen was connected to the load frame via a pair of clearance-free spherical hinge assemblies (one at each end). Each assembly was rated to transmit ±1000 kN and allowed rotations of −30° to +90° about the pitch axis and ±8° about the roll axis. Within these limits, the free rotation facilitated alignment prior to loading and mitigated bending moments induced by minor eccentricities and assembly tolerances, thereby approaching ideal axial-tension boundary conditions. The specimen was connected to each hinge seat via anchor bolts pre-embedded in the specimen. At each end, a steel plate on the specimen and a mating plate on the hinge seat were first fastened with eight M24 bolts each; the two plates were then joined with ten M24 bolts, completing installation of the uniaxial tensile specimen on the testing machine.

Flexural tests employed a custom-designed fixture and were conducted under two equal loads applied at the one-third points. The two loading heads and the two end supports used roller shafts with a diameter of 60 mm that were free to rotate, thereby avoiding parasitic constraints during loading. The up-per loading plate was connected via the spherical hinge assembly used in the uniaxial tensile test setup to eliminate potential end moments at the loading interface.

For the splitting tensile test, a pair of steel bearing strips with a length of 510 mm and a section of 15 mm × 15 mm was used to apply a concentrated line load along the specimen midline. A laser level was employed to align the upper and lower strips in the same vertical plane.

3. Mesoscale Modeling Methodology

3.1. Model Construction

A two-dimensional random-aggregate model for dam concrete was constructed using a parametric modeling method. Only coarse aggregates with particle sizes greater than 5 mm were considered. The aggregate gradation was derived by converting the measured coarse-aggregate gradation of the project into an equivalent 2D gradation curve following Walraven’s method [25]. The principal workflow of the model generator is as follows:

Step I: Parameter specification. First of all, the following inputs are prescribed: (i) the concrete outer boundary; (ii) the coarse-aggregate size range, target gradation curve, and volume fraction; and (iii) the minimum clear spacing $\gamma$, for both aggregate-to-aggregate and aggregate-to-boundary proximity.

Step II: Aggregate generation. Random polygonal aggregates are placed sequentially from larger to smaller sizes to match the target gradation based on grade curve. At each stage, the nominal radius $R^{\prime }$ of the current size class is determined from the gradation curve given the fraction already generated. For each particle, an $n$-gon is created by sampling vertex polar coordinates $\left\{\right(r_i,{\theta }_i){\}}_{i=1}^n$ within prescribed perturbation bounds:

$$\begin{array}{c}{r}_{\mathrm{i}}=R^{\prime }\times \left(1+{\xi }_{\mathrm{i}}\right)\\ {\theta }_{\mathrm{i}}=\mathrm{i}\times {\displaystyle \frac{2\pi }{n}}+{\mu }_{\mathrm{i}},\end{array}$$

(1)

where the radial and azimuthal perturbations satisfy ${\xi }_{\mathrm{i}}\in [-\alpha ,\alpha ]$ and ${\eta }_{\mathrm{i}}\in [-{\displaystyle \frac{\pi }{n}}\beta ,{\displaystyle \frac{\pi }{n}}\beta ]$, respectively, with $\alpha$ and $\beta$ denoting the preset bounds on radius and angle fluctuations. The polar coordinates are then converted to Cartesian coordinates:

$$\begin{array}{c}{x}_{\mathrm{i}}=r_{\mathrm{i}}\cos \left({\theta }_{\mathrm{i}}\right),\\ y_{\mathrm{i}}=r_{\mathrm{i}}\sin \left({\theta }_{\mathrm{i}}\right).\end{array}$$

(2)

Step III: Aggregate placement. Using a background grid, a candidate aggregate is placed at a random center coordinate in the unoccupied domain and assigning a random orientation. The candidate aggregate is accepted only if: (1) it lies entirely within the prescribed concrete boundary and its minimum clearance to the boundary is at least $\gamma$; and (2) it does not intersect any previously placed aggregate and the minimum inter-aggregate spacing is at least $\gamma$. Otherwise, a new center coordinate and orientation are sampled and the trial is repeated until placement succeeds.

Step IV: Termination criterion. Repeat Steps II and III until the aggregate content reaches the prescribed target area fraction.

Upon generating the mesoscale concrete geometry, the domain was discretized into triangular elements. Zero-thickness cohesive interface elements were then inserted along all element edges. An in-house routine was developed to automate this process: the algorithm loops over all solid elements and, for the $i$-th element, creates a new zero-thickness cohesive element on each of its edges using the edge-endpoint coordinates. Material assignment is performed by inspecting the two solid elements adjacent to each edge: if both are of the same material (e.g., mortar–mortar), the interface is assigned the mortar cohesive properties; if the neighbors are mortar and aggregate, the interface is assigned ITZ (interfacial transition zone) cohesive properties. To prevent duplicate insertion, each edge is uniquely keyed by the IDs of its endpoint nodes; edges already tagged as having a cohesive element are skipped.

