Preamble

Simulation Study on the Eccentric Compression Performance of Reinforced Concrete Columns Based on Multi-scale Homogenization

College of Civil Engineering, Huaqiao University; Fujian Key Laboratory of Smart Infrastructure and Monitoring (Huaqiao University)

Abstract

Based on equivalent homogenization theory, this study investigates the influence of mesoscopic factors of concrete aggregates on the mechanical behavior of reinforced concrete (RC) columns under eccentric compression. This is achieved through the development of both mesoscopic numerical models and mesoscopic homogenization numerical models for multi-scale simulation. The results indicate that the simulation outcomes are in good agreement with reference experimental data, thereby validating the effectiveness of the mesoscopic homogenization model at the structural component level. Both the mesoscopic numerical model and the mesoscopic homogenization model effectively reflect the impact of random aggregate distribution on the mechanical performance and damage distribution of RC columns under eccentric compression. Compared to the direct mesoscopic numerical model, the mesoscopic homogenization model exhibits lower sensitivity to variations in aggregate shape. Furthermore, the mesoscopic homogenization model significantly reduces computational memory requirements and improves overall computational efficiency.

1. Introduction

[Text omitted in source]

2. Methodology and Modeling

The mechanical behavior of reinforced concrete is fundamentally influenced by its internal mesostructure. In this study, we employ a multi-scale approach to bridge the gap between material-level heterogeneity and component-level structural response.

2.1 Mesoscopic Numerical Model

The mesoscopic numerical model explicitly accounts for the three-phase composition of concrete: the coarse aggregates, the cement mortar matrix, and the Interfacial Transition Zone (ITZ). Aggregates are modeled with stochastic distributions to capture the inherent variability of the material.

2.2 Mesoscopic Homogenization Model

To address the high computational cost associated with fully resolved mesoscopic models, an equivalent homogenization theory is applied. This approach derives effective constitutive relationships by analyzing representative volume elements (RVEs), allowing for a more efficient simulation of large-scale RC columns while retaining the influence of mesoscopic characteristics.

[FIGURE:1]

3. Results and Discussion

3.1 Validation against Experimental Data

The proposed models were validated by comparing the simulated load-displacement curves and failure modes with existing experimental results for RC columns under eccentric compression. The high degree of correlation confirms that the homogenization approach accurately captures the macroscopic response of the components.

3.2 Influence of Aggregate Distribution and Shape

The simulations demonstrate that the random distribution of aggregates leads to localized stress concentrations and governs the initiation of damage. While the mesoscopic numerical model is highly sensitive to the specific geometry of the aggregates (e.g., angular vs. spherical), the mesoscopic homogenization model provides a more generalized response. Although slightly less sensitive to exact aggregate shapes, the homogenization model remains robust in predicting the overall capacity and damage evolution of the column.

[TABLE:1]

3.3 Computational Efficiency

One of the primary advantages of the mesoscopic homogenization model is its efficiency. By simplifying the internal geometry into equivalent homogeneous regions, the model significantly reduces the number of finite elements and degrees of freedom required for the simulation. This leads to a substantial reduction in CPU time and memory usage compared to the fully resolved mesoscopic model, making it more suitable for engineering applications involving complex structural members.

4. Conclusion

This research demonstrates the feasibility of using multi-scale homogenization to study the eccentric compression performance of RC columns. The key findings are summarized as follows:
- The mesoscopic homogenization model is validated at the component level, showing excellent agreement with experimental benchmarks.
- Both modeling approaches successfully capture the stochastic nature of concrete damage induced by aggregate distribution.
- The homogenization method offers a superior balance between physical accuracy and computational economy, providing a practical tool for the large-scale simulation of heterogeneous civil engineering structures.

关键词

Reinforced concrete columns under eccentric compression; Mesoscopic numerical model; Mesoscopic homogenization numerical model; Mesoscopic factors of aggregate
CLC number: TU375.3 Document code: A

Article ID: Influence of aggregate parameters on mechanical properties of reinforced concrete columns under eccentric compression
LI Weikang, WANG Jiang, XU Bin

361021 Xiamen

China

2. Key Laboratory for

Intelligent Infrastructure and Monitoring of Fujian Province Huaqiao University

361021 Xiamen

China

Abstract

The influence of meso-factors of concrete aggregate on the mechanical behavior of reinforced concrete columns under eccentric compression is studied by establishing a meso-numerical model and a meso-homogenization numerical model which is based on the equivalent homogenization theory. The results show that the simulation results are in good agreement with the reference test results which verifies the ef- fectiveness of the mesoscopic homogenization numerical model at the component level. Both mesoscopic numerical model and mesoscopic homogenization numerical model can effectively reflect the influence of random distribution of aggregate on mechanical properties and damage distribution of reinforced concrete under eccentric compression. Compared with the mesoscopic numerical model the mesoscopic homogeniza- tion numerical model is less sensitive to different shapes of aggregates. The meso-homogeneous numerical model can save computational space and improve computational efficiency.

Key words eccentrically compressed reinforced concrete column meso-numerical model meso-homoge- nization numerical model meso-factor of aggregate

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Concrete is a multiphase composite material composed of aggregates, mortar, micro-cracks, and air bubbles. The random distribution and varying shapes of these aggregates are the primary causes of the stochastic nature and heterogeneity of concrete. Furthermore, the composition at the mesoscopic scale significantly influences the mechanical behavior of reinforced concrete (RC) members. Therefore, investigating the impact of mesoscopic factors of concrete aggregates on the mechanical performance of RC components is of great scientific importance.

