Numerical Simulation Study on the Constitutive Model of Fully-Graded Concrete Based on Statistical Damage Theory

Abstract

A statistical damage model (SDM) of fully-graded concrete was created using statistical damage theory, based on the mechanical properties of axial tension and axial compression of the material. The SDM considers two damage modes, fracture and yield, and explains the intrinsic connection between the mesoscopic damage evolution mechanism and the macroscopic nonlinear mechanical behavior of fully-graded concrete. The artificial bee colony (ABC) algorithm was used to obtain the optimal parameter combination through an intelligent search of parameters ${\epsilon }_a$, ${\epsilon }_h$, ${\epsilon }_b$ and $H$ in the constitutive model by taking the test data as the target value, and the sum of the squares of the differences between the target value and the predicted value as the objective function. The SDM numerical simulation model of fully-graded concrete is proposed by compiling subroutines in FORTRAN by constructing two modules of data model and damage analysis. The numerical results under uniaxial and biaxial forces are in agreement with the experimental results, which verifies the accuracy of the program. The model also analyzes the characteristics of mesoscopic damage evolution and predicts the mechanical properties under triaxial forces. The results show that the proposed numerical simulation model can reflect the salient features for fully-graded concrete under uniaxial, biaxial and triaxial loading conditions, and the evolution law of mesoscopic parameters. Therefore, the proposed model serves as a basis for the refined finite element analysis of hydraulic fully-graded concrete structures and reveals the mesoscopic damage mechanism of concrete under different load environments.

1. Introduction

Fully-graded concrete is a widely used complex nonhomogeneous material in the construction of water conservancy infrastructure—such as gravity dams and arch dams—which consists of cement, aggregate, water and admixture. Compared to conventional concrete particle size, the aggregate size of fully-graded concrete can be up to 150 mm [1]. Concrete has good compressive properties; however, fully-graded concrete structures in real environments are often under multiaxial stresses. The mechanical properties under multiaxial stresses are different compared to uniaxial mechanical properties. The mechanical properties of fully-graded concrete under multidirectional compression have received attention from scholars, and relevant experimental and theoretical research has been carried out [2,3,4]. The full curve of concrete stress–strain characterizes the whole process of concrete from the beginning of deformation—gradual damage to the final destruction—which can comprehensively reflect the strength and deformation properties of concrete. The multiaxial stress–strain constitutive relationship is the basis for the refined numerical analysis of the entire process of concrete structures subjected to external loads [5,6]. Since specimens with large-size aggregates are usually larger in size and have higher requirements for the corresponding testing equipment, there are only a limited number of researchers who can carry out the relevant tests. Adopting numerical simulation methods to study the properties of concrete structure, in combination with the relevant test results, to establish a contact between the mesoscopic damage evolution mechanism and the macro mechanical properties, has become one of the research directions of interest to current scientific workers. Therefore, the study of numerical simulation models that can reflect the multiaxial force state of fully-graded concrete is significant for more effective nonlinear analysis and the structural design of large-scale water conservancy projects.

Concrete is a quasi-brittle material, and a scientific and reasonable constitutive model is the key to numerical simulation. Many researchers have investigated the mechanical properties and mesoscopic damage mechanisms of concrete under complex external loads (static, earthquake, explosion [7,8,9,10,11], etc.) through mechanical property tests, and then established a variety of constitutive models. Evabs and Marathe [12] obtained the complete tension stress–strain curve of the concrete for the first time in 1968. Du et al. [13] proposed a special loading machine to obtain the stress-deformation curve and mechanical parameters of concrete and gave an empirical formula. Zhang et al. [14] found that the damage of the specimens manifested mainly in the form of damage at the interface of the aggregate and the mortar bonding, and investigated the law of change of its mechanical properties with external factors. Wang et al. [2] and Shi et al. [3,4] investigated the stress–strain curves and the variation rules of key parameters of dam concrete under different working conditions through a series of uniaxial, biaxial and triaxial tests, and established the constitutive relationship of concrete without an obvious yield surface by applying internal time theory and damage mechanics. The damage mechanic of concrete, which has been developed in recent decades, provides a feasible way for the study of the constitutive relationship of concrete, the typical characteristics of the mechanical properties of concrete can be clearly expressed through the introduction of damage variables, and the damage constitutive model of elastic-plastic damage, which can describe the non-linear behavior of concrete in a more comprehensive way, has been established [15]. Currently, most constitutive models are derived based on experimental fitting of macroscopic stress–strain relationships, and the empirical damage evolution functions are widely adopted in most material models, in which many empirical parameters without clear physical meanings are required [6]. Furthermore, Krajcinovic [16] stated that “an internal variable inferred from phenomenological evidence and selected to fit a particular stress–strain curve may provide a result that pleases the eye, but seldom contributes to understanding of the processes represented by the fitted curve.” Among the studies of localized softening of materials, the cohesive cracking model has been promoted as the most fundamental model and is the criterion for judging the correctness of other models. In the study of concrete fracture behavior, Hillerborg et al. [17] were the first to propose the cohesive cracking model, which was later developed into the coated crack zone model [18], but failed to explain what kind of destructive process the material undergoes between cracks. As a typical nonhomogeneous composite material, the essence of concrete damage is the continuous evolution process of microcracks that sprout, expand and nucleate, which is caused by local mesoscopic tensile strains, and the constitutive nonlinear stress–strain relationship shown is the macroscopic manifestation of the nonhomogeneous mechanical properties of mesoscopic components [5]. In order to analyze the mesoscopic characteristics of concrete more deeply, other scholars have introduced advanced methods such as nuclear magnetic resonance (NMR), scanning electron microscopy (SEM), CT technology and acoustic emission technology [19,20,21,22,23] to the study of concrete material damage. This reveals the mesoscopic characteristics of concrete from the aspects of interface transition zone microstructures, pore distribution characteristics, microcrack densities, etc. However, these microscopic analyses have not been able to respond directly to the concrete damage constitutive model.