Based on the above modeling approach, two-dimensional models were established for the fully graded concrete four-point bending test, uniaxial tensile test, and splitting tensile test, and all analyses were carried out under the plane stress assumption.

The model for the four-point bending test of the fully graded concrete specimen is shown in Figure 2a. Generally, failure occurred within the 450 mm pure bending segment in the middle of the beam. Therefore, aggregates were placed only within a 600 mm central region of the specimen, while the regions on both sides were modeled using solid elements with linear elastic properties only. The aggregate placement zone and the elastic zones at both ends were connected by cohesive elements assigned elastic properties, ensuring continuity between the different regions.

The model for the uniaxial tensile test is shown in Figure 2b. A 450 mm × 600 mm model was established, corresponding to the central 600 mm test region of the specimen in the experiment. A vertical displacement constraint was applied at the bottom of the specimen, and an upward displacement load was applied at the top of the specimen.

The model for the splitting tensile test is shown in Figure 2c. Square discrete rigid plates with a side length of 15 mm were introduced at the midpoints of the top and bottom surfaces of the specimen to represent the loading strips. A fixed constraint was applied to the lower loading plate, while the upper loading plate was constrained in the horizontal and rotational degrees of freedom and subjected to a downward displacement load. A general contact interaction was defined between the loading plates and the specimen, with the friction coefficient set to 0.3.

A cross-sectional photograph of the flexural tensile specimen after failure is shown in Figure 3. By comparing the two-dimensional meso-scale concrete model with the cross-section of the flexural specimen, it can be seen that the aggregate shapes generated using the above method agree well with the actual aggregate cross-sections. It is also observed that, in the real cross-section, some aggregates are in direct contact with each other. However, in the modeling process a minimum spacing γ was imposed both between adjacent aggregates and between aggregates and the outer concrete boundary. This was done to avoid excessively small or highly distorted finite elements during meshing, which could otherwise lead to an excessive computational cost or even non-convergence of the numerical analysis.

3.2. Constitutive Model and Parameter Selection

This study simulates concrete cracking with cohesive elements; accordingly, the solid elements (mortar and aggregates) are assigned linear-elastic properties only.

3.2.1. Cohesive Model

Cohesive elements are characterized by a bilinear traction–separation relation as shown in Figure 4. The response is linear elastic prior to damage onset; once the nominal traction state satisfies the damage-initiation criterion, a damage-evolution law governs the ensuing softening. The cohesive model adopted herein is briefly summarized below:

(1)

Linear-Elastic Behavior

The elastic response is defined by an uncoupled normal–tangential stiffness matrix:

$$\mathit{t}=\left\{\begin{array}{c}{t}_{\mathrm{n}}\\ t_{\mathrm{s}}\end{array}\right\}=\left[\begin{array}{cc}{E}_{\mathrm{n}\mathrm{n}}& \\ & E_{\mathrm{s}\mathrm{s}}\end{array}\right]\left\{\begin{array}{c}{\epsilon }_{\mathrm{n}}\\ {\epsilon }_{\mathrm{s}}\end{array}\right\}=\mathit{E}\mathit{\epsilon },$$

(3)

where $\mathbf{t}=(t_{\mathrm{n}}, t_{\mathrm{s}}{)}^{\mathsf{T}}$ is the nominal traction vector with normal and shear components; $\mathbf{E}$ is the elastic modulus matrix; and $\mathit{\epsilon }=({\epsilon }_n, {\epsilon }_s{)}^{\mathsf{T}}$ are the corresponding strains. The separations associated with these strains are ${\delta }_n$ and ${\delta }_s$, related by

$${\epsilon }_{\mathrm{n}}={\displaystyle \frac{{\delta }_{\mathrm{n}}}{T_0}},{\epsilon }_{\mathrm{s}}={\displaystyle \frac{{\delta }_{\mathrm{s}}}{T_0}},$$

(4)

where ${\delta }_{\mathrm{n}}$ is the normal separation displacement and ${\delta }_{\mathrm{s}}$ is the tangential separation displacement. $T_0$ denotes the initial constitutive thickness. For zero-thickness cohesive elements, we set $T_0=1$, so that the nominal strains are identical to the relative separations across the interface in the corresponding directions.