In recent years, researchers both domestically and internationally have established various mesoscopic numerical models to study the mechanical behavior of concrete components from a mesoscale perspective. These include lattice models, random particle models, stochastic mechanical property models, and random aggregate models. Gao et al. \cite{} proposed an allocation algorithm for two-dimensional polygonal and convex polyhedral random aggregates in concrete. Liu et al. \cite{} utilized a random aggregate model to simulate the process from initial damage to fracture failure in single-sided notched tensile specimens. Du et al. \cite{} developed an aggregate placement algorithm tailored to the aggregate grading and volume fractions found in practical engineering, subsequently conducting research on the compressive failure of three-dimensional numerical specimens of concrete with random aggregates.

To deeply investigate the influence of mesoscopic components on mechanical behavior at both the material and component levels, many scholars have employed mesoscopic numerical simulations to study the macroscopic mechanical properties of concrete under uniaxial tension, compression, shear, and bending, as well as the behavior of RC columns under axial and eccentric compression. Furthermore, numerical simulations of RC bridge columns with non-contact lap splices have been conducted based on mesoscopic models. However, mesoscopic numerical models for concrete require fine meshes to partition the interface layers of different materials to satisfy deformation compatibility requirements, which leads to high computational costs and low efficiency. To reflect the heterogeneity of concrete while improving computational efficiency, Ma et al. \cite{} proposed a parallel multi-scale modeling method for bridge structures based on homogenization principles. Jin \cite{} simulated the axial compressive behavior of rectangular concrete and RC columns based on a mesoscopic equivalent model for concrete.

Based on the mesoscopic homogenization model, simulations were conducted to study the effects of random distribution and different shapes of aggregates on the macroscopic mechanical properties of concrete under uniaxial tension. The aforementioned mesoscopic homogenization models for concrete are generally based on elastic damage constitutive laws and do not account for factors influencing the elastoplastic stage of concrete. In this study, a concrete homogenization mechanical model is proposed based on the $CDP$ model and composite micromechanics. Considering the random distribution and diverse shapes of aggregates, mesoscopic models and corresponding mesoscopic homogenization models for RC columns—incorporating randomly distributed circular, elliptical, and polygonal aggregates—were established. These models efficiently simulate the mechanical behavior and damage distribution of RC columns under eccentric compression. The results were compared with existing experimental data \cite{}, verifying the feasibility and effectiveness of the mesoscopic homogenization model. This work provides an important methodology and simulation foundation for the multi-scale modeling and efficient mechanical analysis of reinforced concrete columns.

1 均匀化理论及本构关系

Aggregates, Mortar, and Reinforcement

Aggregates are typically described using a linear elastic constitutive model to characterize their mechanical performance, where the stress-strain relationship is represented by a straight line. The slope of this line represents the elastic modulus. The parameters $f_{c.ag.max}$ and $f_{t.ag.max}$ denote the peak compressive and tensile strengths of the aggregate, respectively. In contrast, mortar contains defects such as micro-cracks and air bubbles, which lead to strain softening during the loading process. Consequently, a plastic damage constitutive model is employed to describe the mechanical properties of the mortar.

In this model, $E_{mo}$ represents the elastic modulus of the mortar, while $E_{c.mo.u}$ denotes the ultimate compressive elastic modulus. The peak compressive and tensile points of the mortar are defined by $f_{c.mo.max}$ and $f_{t.mo.max}$, respectively. The plastic damage factors are calculated based on Sidoroff’s energy equivalence principle. The mortar's plastic damage is represented by the compressive damage factor $d_c$ and the tensile damage factor $d_t$, which indicate the degree of degradation under compression and tension, respectively. These factors typically range from 0 to 1. A value of $d = 0$ indicates that the mortar is undamaged, while $d = 1$ signifies that the mortar has completely failed due to damage.

The expression for the compressive plastic damage factor is as follows:

$$\begin{aligned} d_c = 1 - \frac{f_c^*}{E_0 \epsilon_c [\alpha_a + (3 - 2\alpha_a)x + (\alpha_a - 2)x^2]} \end{aligned} \tag{1}$$
When $x > 1$:

The expression for the tensile plastic damage factor is as follows:

$d_t = 0$ (3)
When $x > 1$:

In these expressions, $E_0$ is the initial elastic modulus of the mortar; $f_c^*$ is the compressive strength of the mortar; $\alpha_a$ and $\alpha_d$ are the parameters for the rising and falling segments of the mortar's compressive plastic damage constitutive curve, respectively; and $\alpha_t$ is the parameter for the falling segment of the tensile plastic damage constitutive curve.

Based on existing experimental data \cite{1}, the material parameters for the aggregate and mortar were obtained through mesoscopic simulation inversion, as shown in [TABLE:1], which details the mechanical parameters of the concrete mesoscopic components.

d t = 1 -

[TABLE:1]

Table 1: Mechanical parameters of concrete micro-components

Mechanical Parameters Aggregate Mortar Mesoscopic Simulation Results Experimental Results Compressive strength $\sigma_c$ / MPa - 62 65 67.5 Tensile strength $\sigma_t$ / MPa - 5.5 - - Elastic modulus $E$ / GPa 56 26 38 38.5 Poisson's ratio $\nu$ 0.2 0.2 0.2 -

The mechanical properties of the reinforcement are described using an ideal elastoplastic constitutive model. As a classic model for characterizing the mechanical behavior of steel reinforcement, the stress-strain curve initially appears as an inclined straight line, the slope of which represents the elastic modulus of the steel. Upon reaching the compressive or tensile yield point, the curve transitions into a horizontal line, as illustrated in [FIGURE:1].