Many studies have been conducted that explore the complex mechanisms of concrete behavior through experimental and numerical methods. Many scholars have used numerical simulations based on mesostructure to represent the internal structure of the material to explore the concrete damage mechanism. Bažant et al. [24] proposed a random particle model to study the fracture damage of concrete material. This model discretizes the material into a series of particles with only axial contact forces between the particles, and simulates the cracking problem through the tensile destruction of the units. Schlangen et al. [25] proposed a Lattice model to simulate the concrete fracture, and investigated the mesoscopic cracking mechanism of brittle materials in conjunction with experiments. Xu et al. [26] designed a comprehensive numerical framework to statistically characterize the proportion of interfacial transition zones around non-convex aggregates in fully-graded concrete, which was used to assess the physicomechanical properties of fully-graded concrete. Ma et al. [27] developed a three-phase, fully-graded concrete mesoscopic structure model consisting of anisotropic high-content aggregate, cement paste and its interfacial bond, which can qualitatively simulate and explain the influence of mesoscopic structure properties on the macroscopic mechanical properties of fully-graded concrete. Qin et al. [28] used a method that considers the randomness of aggregate geometry and spatial distribution in real engineering to investigate the mesoscopic fracture behavior of fully-graded concrete. Guo et al. [1] investigated the applicability and rationality of the phase field method in the mesoscopic study of fully-graded concrete under uniaxial tension. Xu et al. [29] developed a mesoscale fracture simulation method for fully-graded concrete. Xu et al. [30] proposed a method for generating three-dimensional mesoscale structures of fully-graded concrete: the hierarchical point-cloud method. These studies have played an important role in deepening the understanding of the cracking and damage mechanisms of brittle materials such as concrete, but it is difficult to perform large-scale structural finite element analysis, due to the large number of elements. For the finite element analysis of hydraulic structures such as dams, the Concrete Damaged Plasticity (CDP) model is often used [31,32,33], which can reflect the macroscopic mechanical behavior of concrete but cannot reveal the mesoscale evolution law. The mesoscopic SDM was developed from continuum damage mechanics to explore the connection between the mesoscopic mechanism of materials and the macroscopic mechanical behavior. Based on the parallel bar system (PBS) proposed by Krajcinovic and Silva [34], which has been improved by many scholars such as Breysse [35], Kandarpa et al. [36] and Bai et al. [37,38,39,40,41,42], the mechanical properties of recycled concrete have been investigated under the factors of freeze-thaw, age, and strain rate, and the proposed SDM can effectively reveal the intrinsic connection between the mesoscopic damage evolution mechanism and the macroscopic non-linear constitutive behavior of concrete materials.

Despite the significant research carried out in recent years, the task of nonlinear finite element analysis of fully-graded concrete structures is still quite challenging. The accurate modeling of concrete performances is the key issue to numerical simulation. Parameters of the constitutive relationship obtained by fitting the experimental curves do not have clear physical meanings [6,16]; the analysis results based on NMR, SEM, CT, etc. [19,20,21,22,23] cannot be reflected in the constitutive model; it is difficult to analyze large-scale concrete structures using a microscopic model [24,25,26,27,28,29,30] that considers mortar matrix, coarse aggregate and interface transition zone; the SDM, which can reflect the microscopic damage mechanism, is currently mainly used to study uniaxial mechanical performance tests of ordinary concrete. Therefore, it is significant to further extend the statistical damage theory to fully-graded concrete, and apply it to numerical simulations. Related research results will contribute to refined finite element analysis of hydraulic structures, reveal mesoscopic damage mechanism and optimize structure design, etc.

In this paper, based on statistical damage theory, combined with the full stress–strain curve of axial tension and axial compression tests, an SDM of fully-graded concrete was established. The ABC method is used to obtain the optimal parameter combinations by intelligent search of the parameters with the test data as the target value, the mesoscopic parameters in the SDM as the variables and the sum of the squares of the differences between the predicted values and the target value as the objective function. FORTRAN was used to compile the subroutine, two modules of the data model and damage analysis were constructed, and a numerical simulation model of SDM was proposed for fully-graded concrete was proposed. The accuracy of the model was verified by comparing the numerical results with the experimental results, and the characteristics of the evolution of the mesoscopic damage were also analyzed and the mechanical properties were predicted under triaxial forces.

2. Experimental Study on the Mechanical Properties of Fully-Graded Concrete

2.1. Fully-Graded Concrete Specimens

Full-grade concrete specimens were made and tests were carried out according to the Test Procedure for Hydraulic Concrete (SL/T 352-2020) [43]. Ordinary silicate cement—P.O 42.5—was used with a water–cement ratio of 0.42, and the fully-graded concrete mix proportions are shown in

Table 1. The mix proportion of fully-graded concrete (kg/m³).

Cement	FLY Ash	Sand	Small Stone	Medium Stone	Large Stone	Extra Large Stone
315	230	266	272	273	274	276

. The dry absorption of saturated surface of sand, small stone, medium stone, large stone and extra-large stone was 1.37%, 0.85%, 0.66%, 0.56% and 0.38%, respectively. In order to improve the workability of concrete mixtures and mitigate the impact of increased hydration temperature on the water–cement ratio, air entraining agents and superplasticizers are incorporated into the mixtures.

