Experiment and phase-field simulation of uniaxial compression of ultra-high-performance concrete with coarse aggregate

Abstract

Ultra-high-performance concrete with coarse aggregate (UHPC-CA) is a newly developed material. To explore the influence of aggregate volume fraction and maximum aggregate particle size on the compressive strength of UHPC-CA, a batch of UHPC-CA numerical samples as well as some experiments is carried out. The phase-field method is introduced to simulate the uniaxial compressive strength, and the simulation results show good agreement with experimental results. Both the simulation results and experiment results show that the compressive strength decreases with the aggregate volume fraction and increases with the maximum aggregate particle size. A size effect model of UHPC-CA is established by using its aggregate volume fraction and the maximum aggregate particle size.

Keywords

Ultra-high-performance concrete with coarse aggregate phase-field method compression strength Monte Carlo simulation stress–strain curves

Introduction

Ultra-high-performance concrete (UHPC) is a new generation of concrete developed in the past 30 years (Yoo & Banthia, 2016; Richard and Cheyrezy, 1995), among which the ultra-high-performance concrete with coarse aggregate (UHPC-CA) has a wide application prospect in engineering (Amin et al., 2020; Kim et al., 2018; Li et al., 2021; Zavickis et al., 2020). Adding the coarse aggregates to the UHPC, UHPC-CA has a high economy and environmental protection while only 20% of the compressive strength is lost (Jiang et al., 2019; Teng et al., 2019). However, the coarse aggregates increase the meso-heterogeneity of the material, and the meso-heterogeneity has a pronounced influence on the macroscopic property of concrete (Sahmaran et al., 2009; Wille and Boisvert-Cotulio, 2015; Sobuz et al., 2016). Many experiments were carried out to explore the influence of aggregate on the mechanical property of UHPC-CA.

Reddy and Perumal (2020a, 2020b) put the coarse aggregate with diameters of 6 mm and 10 mm into UHPC. The results show that adding coarse aggregate slightly reduces the mechanical property of UHPC. Li et al. (2019) conducted experiments by controlling the size of coarse aggregates of UHPC-CA and found that when the maximum size of coarse aggregate increases from 8 mm to 25 mm, the reduction of the compressive strength decreases from 14% to 12%. Deng et al. (2020) designed a series of experiments and concluded that adding coarse aggregates increase the bulk density and form a stronger “skeleton,” moreover, compared with UHPC, the elastic modulus of UHPC-CA increases slightly, but the deformability and toughness of UHPC-CA decrease. Huang and Li (2018) studied the influence of the coarse aggregate content and steel bar on the mechanical property of UHPC-CA. The results showed that, with the increase of the coarse aggregate content, the compressive strength of UHPC increases first and then decreases, and the elastic modulus increases almost linearly. Peng et al. (2012,2013) designed a series of experiments to study the high-temperature resistance and explosive peeling behavior of UHPC and UHPC-CA. The results showed that UHPC-CA has better high-temperature resistance than UHPC, and the coarse aggregates have a beneficial effect on reducing explosion spalling of UHPC. Ma et al. (2004) found through experiments that UHPC and UHPC-CA exhibit similar behaviors under compressive stress. The elastic modulus and the peak strain are related to the stiffness of the aggregate used. Besides that, the lower aggregate volume fraction results in a lower auto-shrinkage of UHPC-CA.

In addition to the study of aggregate content, some scholars also focus on the types of coarse aggregates. UHPC-CA with different types of coarse aggregates is designed and tested. For instance, Liu and Wei (2021) used the basalts and the calcined bauxite as coarse aggregates of UHPC-CA. The results showed that UHPC-CA has a lower resistance than UHPC without coarse aggregates. Selecting the basalts with particle sizes of 5–15 mm as coarse aggregate, Lizhi (2020) concluded from a series of experiments that adding aggregates can improve the flexural toughness of UHPC-CA.

Almost all studies show that the addition of coarse aggregates affects the mechanical property of UHPC, but how the coarse aggregates affect the mechanical property is still unclear. For this reason, Lv et al. (2020) discussed the internal mechanism of UHPC-CA through experiments and mercury intrusion scanning and concluded that the type and maximum particle size of coarse aggregate have a significant impact on the workability and mechanical property of UHPC-CA.

Obviously, due to the limitations of experiments in the laboratory, it is not enough to analyze the internal component effects of UHPC-CA only by experimental approaches. Thus, some scholars try to explore the property of UHPC-CA by numerical simulations. Traditional concrete compression failure simulation methods include the element deletion method (Liu et al., 2014; Song et al., 2008), interface element method (Song et al., 2008; Xu and Needleman 1994; Camacho and Ortiz, 1996; Wang et al., 2015), and extended finite element method (XFEM) (Rybczyński et al., 2020). As is known to all, the element deletion method and interface element method have an obvious mesh dependence (Song et al., 2008). The extended finite element method (XFEM) allows the crack to occur within the element (Rybczyński et al., 2020; Zhang and Teng, 2014) so that the entire process of crack propagation can be traced. However, XFEM is difficult to treat the multi-crack problems (Song et al., 2008; Xu and Needleman 1994; Camacho and Ortiz, 1996; Rybczyński et al., 2020; Xu et al., 2014). A large number of complex cracks will produce in the compression test of UHPC-CA. The traditional finite element method cannot well simulate this phenomenon (Liu et al., 2014; Song et al., 2008; Xu and Needleman 1994; Camacho and Ortiz, 1996; Wang et al., 2015; Rybczyński et al., 2020). The phase-field method (Miehe et al., 2010; Wu, 2017; 2020; Bilgen et al., 2019), which has no mesh dependence and excellent performance in crack propagation, is undoubtedly more suitable to simulate the compression of UHPC-CA.