3 所示，钢筋材料参数见表

[TABLE:2] Mechanical property parameters of reinforcement

The connection between the reinforcement and the concrete is modeled using the Spring2 nonlinear spring element, which simulates the bond-slip effect between the two materials. The bond-slip constitutive relationship is referenced from the Standard for Design of Concrete Structures (GB/T 50010). The bond stress-slip constitutive curve between the reinforcement and concrete is primarily determined by key characteristic points, including the splitting point, the peak point, and the residual point. These characteristic points are derived from the relative slip curve. [TABLE:3] presents the characteristic values of the bond-slip constitutive model, where $f_t$ represents the tensile strength of the concrete and $d$ denotes the diameter of the reinforcement.

Constitutive Relationship of the Homogenized Element

The constitutive model for the meso-scale homogenized element is based on homogenization theory and the parallel model of composite materials. By combining semi-mechanical empirical methods from the mechanics of materials with plastic damage constitutive laws, the model uses the area fraction of the aggregate as the independent variable. This approach sequentially equivalentizes the elastic modulus and strength of the aggregate and mortar within the homogenized grid. Specifically, the first step is based on the parallel model of composite materials to...

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effective elastic modulus; the second step is based on semi-mechanical empirical methods from the mechanics of materials to determine the equivalent tensile and compressive strengths; the third step involves deriving the corresponding plastic damage constitutive model based on the equivalent elastic modulus and peak strength.

The first step of elastic modulus equivalence is based on the Voigt hypothesis, which assumes that the aggregate and mortar undergo the same displacement during the loading process, as shown in [FIGURE:1]. Through Eq. (5), the equivalent elastic modulus can be obtained.

S ag + S mo = 1 （ 5 ）

$E'$ and $E$ represent the elastic modulus of the aggregate and mortar, respectively, along with their corresponding area fractions within the equivalent grid element. The equivalence of peak strength is based on a semi-mechanical empirical method from material mechanics, which posits that the strength of concrete composites during the elastic phase can be categorized into two distinct scenarios.

The first scenario occurs when the aggregate content is relatively high. In this case, the ultimate strain of the bulk concrete is close to the failure strain of the aggregate ($\epsilon_{ag.max}$). Consequently, the strength of the concrete is primarily governed by the aggregate. This phase is referred to as the aggregate-controlled strength stage. The expressions for the peak tensile and compressive stresses of the concrete during this stage are given in Eq. (6).

The second scenario occurs when the aggregate content is low. In this instance, the ultimate strain of the bulk concrete is significantly greater than the failure strain of the aggregate ($\epsilon_{ag.max}$). By the time the bulk concrete reaches its ultimate strain, the sparse aggregate has already failed, and the concrete strength is entirely controlled by the mortar matrix. This phase is termed the mortar-controlled strength stage. The expressions for the peak tensile and compressive stresses of the concrete in this stage are provided in Eq. (7).

V ag = S ag S ag + S mo （ 7 ）

$\eta_{ag.max}$ and $\eta_{mo.max}$ represent the proportions occupied by aggregate and mortar within the equivalent grid, respectively. These values determine the peak tensile and compressive stresses of the material element grid during the elastic phase. Taking the equivalent peak tensile stress of concrete as an example, $\sigma_{mo.m}$ represents the tensile or compressive stress corresponding to the mortar when the aggregate reaches its peak tensile or compressive strain. [FIGURE:1] illustrates the stress-strain relationship curves for the aggregate and mortar under tension. It can be observed that the peak tensile strain of the aggregate, $\epsilon_{ag.max}$, is significantly smaller than that of the mortar, $\epsilon_{mo.max}$. Based on the aforementioned formulas, by using the aggregate proportion as the abscissa and the overall peak tensile stress of the concrete as the ordinate, the two cases can be plotted as two intersecting straight lines, as shown in [FIGURE:2]. The concrete strength curve during the aggregate-controlled strength phase is a rising straight line with a positive slope, while the curve during the mortar-controlled strength phase is a falling straight line with a negative slope. Taking the red-marked portion as the effective value, the peak stress of the equivalent grid first decreases and then increases, essentially exhibiting a "V-shaped" development trend.

It is worth mentioning that the ascending branch of the stress-strain relationship curve for concrete under tension is a straight line, which is fully applicable to the aforementioned peak tensile stress equivalence of the homogenized concrete grid. However, the ascending branch of the stress-strain relationship for concrete under compression primarily consists of an elastic phase and a plastic phase. The method described above can only determine the peak compressive stress of the concrete within the elastic phase. To obtain the ultimate peak compressive stress of the homogenized concrete grid, the peak compressive stress from the elastic phase must be multiplied by an expansion coefficient.

K i = α a + 3 - 2 α ( ) a 2 + α a - ( ) 2 3

In the stress-strain relationship curve for concrete under compression, the parameter for the ascending branch represents the boundary point between the elastic and plastic phases; this characterizes the compressive behavior of concrete during the elastic stage.

The term denotes the peak stress, while represents the ultimate peak compressive stress of the concrete.

2 细观及细观均匀化模型的建立

Establishment of the Mesoscopic Model

Establishment of the Random Aggregate Model

Based on the reference experiments, the dimensions of the eccentric column specimens are $960 \text{ mm} \times 150 \text{ mm}$. The total volume fraction of concrete aggregate is determined accordingly. By applying the Fuller distribution and the Raven transformation, the grading parameters for the two-dimensional random aggregate model were obtained, as shown in [TABLE:4]. Treating concrete as a two-phase composite material consisting of aggregates and mortar, a Monte-Carlo method was implemented via a Matlab program to randomly distribute circular, elliptical, and polygonal aggregates. A total of several two-dimensional geometric models of the eccentric columns were generated, as illustrated in [FIGURE:N].

