The Properties and Behavior of Ultra-High-Performance Concrete: The Effects of Aggregate Volume Content and Particle Size

Abstract

Ultra-High-Performance Concrete (UHPC) and Ultra-High-Performance Fiber-Reinforced Concrete (UHPFRC) represent promising materials in the field of construction, offering exceptional strength and durability, making them ideal for the development of a wide range of infrastructure projects. One of the goals is to better understand the impact of each component of the materials on their key properties in the hardened state. This work examines the effect of the aggregate on the properties of UHPC and UHPFRC. This article provides test results for five compositions without fiber, and five compositions with 2% corrugated steel fiber. Three aggregate concentrations (0, 0.2, and 0.4 ${\mathrm{m}}^3$) and quartz sand with different maximum particle sizes (0.4 and 0.8 mm) were selected. It was found that the mechanical properties of the material, such as the steel fiber bond strength, compressive and axial tensile strength, fracture energy, and critical stress intensity factor, depend on both the concentration of the aggregate and the size of its particles. A novel mix-design parameter was proposed, which reflects the total surface area of the aggregate in the composition ($S_{agg,tot}$). The relationships between the parameter $S_{agg,tot}$ and the mechanical characteristics of UHPC and UHPFRC were established. The steel fiber bond strength, axial tensile strength, and fracture energy-related parameters grew non-linearly when the parameter $S_{agg,tot}$ increased. When the parameter $S_{agg,tot}$ was changed from 0 to 12.38 · ${10}^3{ \mathrm{m}}^2$, the fiber bond strength increased by 1.38 times. The axial tensile strength and total fracture energy of the UHPFRC increased by 1.48 and 1.63 times, respectively. The compressive strength changed linearly and increased by 1.12 times. The improvement in the mechanical properties of the material was associated with an increase in the friction force between the fiber and the matrix, which was confirmed by the formation of a greater number of scratches on the surface of the fiber with an increasing value of the parameter $S_{agg,tot}$. The deformation characteristics, such as modulus of elasticity, Poisson’s ratio, and drying shrinkage strain, were determined solely by the volumetric concentration of the aggregate, as in conventional concrete. An increase in the aggregate volume content from 0 to 0.4 ${\mathrm{m}}^3$ led to an increase in the modulus of elasticity of 1.41–1.44 times, and a decrease in the ultimate shrinkage strain of almost 2 times. The dependencies obtained in this work can be used to predict the properties of UHPC and UHPFRC, taking into account the type and volume concentration of the aggregate.

1. Introduction

Ultra-High-Performance Concrete (UHPC) is a relatively new structural material, which, recently, has been actively used in the construction of bridges [1,2,3], road pavements [4], filling joints between precast reinforced concrete elements [5,6], and also for repairing damaged structures [7,8]. The main advantage of UHPC is its low porosity, which ensures its high strength and durability, even when used in aggressive environments. The special properties of UHPC are achieved by carefully optimizing the particle packing density, using active mineral additives, and reducing the water/cement ratio to less than 0.25 [9,10,11]. To avoid brittle fracture and increase the tensile strength, UHPC is reinforced with steel fibers of various shapes and geometric sizes, resulting in Ultra-High-Performance Fiber-Reinforced Concrete (UHPFRC) [12,13]. By fine-tuning the characteristics of the concrete matrix, as well as the bond strength and geometry of the fibers, it is feasible to develop composites that exhibit an axial tensile strength ranging from 10 to 20 MPa [14,15].

The high cost of UHPC and UHPFRC compared to conventional concrete requires a more in-depth understanding of how the content and properties of the individual components affect the key properties of the material or matrix in the hardened state. Such knowledge will make it possible to use locally available raw materials more efficiently and to obtain the most optimal compositions from the variety of possible ones. To date, the impact of the volume fraction and the geometry of steel fibers on the mechanical properties and deformation behavior of UHPFRC have been extensively investigated [16,17,18,19,20,21,22,23,24,25,26,27,28]. Articles [29,30,31,32,33] also presented studies on UHPFRC using non-metallic fiber. In works [34,35,36,37,38,39,40,41,42], studies were carried out on compositions with various active mineral additives. Quartz powder is often used as an inert filler in order to increase the packing density of particles. The effect of the average size of quartz powder particles on compressive strength was examined in [43]. In articles [44,45,46,47], the authors explored the possibility of replacing the traditionally used quartz powder with cheaper limestone powder. Works [48,49,50] presented experimental studies of compositions with different types of cement.

The aggregate content in UHPC may range from 20% in fine-grained mixtures to 46–50% in mixtures with a coarse aggregate [51,52]. Only a limited number of publications have been devoted to understanding the effects of aggregate on the properties of a material in its hardened state. The scientific literature shows that changing the particle size or aggregate content is one way to control the properties of UHPC or UHPFRC. One of the fundamental properties that determines the tensile behavior of fiber-reinforced concrete is the bond strength of the fiber to the concrete matrix, as the composite is a mixture of fiber, matrix, and filler. It has been reported that increasing the sand to binder ratio (S/B) or decreasing the particle size leads to an increase in the bond strength of straight fiber to UHPC [53,54,55]. A similar trend has been observed in HPC for deformed fibers [56], and in mortars of medium strength [57]. The macroscopic properties of fine-grained UHPC and UHPFRC, such as their compressive, flexural, and tensile strengths, also depend on the particle size and S/B ratio [58,59,60,61]. The same has been found for UHPC with coarse aggregate [62,63].

Table 1. Summary of literature review.