(2)

Damage Initiation Criterion

Damage initiation denotes the onset of stiffness degradation once the traction state reaches a prescribed threshold. In this study, damage onset is detected using the quadratic nominal-stress criterion:

$${\left\{{\displaystyle \frac{⟨t_{\mathrm{n}}⟩}{t_{\mathrm{n}}^0}}\right\}}^2+{\left\{{\displaystyle \frac{t_{\mathrm{s}}}{t_{\mathrm{s}}^0}}\right\}}^2=1,$$

(5)

where $t_{\mathrm{n}}^0$ is the peak nominal normal traction under purely normal separation, $t_{\mathrm{s}}^0$ is the peak nominal shear traction under purely tangential sliding, and $t_{\mathrm{n}},t_{\mathrm{s}}$ are the current nominal tractions. The Macaulay bracket on the normal traction, $⟨t_{\mathrm{n}}⟩$, enforces that normal-direction damage is activated only in tension; in compression ($t_{\mathrm{n}}<0$) the normal term does not contribute to the criterion.

$$⟨t_{\mathrm{n}}⟩=\left\{\begin{array}{c}{t}_{\mathrm{n}}, t_{\mathrm{n}}\ge 0\\ 0, t_{\mathrm{n}}<0\end{array}\right..$$

(6)

(3)

Damage Evolution

To capture the joint contribution of normal and tangential separations to damage evolution, Camanho [26] introduced an effective displacement defined as

$${\delta }_{\mathrm{m}}=\sqrt{{〈{\delta }_{\mathrm{n}}〉}^2+{\delta }_{\mathrm{s}}^2},$$

(7)

where ${\delta }_m$ denotes the effective separation. The corresponding equivalent nominal strain is

$${\epsilon }_{\mathrm{m}}={\displaystyle \frac{{\delta }_{\mathrm{m}}}{T_0}}=\sqrt{{〈{\epsilon }_{\mathrm{n}}〉}^2+{\epsilon }_{\mathrm{s}}^2}.$$

(8)

Define a scalar damage variable $D\in \left[\mathrm{0,1}\right]$ as a function of the effective separation ${\delta }_{\mathrm{m}}$:

$$D={\displaystyle \frac{{\delta }_{\mathrm{m}}^{\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}}({\delta }_{\mathrm{m}}^{\mathrm{m}\mathrm{a}\mathrm{x}}-{\delta }_{\mathrm{m}}^{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}})}{{\delta }_{\mathrm{m}}^{\mathrm{m}\mathrm{a}\mathrm{x}}({\delta }_{\mathrm{m}}^{\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}}-{\delta }_{\mathrm{m}}^{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}})}}.$$

(9)

where ${\delta }_{\mathrm{m}}^{\mathrm{m}\mathrm{a}\mathrm{x}}$ is the maximum effective separation attained in the loading history, ${\delta }_{\mathrm{m}}^{\mathrm{initial}}$ is the value at damage initiation, and ${\delta }_{\mathrm{m}}^{\mathrm{final}}$ is the value at complete decohesion. An energy-based linear damage evolution is adopted in this research:

$$G_{\mathrm{C}}={\int }_0^{{\delta }_{\mathrm{m}}^{\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}}}t\left(\delta \right)d\delta ={\displaystyle \frac{1}{2}}{t}^0{\delta }_{\mathrm{m}}^{\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}},$$

(10)

where $t^0$ is the nominal traction at damage initiation and $G_{\mathrm{C}}$ is the critical fracture energy. The $G_{\mathrm{C}}$ is determined by the Benzeggagh–Kenane (B–K) criterion, which is particularly suitable when the two shear directions share the same critical energy, $G_{\mathrm{s}}^{\mathrm{c}}=G_{\mathrm{t}}^{\mathrm{c}}$:

$$G^{\mathrm{C}}=G_{\mathrm{n}}^{\mathrm{C}}+\left(G_{\mathrm{s}}^{\mathrm{C}}-G_{\mathrm{n}}^{\mathrm{C}}\right){\left\{{\displaystyle \frac{G_{\mathrm{S}}}{G_{\mathrm{T}}}}\right\}}^{\eta },$$

(11)

where $G^{\mathrm{C}}$ is the mixed-mode critical fracture energy; $G_{\mathrm{S}}=G_{\mathrm{s}}+G_{\mathrm{t}}$ is the total tangential fracture energy in 3 dimension situation; $G_{\mathrm{T}}=G_{\mathrm{n}}+G_{\mathrm{S}}$ is the total fracture energy; $G_{\mathrm{n}}^{\mathrm{C}}$ is the pure-mode normal critical fracture energy under purely normal opening; $G_{\mathrm{s}}$ and $G_{\mathrm{t}}$ are the pure-mode tangential critical fracture energies under sliding in the first and second shear directions, respectively; and $\eta$ is a material parameter.

3.2.2. Parameter Selection

As no direct tests were conducted to determine the mechanical properties of the mortar, aggregates, and interfacial transition zone (ITZ), the simulation parameters were inferred from the fully graded concrete test results and prior studies, and then iteratively calibrated until the simulated stress–strain response closely matched the experiments.