[TABLE:4] Gradation of the random aggregate model
Note: For elliptical and polygonal aggregates, the major axis is defined as the particle diameter.

Finite Element Settings for the Mesoscopic Model

The mesoscopic model of the concrete was developed using a "topology partitioning method" to separately create aggregate and mortar regions and assign their respective material properties. It is assumed that the bond between the aggregate and mortar remains intact throughout the loading process. Consequently, the interaction between the aggregate and mortar regions is defined using a tie constraint, with the mortar boundary designated as the master surface and the aggregate boundary as the slave surface. To improve computational convergence, reduce the number of iterations, and control mesh quality, the aggregate and mortar meshes were configured as triangular and predominantly quadrilateral, respectively. Specifically, three-node plane stress triangular elements and CPS4R four-node bilinear plane stress quadrilateral elements were employed. To prevent stress concentration during the simulated loading of the eccentric column, rigid loading plates were placed at both ends of the model and tied to the concrete. Reference points were established on both the loading and constraint sides of the eccentric column model, with "point-surface" coupling constraints applied between these reference points and the surfaces of the rigid plates.

Establishment of the Mesoscopic Homogenization Model

Taking the $960 \text{ mm} \times 150 \text{ mm}$ concrete eccentric column with a circular random aggregate model as an example, and considering the maximum aggregate size and specimen dimensions, the random aggregate model was partitioned into several homogenization units using a $20 \text{ mm} \times 15 \text{ mm}$ quadrilateral grid, as shown in [FIGURE:N]. Using Matlab and Python scripts, the constitutive relationships of the aggregate and mortar within each homogenization unit were sequentially equivalentized. These parameters were then imported into ABAQUS to establish the mesoscopic equivalent homogenization model of the reinforced concrete eccentric column. The elastic modulus and the tensile and compressive strengths of each homogenization unit are presented in [FIGURE:N]. As seen in the figure, the elastic modulus, compressive strength, and tensile strength of the homogenization units fundamentally follow a normal distribution. This distribution reflects the randomness of the aggregate placement and validates the rationality of the chosen homogenization unit size.

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Finite Element Modeling of Reinforcement

The reinforcement mesh consists of two-node 2D truss elements, with a controlled mesh size of $10 \text{ mm}$. The material parameters for the reinforcement are detailed in [TABLE:1]. The dimensions and reinforcement configurations of the column components are illustrated in [FIGURE:1]. The stirrup spacing is set at 150 mm, with reinforcement densification at both ends to prevent local failure during the loading process. Spring2 elements are employed to connect the reinforcement and concrete nodes, thereby simulating the bond-slip interaction between the two materials.

Consistent with the meso-scale model, rigid pads are positioned on both the loading and constrained sides of the eccentric column. These pads serve to prevent stress concentration in the concrete during loading, which could otherwise compromise the numerical simulation of the column's mechanical behavior. A "tie" constraint is applied between the concrete and the pads, while a "point-to-point" constraint is used between the reinforcement and the pads. Reference points are established at the eccentric positions on the upper and lower sides of the pads, with "point-to-surface" coupling constraints applied between these reference points and the pad surfaces. A vertical eccentric displacement load is applied to the upper reference point, with an eccentricity ratio of $e = 0.1 \times h$, where $h$ represents the section depth of 150 mm. At the lower reference point's eccentric position, displacements in all directions are constrained, while the rotational degrees of freedom remain free.

3 偏心柱数值分析

Validation of Meso-scale and Meso-homogenization Models: Stress Field Analysis

[FIGURE:1] illustrates the stress distribution of the meso-scale and meso-homogenization models for a circular aggregate eccentric column. From left to right, the figures represent the elastic-plastic stage, the peak stage, the descending stage, the residual stage, and the reinforcement stress distribution at the point of failure. During the elastic-plastic stage, a large compressive stress zone is observed on the eccentrically loaded side of the column, accompanied by a few red regions indicating high stress values. The primary transmission path for these high stresses is characterized by "aggregate-to-aggregate" interaction.

As the column enters the peak stage, the red regions representing high stress values on the eccentrically loaded side increase significantly. In the meso-scale model, the maximum stress value within the aggregate regions is approximately...

7 MPa

The maximum stress value in the mortar region is approximately $62 \text{ MPa}$.

62 MPa

During the post-peak stage, the load-bearing capacity of the mortar elements decreases due to failure following damage, leading to a reduction in the high-stress red regions on the eccentric loading side. In the residual stage of the eccentrically loaded column, the high-stress red regions disappear entirely. At this point, the column has undergone failure, and the stress transmission mechanism is completely altered, with new stress paths forming in previously undamaged areas. The reinforcement stress distribution indicates that the longitudinal bars on both sides are in a state of compression; the longitudinal bars on the eccentric side reach compressive yield first, while the stirrups exhibit an outward buckling tendency. The stress distribution patterns and evolution processes of the meso-scale model and the meso-scale homogenization model are fundamentally consistent. This demonstrates that the meso-scale homogenization model can effectively reflect the stress distribution process, thereby validating the feasibility of the equivalent homogenization approach at the structural component level.