The size of the axial tension specimen is Φ450 mm × 1470 mm. The specimen is embedded with eight rebar with 32 mm diameter at each end, and the embedding depth is 300 mm. The length of exposed rebar was 60 mm and the screw was machined to a diameter of 22 mm. The arrangement of strain gauges and high-precision extensometers was used to measure the deformation of the specimen, as shown in Figure 1a,c. The strain gauges were arranged on the four sides of the specimen, which were pasted in the axial direction, and each group consisted of 4 strain gauges with a length of 150 mm and the total measuring distance of 660 mm. Four extensometers were arranged in the vicinity of each group of strain gauges, and a total measuring distance was 660 mm. The size of the axial compression specimen used in this study is 450 mm × 450 mm × 450 mm, and strain gauges were used to measure the deformation, as shown in Figure 1b,d. Three 150 mm-long strain gauges and one laser displacement sensor were arranged vertically on each side, for a total of twelve strain gauges and four laser displacement sensors, where the end constraints were as removed by two layers of plastic film and three layers of butter constraints.

2.2. Test Equipment and Loading

In this test, the 15 MN large-scale MTS material testing machine of the China Institute of Water Resources and Hydropower Research is used, as shown in Figure 2, which can meet the requirements of static and dynamic testing of fully-graded concrete.

In axial tension and compression tests, the main test set-up consists of the following components: (1) the actuator is mounted on a crossbeam with a full travel of ±300 mm, hydraulic lift, hydraulic locking and an emergency switch control button; (2) articulated seat assemblies for static and cyclic tests, connected to the actuator and the base, with a rated force of ±1000 kN, swivel angle of +90° or −30° and a tilt angle of ±8°; (3) a 2.5 MN load sensor is used for load measurement and control, with the measurement range of 10~100%, a measurement error less than ±0.5% of the display value and the value variation less than ±0.1% of the display value; (4) temposonic digital displacement transducer is used for deformation measurement, with a stroke of 500 mm, resolution of 0.003 mm. In addition, there are 1500 lpm shunt, coupled accumulator, double-loop inspection valve, etc.

The MTS FlexTest digital control system is used in this test, which supports a 32 measurement channel return signal control. The system uses a full digital closed-loop control. In order to satisfy different test requirements, the China Institute of Water Resources and Hydropower Research has built a multi-purpose test (MTP) software based on the Level Cross Acquirement function provided by MTS.

For the axial tension test, in order to obtain a complete fully-graded concrete stress–strain curve and capture the damage location, the maximum strain control mode was used in this paper. The maximum value of the 16 strain gauges was used as a control signal to control the loading rate. The schematic diagram of axial tension test is shown in Figure 3. For axial compression test, in order to obtain a complete fully-graded concrete stress–strain curve, the maximum strain control mode was also used in this paper. The maximum value of the 12 strain gauges was used as a control signal to control the loading rate. The schematic diagram of axial compression test is shown in Figure 4.

2.3. Test Results

For the axial tension test, with gradual increase in load, the specimen was fractured near its mid-upper position. Figure 5 shows the location of the fracture, typical fracture surfaces and full stress–strain curves. The fracture surface shows damage to the aggregate-mortar bond interface, and the peak strain is 99.87 × 10⁻⁶. The peak stress is 1.73 MPa.

For the axial compression test, the specimen was crushed with a gradual increase of load. The damage state of the compression specimen and the full stress–strain curve are shown in Figure 6. The peak strain is 1248.13 × 10⁻⁶ and the peak stress is 31.04 MPa.

Through the test results, the macroscopic damage characteristics and the full stress–strain curve of the axial tension and compression tests can be clarified. However, the mesoscopic damage destruction mechanism cannot be revealed.

3. SDM of Fully-Graded Concrete

3.1. Mesoscopic Damage Mechanism and Macroscopic Constitutive Behavior

Bai et al. [5,37,38,39,40,41,42] proposed a statistical damage theory for concrete, and the concrete is considered a complex system with self-organized behavior. Concrete deformation damage is a process of “adapting” to the external loading environment by further releasing the potential load-bearing capacity through “damage” (microcrack initiation and expansion).

Figure 7 shows the concrete axial tension deformation damage process, which can be divided into two stages of homogeneous damage and localized damage, and contains three typical states: A–C, of which A is the proportional limit state. The direction of stretching is noted as 1-direction, the corresponding nominal and effective stresses are ${\sigma }_{1\mathrm{A}}$, ${\sigma }_{1\mathrm{B}}$, ${\sigma }_{1\mathrm{C}}$ and ${\sigma }_{1\mathrm{E},\mathrm{A}}$, ${\sigma }_{1\mathrm{E},\mathrm{B}}$, ${\sigma }_{1\mathrm{E},\mathrm{C}}$, respectively. Due to the non-homogeneity of the mesoscopic composition, the stress state of the internal parts of concrete is different. In the early stage of loading, the load skeleton in the microstructure is not optimized, and some parts with high strength may not even participate in the stressing. With the increase of tensile deformation, a series of microcracks are randomly generated in weak parts such as aggregate and mortar interfaces, and the direction is roughly perpendicular to the tensile direction. In this process, the weak parts gradually withdraw from the loading, and stress redistribution can be realized. The stronger parts are more involved in the load, and some parts that were not involved in the loading at the beginning are successively added to the load skeleton. By the above method, the effective load skeleton of the microstructure can be further optimized and adjusted, and then the load carrying capacity can be obtained that is consistent with each stress state.