For the solid elements, the mortar was assigned an elastic modulus of $1.8\times {10}^4 \mathrm{M}\mathrm{P}\mathrm{a}$, Poisson’s ratio of 0.24, and a density of $2.2\times {10}^{-9} \mathrm{t}/\mathrm{m}{\mathrm{m}}^3$. The aggregates were assigned an elastic modulus of $5\times {10}^4 \mathrm{M}\mathrm{P}\mathrm{a}$, Poisson’s ratio of 0.22, and a density of $2.5\times {10}^{-9} \mathrm{t}/\mathrm{m}{\mathrm{m}}^3$.

For the cohesive parameters, the tensile strength of mortar typically falls in the range 2~6 MPa [27]; a value of 4 MPa was adopted. The ITZ cohesive strength, stiffness, and fracture energy were each taken as one-half of the mortar values [28], giving an ITZ tensile strength of 2 MPa, which is consistent with the cracking stress inferred from the experimental stress–strain curve. Reported tensile strengths for granodiorite are 6–20 MPa; because aggregate rupture was observed under static loading in our tests, the aggregate tensile strength was set to 6 MPa. Unless otherwise specified, shear parameters were taken as three times the corresponding normal parameters.

The cohesive penalty stiffness has limited influence on the global response but must be set reasonably to avoid interpenetration and numerical ill-conditioning [29]. Typical values lie between $1\times {10}^4$ and $1\times {10}^9$ MPa/mm [21,30]. In this study, the mortar–mortar and aggregate–aggregate cohesive stiffnesses were taken equal to those of the corresponding bulk phases (consistent with $K=E/T_0$ and $T_0=1$). All parameter values are summarized in

Table 2. Material Properties for the Mesoscale Concrete Model.

Parameter	Aggregate	Mortar	ITZ
Solid Elements
$\mathrm{Density}, \rho (\mathrm{t}/{\mathrm{m}\mathrm{m}}^3)$	2.5 × 10−9	2.2 × 10−9
$\mathrm{Elastic} \mathrm{modulus}, E_0\left(\mathrm{M}\mathrm{P}\mathrm{a}\right)$	50,000	18,000
$\mathrm{Poisson}’\mathrm{s} \mathrm{ratio}, \upsilon$	0.22	0.24
Cohesive Parameters
$\mathrm{Normal} \mathrm{stiffness}, k_n(\mathrm{M}\mathrm{P}\mathrm{a}/\mathrm{m}\mathrm{m})$	50,000	18,000	12,000
$\mathrm{Shear} \mathrm{stiffness}, k_s(\mathrm{M}\mathrm{P}\mathrm{a}/\mathrm{m}\mathrm{m})$	150,000	54,000	36,000
$\mathrm{Tension} \mathrm{strength}, t_n^0\left(\mathrm{M}\mathrm{P}\mathrm{a}\right)$	6	4	2
$\mathrm{Shear} \mathrm{strength}, t_s^0\left(\mathrm{M}\mathrm{P}\mathrm{a}\right)$	18	12	6
$\mathrm{Normal} \mathrm{energy}, G_n^c(\mathrm{N}/\mathrm{m}\mathrm{m})$	0.3	0.18	0.09
$\mathrm{Shear} \mathrm{energy}, G_s^c(\mathrm{N}/\mathrm{m}\mathrm{m})$	0.9	0.54	0.27

.

4. Results and Discussion

For each of the flexural, direct-tension, and splitting-tensile test simulation, ten random aggregate models were generated for the mesoscale simulations. For the fully graded concrete, three specimens were successfully tested in flexural test, three specimens in splitting tension test, and two specimens in uniaxial tension test. The corresponding experimental results are presented together with the numerical simulation results in the subsequent analysis.

4.1. Mesh Sensitivity Analysis

When the overall dimensions of the numerical model are fixed, the mesh size determines the number of nodes and elements in the model. Moreover, since fracture in the simulation is restricted to propagate along the cohesive elements, different mesh discretization may alter the crack path. Therefore, a comparison of different mesh densities is required to identify a reasonable mesh size while also accounting for computational efficiency.

Using the splitting tensile model, analyses were performed with mesh sizes of 2 mm, 3 mm, 4 mm, 5 mm, 6 mm, and 7 mm. The resulting failure modes are shown in Figure 5, and the corresponding stress–strain curves are plotted in Figure 6. When the mesh size is smaller than 7 mm, both the primary cracking pattern and the peak stress are relatively consistent. Considering the balance between computational efficiency and model accuracy, a mesh size of 5 mm is adopted in the subsequent analyses.

4.2. Failure Modes

The mesoscale failure pattern under flexural load agreed well with the experimental observations as shown in Figure 7. The numbers in the figure are specimen identification codes used for convenient recording during the tests. The circles next to the strain gauges are numbered stickers, also used to identify the corresponding gauges. Similar numbers and circles appear in the subsequent figures and are not explained again. In all ten random aggregate models, cracking initiated within the 450 mm mid-span region; cracks propagated predominantly along the interfacial transition zones (ITZs) and traversed the mortar to link adjacent ITZs, ultimately coalescing into a through-crack across the specimen.