[FIGURE:N] compares the compressive damage distribution of the meso-scale and corresponding homogenized models for circular aggregate eccentric columns with experimental failure results. Compressive damage primarily occurs in the mortar regions above and below the aggregates. The final failure mode is characterized by localized compressive crushing on the eccentric side of the column mid-section. The simulated damage extent and morphology are in good agreement with the reference experiments. A comparison between the meso-scale and meso-scale homogenization results reveals that damage in the meso-scale model primarily initiates around the aggregates and covers a relatively wide area, concentrated at the mid-height of the eccentric side. The damage distribution in the meso-scale homogenization model is essentially the same as that in the meso-scale model, effectively reflecting the damage distribution and failure modes of eccentrically loaded concrete columns. For the circular aggregate concrete column, the peak load of the meso-scale model is $1021.3 \text{ kN}$ with a corresponding mid-height deflection of $2.77 \text{ mm}$. The peak load of the meso-scale homogenization model is

$1023.4 \text{ kN}$, with a corresponding mid-height deflection of $l_1 = 2.83 \text{ mm}$. The peak load from the reference experiment is $F = 1007.0 \text{ kN}$, with a corresponding mid-height deflection of $l = 2.71 \text{ mm}$.

The discrepancy between these results does not exceed $5\%$, indicating that the simulation results are in good agreement with the experimental data. The load-deflection curves for the meso-scale and meso-scale homogenization models nearly coincide during the elastic stage. In the elasto-plastic stage, the degradation rate of the elastic modulus in the meso-scale homogenization model is slower, which is attributed to the regular rectangular mesh partitioning that results in a more uniform stress distribution. Consequently, the load-bearing capacity of the meso-scale homogenization model is slightly higher than that of the meso-scale model. In the descending branch, the load-bearing capacity of the meso-scale homogenization model drops more rapidly than that of the meso-scale model. This is because the homogenized mesh uses a plastic damage constitutive law to describe all material properties, lacking the purely elastic aggregate components, which leads to a more concentrated damage distribution compared to the meso-scale model.

The comparative analysis of the mid-height deflection curves for the meso-model and homogenization model shows the compressive damage distribution for several meso-scale models with randomly distributed circular aggregates and their corresponding homogenized models. From the compressive damage distribution of the meso-scale models, it can be observed that the locations of concentrated plastic damage are fundamentally consistent, with the final failure mode being compressive crushing on the eccentric side of the concrete column.

Variations in aggregate distribution lead to shifts in the damage location. In Model 1, the damage location is significantly higher; in Model 2, the damage is located exactly at the center of the eccentric side; and in Model 3, damage appears in both the upper and lower regions. These distinct damage distributions across models are, upon analysis, caused by differences in aggregate spacing. Regions with smaller aggregate spacing are more prone to forming "aggregate-to-aggregate" stress transmission paths. Therefore, the random distribution of aggregates has a significant impact on the spatial distribution of compressive damage in eccentrically loaded columns. Because the homogenization method essentially performs an "averaging" treatment of the elastic modulus and strength of the aggregates and mortar within the equivalent elements, the high-stress "aggregate" regions in the meso-scale homogenization model are reduced while the low-stress "mortar" regions increase. This causes some compressive damage observed in the meso-scale model to not be explicitly displayed in the homogenized model. Furthermore, since the meso-scale homogenization model lacks purely elastic aggregates and instead describes all mechanical properties via a plastic damage constitutive model, it is more susceptible to damage localization during loading, which influences the damage distribution.

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The relationship between the load and mid-column deflection for the mesoscopic models with a random distribution of circular aggregates and the mesoscopic homogenization models is shown in [FIGURE:14]. As illustrated in the figure, the simulation curves for the group of mesoscopic models exhibit small fluctuations, and the ascending branch of the curves aligns well with the experimental data. The peak values of the mesoscopic models are close to the experimental values, although the mesoscopic models are slightly smaller. Under eccentric compression, high-stress regions typically appear on the biased side of the column. Dense stress transmission paths are prone to forming in areas where the aggregate spacing is small on the eccentric side. When the stress increases to a certain level, these "dense" regions are the first to undergo damage. The more numerous and dense these damage zones are, the greater the bearing capacity of the eccentric column.

The influence of aggregate shape on the mechanical properties of eccentric columns is analyzed by comparing the plastic damage distributions of reinforced concrete columns under eccentric compression. These comparisons involve mesoscopic models and mesoscopic homogenization models established using circular, elliptical, and polygonal aggregate shapes.

Based on the damage distribution of the eccentric column mesoscopic models, the overall damage patterns for circular, elliptical, and polygonal aggregates are roughly the same, which largely eliminates the influence of different aggregate distributions. However, different aggregate shapes lead to variations in stress transmission, resulting in subtle differences in damage distribution. Compared to circular and elliptical aggregates, polygonal aggregates are prone to stress concentration at their corners during eccentric force transmission. Damage paths often develop from the sharp corners of polygonal aggregates, causing the damage distribution to extend linearly and penetrate easily. Compared to circular aggregates, elliptical aggregates exhibit two modes of force transmission: first, when the eccentric force acts toward the long axis of the ellipse, the long side bears a greater force while the short side bears less, resulting in compression damage on both sides of the long axis; second, when the force acts toward the short axis, the short side bears a greater force, resulting in compression damage on both sides of the short axis. Consequently, the damage distribution in elliptical aggregate models is more fragmented than in circular ones, forming intersecting damage zones around the aggregates. From the perspective of the mesoscopic model, the aggregate shape has no significant impact on the overall shape of the damage distribution in reinforced concrete columns under eccentric compression. Due to these differences in stress transmission, subtle variations exist between the models with different aggregate shapes. It is difficult to determine the influence of aggregate shape on the mechanical properties of reinforced concrete columns solely from the compression damage distribution of the mesoscopic homogenization model. Analysis suggests two main reasons: first, the homogenization method uses the area percentage of aggregate and mortar within a grid as a variable, and different mesoscopic aggregate shapes do not significantly affect these percentages in the equivalent grid; second, the grid shape of the mesoscopic homogenization model is regular (square or rectangular). Even if elements with higher strength and elastic moduli exist, they cannot represent the specific shape of the aggregate, only the influence of square or rectangular aggregate units. Given these combined factors, studying the influence of aggregate shape via mesoscopic homogenization models has certain limitations.