In this paper, the phase-field method is introduced to simulate the uniaxial compressive strength of UHPC-CA. The simulation results are used to establish a size effect model of UHPC-CA in which the relationship between UHPC-CA compressive strength and the aggregates is described.

The outline of the paper is as follows. In the section Model and method, the phase-field method, the random aggregate model of UHPC-CA, and the preparation of experimental samples are introduced. The results of experiments and numerical simulations are shown in the Results section. In the section Discussion, the relationship between UHPC-CA compressive strength and the aggregates is established, and the size effect of UHPC-CA compressive strength is further discussed. The main conclusions are drawn in the section Conclusion.

Model and method

Phase-field method

The phase-field method is a mathematical tool to predict the evolution of the microstructure (Bourdin et al., 2000). The key step of the phase-field fracture simulation is to determine the free energy of the target system and to construct the free energy function of the system. According to the microstructure of the material evolving in the direction of minimizing the free energy, the change of the structure is predicted by the phase-field variables.

The total potential energy of concrete fracture is composed of three parts (Bourdin et al., 2000): volume storage energy, crack surface energy, and external force work

\begin{matrix} Π = Y + U + W = \int_{Γ} G_{c} d Γ + \int_{V} ψ d V \\ - (\int_{S_{σ}} T_{i} u_{i} d S + \int_{V} f_{i} u_{i} d V) \end{matrix}

(1)

where П is the total fracture potential energy, Y is the surface energy of the crack, U is the volume storage energy, W is the external force work, G_c is the Griffith critical fracture energy constitutive threshold, ψ is the volume storage energy density, V is the accumulated volume, Γ represents for the crack surface, S_σ is the stress boundary, u_i is the displacement, T_i is surface forces, and f_i is body forces.

Based on the fracture variational theory (Bourdin et al., 2000), a scalar variable ϕ is defined in the interval [0, 1] as the crack phase-field (order parameter). When ϕ = 1, it means that the material is destroyed completely, and when ϕ = 0, it means that the material is intact. The crack surface energy in the form of phase-field can be expressed as

Y (ϕ) = \int_{Γ} G_{c} d Γ = G_{c} \int_{V} γ (ϕ) d V

(2)

For a two-dimensional problem, the form of the crack surface density function per unit volume is

γ (ϕ) = \frac{1}{2 l_{0}} (ϕ^{2} + l_{0}^{2} \frac{\partial ϕ}{\partial x_{i}} \frac{\partial ϕ}{\partial x_{i}}) i = 12

(3)

where l₀ is a length scale parameter that controls the crack “diffusion.”

The volumetric storage energy density ψ is determined by the strain tensor ε_ij and the order parameter ϕ

ψ (ε_{ij}, ϕ) = g (ϕ) H (ε_{ij}) = \frac{1}{2} g (ϕ) ε_{ij} D_{ijkl} ε_{ij}

(4)

where H(ε_ij) is the elastic strain energy density, D_ijkl is the elastic tensor, ε_ij is the strain tensor, and g(ϕ) is the stress degradation function

g (ϕ) = {(1 - ϕ)}^{2} + k

(5)

where k≈10⁻⁶ so that the stiffness matrix maintains good non-singularity during a fracture simulation.

To consider both tensile and compression failures of UHPC-CA, a unified phase-field model (Wu et al., 2020) is introduced, in which a modified reference energy y in terms of equivalent effective stress ${\bar{σ}}_{eq}$ is,

\begin{matrix} y = \frac{1}{2} ε_{ij} D_{ijkl} ε_{ij} = \frac{1}{2 E_{0}} {\bar{σ}}_{eq}^{2}, {\bar{σ}}_{eq} \\ = \frac{1}{1 + β_{c}} (β_{c} 〈 {\bar{σ}}_{1} 〉 + \sqrt{3 {\bar{J}}_{2}}) \end{matrix}

(6)

where y is the elastic strain energy density, E₀ is the initial elastic stiffness, the Macaulay brackets <•> are defined as <x> = max{x,0}, ${\bar{σ}}_{1}$ and ${\bar{J}}_{2}$ denote the first major principle value of the effective stress tensor ${\bar{σ}}_{ij} = D_{ijkl} ε_{kl}$ and the second invariant of the deviatoric one ${\bar{s}}_{ij} = {\bar{σ}}_{ij} - {\bar{σ}}_{ii} / 3$ . The material parameter β_c = f_c/f_t−1 is related to the ratio of the uniaxial compressive strength f_c to tensile one f_t, and the β_c is set to 10 (Wu 2017).

From equation (5) and equation (6), the volume storage energy U can be expressed as

U (ε_{ij}, ϕ) = \int_{V} ψ (ε_{ij}, ψ) d V = \int_{V} [{(1 - ϕ)}^{2} + k] y d V

(7)

When surface forces are not considered, by substituting equation (2), equation (3), and equation (7) into equation (1), the total fracture potential energy can be expressed as

\begin{matrix} Π (u_{i}, ϕ) = \int_{V} [{(1 - ϕ)}^{2} + k] y d V \\ + \int_{V} \frac{G_{c}}{2 l_{0}} (ϕ^{2} + l_{0}^{2} \frac{\partial ϕ}{\partial x_{i}} \frac{\partial ϕ}{\partial x_{i}}) d V - \int_{V} f_{i} u_{i} d V \end{matrix}

(8)