As shown in Figure 8a, the stress–strain curves from the mesoscale flexural test simulations agree well with the experiments. Before the peak, the experimental curves for fully graded concrete nearly coincide, and the random-aggregate models reproduce the same behavior, indicating that aggregate randomness has a negligible effect on the pre-peak response. The simulated peak stresses range from 4.60 to 5.26 MPa, with a mean of 4.95 MPa, close to the experimental mean of 4.82 MPa. Figure 8a further indicates that the influence of aggregate distribution is concentrated at the peak strength and along the post-peak softening branch.

As shown in Figure 9, the typical mesoscale failure pattern for fully graded concrete under the splitting tensile test agrees well with the experimental observations. In all ten random-aggregate realizations, a through-crack formed along the line connecting the upper and lower steel bearing strips, and localized crushing occurred in the vicinity of the strips.

As shown in Figure 8b, the stress–strain curves from the mesoscale splitting-tension model of fully graded concrete compare well with the experimental results. The elastic ascending branches nearly coincide, indicating good agreement in the initial stiffness. The simulated peak stress ranges from 3.92 to 4.68 MPa, with a mean of 4.33 MPa, which is close to the experimental mean of 4.37 MPa. These results suggest that the selected model parameters and boundary conditions are appropriate.

A mesoscale uniaxial tension simulation was performed for fully graded concrete. As shown in Figure 10, the predicted crack orientation and location agree well with the experimental observations: a through-crack formed approximately perpendicular to the loading direction and propagated across the entire specimen.

As shown in Figure 8c, the stress–strain curve from the mesoscale uniaxial-tension simulation compares well with the experimental result. The computed and experimental curves nearly coincide along the elastic ascending branch.

4.3. Damage Progression

To investigate the damage-evolution process, the scalar damage variable $D$ of the cohesive elements associated with the mortar, aggregate, and ITZ phases was recorded (with $D$ defined in Equation (8)). The total damage of a given phase is defined as the sum of the damage variables over all cohesive elements belonging to that phase:

$$D_{\mathrm{m}\mathrm{a}\mathrm{t}}^{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}=\sum _{\mathrm{i}=1}^{\mathrm{n}}{D}_{\mathrm{m}\mathrm{a}\mathrm{t}}^{\mathrm{i}},$$

(12)

where $D_{\mathrm{m}\mathrm{a}\mathrm{t}}^{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}$ denotes the damage measure of one material, $D_{\mathrm{m}\mathrm{a}\mathrm{t}}^i$ is the element-wise damage variable of the $i$-th cohesive element in that phase, and $n$ is the number of cohesive elements belonging to that phase. The damage of a specimen is obtained by summing over the three phases:

$$D^{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}=D_{\mathrm{I}\mathrm{T}\mathrm{Z}}^{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}+D_{\mathrm{m}}^{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}+D_{\mathrm{a}}^{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}},$$

(13)

where $D_{\mathrm{I}\mathrm{T}\mathrm{Z}}^{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}$, $D_{\mathrm{m}}^{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}$, and $D_{\mathrm{a}}^{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}$ denote the cumulative damage formed within the ITZ, the mortar, and the aggregate, respectively. The total damage $D^{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}$ is defined as the sum of the damage variables of all cohesive elements at the load stage where the post-peak stress has dropped to zero. The damage contribution of each phase is then obtained by dividing its cumulative damage by the total damage.

$$D_{\mathrm{I}\mathrm{T}\mathrm{Z}}={\displaystyle \frac{D_{\mathrm{I}\mathrm{T}\mathrm{Z}}^{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}}{D^{\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}}}},$$

(14)

$$D_{\mathrm{m}}={\displaystyle \frac{D_{\mathrm{m}}^{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}}{D^{\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}}}},$$

(15)

$$D_{\mathrm{a}}={\displaystyle \frac{D_{\mathrm{a}}^{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}}{D^{\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}}}}.$$

(16)

In the three expressions above, $D_{\mathrm{ITZ}}$, $D_{\mathrm{m}}$, and $D_{\mathrm{a}}$ denote the damage variables associated with the ITZ, mortar, and aggregate phases, respectively. At specimen failure, the sum of these three phase-wise damage measures equals one. Since the mesh size in the model is already sufficiently small and the lengths of all cohesive elements are similar, the influence of element length has not been taken into account in the above statistical evaluation of the damage level.