The corresponding load-deflection curves at the mid-column show that the peak load of the mesoscopic homogenization model is slightly larger than that of the mesoscopic model. The mesoscopic simulation results indicate that the load-deflection curves for circular and polygonal aggregate models are similar, while the elliptical aggregate model yields a slightly higher peak load. The fact that different aggregate shapes affect the action and transmission of stress confirms the aforementioned compression damage analysis from the perspective of macroscopic mechanical performance.

A comparison of the computational requirements for the concrete column mesoscopic model and the mesoscopic homogenization model shows that the ratio of meshes, nodes, and degrees of freedom between the two is approximately 1:110. After applying the homogenization treatment to the eccentric column mesoscopic model, the number of mesh elements, nodes, and degrees of freedom is significantly reduced. This allows the mesoscopic homogenization model to save computational space and improve calculation efficiency.

[TABLE:5] Finite element calculation of eccentric column. Model types and dimensions: 960 mm × 150 mm × 150 mm; 960 mm × 20 mm × 15 mm.

4 结

In this study, through meso-scale numerical simulations and meso-scale homogenization numerical simulations, the following primary conclusions have been drawn:

1) Both the meso-scale model and the meso-scale homogenization model can effectively reflect the macroscopic mechanical properties and meso-scale damage distribution of eccentric columns.

2) Both the meso-scale model and the meso-scale homogenization model can accurately capture the influence of random aggregate distribution on the mechanical behavior of eccentric columns. The random distribution of aggregates affects the macroscopic mechanical properties and the distribution of meso-scale damage by altering the spacing between aggregates.

3) Compared to the meso-scale model, the meso-scale homogenization model of the eccentric column exhibits lower sensitivity to aggregate shape due to its larger mesh size and the uniform application of elastoplastic constitutive relations. Aggregate shape influences the macroscopic mechanical properties and meso-scale damage distribution of eccentric columns by altering the stress paths and loading mechanisms. Specifically, models with elliptical aggregates exhibit higher load-bearing capacity, while models with circular and polygonal aggregates show lower load-bearing capacity.

4) The number of mesh elements, nodes, and degrees of freedom in the two-dimensional meso-scale model is approximately $N$ times that of the meso-scale homogenization model. Consequently, the meso-scale homogenization model significantly reduces computational storage requirements and improves computational efficiency.

References:
SCHLANGEN E, GARBOCZI E J. Fracture simulations of concrete using lattice models: computational aspects \cite{...}. Engineering Fracture Mechanics...

ture mechanics ， 1997 ， 57 （ 2 / 3 ）： 319-332. [ 2 ] BAŽANT Z P ， TABBARA M R ， KAZEMI M T ， et al. Random parti-

Numerical Simulation of the Mesoscopic Mechanical Properties of Roller Compacted Concrete

Abstract

This study investigates the mechanical behavior of Roller Compacted Concrete (RCC) at the mesoscopic scale. By considering the heterogeneous nature of the material, including the distribution of aggregates, mortar matrix, and the interfacial transition zone (ITZ), a numerical model is developed to simulate fracture processes. The research builds upon established models for aggregate and fiber composites to provide a detailed analysis of how mesoscopic structures influence the macroscopic strength and failure patterns of RCC.

1. Introduction

Roller Compacted Concrete (RCC) is widely utilized in hydraulic engineering due to its cost-effectiveness and rapid construction capabilities. However, its mechanical properties are highly dependent on its complex internal structure. To accurately predict the performance of RCC structures, it is essential to move beyond macroscopic homogenized models and examine the material at the mesoscopic level. At this scale, RCC can be viewed as a three-phase composite material consisting of coarse aggregates, a mortar matrix, and the interfacial transition zones (ITZ) between them.

2. Mesoscopic Modeling Approach

The numerical simulation of RCC requires a precise representation of its internal geometry. In this study, we adopt a stochastic approach to model the distribution and shape of aggregates. The mechanical properties of each phase—aggregate, mortar, and ITZ—are assigned based on experimental data and micromechanical theories.

2.1 Material Phase Characterization

The aggregate phase is typically modeled as a linear elastic material, as its strength significantly exceeds that of the mortar and the ITZ. The mortar matrix is treated as a quasi-brittle material, where damage evolution and softening behavior are considered. The ITZ, often the weakest link in the composite, is modeled with reduced stiffness and strength parameters compared to the bulk mortar.

[FIGURE:1]

2.2 Fracture Mechanics Framework

To simulate the initiation and propagation of cracks, we utilize a damage mechanics approach combined with the principles of fracture mechanics for aggregate composites. The constitutive relationship for the $i$-th phase can be expressed as:

$$\sigma = (1 - d) \mathbf{C} : \epsilon$$

where $\sigma$ is the stress tensor, $d$ is the scalar damage variable, $\mathbf{C}$ is the initial elastic stiffness tensor, and $\epsilon$ is the strain tensor. The evolution of $d$ is governed by the local strain state and the specific fracture energy of the material phase.

level mechanical properties of roller compacted concrete [ J ] . Jour-

References

LIU Guangting, WANG Zongmin. Numerical simulation study of fracture of concrete materials using random aggregate model. Journal of Tsinghua University (Science and Technology).