The first-order variation of the total fracture potential energy is

\begin{matrix} δ Π (u_{i}, φ) = \frac{\partial Π}{\partial u_{i}} δ u_{i} + \frac{\partial Π}{\partial φ} δ φ \\ = \int_{V} [{(1 - φ)}^{2} + k] ε_{kl} D_{ijkl} δ ε_{ij} d V - \int_{V} f_{i} δ u_{i} d V \\ + \int_{V} [- 2 (1 - φ) y + \frac{G_{c}}{l_{0}} φ] δ φ d V + \int_{V} G_{c} l_{0} \frac{\partial φ}{\partial x_{i}} \frac{\partial δ φ}{\partial x_{i}} d V \\ = \int_{\partial V} σ_{ij} n_{j} δ u_{i} d \partial V - \int_{V} (σ_{ij, j} + f_{i}) δ u_{i} d V \\ + \int_{\partial V} G_{c} l_{0} \frac{\partial φ}{\partial x_{j}} n_{j} δ φ d \partial V \\ + \int_{V} [- G_{c} l_{0} \nabla^{2} φ + (\frac{G_{c}}{l_{0}} + 2 y) φ - 2 y] δ φ d V \end{matrix}

(9)

where n_j is the direction cosine of the outer normal of the boundary ∂V, σ_ij =[(1–ϕ)²+k] •D_ijkl•ε_kl,

Fracture variational theory believes that any dynamic growth state of cracks at any position should minimize the total potential energy. That is, δП = 0 holds for any δu_i and δϕ in equilibrium, the governing equation of the fracture problem is calculated by equation (9)

σ_{ij, j} + f_{i} = 0

(10)

The phase-field evolution equation of the fracture problem is

- G_{c} l_{0} \nabla^{2} ϕ + (\frac{G_{c}}{l_{0}} + 2 y) ϕ - 2 y = 0

(11)

The boundary conditions of force are

σ_{ij} n_{j} = 0

(12)

The natural boundary conditions of the phase-field are

\frac{\partial ϕ}{\partial x_{j}} n_{j} = 0

(13)

The phase-field fracture variational theory inherits and develops the traditional Griffith theory, and solves the problems of crack initiation, propagation path, and instability bifurcation that cannot be solved by Griffith theory, providing a new perspective for the study of fracture problems.

The phase-field simulation is performed through the secondary development of commercial software. In addition to the two degrees of freedom of displacement, a phase-field degree of freedom is added in FEM (Miehe et al., 2010).

Numerical samples

A two-dimensional model is established, in which UHPC-CA is considered to be a three-phase composite material composed of aggregate, cementitious paste, and interfacial transition zone (ITZ). Both the length and width of the UHPC-CA numerical sample are set as L = 100 mm, and the coarse aggregate is assumed to be round. The uniform horizontal compressive stress q is distributed on the top boundary, the type of load is displacement loading and the total displacement loading is set as 0.6 mm, and the loading step is set as 100. The geometric shape and boundary conditions of the numerical model are shown in Figure 1(a).

Figure 1.

Numerical sample: (a) Geometry and boundary conditions, (b) mesh of the model.

The diameter of the coarse aggregate (ϕ) are set as ϕ₁ = 4 mm, ϕ₂ = 8 mm, ϕ₃ = 13 mm and ϕ₄ = 16 mm respectively. The Monte Carlo method is used to generate the random aggregate model of UHPC-CA, and the number of the aggregate of different sizes is determined according to the actual mix proportion. A four-node plane strain square element is used to mesh the UHPC-CA numerical model. There are three types of elements: aggregate element, cementitious paste element, and ITZ element. To reduce the influence of network density, all samples are divided into 320 × 320 elements, and all elements have the same size, as shown in Figure 1(b). To simplify the model, and considering that the main analysis objective is the compressive strength f_c which is little affected by steel fibers, so the samples prepared in this article are not mixed with steel fibers.

To explore the influence of the aggregate volume fraction (P) and gradation (G) of the coarse aggregate in UHPC-CA samples on its compressive strength, a batch of UHPC-CA numerical samples with different P and G are prepared. There are four types of G, as shown in Table 1.

Table 1.

Size and gradations of coarse aggregates.

Aggregate	Radius (mm)	G1	G2	G3	G4
ϕ ₁	4	√	√	√	√
ϕ ₂	8	—	√	√	√
ϕ ₃	13	—	—	√	√
ϕ ₄	16	—	—	—	√

An orthogonal experiment is designed, as shown in Figure 2. The aggregate volume fraction P of the orthogonal experiments ranges from 20% to 40% and the gradation G of the orthogonal experiments ranges from G1 (ϕ₁) to G4 (ϕ₁+ϕ₂+ϕ₃+ϕ₄). The detailed parameters are shown in Table 2.

Figure 2.

Numerical samples of different P and G.

Table 2.

Summary of numerical samples.

Sample IDs	Size (m²)	Aggregate volume fraction, %	Aggregate gradation	Number of aggregates
Sample IDs	Size (m²)	Aggregate volume fraction, %	Aggregate gradation	ϕ ₁	ϕ ₂	ϕ ₃	ϕ ₄
Pf20-G1	0.1 × 0.1	20	G1	159	0	0	0
Pf20-G2	0.1 × 0.1	20	G2	66	23	0	0
Pf20-G3	0.1 × 0.1	20	G3	36	15	6	0
Pf20-G4	0.1 × 0.1	20	G4	22	9	5	3
Pf25-G1	0.1 × 0.1	25	G1	199	0	0	0
Pf25-G2	0.1 × 0.1	25	G2	82	29	0	0
Pf25-G3	0.1 × 0.1	25	G3	47	17	8	0
Pf25-G4	0.1 × 0.1	25	G4	28	11	6	4
Pf30-G1	0.1 × 0.1	30	G1	239	0	0	0
Pf30-G2	0.1 × 0.1	30	G2	99	35	0	0
Pf30-G3	0.1 × 0.1	30	G3	53	20	10	0
Pf30-G4	0.1 × 0.1	30	G4	33	13	7	5
Pf35-G1	0.1 × 0.1	35	G1	279	0	0	0
Pf35-G2	0.1 × 0.1	35	G2	116	41	0	0
Pf35-G3	0.1 × 0.1	35	G3	67	24	11	0
Pf35-G4	0.1 × 0.1	35	G4	38	15	8	6
Pf40-G1	0.1 × 0.1	40	G1	319	0	0	0
Pf40-G2	0.1 × 0.1	40	G2	132	47	0	0
Pf40-G3	0.1 × 0.1	40	G3	73	27	13	0
Pf40-G4	0.1 × 0.1	40	G4	44	17	9	7