By compiling the damage status of all cohesive elements in flexural simulation of the fully graded concrete model, the damage evolution is shown in Figure 11. The process is divided into four stages, labeled A–D. Stage A: virtually no elements are damaged and the specimen is fully elastic; the stress is about 40% of the peak, consistent with the code definition of elastic modulus (evaluated from 0.05 MPa to 40% of the peak stress). Stage B: damage initiates in the ITZ cohesive elements, and the overall modulus begins to decrease, that is, softening starts. Stage C: damage emerges in the mortar cohesive elements, and the softening rate accelerates. Stage D: the accumulation of ITZ and mortar damage slows, some cohesive elements reach complete damage (i.e., the damage variable attains unity) and are removed, macroscopic cracking develops, propagates through the specimen, and ultimately leads to complete failure.

By compiling the damage status of all cohesive elements during splitting-tension loading of the fully graded dam concrete, the damage evolution is shown in Figure 12. In Figure 12, the red lines in the meso-scale concrete model indicate elements that have entered the damage or failure state. The damage process is divided into four stages labeled A–D: Stage A: The specimen responds elastically with virtually no damaged elements. Stage B: Damage initiates successively in the ITZ, mortar, and aggregate; macroscopic response becomes nonlinear. Stage C: A macrocrack forms near the specimen mid-section and extends along the ITZ toward the top and bottom. Because cracking concentrates in the ITZ and mortar, stresses in the aggregates are relieved and aggregate damage ceases to accumulate. Stage D: The specimen exhibits brittle failure. As illustrated by the stress–strain history in Figure 12, the stress drops rapidly to zero; the subsequent “strain” mainly reflects the relative displacement of the two separated fragments, and the stress–strain curve no longer represents a load-bearing state—the specimen has completely lost its capacity.

By compiling the damage states of all cohesive elements during uniaxial-tension loading of the dam fully graded concrete, the resulting damage evolution is shown in Figure 13. As seen in Figure 13, damage occurs predominantly in the ITZ, consistent with the experimental observations: cracks develop mainly along the ITZ and ultimately penetrate the mortar.

4.4. Parameter Sensitivity Analysis

To assess how cohesive parameters affect the mechanical response of dam concrete, a one-at-a-time sensitivity analysis was conducted. In each run, only a single parameter was varied while all others were held fixed. For the mortar, ITZ, and aggregate cohesive interfaces, the parameters examined were: the normal traction at damage initiation $T_n^0$, the shear traction at damage initiation $T_s^0$, the normal critical fracture energy $G_n^C$, and the shear critical fracture energy $G_s^C$.

4.4.1. Sensitivity Analysis of Mortar Cohesive-Element Parameters

The parameter settings for the mortar cohesive elements are listed in

Table 3. Parameter Values for Mortar Cohesive Elements.

Parameters	Case 1	Case 2	Case 3	Case 4	Case 5
$T_n^0$ (MPa)	1	2	4	6	8
$T_s^0$ (MPa)	6	9	12	18	24
$G_n^C$ (N/mm)	0.12	0.15	0.18	0.21	0.24
$G_s^C$ (N/mm)	0.26	0.39	0.52	0.65	0.78

. Each parameter was assigned five cases; case 3 served as the baseline, and a one-factor-at-a-time comparison was performed by varying a single parameter to another level while keeping all others at their baseline.

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Normal Traction at Damage Initiation $T_n^0$ of mortar cohesive elements

Figure 14 presents the stress–strain curves for different values of the mortar cohesive normal strength $T_n^0$. Increasing $T_n^0$ leads to a pronounced rise in the specimen’s peak stress.

For flexural test simulation, when the mortar $T_n^0$ is set below the ITZ value, the response changes markedly: the specimen first reaches the lower mortar strength, causing a load drop, and then exhibits a secondary rise governed by the higher ITZ strength. In practice, however, the mortar $T_n^0$ is not lower than that of the ITZ; the anomalous curve is therefore an artifact of a nonphysical parameter ordering. Excluding the pathological case $T_n^0=1$, variations in $T_n^0$ do not change the qualitative shape of the flexural stress–strain curve; they chiefly elevate the peak strength and correspondingly raise the descending branch.

For the splitting-tension test simulation, increasing $T_n^0$ while holding $G_n^c$ constant reduces the failure separation ${\delta }_m^{\mathrm{final}}$, thereby making the response more brittle: the post-cracking plateau shortens. The far post-peak tail, however, tends to collapse onto a common trajectory; thus, $T_n^0$ chiefly influences the curve in the vicinity of the peak.

For uniaxial tension test simulation, varying the mortar $T_n^0$ has little influence on the crack-initiation stress, which is governed primarily by the ITZ $T_n^0$. However, the subsequent nonlinear hardening branch becomes noticeably longer as the mortar $T_n^0$ increases. This is because cracking initiates in the ITZ and then propagates into the mortar; a higher mortar $T_n^0$ requires a larger effective initiation separation ${\delta }_m^{\mathrm{initial}}$ for the crack to transfer from the ITZ to the mortar, thereby prolonging the hardening stage. Similarly to the splitting-tension test simulation, the post-peak response becomes more brittle as $T_n^0$ increases.