GAO Zhengguo, LIU Guangting. Research on two-dimensional random aggregate models for concrete. Journal of Tsinghua University (Science and Technology).

structure for concrete [ J ] . Journal of Tsinghua University （ science

A Generating Approach for Random Convex Polygon Aggregate Models

Ma Huaifa, Mi Shuzhen, Chen Houqun
Journal of China Institute of Water Resources and Hydropower Research

Du Chengbin, Sun Liguo
Numerical Simulation of Arbitrarily Shaped Concrete Aggregates and Its Applications, Journal of Hydraulic Engineering

Abstract

Concrete is a complex composite material consisting of mortar, aggregates, and the interfacial transition zone (ITZ). The geometric distribution and shape of aggregates significantly influence the mechanical properties and failure mechanisms of concrete. This paper presents a numerical approach for generating random convex polygon aggregate models to simulate the meso-structure of concrete. By employing stochastic mathematical methods and geometric constraints, the proposed model effectively captures the random distribution and morphological characteristics of real-world aggregates.

1. Introduction

In the study of concrete at the meso-scale, the material is no longer treated as a homogeneous continuum but as a three-phase composite. Accurate numerical simulation requires the generation of a representative volume element (RVE) that reflects the actual grading and spatial distribution of aggregates. Traditional models often simplify aggregates as circles or spheres; however, natural aggregates are typically angular and irregular. Therefore, developing a robust algorithm for generating random convex polygons is essential for improving the fidelity of meso-scale finite element analysis.

2. Generation of Random Convex Polygons

The generation process begins with the definition of a base geometry, which is then perturbed using stochastic parameters. To ensure the physical validity of the model, several geometric constraints must be satisfied:

1. Area and Grading Requirements: The total area of the generated aggregates must comply with the prescribed volume fraction and the Fuller grading curve or relevant experimental sieving data.
2. Convexity: For computational efficiency in contact detection and mesh generation, the polygons must remain convex. This is achieved by ensuring all internal angles are less than $180^\circ$.
3. Non-overlap Constraints: A rigorous check is performed to ensure that no two aggregates overlap and that a minimum clear distance is maintained between them to allow for the representation of the mortar phase and ITZ.

The placement of aggregates follows a "take-and-place" strategy. For each aggregate, a random position $(x, y)$ and orientation $\theta$ are generated within the specimen boundaries. If the placement satisfies the non-overlap condition with previously placed aggregates, it is accepted; otherwise, a new position is sought.

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References

DU Chengbin, SUN Liguo. Numerical simulation of concrete aggregates with arbitrary shapes and its application. Journal of Hydraulic Engineering (in Chinese).

PENG Yijiang, LI Baokun, QU Yanling. Numerical study of the shear strength of matrix layer in rolled compact concrete on meso-level. China Safety Science Journal (in Chinese).

REN Zhaojun, DU Chengbin, DAI Chunxia. Meso-structure numerical simulation of uniaxial failure in three-graded concrete. Journal of Hohai University (Natural Sciences) (in Chinese).

cal simulation of uniaxial failure of three-graded concrete [ J ] . Jour-

Study on Three-Dimensional Mesoscopic Mechanical Numerical Models of Dam Concrete

Authors: MA Huaifa, CHEN Houqun, WU Jianping, et al.
Source: Journal of Hohai University (Natural Sciences) / Chinese Journal of Computational Mechanics

Abstract

This study investigates the three-dimensional mesoscopic mechanical behavior of dam concrete through numerical modeling. By considering the heterogeneity of concrete at the mesoscale—comprising aggregate, mortar matrix, and the interfacial transition zone (ITZ)—the research develops a robust computational framework to simulate damage evolution and failure mechanisms under various loading conditions.

1. Introduction

Concrete is a complex composite material characterized by significant heterogeneity at the mesoscopic level. For large-scale structures such as dams, the macroscopic mechanical properties and failure modes are fundamentally governed by the interactions between its internal constituents. Traditional macroscopic models often fail to capture the localized cracking and energy dissipation processes inherent in dam concrete. Therefore, developing a three-dimensional mesoscopic numerical model is essential for a deeper understanding of the structural integrity and safety of hydraulic engineering projects.

2. Mesoscopic Numerical Modeling

2.1 Generation of Three-Dimensional Random Aggregates

To accurately reflect the internal structure of dam concrete, a random aggregate model is employed. The aggregates are treated as discrete phases embedded within a continuous mortar matrix. The generation process accounts for the volume fraction, particle size distribution (grading), and spatial distribution of the aggregates.

[FIGURE:1]

The geometric representation of aggregates often utilizes simplified spheres or polyhedra to balance computational efficiency with physical realism. The placement of these aggregates follows a "take-and-place" algorithm, ensuring no overlap between particles while satisfying the prescribed grading curves.

2.2 Material Constitutive Relations

At the mesoscale, concrete is modeled as a three-phase composite material consisting of:
1. Aggregates: Generally modeled as linear elastic bodies, as their strength is significantly higher than the other phases.
2. Mortar Matrix: Modeled using an elastoplastic damage constitutive law to capture its nonlinear behavior and cracking.
3. Interfacial Transition Zone (ITZ): Represented as a thin layer surrounding the aggregates. The ITZ is typically the weakest link in concrete, and its mechanical properties are assigned as a scaled-down version of the mortar matrix properties.

The damage evolution is governed by the following relation:
$$\begin{aligned} \sigma = (1 - d) \mathbf{C} : \epsilon \end{aligned}$$

cal algorithm of 3D meso-mechanics model of dam concrete [ J ] .