Figure 8 shows the damage process of the axial compression deformation of concrete, and the process is also divided into uniform damage and localized destruction of two stages, including the four typical states A–D, of which A is the proportion of the limit state. The direction of compression is noted as the 3-direction, and the two lateral directions as the 1-direction and 2-direction. The nominal and effective stresses corresponding to the compression direction are ${\sigma }_{3\mathrm{A}}$, ${\sigma }_{3\mathrm{B}}$, ${\sigma }_{3\mathrm{C}}$, ${\sigma }_{3\mathrm{D}}$ and ${\sigma }_{3\mathrm{E},\mathrm{A}}$, ${\sigma }_{3\mathrm{E},\mathrm{B}}$, ${\sigma }_{3\mathrm{E},\mathrm{C}}$, ${\sigma }_{3\mathrm{E},\mathrm{D}}$, respectively. Due to the influence of the Poisson effect, the concrete specimen produces transverse tensile strain in the 1 and 2 lateral directions. When some weak parts of the local tensile strain exceed the ultimate tensile strain, this will lead to the generation of local microcracks. Microcracks first arise and expand at the junction of aggregate and mortar in a direction roughly parallel to the direction of pressure. The microcrack density increases gradually with increasing pressure.

3.2. Statistical Damage Constitutive Model

Bai et al. [5,37,38,39,40,41,42] established an improved parallel bar system (IPBS) to simulate the axial tensile and compressive deformation and failure processes of concrete, where the compressive direction damage is controlled by the lateral tensile damage process caused by the Poisson effect. In this model, the representative volume element (RVE) in the fracture process zone of material, is assumed as a system composed of T (T→∞) parallel bars. Each micro-bar is composed of spring, cementation bar and sliding block. Each micro-bar has two feature strains, i.e., the fracture strain and the yield strain, and it may have two kinds of failure modes (fracture and yield). The material disorder is introduced by strain and the yield strain to each bar, which are defined by the independent probability density function $q\left(\epsilon \right)$ and $p\left(\epsilon \right)$, respectively. Therefore, the model considers two damage modes, microscopic fracture and yield, which correspond to the “degradation” and “strengthening” effects of the microstructure, and can be characterized by the fracture and yield of the microrods, respectively. The mesoscopic inhomogeneity of the material is introduced by assigning random fracture and yield strengths to the microrods. The macroscopic nonlinear stress–strain behavior of concrete is controlled by the evolution of the two damage modes of mesoscopic fracture and yield (in Figure 7b and Figure 8b). The constitutive relationships of axial tensile and compressive statistical damage can be expressed uniformly as follows:

$$\sigma =E_0(1-D_{\mathrm{y}})(1-D_{\mathrm{R}})\epsilon$$

(1)

$${\sigma }_{\mathrm{E}}=E_0(1-D_{\mathrm{y}})\epsilon$$

(2)

$$D_{\mathrm{y}}={\displaystyle {\int }_0^{{\epsilon }^{+}}p({\epsilon }^{+})\mathrm{d}}{\epsilon }^{+}-\frac{{\displaystyle {\int }_0^{{\epsilon }^{+}}p({\epsilon }^{+}){\epsilon }^{+}\mathrm{d}}{\epsilon }^{+}}{{\epsilon }^{+}}$$

(3)

$$D_{\mathrm{R}}={\displaystyle {\int }_0^{{\epsilon }^{+}}q({\epsilon }^{+})}\mathrm{d}{\epsilon }^{+}$$

(4)

$$E_{\mathrm{v}}={\displaystyle {\int }_0^{{\epsilon }^{+}}q({\epsilon }^{+})}\mathrm{d}{\epsilon }^{+}$$

(5)

$$p({\epsilon }^{+})=\{\begin{array}{ll}0& ({\epsilon }^{+} \le {\epsilon }_{\mathrm{a}})\\ \frac{2({\epsilon }^{+}-{\epsilon }_{\mathrm{a}})}{({\epsilon }_{\mathrm{h}}-{\epsilon }_{\mathrm{a}})({\epsilon }_{\mathrm{b}}-{\epsilon }_{\mathrm{a}})}& ({\epsilon }_{\mathrm{a}}<{\epsilon }^{+}\le {\epsilon }_{\mathrm{h}} )\\ \frac{2({\epsilon }_{\mathrm{b}}-{\epsilon }^{+})}{({\epsilon }_{\mathrm{b}}-{\epsilon }_{\mathrm{h}})({\epsilon }_{\mathrm{b}}-{\epsilon }_{\mathrm{a}})}& ({\epsilon }_{\mathrm{h}}<{\epsilon }^{+}\le {\epsilon }_{\mathrm{b}})\end{array}$$

(6)

$$q({\epsilon }^{+})=\{\begin{array}{ll} 0& ({\epsilon }^{+} \le {\epsilon }_{\mathrm{a}})\\ \frac{2H({\epsilon }^{+} - {\epsilon }_{\mathrm{a}})}{{({\epsilon }_{\mathrm{b}}-{\epsilon }_{\mathrm{a}})}^2}& ({\epsilon }_{\mathrm{a}}<{\epsilon }^{+}\le {\epsilon }_{\mathrm{b}})\end{array}$$

(7)

$$H=D_{\mathrm{R}}({\epsilon }_{\mathrm{b}})$$

(8)

$E_{\mathrm{v}}$ characterizes the degree of optimized adjustment of the microstructure load skeleton and the potential load-bearing capacity. When $E_{\mathrm{v}}$ = 0, it corresponds to the initial state without damage. And when $E_{\mathrm{v}}$ = 1, it corresponds to the critical state with all microbars in the model yield, and the adjustment of the load skeleton is optimized to reach the maximum value. When ${\sigma }_{\mathrm{E}}$ reaches its maximum value, the model is about to enter the stage of local destruction. It should be noted that the relevant parameter in the above equation is subscripted as 1 for axial tension, and the relevant parameter in the above equation is subscripted as 3 for axial compression.