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Shear Traction at Damage Initiation $T_s^0$ of mortar cohesive elements

Figure 15 presents the stress–strain curves for different values of the mortar cohesive shear strength $T_s^0$.

For flexural test simulation, the effect of $T_s^0$ on flexural strength is non-monotonic. Although the mid-span is nominally a pure-bending zone, material heterogeneity induces nonuniform deformation and attendant shear stresses; thus $T_s^0$ influences the peak strength through the mixed-mode damage-initiation criterion. When $T_s^0=6 \mathrm{M}$Pa, shear damage is triggered prematurely, producing a lower peak. At $T_s^0=9 \mathrm{M}$Pa, the ratio of shear traction to its strength is low enough to suppress early shear slip, allowing fuller pre-peak development and more coordinated damage evolution, yielding the highest peak. For $T_s^0=12$ and 18 MPa, failure is governed mainly by normal separation and the nominal peak declines. At $T_s^0=24 \mathrm{M}$Pa, the response is almost entirely controlled by normal separation; however, the excessively high shear strength causes local strengthening, and the peak rises again.

For the splitting-tension test simulation, the measured strength shows a modest dependence on the cohesive shear strength $T_s^0$. Because loading is applied through finite-width steel bearing strips rather than an idealized point/line load, shear components are introduced: at low $T_s^0$, failure may initiate in shear beneath the loading strips; as $T_s^0$ increases, the failure mode shifts to tensile splitting through the specimen mid-section, so the incremental strength gain with further increases in $T_s^0$ diminishes.

For uniaxial tension test simulation, the stress state is essentially Mode I; the response is governed by the normal parameters, and variations in $T_s^0$ have negligible influence.

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Normal fracture energy $G_n^C$ of mortar cohesive elements

Figure 16 presents the stress–strain curves for different values of the normal fracture energy $G_n^C$ of the mortar cohesive elements. The flexural, splitting-tensile, and uniaxial tensile strengths are essentially insensitive to $G_n^C$. However, in flexural and uniaxial tension test simulation the post-peak softening branch becomes progressively more gradual as $G_n^C$ increases. This trend accords with the role of $G_n^C$ in the cohesive law: with other parameters fixed, a larger $G_n^C$ increases the displacement (and energy) required for complete decohesion, leaving the pre-peak response nearly unchanged while delaying crack growth after the peak.

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Shear fracture energy $G_s^C$ of mortar cohesive elements

Figure 17 presents the stress–strain curves for different values of the shear fracture energy $G_s^C$ of the mortar cohesive elements. The uniaxial-tension response is virtually insensitive to $G_s^C$, for the same reason as with $T_s^0$: axial loading is essentially Mode I and introduces negligible shear.

By contrast, in flexural test simulation and splitting tension test simulation, where shear components are present markedly affects the post-peak softening, with a stronger sensitivity in the splitting test. In the splitting tension test simulation, the descending branch becomes progressively more gradual as $G_s^C$ increases, indicating that post-peak failure is influenced by the mortar’s shear resistance. Because crack growth from the ITZ into the mortar involves shear, larger $G_s^C$ increases the tangential failure displacement (${\delta }_s^{\mathrm{final}}$) and, macroscopically, the specimen must dissipate more energy before reaching final failure under splitting.

4.4.2. Sensitivity Analysis of ITZ Cohesive-Element Parameters

The parameter settings for the ITZ cohesive elements are listed in

Table 4. Parameter Values for ITZ Cohesive Elements.

Parameters	Case 1	Case 2	Case 3	Case 4	Case 5
$T_n^0$ (MPa)	1	1.5	2	2.5	3
$T_s^0$ (MPa)	3	4.5	6	7.5	9
$G_n^C$ (N/mm)	0.03	0.045	0.06	0.075	0.09
$G_s^C$ (N/mm)	0.1	0.15	0.2	0.25	0.3

. Each parameter was assigned five cases; case 3 served as the baseline, and a one-factor-at-a-time comparison was carried out by varying a single parameter to another level while keeping all others at their baseline.

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Normal Traction at Damage Initiation $T_n^0$ of ITZ cohesive elements

Figure 18 presents the stress–strain curves for different ITZ cohesive normal strengths $T_n^0$. The three loading configurations exhibit similar sensitivity to $T_n^0$. As $T_n^0$ increases, both the cracking stress and the peak strength rise most markedly under uniaxial tension, followed by flexural tension, and least under splitting tension. In all cases, the crack-initiation stress increases with $T_n^0$; under uniaxial tension it is approximately equal to $T_n^0$, indicating that the ITZ normal strength governs the overall tensile performance primarily through its control of crack initiation.