Experimental samples

To verify the numerical model, a comparative physical experiment is designed for the numerical model with an aggregate volume fraction of 30% or a gradation of G3. Among them, the composition of UHPC-CA is shown in Table 3, and frosted glass beads with the same particle size as the numerical experiment are used as the coarse aggregate. For each numerical sample, three comparative experimental samples are set. In addition to the above samples, three samples without coarse aggregate (UHPC-NCA) are also prepared to obtain the material parameters of the concrete. All the experimental samples are listed in Table 4.

Table 3.

Composition of UHPC-CA.

Cement	Silica fume	Water	Aggregate
1	0.266	0.1	20–40% by vol

Table 4.

Summary of experimental samples.

Number	Samples ID	Size (m³)	Aggregate volume fraction	Aggregate gradation	Number of aggregates at all gradation				No. of samples
Number	Samples ID	Size (m³)	Aggregate volume fraction	Aggregate gradation	ϕ1	ϕ2	ϕ3	ϕ4	No. of samples
1	Pf20-G3	0.1 × 0.1 × 0.1	20%	G3	3311	171	37	0	3
2	Pf25-G3	0.1 × 0.1 × 0.1	25%	G3	4138	214	47	0	3
3	Pf30-G3	0.1 × 0.1 × 0.1	30%	G3	4966	257	56	0	3
4	Pf35-G3	0.1 × 0.1 × 0.1	35%	G3	5794	300	66	0	3
5	Pf40-G3	0.1 × 0.1 × 0.1	40%	G3	6621	343	75	0	3
6	Pf30-G1	0.1 × 0.1 × 0.1	30%	G1	8952	0	0	0	3
7	Pf30-G2	0.1 × 0.1 × 0.1	30%	G2	6330	328	0	0	3
8	Pf30-G4	0.1 × 0.1 × 0.1	30%	G4	4476	231	50	13	3
9	UHPC-NCA	0.1 × 0.1 × 0.1	/	/	/	/	/	/	3
Total									13

Some of the prepared samples are shown in Figure 3(a). A uniaxial compression test is carried out on this batch of samples. The experiment adopts displacement-based loading the same as numerical simulation, the total load is set to 0.6 mm, and the load is completed in 100 steps. The specific experimental setup is shown in Figure 3(b).

Figure 3.

Uniaxial compression tests of UHPC-CA: (a) Samples of UHPC-CA, (b) test setup.

Numerical parameters

The elastic modulus of the cementitious paste is calculated from the uniaxial compression test results of the UHPC-NCA sample prepared in the section Experimental samples. The stress–strain curves of the UHPC-NCA sample are shown in Figure 4. The modulus of elasticity is calculated as follows (Bujnakova et al., 2019)

E = \frac{Δ σ}{Δ ε} = \frac{σ_{B} - σ_{A}}{ε_{B} - ε_{A}}

(14)

where, σ_A and ε_A are the stress and strain of point A in Figure 4, σ_B and ε_B are the stress and strain of point B in Figure 4, point A and point B should be before 1/3 ultimate strain of the stress-strain curve (Bujnakova et al., 2019).

Figure 4.

Average stress–strain curve of UHPC sample in a uniaxial compression test.

The Poisson’s ratio of cementitious paste is set as 0.2, and the phase-field parameters G_c of cementitious paste is set as 49.34 Pa·m⁻¹ (Wu et al., 2020). The corsage aggregate used in the experiment is ground glass beads, of which the Poisson’s ratio and the elastic modulus are supported by the manufacturer and be set as 0.25 and 80 GPa. The G_c of corsage aggregate is set to 144.00 Pa·m⁻¹. The elastic modulus and G_c of ITZ are determined through a series of numerical simulations based on the shooting method and be set as 22.6 Gpa and 18.10 Pa·m⁻¹ (Kotova et al., 2021), Poisson’s ratio is the same as cement. l₀ is related to the crack width of the material failure (Wu & Nguyen, 2018). Several numerical simulations are carried out and when l₀ of the three materials are taken as 0.0015 m, 0.001 m and 0.0008 m, the cracks shown by the numerical model are consistent with the experiments. All material parameters are summarized in Table 5.

Table 5.

Material parameters and phase-field parameters in numerical samples.

Material parameters	Aggregate	Cementitious paste	ITZ
Modulus of elasticity E (GPa)	80.0	45.2	22.6
Poisson’s ratio ν (−)	0.25	0.2	0.2
Parameter G_c (Pa·m⁻¹)	144.00	49.34	18.10
Parameter l₀ (m)	0.0015	0.001	0.0008

Results

Compare of experiment and numerical simulation

The results of an experimental sample (Pf30-G3) under the uniaxial compression are shown in Figure 5. The evolution simulated for a typical numerical sample of Pf30-G3 is illustrated in Figure 6. Comparing the experimental results with the numerical results, it is not difficult to see that the crack positions are all consistent, and the cracks generally occur first in the ITZ between the cementitious paste and the coarse aggregate, and gradually spread along the gap between the particles.

Figure 5.

Typical results of an experimental sample (Pf30-G3) under the uniaxial compression.

Figure 6.