The uniaxial SDM of concrete is given above. Considering that concrete will be in two-dimensional or three-dimensional stress state, it is necessary to extend the constitutive model to three-dimensional, which is more conducive to the numerical simulation study.

Based on the assumption of Lamitre equivalent strain, the strain corresponding to the damage state of the material is considered to be equivalent to the ideal elastic state. According to the generalized Hooke’s law, the strain components of an elastomer satisfy the following relationship:

$${\epsilon }_i={\epsilon }_{ii}+{\epsilon }_{ij}+{\epsilon }_{ik}, i,j,k=1,2,3 \mathrm{and} i\ne j\ne k$$

(9)

$${\epsilon }_{ii}=\frac{1}{\left(1+\nu \right)\left(1-2\nu \right)}\left[\left(1-\nu \right){\epsilon }_i+\nu \left({\epsilon }_j+{\epsilon }_k\right)\right]$$

(10)

$${\epsilon }_{ij}=\frac{\nu }{\left(1+\nu \right)\left(1-2\nu \right)}\left[\left(1-\nu \right){\epsilon }_i+\nu \left({\epsilon }_j+{\epsilon }_k\right)\right]$$

(11)

Therefore, the three-dimensional damage constitutive equation for concrete can be obtained:

$$\left\{\begin{array}{c}{\sigma }_1\\ {\sigma }_2\\ {\sigma }_3\end{array}\right\}=E_0\left[\begin{array}{ccc}\left(1-D_1\right)& 0& 0\\ 0& \left(1-D_2\right)& 0\\ 0& 0& \left(-D_3\right)\end{array}\right]\left\{\begin{array}{c}{\epsilon }_{11}\\ {\epsilon }_{22}\\ {\epsilon }_{33}\end{array}\right\}$$

(12)

and

$$D_i=\{\begin{array}{c}{D}_i^{+}, {\epsilon }_{ii}\ge 0\\ D_i^{-}, {\epsilon }_{ii}<0\end{array} i=1,2,3$$

(13)

According to the assumption of equivalent strain, each strain component in the main direction in the damage state is equivalent to the corresponding strain component in the ideal elastic state, and the following expression can be established as follows.

$${\epsilon }_i^{+}=\langle {\epsilon }_{ii}+k_1\left({\epsilon }_{ij}+{\epsilon }_{ik}\right)\rangle \left({\epsilon }_{ii}\ge 0\right)$$

(14)

where, $\langle \rangle$ is a positive sign, $\langle x\rangle =\{\begin{array}{c}x, x\ge 0\\ 0, x<0\end{array}$.

$$k_1=\{\begin{array}{c}{k}_{10 } {\epsilon }_{ij}+{\epsilon }_{ik}\ge 0\\ k_{11} {\epsilon }_{ij}+{\epsilon }_{ik}<0\end{array}$$

(15)

$${\epsilon }_i^{*}=\max \left\{\langle {\epsilon }_{ij}+k_2{\epsilon }_{ii}+k_3{\epsilon }_{ik}\rangle \right\} \left(i,j,k=1,2,3; i\ne j\ne k\right)$$

(16)

$$k_2=\{\begin{array}{c}{k}_{20 } {\epsilon }_{ij}\ge 0\\ k_{21} {\epsilon }_{ii}<0 \end{array}; k_3=\{\begin{array}{c}{k}_{30 } {\epsilon }_{ik}\ge 0\\ k_{31} {\epsilon }_{ik}<0\end{array}$$

(17)

$${\epsilon }_i^{-}=\sqrt{\frac{1}{2}\left[{\left({\epsilon }_j^{*}\right)}^2+{\left({\epsilon }_k^{*}\right)}^2\right]} \left(i,j,k=1,2,3; i\ne j\ne k\right) \left({\epsilon }_{ii}<0\right)$$

(18)

In order to be consistent with the intrinsic direction of uniaxial compression, this function is written in this paper in the form of Equation (18).

It should be noted that axial tensile and compression constitutive models are special cases of the three-dimensional constitutive model.

3.3. Parameterization Method Based on ABC

SDM can establish an effective link between the mesoscopic damage mechanisms of concrete and the nonlinear macroscopic behavior. The determination of the ${\epsilon }_a$, ${\epsilon }_h$, ${\epsilon }_b$ and $H$ parameters is the key to obtaining the constitutive relationship of fully-graded concrete. The ABC method is used to obtain the optimal parameter combinations by intelligent search of the parameters with the test data as the target value, with the mesoscopic parameters ε_a, ε_h, ε_b and H in the SDM as the variables, and the objective function is constructed by solving the sum of the squares of the differences between the target and predicted values corresponding to all data points. Then the SDM of concrete can be obtained (Figure 9).

The detailed steps are as follows:

① Obtain stress–strain curves from axial tensile or compressive test of fully-graded concrete specimens.