Chinese journal of computational mechanics ， 2008 ， 25 （ 2 ）： 241-

3D Numerical Simulation of the Failure Process of Concrete

Concrete is a complex quasi-brittle material characterized by significant heterogeneity at the meso-scale. Understanding its failure mechanism is crucial for ensuring the structural integrity of civil engineering projects. This study develops a three-dimensional numerical model to simulate the progressive cracking and ultimate failure of concrete under various loading conditions.

By treating concrete as a three-phase composite material consisting of aggregate, mortar matrix, and the interfacial transition zone (ITZ), we can capture the initiation and propagation of micro-cracks. The numerical approach utilizes a damage mechanics framework to describe the constitutive behavior of each phase. Our results demonstrate that the spatial distribution of aggregates and the mechanical properties of the ITZ play a decisive role in the macroscopic strength and ductility of the concrete specimen.

[FIGURE:1]

Meso-scale Numerical Study on Axial Compression Performance and Size Effect of Reinforced Concrete Columns

The mechanical behavior of reinforced concrete (RC) columns under axial compression is a fundamental problem in structural engineering. However, the "size effect"—the phenomenon where the nominal strength of a structural member decreases as its characteristic size increases—remains a challenge for traditional design codes. This research employs a meso-scale numerical simulation to investigate the axial compressive performance of RC columns, accounting for the heterogeneity of the concrete and the interaction between the reinforcement and the matrix.

1.1 Methodology and Modeling

The numerical model explicitly represents the coarse aggregates and the steel reinforcement cages. The concrete matrix is modeled using a plastic-damage constitutive law, while the steel bars are treated as elastoplastic materials. By varying the cross-sectional dimensions of the columns while maintaining a constant reinforcement ratio, we analyze the sensitivity of the ultimate bearing capacity to the specimen size.

[TABLE:1]

1.2 Results and Discussion

The simulation results indicate that the size effect in RC columns is significantly influenced by the confinement provided by the stirrups. As the column size increases, the relative effectiveness of the core concrete confinement changes, leading to a non-linear reduction in nominal strength. These findings provide a theoretical basis for refining size effect laws in the design of large-scale hydraulic structures and high-rise buildings.

$$\begin{aligned} \sigma_n = \frac{f_t}{\sqrt{1 + \frac{D}{\lambda d_0}}} \end{aligned}$$

Where $\sigma_n$ represents the nominal strength, $f_t$ is the tensile strength, $D$

LI Dong ， JIN Liu ， DU Xiuli. Mesoscopic simulation of the mechani-

References

Jin Liu, Du Xiuli. Mechanical properties and the size effect of reinforced concrete columns subjected to axial compressive loading. Journal of Hydraulic Engineering (in Chinese).

Jin Liu, Du Xiuli. Meso-numerical simulation analysis of reinforced concrete members. Journal of Hydraulic Engineering (in Chinese).

Du Xiuli, Lu Aizhen, Zhao Jun. Experimental study on the size effect of small eccentric reinforced concrete columns under compressive loads. Journal of Architecture and Civil Engineering (in Chinese).

Li Dong, Jin Liu, Du Xiuli, et al. Mesoscopic numerical study on the mechanical properties of reinforced concrete columns under eccentric compression. Engineering Mechanics (in Chinese).

ted to eccentric compressive loading [ J ] . Engineering mechanics ，

References and Related Works

The research landscape regarding the structural integrity of bridge components has been significantly advanced by several key studies. Chen et al. \cite{Ref1} conducted a multiscale analysis of non-contact splices at the interface between drilled shafts and bridge columns, providing critical insights into the mechanical behavior of these connections. This work was further expanded in a subsequent parametric study \cite{Ref2}, which utilized a multiscale modeling approach to evaluate the influence of various design parameters on the performance of non-contact splices at the drilled shaft-to-column interface.

In the field of constitutive modeling, Wang \cite{Ref3} performed extensive research at the Beijing University of Technology regarding meso-scale concrete analysis models and methodologies. Building upon these foundational theories, Wang, Xu, and Chen \cite{Ref4} developed a numerical simulation framework for investigating the tensile behavior of concrete. Their approach employs multi-scale homogenization techniques to bridge the gap between microstructural characteristics and macroscopic mechanical responses, allowing for a more accurate prediction of concrete performance under tensile loading.

Numerical Simulation of Concrete Tensile Behavior Using Multi-scale Homogenization

The mechanical properties of concrete are inherently governed by its complex internal structure across multiple length scales. To capture this complexity, multi-scale homogenization methods are employed to derive macroscopic material laws from the analysis of representative elementary volumes (REVs) at the meso-scale.

Methodology

The homogenization process involves defining a meso-scale model that explicitly accounts for the distribution of aggregates, cement paste, and the interfacial transition zone (ITZ). By applying periodic boundary conditions to the REV, the effective macroscopic stress $\bar{\sigma}$ and strain $\bar{\epsilon}$ can be related through the homogenization operator.

For a given displacement field $u$, the macroscopic strain is defined as:
$$\bar{\epsilon} = \frac{1}{V} \int_V \epsilon(u) dV$$
where $V$ represents the volume of the meso-scale domain. The corresponding macroscopic stress is determined by the volume average of the local stress fields:
$$\bar{\sigma} = \frac{1}{V} \int_V \sigma(x, \epsilon) dV$$

Results and Discussion

The numerical simulations of tensile behavior demonstrate that the multi-scale approach effectively captures the softening response and localization of damage within the concrete matrix. By incorporating the stochastic nature of aggregate distribution, the model provides a realistic representation of crack initiation and propagation.

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