Typical results of a numerical sample (φ = 0 means the material is intact, φ = 1 means completely destroyed): (a) Displacement load u = 0 mm, (b) u = 0.270 mm, (c) u = 0.324 mm, (d) u = 0.384 mm, (e) u = 0.402 mm, (f) u = 0.414 mm, (g) u = 0.432 mm, (h) u = 0.600 mm.

Extensive Monte Carlo simulations of numerical samples of different P and G (P = 20%–40%, G = G1∼G4, see Figure 2) with different aggregates distribution were carried out. The samples have the same aggregate volume fraction and aggregate gradation, but different aggregate distribution locations (Figures 1 and 2). For each type of UHPC-CA, 50 samples were modeled to ensure that the results are statistical convergence (Figure 7). The mean compressive strength of the UHPC-CA model for each P and G was obtained by averaging the Monte Carlo simulation results (Rybczyński et al., 2020).

Figure 7.

Samples of Monte Carlo simulation (take Pf30-G3 as an example).

The stress–strain curves of the experiment and numerical simulation with the mean curve are shown in Figure 8 (taking Pf30-G3 as an example). The mean compressive strength is compared with experimental strength, as shown in Table 6. The physical experiment was only carried out for aggregate volume fraction of 30% and aggregate gradation of G3, while the numerical experiments are carried out for the models of all gradations and all aggregate volume fractions.

Figure 8.

Results of the experiment and Monte Carlo simulation (taking Pf30-G3 as an example).

Table 6.

Compressive strength of simulation and experiment.

Sample ID	Mean compressive strength of Monte Carlo simulation (MPa)	Experimental compressive strength (MPa)				Error
Sample ID	Mean compressive strength of Monte Carlo simulation (MPa)	Exp1	Exp2	Exp3	Average	Error
Pf20-G1	120.79	—	—	—	—
Pf20-G2	125.85	—	—	—	—
Pf20-G3	127.12	127.25	129.78	131.56	129.46	1.84%
Pf20-G4	128.51	—	—	—	—
Pf25-G1	117.87	—	—	—	—
Pf25-G2	123.40	—	—	—	—
Pf25-G3	125.26	125.51	127.72	126.35	126.53	1.01%
Pf25-G3	126.70	—	—	—	—
Pf30-G1	115.30	113.05	114.19	109.15	112.12	2.70%
Pf30-G2	120.15	116.9	120.83	118.16	118.63	1.27%
Pf30-G3	122.51	122.21	123.35	117.25	120.94	1.30%
Pf30-G4	123.82	125.43	122.85	127.85	125.38	1.26%
Pf35-G1	113.72	—	—	—	—
Pf35-G2	118.16	—	—	—	—
Pf35-G3	120.21	120.33	118.84	118.21	119.13	0.90%
Pf35-G4	121.78	—	—	—	—
Pf40-G1	111.59	—	—	—	—
Pf40-G2	116.22	—	—	—	—
Pf40-G3	118.46	117.52	119.33	113.15	116.67	1.51%
Pf40-G4	119.53	—	—	—	—
Average						1.47%

It can be seen from Figure 8 that the stress–strain curve obtained from the experiment is within the stress–strain curve domain of 50 kinds of numerical simulations. In the ascending and descending sections of the stress–strain curve, the experiment and numerical simulation are similar. At the end of the stress–strain curve, the experiment results show that the samples still have residual stress, while the numerical simulation result shows that the stress is zero. This may be because the elements of the phase-field method will lose all the bearing capacity after failure.

The mean compressive strength of the Monte Carlo numerical simulation is compared with the experimental compressive strength f_c in Table 6. It can be seen that the average error between the results of the numerical simulation and the experiment is 1.47%. It is small enough to prove that the results of our numerical simulation are credible.

Influence of aggregate volume fraction and gradation

Based on the numerical simulation results in 3.1, the influences of P and G on the compressive strength of UHPC-CA are explored in this section.

Influence of aggregate volume fraction P

The mean stress–strain curves of UHPC-CA samples with different P and same G are compared in Figure 9. The compressive strength f_c of different P is compared in Figure 10. It can be seen that the uniaxial compressive strength of concrete decreases linearly with the aggregate volume fraction P. It is consistent with experimental observations, as shown in Table 6.

Figure 9.

Influence of aggregate volume fraction on the uniaxial compression compressive strength of UHPC-CA: (a) mean stress-strain curves of G1, (b) mean stress–strain curves of G2, (c) mean stress–strain curves of G3, (d) mean stress–strain curves of G4.

Figure 10.

Fitting curve of compressive strength with aggregate volume fraction.

The reason for this phenomenon may be that the addition of coarse aggregate causes defects in the ITZ of UHPC-CA, and the increase of defects leads to the weakening of the compressive strength of UHPC-CA.

In addition, although coarse aggregate has a higher modulus of elasticity, and has a strengthening effect on the elastic modulus of UHPC, the addition of coarse aggregate also causes the increase in the ITZ, which has a greater impact on the modulus of UHPC-CA. Therefore, the addition of coarse aggregate will reduce the elastic modulus of UHPC-CA.

It can be seen from Figure 10 that the trend of compressive strength f_c with aggregate volume fraction P is an approximately linear negative correlation. The fitting curve between f_c and P is drawn in Figure 10. For the same G, the fitting expression is as follows

f_{c} = A_{f} + B_{f} P

(15)

where A_f and B_f are fitting parameters and are shown in Table 7, f_c is the uniaxial compressive strength of the UHPC-CA samples.

Table 7.

Fitting parameters.