② Establish the objective function. The sum of the squares of the differences between the target and predicted values corresponding to all data points is taken as the objective function, as shown in Equation (19):

$$F={\displaystyle \sum _{n=1}^N{\left(x_{tn}-x_{en}\right)}^2}$$

(19)

where $n=1,2,3,\dots ,N$. The objective is to minimize the objective function. The predicted values are obtained by Equations (1)–(8) under a set of values for ${\epsilon }_a$, ${\epsilon }_h$, ${\epsilon }_b$ and $H$.

③ Determine the fitness function. The ABC algorithm will always choose a larger fitness value when searching, so the inverse of the objective function $F$ is chosen as the fitness function, as shown in Equation (20):

$$f=\frac{1}{{\displaystyle \sum _{n=1}^N{\left(x_{tn}-x_{en}\right)}^2}}$$

(20)

④ Determine the maximum number of searches L and the maximum number of cycles C.

⑤ Determine the parameter search range. For the four parameters, the ranges of ${\epsilon }_a$, ${\epsilon }_h$, ${\epsilon }_b$ and $H$ are given by combining the ontological relationship and experimental data, and each parameter is assigned a value to $X_r$ according to Equation (21):

$$x_{tr}=x_{\min ,r}+rand\left(0,1\right)\left(x_{\max ,r}-x_{\min ,r}\right)$$

(21)

where, $r=1,2,3,4$, and $X_1={\epsilon }_a,X_2={\epsilon }_h,X_3={\epsilon }_b,X_4=H$. $rand\left(0,1\right)$ is a random number in the range (0,1).

⑥ Search parameters. For the four parameters, a neighborhood search is performed according to Equation (22):

$$x_m’=x_{tr}+{\phi }_{tr}\left(x_{tr}-x_{sr}\right)$$

(22)

where, $m=1,2\dots L$, and $t\ne s$.

⑦ Calculate the current optimal solution. Calculate the selection probability according to Equation (23):

$$p=f_z/{\displaystyle \sum _{u=1}^z{f}_u}$$

(23)

where, $u=1,2,3,\dots ,z$.

Neighborhood search of $x_{tr}$ according to Equation (22) yields the parameter $x_1^{’}$, and the associated fitness $f_1^{’}$ and selection probability $p_1^{’}$ are calculated according to Equations (20) and (23).

If $p_1’ > p$, then the current optimal solution is $x_{tr}{}^d=x_1’$. $p$ denotes the selection probability corresponding to $x_{tr}$.

If $p_1’\le p$ and the maximum number of searches has not been reached, step ⑦ is repeated to rerun the neighborhood search.

If the maximum number of searches is reached without obtaining a greater probability of selection, then $p_1’\le p$ is the current optimal solution.

⑧ Calculate the global optimal solution. Calculate the fitness $f_{}^d$ and selection probability $p_{}^d$ corresponding to all current optimal solutions $x_{tr}{}^d$ according to Equations (20) and (23). Compare all selection probabilities $p_{}^d$ and take the current optimal solution $x_{tr}{}^d$ with the largest selection probability $p_{}^d$ as the optimal solution and output $X_r=x_{tr}{}^d$.

According to statistical damage theory, a reasonable range of values for the four parameters needs to be determined. Based on the axial tensile test data of fully-graded concrete, the value ranges of ${\epsilon }_a$, ${\epsilon }_h$, ${\epsilon }_b$ and $H$ were selected as [0.0001, 0.2], [0.2001, 1], [1.0001, 3] and [0.0001, 1], respectively. Based on the axial compression test data, the value ranges of ${\epsilon }_a$, ${\epsilon }_h$, ${\epsilon }_b$ and $H$ were selected as [0.0001, 1], [1.0001, 3], [3.0001, 5] and [0.0001, 1], respectively. The maximum number of searches is set at 50 and the maximum number of loops is set at 100, then the parameter optimization is carried out according to the method proposed in Figure 9.

For the axial tension test, the modulus of elasticity is 3.55 × 10⁴ MPa, and the optimal parameters of ${\epsilon }_a$, ${\epsilon }_h$, ${\epsilon }_b$ and $H$ are 0.0495, 0.2488, 1.4423 and 0.2663, respectively; for the axial compression test, the modulus of elasticity is 3.55 × 10⁴ MPa, the optimal parameters of ${\epsilon }_a$, ${\epsilon }_h$, ${\epsilon }_b$ and $H$ are 0.6548, 2.6848, 3.6711 and 0.4965, respectively. Figure 10 and Figure 11 show the optimization process of axial tensile and compression parameters, and the comparison curves of test and fitting results. It can be seen that the ABC method is able to converge to the optimal result very quickly, and the nominal stress–strain curves predicted by the model are in good agreement with the experimental curves. This validates the effectiveness of the methodology in this article.

4. Numerical Implementation of SDM

The SDM developed in this paper is capable of characterizing the mesoscopic fracture and yield damage evolution process of concrete and revealing the relationship between the mesoscopic damage mechanism and the macroscopic nonlinear stress–strain behavior. In order to apply the model to the analysis of fully-graded concrete structures, numerical implementation is necessary.

The numerical implementation process of the model can be divided into two parts: (1) the establishment of the physical model and attribute assignment; (2) the finite element method solution, including the displacement field, stress field, strain field, etc. Then the evolution of the mesoscopic damage of the element is analyzed. Figure 12 shows the numerical analysis process and calculation scheme. The detailed steps are as follows:

① Establish a finite element model;

② Obtain model data for the S-th loading step;

③ Perform the w-th sub-step calculation (w initial value is 0);

④ Solve the finite element equations and update the element elastic strain;

⑤ Update the element stresses;

⑥ Calculate the three principal strains;

⑦ According to the SDM, determine whether the element produces damage or not: if there is damage, calculate the damage value, carry out the stiffness degradation treatment, and repeat step ④; if there is no damage, carry out the next step;

⑧ According to the load applied in sub-step w, determine whether the end of the sub-step: if the end requirements are not met, make w = w + 1 and repeat step ③; if the end requirements are met, go to the next step;

⑨ According to the overall load application, judge whether to end the load step: if the end requirements are not met, make S = S + 1 and repeat step ②; if the end requirements are met, end the calculation.