As the weakest link in concrete, the ITZ controls the onset of tensile failure. Increasing the ITZ normal cohesive strength ($T_n^0$) delays macrocrack formation and strain localization within the ITZ. When $T_n^0$ is low, premature macrocracking induces stress concentrations that rapidly drive cracks through the mortar between aggregates, thereby reducing the overall tensile strength. If only $T_n^0$ is varied while the normal critical fracture energy ($G_n^C$) is held fixed, the normal failure separation decreases; cracks then extend at smaller displacements and the response becomes more brittle.

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Shear Traction at Damage Initiation $T_s^0$ of ITZ cohesive elements

Figure 19 shows the stress–strain curves for different ITZ shear strengths $T_s^0$. The sensitivity is strongest in the splitting-tension test simulation, weaker in flexural test simulation, and negligible in uniaxial tension test simulation. Under splitting loads, increasing $T_s^0$ raises the splitting tensile strength; the small plateau near the peak shortens and the post-peak drop steepens (i.e., the response becomes more brittle when other parameters, including fracture energy, are fixed). In flexural test simulation, a higher $T_s^0$ suppresses early shear slip and slightly elevates the peak, but the overall curve shape remains governed mainly by normal opening; the marginal gain diminishes at larger $T_s^0$. In uniaxial tension, the response is essentially Mode I, so varying $T_s^0$ has almost no effect.

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Normal fracture energy $G_n^C$ of ITZ cohesive elements

Figure 20 presents the stress–strain curves for different ITZ normal fracture energies $G_n^C$. The splitting-tension response is virtually insensitive to changes in $G_n^C$. By contrast, in flexural test simulation the post-cracking trajectory is strongly affected: flexural failure typically involves crack development within the ITZ up to near mid-depth before reaching capacity; comparison with the mortar results indicates that the evolution from crack initiation to the peak is governed primarily by ITZ cracking, after which the macrocrack formed in the ITZ extends into—and eventually cuts through—the mortar.

Under uniaxial tension load, the response shows higher sensitivity to the ITZ $G_n^C$: increasing $G_n^C$ elevates the tensile strength and the peak strain, whereas the ultimate failure strain is essentially unchanged. This indicates that $G_n^C$ chiefly controls the resistance to crack propagation along the ITZ (i.e., the crack-advance rate), thereby improving tensile performance without altering the terminal separation threshold set by other factors.

(4)

Shear fracture energy $G_s^C$ of ITZ cohesive elements

Figure 21 presents the stress–strain curves for different ITZ tangential (shear) critical fracture energies $G_s^C$. Under flexural load, the mechanical response is virtually insensitive to $G_s^C$. In uniaxial tension test simulation, however, the post-peak softening branch becomes progressively more gradual as $G_s^C$ increases. A plausible explanation is that, when a crack must bypass aggregates, shear deformation is required to continue propagation if the ITZ orientation is oblique to the loading direction; increasing $G_s^C$ raises the tangential failure displacement, thereby increasing the ultimate displacement of the specimen. Under splitting-tension loading, a higher $G_s^C$ also leads to an increase in the splitting tensile strength.

5. Conclusions

This study investigated the mesoscale mechanisms governing the tensile failure of dam fully graded concrete, focusing on parameter sensitivity, failure modes, and damage evolution. The main findings are:

• A mesoscale concrete model with globally inserted cohesive interfaces reproduces the tensile failure process of dam concrete with good fidelity. The simulated stress–strain curves agree well with the tests and capture the cracking process.
• The one-at-a-time sensitivity analysis clarifies how the three phases (mortar, aggregates, ITZ) affect the macroscopic response. The mortar tensile strength primarily controls the peak strength; increasing it raises the peak but enhances brittleness. The ITZ properties govern crack nucleation and propagation: the stress at crack initiation is generally assumed to be approximately equal to the tensile strength of the ITZ, the higher the shear strength of the ITZ, the higher the splitting tensile strength of the concrete, and the fracture energies mainly delay post-peak softening.
• A quantitative analysis of tensile damage evolution reveals three distinct stages: (i) elastic deformation; (ii) crack initiation and propagation along the interfacial transition zone (ITZ); and (iii) crack penetration through the mortar matrix. The results indicate that more than 80% of the tensile damage is concentrated in the ITZ. Cracks preferentially extend along the ITZ network and eventually cut through the mortar, while the spatial arrangement of aggregates redirects ITZ paths and thus governs the overall crack trajectory.
• From a mesoscale viewpoint, the flexural and uniaxial-tension capacities are governed mainly by the tensile strengths and normal fracture energies of the mortar and ITZ, with limited sensitivity to shear parameters. In contrast, the splitting tensile capacity increases with the tensile strength, shear strength, and tangential fracture energy of the mortar and the ITZ and is nearly insensitive to the normal fracture energy.