	G1	G2	G3	G4	Average
Intercept (A_f)	130.28369	135.53526	136.71901	138.30116	—
Slope (B_f)	−0.47107	−0.48002	−0.48747	−0.46784	−0.47660
Adj. R-square	0.98364	0.97429	0.9883	0.99504	—

It can be seen from Table 7 that the slopes of several fitting curves are relatively close but the intercepts are different. It can be considered that the B_f is a constant and the A_f decreases with G. It can be concluded that adding aggregate decreases the compressive strength of UHPC-CA.

Influence of aggregate gradation G

For UHPC-CA samples with the same aggregate volume fraction P, the maximum aggregate particle size (ϕ_m) increases with the aggregate gradation G, as shown in Table 1. In this paper, the relationship between ϕ_m and f_c is discussed to reveal the influence of aggregate gradation G on the f_c of UHPC-CA.

For each group of UHPC-CA numerical models with the same P and different G, the uniaxial compression stress-strain curves are shown in Figure 11. The f_c of different ϕ_m is compared in Figure 12(a). It can be seen from Figure 11 that the elastic modulus (slopes of ascending sections of the curve) increases with the G. Also, it can be seen from Figure 12(a) that f_c increases with ϕ_m.

Figure 11.

Influence of aggregate gradation on the uniaxial compression compressive strength of UHPC-CA: (a) mean stress-strain curves of P = 20%, (b) mean stress–strain curves of P = 25%, (c) mean stress–strain curves of P = 30%, (d) mean stress–strain curves of P = 35%, (e) mean stress–strain curves of P = 40%.

Figure 12.

Relationship between the compressive strength and the ϕ_m: (a) compressive strength with ϕ_m, (b) fitting curve of compressive strength with 1/ϕ_m.

It is not difficult to see from Figure 12(a) that the uniaxial compressive strength of the UHPC-CA sample is positively correlated with the ϕ_m, and the curve of UHPC-CA uniaxial compressive strength and 1/ϕ_m is fitted as shown in Figure 12(b). The fitting relationship can be described as

f_{c} = \frac{C_{f} + D_{f}}{ϕ_{m}}

(16)

where C_f and D_f are fitting parameters and are shown in Table 8.

Table 8.

Fitting parameters.

	20%	25%	30%	35%	40%	Average
Intercept (C_f)	131.44252	129.1239	126.0732	123.68582	121.73591	—
Slope (D_f)	−40.27068	−43.22993	−41.75005	−40.97903	−41.01771	−41.55742
Adj. R-square	0.97868	0.98935	0.98203	0.9689	0.98771	—

It can be seen from Table 8 that D_f is approximate a constant, and C_f is a value related to P. To obtain a unified f_c expression, the relationship between C_f and P is fitted to Figure 13(b), and for the section Influence of aggregate volume fraction P, the relationship between A_f and ϕ_m is fitted in Figure 13(a).

Figure 13.

Relationship between the fitting parameter and the aggregate: (a) C_f and P (b) A_f and ϕ_m.

The fitting relationship is as follows

A_{f} = α_{f}^{1} + β_{f}^{1} / ϕ_{m}, C_{f} = α_{f}^{2} + β_{f}^{2} P

(17)

where $α_{f}^{1}$ = 140.41447, $α_{f}^{2}$ = 141.32305, $β_{f}^{1}$ = −41.55742, and $β_{f}^{2}$ = −0.49703 are fitting parameters; it can be seen from Figure 13 that $α_{f}^{1}$ ≈ $α_{f}^{2}$ , $β_{f}^{1} \approx B_{f}$ and $β_{f}^{2} \approx D_{f}$ .

By coalesce equation (15), equation (16), and equation (17), it is not difficult to get a unified formula

f_{c} = \frac{p_{f} + q_{f} P + w_{f}}{ϕ_{m}}

(18)

where p_f, q_f, and w_f is the fitting parameter related to B_f, D_f, $α_{f}^{1}$ , $α_{f}^{2}$ , $β_{f}^{1}$ , and $β_{f}^{2}$ , in which p_f = 140.86876 ≈ $α_{f}^{1}$ ≈ $α_{f}^{2}$ , q_f = −0.49703, and w_f = −40.45641 when P→ 0, f_c = p_f = 140.86876, which is the same as the compressive strength of UHPC-NCA.

So far, the uniaxial compressive strength can be predicted by its aggregate volume fraction and the maximum aggregate particle size. It can be concluded that when P is constant, a reasonable aggregate gradation can significantly improve the compressive strength of UHPC-CA.

Discussion

In this section, the phase-field model of UHPC-CA which is proposed in this paper is verified by discussing the size effect of UHPC-CA. Based on the UHPC-CA numerical model given in Sect. 2, a series of uniaxial compression numerical samples of UHPC-CA with the side length from 25 mm to 200 mm (as shown in Figure 14) are used to simulate the size effect of UHPC-CA. To ensure the same aggregate volume fraction of the samples of various sizes, the aggregate volume fraction of the samples is 30%, and the G is set as G1 (ϕ_m = 2 mm).

Figure 14.

UHPC-CA samples of different sizes (take Pf30-G1 as an example): (a) L = 25 mm, (b) L = 50 mm, (c) L = 75 mm, (d) L = 100 mm, (e) L = 125 mm, (f) L = 150 mm, (g) L = 175 mm, (h) L = 200 mm.

The stress–strain curves of these samples are shown in Figure 15(a); the fitting curve of $f_{c}^{L} / f_{c}$ and 1/L is shown in Figure 15(b), where $f_{c}^{L}$ is the uniaxial compressive strength of the sample with the side length of L. It can be seen that the compressive strength of UHPC-CA increases linearly with 1/L. Using equation (19) to fit the relationship between $f_{c}^{L} / f_{c}$ and 1/L

\frac{f_{c}^{L}}{f_{c}} = λ + μ / L

(19)

where the fitting parameters λ = 0.9825 and μ = 0.01639, and the adjusted coefficient of determination (adj. R-square) equal to 0.98275, as shown in Figure 15(b).