In order to meet the requirements of the static method and consistent with the quasi-static damage process of the concrete test, the loading step should be controlled in the quasi-static range. In the loading process, it is important to set the loading step reasonably, because too large a loading step will lead to a large one-step error, and too small a loading step will increase the amount of calculation.

In this paper, the numerical simulation of SDM is realized through FORTRAN programming, and the whole process is to embed the FORTRAN program into the finite element software to analyze the damage evolution characteristics of the structure.

5. Results

Figure 13 gives the overall numerical simulation idea of fully-graded concrete constitutive model. First, the full stress–strain curves are obtained by axial tensile and compression tests, then the SDM of fully-graded concrete is established, the main parameters of the constitutive model are determined based on the ABC algorithm, and finally the FORTRAN subroutine is compiled to realize the numerical simulation calculation. In order to verify the accuracy of the proposed numerical simulation method, a finite element model with a single element is established, as shown in Figure 14a, while Figure 14b shows the boundary conditions. The uniaxial, biaxial, and triaxial forces are analyzed. It should be noted that the compression and tension modulus of elasticity are assumed to be the same, and Poisson ratio is 0.2 during the numerical simulation.

5.1. Numerical Simulation of Uniaxial Force

Based on the boundary conditions in Figure 14b, the axial tensile case is simulated by applying the positive displacement in the Y-direction, and the axial compression case is simulated by applying the negative displacement in the Y-direction. Comparisons of the axial tension and compression simulation results with the test results are given in Figure 15. It can be seen that the nominal stress–strain full curves of the numerical simulation match well with the test results, which also shows that the proposed simulation method can simulate the uniaxial stress situation well. Figure 16 shows the evolution curves of the fracture damage variables $D_{1\mathrm{R}}$ and $D_{3\mathrm{R}}$. For axial tensile, microcracks begin to develop in small strains and gradually expand until the specimen enters a state of destruction. For axial compression, microcracks begin to develop at a strain of 400 × 10⁻⁶ and gradually expand, and $D_{3\mathrm{R}}$ is 0.47 at a strain ${\epsilon }_3$ of 1800 × 10⁻⁶. Figure 17 shows the evolution curves of evolutionary factors $E_{1\mathrm{v}}$ and $E_{3\mathrm{v}}$. For axial tension and axial compression, the value of $E_{\mathrm{v}}$ ranges from 0 to 1 and the evolution curve is S-shaped.

5.2. Numerical Simulation of Biaxial Forces

In practice, hydraulic structures are not only under simple uniaxial loads, but also under complex multiaxial loads. Wang et al. [2] investigated the stress characteristics of fully-graded concrete under different stress ratios through experiments. Referring to the relevant data of Wang et al. [2], the mesoscopic parameters of the SDM were determined by the ABC algorithm and then combined with the FORTRAN subroutine to perform the biaxial forces state analysis. Based on the boundary conditions in Figure 14b, the biaxial compression state is realized by applying negative displacement in the Y direction and negative displacement in the X direction, and the analysis is carried out for four working conditions with stress ratios of 0, 0.25, 0.5 and 1. It should be noted that loading is controlled by the strain ratio during the numerical simulation, but Wang et al. [2] controlled the loading by the stress ratio during the experiment. Therefore, the corresponding stress ratio needs to be obtained by trial calculation. For stress ratios $\alpha$ of 0 and 1, the strain ratios and stress ratios are identical. For the stress ratio of 0.25, the X-direction to Y-direction strain ratio of 0.053 is chosen, and the calculated stress ratio is in the range of [0.2497, 0.2503]; for the stress ratio of 0. 5, the X-direction to Y-direction strain ratio of 0.34 is chosen, and the calculated stress ratio is in the range of [0.4994, 0.5068].

Figure 18 shows the comparison between the simulation results and the test results for the four working conditions. It can be seen that the nominal stress–strain curves calculated by simulation are in good agreement with the test curves. The peak stresses corresponding to the four conditions are 20.67 MPa, 28.51 MPa, 27.83 MPa and 26.57 MPa, respectively. The condition with stress ratio of 0 is an axial compression state. For the other three conditions, the peak stress is found to decrease with an increase in the stress ratio. In order to further clarify the change rule of biaxial forces state, biaxial compression-compression, biaxial tension-compression and biaxial tension-tension and other working conditions are analyzed, the biaxial relative strength envelope in the stress space is plotted, as shown in Figure 19. The evolution curves of $D_{3\mathrm{R}}$ and $E_{3\mathrm{v}}$ for different stress ratios are shown in Figure 20. For the stress ratio $\alpha$ of 0, the evolution curves are consistent with the characteristics of axial compression.