Figure 15.

Numerical simulation results of UHPC-CA different sizes (P = 30%, G = G1, ϕ_m = 2 mm): (a) Stress–strain curves, (b) fitting curve of $f_{c}^{L} / f_{c}$ and 1/L.

The substitution of equation (18) into equation (19) yields

f_{c}^{L} = (λ + μ / L) (\frac{p_{f} + q_{f} P + w_{f}}{ϕ_{m}})

(20)

It can be seen from equation (20) that the compressive strength of UHPC-CA sample with side length L can be predicted by the aggregate volume fraction (P), the maximum aggregate particle size (ϕ_m) of UHPC-CA sample, and the compressive strength of UHPC-CA sample with a side length of 100 mm.

In spite that equation (20) is derived from the samples of P = 30% and ϕ_m = 2 mm, equation (20) can be used in the samples with different side lengths, different aggregate volume fraction, and different maximum aggregate particle sizes. Some numerical samples of UHPC-CA with different parameters are established to verify the expression. The compressive strength of these samples is calculated by numerical simulation and equation (20), as shown in Table 9.

Table 9.

Compare of compressive strength calculated by numerical simulation and equation (20).

Sample size(L)	P %	ϕ_m, mm	By numerical simulation	By equation (20)	Error, %
150 mm	20	2	127.7173	128.5662	0.66
25 mm	25	4	121.3250	124.1967	2.31
75 mm	30	6.5	128.0689	128.4058	0.26
125 mm	20	4	133.2523	133.416	0.12
175 mm	40	8	127.9482	126.4379	1.19
50 mm	35	6.5	126.5142	124.1027	1.94
200 mm	40	8	126.6901	126.7305	0.03
Average					0.93

It can be seen from Table 9 that the error of the compressive strength of these different samples is about 1%. It can be concluded that the size effect can be described by the phase-field model of UHPC-CA proposed in this paper. The phase-field model of this paper has sufficient precision.

Conclusion

The compressive strength of UHPC-CA is studied by the phase-field method and the uniaxial compression testing. The Monte Carlo simulation of UHPC-CA shows that the distribution of the aggregates has little influence on the compressive strength of UHPC-CA. The effects of the volume fraction and particle size of the coarse aggregate on the compressive strength of UHPC-CA are investigated. Both the experiment and simulation results show that the compressive strength decreases with the aggregate volume fraction, and increases with the maximum aggregate particle size. A size effect model of UHPC-CA is established to describe the relationship between the UHPC-CA compressive strength and the volume fraction and particle size of the coarse aggregate. For a UHPC-CA sample, adding coarse aggregates decrease the compressive strength, but a reasonable aggregate gradation can help reduce this decrease.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Bing Zhao

Zhenhao Zhang

References

Amin

Tayeh

Agwa

(2020) Effect of using mineral admixtures and ceramic wastes as coarse aggregates on properties of ultrahigh-performance concrete. Journal of Cleaner Production 273: 123073. DOI: 10.1016/j.jclepro.2020.123073.

Bilgen

Homberger

Weinberg

(2019) Phase-field fracture simulations of the Brazilian splitting test. International Journal of Fracture 220: 85–98. DOI: 10.1007/s10704-019-00401-w.

Bourdin

Francfort

Marigo

(2000) Numerical experiments in revisited brittle fracture. Journal of the Mechanics and Physics of Solids 48(4): 797–826. DOI: 10.1016/S0022-5096(99)00028-9.

Bujnakova

Jost

Farbak

, et al. (2019) Experimental study of the modulus of elasticity of concrete at different ambient temperature. IOP Conference Series: Materials Science and Engineering 549(1): 012049. DOI: 10.1088/1757-899X/549/1/012049.

Camacho

Ortiz

(1996) Computational modelling of impact damage in brittle materials. International Journal of Solids and Structures 33(20/21/22): 2899–2938.

Deng

Chi

, et al. (2020) Effect of steel-polypropylene hybrid fiber and coarse aggregate inclusion on the stress-strain behavior of ultra-high performance concrete under uniaxial compression. Composite Structures 252: 112685. DOI: 10.1016/j.compstruct.2020.112685.

Huang

(2018) Study on mechanical properties of ultra high performance concrete with coarse aggregate. Journal of Hunan University Natural Sciences 45(3): 47–54. DOI: 10.16339/j.cnki.hdxbzkb.2018.03.006.

Jiang

Zhou

Chu

, et al. (2019) Design of eco-friendly ultra-high performance concrete with supplementary cementitious materials and coarse aggregate. Journal of Wuhan University of Technology (Materials Science) 34(6): 1350–1359. DOI: 10.1007/s11595-018-2198-4.

Kim

Bae

Pyo

(2018) Concrete sleeper using ultra high performance concrete(UHPC) incorporating aggregates with maximum particle size of 5mm. Magazine of the Korea Concrete Institute 30(4): 53–57.

10.

Kotova

Nosikov

Klimenko

, et al. (2021) Comparison of shooting method and variational approach for two-point ionospheric ray tracing. Bulletin of the Russian Academy of Sciences Physics 85(3): 262–267.

11.

Jensen

(2021) Mechanism of rate dependent behaviour of ultra-high performance fibre reinforced concrete containing coarse aggregates under flexural loading. Construction and Building Materials 301(7): 124055. DOI: 10.1016/j.conbuildmat.2021.124055.

12.

Brouwers

HJH

, et al. (2019) Conceptual design and performance evaluation of two-stage ultra-low binder ultra-high performance concrete. Cement and Concrete Research 125: 105858. DOI: 10.1016/j.cemconres.2019.105858.

13.