5.3. Numerical Simulation of Triaxial Forces

The mechanical properties of fully-graded concrete under the state of triaxial forces are complex. There are relatively few valid experimental results in the literature, and the experimental studies are mainly focused on the triaxial compression-compression-compression state. A numerical simulation of the constitutive behavior of fully-graded concrete under triaxial compression-compression-compression condition is carried out using the model of this paper. Strain-controlled proportional loading is used in two modes: ${\epsilon }_1={\epsilon }_2>{\epsilon }_3$ and ${\epsilon }_1>{\epsilon }_2={\epsilon }_3$. For the two modes, six working conditions are analyzed, and the nominal stress–strain curves corresponding to different working conditions are shown in Figure 21. It can be seen that the peak stress and peak strain are different for different strain ratios, which further shows the complexity of the triaxial stress state. The evolution curves of $D_{3\mathrm{R}}$ and $E_{3\mathrm{v}}$ under different working conditions of the two modes are given in Figure 22 and Figure 23, respectively, and the change rule of $D_{3\mathrm{R}}$ and $E_{3\mathrm{v}}$ reveals the mesoscopic mechanism of the macroscopic stress–strain curve.

6. Discussion

The numerical simulation method proposed in this paper can effectively simulate the uniaxial, biaxial and triaxial stress characteristics of concrete.

$D_{\mathrm{R}}$ is related to the microcrack density and reflects the microcrack sprouting and expansion process. In Figure 16, under the same strain, axial compression has a lower degree of optimization and adjustment of the microstructural force skeleton than axial tension, with a relative lag in the critical state and a higher deformation capacity. $E_{\mathrm{v}}$ is related to yield damage and reflects the degree of optimized adjustment of the concrete microstructure stress skeleton. In the uniform damage stage, $E_{\mathrm{v}}$ varies from 0 to 1 and plays a decisive role in the whole process. In Figure 17, the critical state is reached when $E_{\mathrm{v}}$ = 1, which means that the effective stress skeleton of the microstructure is adjusted to the optimum, the potential bearing capacity of the material is played to the limit, and enters the damage stage characterized by damage localization. Under the same strain, the degree of optimization and adjustment of microstructure effective stress skeleton for axial compression is lower than that for axial tension.

As the main part of the mesoscopic damage cumulative evolution, the uniform damage part of the nominal stress–strain curve consists of an ascending section and a partially descending section. In this section, the nominal stress first increases and then decreases; the effective stress increases monotonically, with the maximum value occurring in the critical state, after which the concrete enters the localized damage stage. In Figure 19, the biaxial relative strength envelope conforms to conventional test characteristics, which also indicates the rationality of the numerical analysis results. In Figure 20, for the biaxial compression-compression state with stress ratio $\alpha$ varying from 0.25 to 1, under the same strain, the larger the stress ratio, the larger the value of $D_{3\mathrm{R}}$, which indicates that the density of microcracks is greater. Meanwhile, the value of $E_{3\mathrm{v}}$ is also increased by the increase of the stress ratio, which indicates that the critical state of the concrete is relatively forward when the stress ratio is larger, resulting in a reduction of its deformation capacity. The characteristics of the $D_{3\mathrm{R}}$ and $E_{3\mathrm{v}}$ curves for different stress ratios correspond to the nominal stress–strain curves.

From the simulation results, it can be seen that the triaxial stress state is complex. For the mode ${\epsilon }_1={\epsilon }_2>{\epsilon }_3$, the peak stress and peak strain of ${\sigma }_3$-${\epsilon }_3$ curve increase as the strain in the other two directions change from tensile strain to compressive strain, and the related $D_{3\mathrm{R}}$ and $E_{3\mathrm{v}}$ curves show regular changes. For the mode ${\epsilon }_1>{\epsilon }_2={\epsilon }_3$, the peak stress and peak strain of ${\sigma }_3$-${\epsilon }_3$ curve increase as the tensile strain ${\epsilon }_1$ decrease, and the related $D_{3\mathrm{R}}$ and $E_{3\mathrm{v}}$ curves also show regular changes. Therefore, it can be concluded that the numerical simulation method proposed in this study is able to effectively calculate the mechanical properties of fully-graded concrete under a triaxial stress state.

7. Conclusions

This paper studies the method of numerical implementation of SDM by proposing a numerical simulation model through the FORTRAN subroutine. The reasonableness of the proposed model was verified through the uniaxial and biaxial tests. The main findings and conclusions are summarized as follows:

1.

The axial tensile and compression tests of fully-graded concrete are the basis for establishing SDM. The determination of ${\epsilon }_a$, ${\epsilon }_h$, ${\epsilon }_b$ and $H$ parameters is significant for establishing the constitutive relationship of fully-graded concrete. The ABC method is able to converge to the optimal parameters very quickly, and the nominal stress–strain curves predicted by the model are in good agreement with the experimental curves.

2.

The numerical simulation model of SDM was proposed through FORTRAN programming. In the calculation process, the impact of damage is reflected by reducing stiffness. The whole process is to embed the FORTRAN program into the finite element software to analyze the damage evolution characteristics of the structure.

3.

Under the same strain, the degree of optimization and adjustment of microstructure effective stress skeleton for axial compression is lower than that for axial tension. For the biaxial compression-compression state with stress ratio $\alpha$ varying from 0.25 to 1, under the same strain, the larger the stress ratio, the density of microcracks is greater; meanwhile, the critical state of the concrete is relatively forward when the stress ratio is larger, resulting in a reduction of its deformation capacity. The triaxial stress state is complex, and the stress characteristics are different under different strain ratios.

The numerical simulation method proposed in this paper is able to apply the SDM of fully-graded concrete to finite element analysis, laying the foundation for refined finite element analysis of hydraulic structures, mesoscopic damage mechanism study and structure design optimization. According to actual engineering requirements, the numerical simulation of SDM considering freeze–thaw and dynamic strain rates will be researched in the near future to further reveal the durability and seismic performance of concrete structures.