Liu

Filonova

, et al. (2014) A regularized phenomenological multiscale damage model. International Journal for Numerical Methods in Engineering 99(12): 867–887. DOI: 10.1002/nme.4705.

14.

Liu

Wei

(2021) Effect of calcined bauxite powder or aggregate on the shrinkage properties of UHPC. Cementand Concrete Composites 118(2): 103967. DOI: 10.1016/j.cemconcomp.2021.103967.

15.

Lizhi

(2020) Summary of Research on Effect of Mixing Coarse Aggregate on Performance of Ultra-high Performance Concrete. Urban Roads Bridges & Flood Control.

16.

Zhang

, et al. (2020) Static mechanical properties and mechanism of C200 ultra-high performance concrete (UHPC) containing coarse aggregates. Science and Engineering of Composite Materials 27(1): 186–195. DOI: 10.1515/secm-2020-0018.

17.

Orgass

Dehn

, et al. (2004) Comparative investigations on ultra-high performance concrete with and without coarse aggregates. International Symposium on Ultra High Performance Concrete (UHPC).

18.

Miehe

Hofacker

Welschinger

(2010) A phase field model for rate-independent crack propagation: Robust algorithmic implementation based on operator splits. Computer Methods in Applied Mechanics and Engineering 199(45–48): 2765–2778. DOI: 10.1016/j.cma.2010.04.011.

19.

Peng

Yang

Gao

, et al. (2012). Factors Influencing Compressive Strength of Ultra-High-Performance Concrete with Coarse Aggregate. Journal of North China Institute of Water Conservancy and Hydroelectric Power.

20.

Peng

Yan

Huang

, et al. (2013) Fire Resistance of Ultra-high-performance Concrete with Coarse Aggregate: A Research Review. Journal of North China Institute of Water Conservancy and Hydroelectric Power.

21.

Teng

Xiang

, et al. (2019) Development and mechanical behaviour of ultra-high-performance seawater sea-sand concrete. Advances in Structural Engineering 22(14): 3100–3120. DOI: 10.1177/1369433219858291.

22.

Reddy

GGK

Perumal

(2020a) Performance evaluation of ultra-high performance concrete designed with alccofine. Innovative Infrastructure Solutions 6(1): 6. DOI: 10.1007/s41062-020-00375-y.

23.

Reddy

GGK

Ramadoss

(2020b) Influence of alccofine incorporation on the mechanical behavior of ultra-high performance concrete (UHPC). Materials Today: Proceedings 33: 789–797. DOI: 10.1016/j.matpr.2020.06.180.

24.

Richard

Cheyrezy

(1995) Composition of reactive powder concretes. Cement and Concrete Research 25(7): 1501–1511. DOI: 10.1016/0008-8846(95)00144-2.

25.

Rybczyński

Dosta

Schaan

, et al. (2020) Numerical study on the mechanical behavior of ultrahigh performance concrete using a three-phase discrete element model. Structural Concrete 23: 548–563. DOI: 10.1002/suco.202000435.

26.

Sahmaran

Lachemi

Hossain

KMA

, et al. (2009) Influence of aggregate type and size on ductility and mechanical properties of engineered cementitious composites. Aci Materials Journal 106(3): 308–316.

27.

Sobuz

Visintin

Ali

, et al. (2016) Manufacturing ultra-high performance concrete utilising conventional materials and production methods. Construction & Building Materials 111: 251–261. DOI: 10.1016/j.conbuildmat.2016.02.102.

28.

Song

Wang

Belytschko

(2008) A comparative study on finite element methods for dynamic fracture. Computational Mechanics 42(2): 239–250. DOI: 10.1007/s00466-007-0210-x.

29.

Wang

Yang

Jivkov

(2015) Monte Carlo simulations of mesoscale fracture of concrete with random aggregates and pores: A size effect study. Construction and Building Materials 80: 262–272. DOI: 10.1016/j.conbuildmat.2015.02.002.

30.

Wille

Boisvert-Cotulio

(2015) Material efficiency in the design of ultra-high performance concrete. Construction and Building Materials 86: 33–43. DOI: 10.1016/j.conbuildmat.2015.03.087.

31.

(2017) A unified phase-field theory for the mechanics of damage and quasi-brittle failure. Journal of the Mechanics and Physics of Solids 103: 72–99. DOI: 10.1016/j.jmps.2017.03.015.

32.

Nguyen

(2018) A length scale insensitive phase-field damage model for brittle fracture. Journal of the Mechanics and Physics of Solids 119(OCT): 20–42.

33.

Huang

Nguyen

(2020) On the BFGS monolithic algorithm for the unified phase field damage theory. Computer Methods in Applied Mechanics and Engineering 360: 112704. DOI: 10.13140/RG.2.2.24218.59843.

34.

Liu

, et al. (2014) Modeling of dynamic crack branching by enhanced extended finite element method. Computational Mechanics 54(2): 489–502. DOI: 10.1007/s00466-014-1001-9.

35.

Needleman

(1994) Numerical simulations of fast crack growth in brittle solids. Journal of the Mechanics and Physics of Solids 42(9): 1397–1434. DOI: 10.1016/0022-5096(94)90003-5.

36.

Yoo

Banthia

(2016) Mechanical properties of ultra-high-performance fiber-reinforced concrete: A review. Cement & Concrete Composites 73: 267–280. DOI: 10.1016/j.cemconcomp.2016.08.001.

37.

Zavickis

Lukasenoks

Macanovskis

, et al. (2020) Optimization of Packing of Local Coarse Aggregates for Use in UHPC (Ultra-High-Performance Concrete). 19th International Scientific Conference Engineering for Rural Development.

38.

Zhang

Teng

(2014) Finite element analysis of end cover separation in RC beams strengthened in flexure with FRP. Engineering Structures 75: 550–560.