Material Property	Variable	Key Findings	Reference
Steel fiber bond strength	Sand to cement ratio (S/C)	Increasing the S/C from 0 to 1.37 resulted in a 38% increase in the straight fiber bond strength.	[53]
Steel fiber bond strength	Sand to cement ratio (S/C)	Increasing the S/C from 0 to 1.38 resulted in a 62% increase in the straight fiber bond strength.	[54]
	Sand grain size	The reduction in the average sand particle size from 0.42 to 0.11 mm resulted in a 52% increase in the bond strength of straight steel fiber.
Steel fiber bond strength	Sand grain size	The reduction in the maximum particle size from 2.36 to 1.18 mm resulted in an increase in the bond strength of straight and hooked-end steel fibers of 18 and 5%, respectively.	[55]
Steel fiber bond strength	Sand to cement ratio (S/C)	The bond strength of the straight and twisted fibers increased by 67–131 and 28–67%, respectively, when the S/C was increased from 1 to 1.5. The strength of the hooked-end fiber remained practically unchanged.	[56]
	Sand grain size	When the particle size was reduced from 0.42 to ~0.04 mm, the bond strength of all types of fibers increased by 20–60%.
Steel fiber bond strength	Sand to cement ratio (S/C)	Increasing the S/C from 0 to 1 resulted in an increase in the bond strength of the straight steel fiber to the mortar (W/C = 0.4) of 129%.	[57]
Tensile and flexural strength	Sand grain size	The reduction in the average particle size from 0.7 to 0.1 mm resulted in an enhancement of the axial tensile and flexural strength of the HPFRCC of 23–32% and 21–31%, respectively, depending on the fiber type.	[64]
Compressive strength	Sand to binder ratio (S/B)	By increasing the S/B from 0.16 to 0.80, the compressive strength of the UHPFRC increased by 16%.	[58]
Tensile strength and strain capacity		The axial tensile strength and strain capacity increased by 39 and 27%, respectively, when the S/B changed from 0.16 to 0.8.
Compressive strength	Sand grain size	A decrease in the average particle size from 1.60 mm to 0.16 mm resulted in an increase in the compressive strength of 10%.	[59]
Tensile strength and strain capacity		The axial tensile strength and strain capacity increased by 34 and 20% when the average particle size was decreased from 1.60 to 0.16 mm, respectively.
Flexural strength	Sand grain size	Reducing the average particle size from 0.6 to 0.3 mm led to an increase in the flexural strength of 15%.	[60]
Critical stress intensity factor		Reducing the average particle size from 0.6 to 0.3 mm led to an increase in the critical stress intensity factor of 15%.
Compressive strength	Sand grain size	Reducing the maximum particle size from 4.75 to 0.6 mm led to an increase in the UHPC compressive strength of 6%.	[61]
Tensile cracking strength	Coarse aggregate content	When the volume fraction of the coarse basalt aggregate in the UHPFRC mixture was increased from 0 to 25%, the tensile cracking strength decreased by 6%.	[62]
Compressive strength, flexural strength	Coarse aggregate size	The compressive and flexural strength of the UHPFRC decreased by 11 and 26% when the maximum size of the basalt aggregate was increased from 10 to 20 mm.	[63]

provides a summary of the effect of the aggregate on the properties of UHPC/UHPFRC, based on the literature data.

It is evident from Table 1 that a reduction in the average/maximum particle size of the aggregate, coupled with an increase in its volumetric concentration in the composition of the UHPC and UHPFRC, lead to an improvement in the mechanical properties of the material. In this regard, a novel mix-design parameter is proposed that reflects both the average size of the aggregate particles and their volumetric content in the material composition, namely $S_{agg,tot}$. A detailed definition of $S_{agg,tot}$ is presented in the next section.

To the authors’ knowledge, there are currently no studies devoted to examining the impact of both the particle size and volumetric content of the aggregate on the primary characteristics of UHPC and UHPFRC. This work examines the composition of UHPC and UHPFRC incorporating 2% corrugated steel fibers with a varying aggregate volume content with different particle sizes. This results in $S_{agg,tot}$ values ranging from 0 (cement paste) up to 12.38 · ${10}^3$ ${\mathrm{m}}^2$. Fiber pullout tests and axial tensile testing of fiber-reinforced concrete samples were carried out to determine the strength and fracture energy characteristics. The compressive strength and critical stress intensity factor were determined as well. The deformation characteristics of the material, including the elastic modulus, Poisson’s ratio, and drying shrinkage strain, were also determined. Based on the data obtained, the relationships between the material’s properties and the parameter $S_{agg,tot}$ were estimated. A description of the test methods and the results obtained are presented in the following subsections.

2. Definition of the Proposed Mix-Design Parameter

The proposed mix-design parameter, designated as $S_{agg,tot}$, is defined as the total surface area of the aggregate particles, according to the following equation:

$$S_{agg,tot}={\sum }_{i=1}^n{V}_{agg,i}·{\rho }_{agg,i}·S_{agg,i}, [{\mathrm{m}}^2]$$

(1)

where

$V_{agg,i}$ is the volume content of the i-th fraction of the aggregate, ${\mathrm{m}}^3$;

${\rho }_{agg,i}$ is the density of the i-th fraction of the aggregate, $\mathrm{kg}/{\mathrm{m}}^3$;

$S_{agg,i}$ is the specific surface area of the i-th fraction of the aggregate, ${\mathrm{m}}^2/\mathrm{kg}$.

The specific surface area of the i-th aggregate fraction is determined by the following equation:

$$S_{agg,i}=f·{\displaystyle \frac{6000}{{\rho }_{agg,i}·d_{agg,i}}}, [{\mathrm{m}}^2/\mathrm{kg}]$$

(2)

where

$f$ is the coefficient taking into account the shape of the aggregate grains, -;

$d_{agg,i}$ is the mean aggregate size of the i-th fraction of the aggregate, mm.

The coefficient f is defined as the ratio of the specific surface area of the irregularly shaped particles to the specific surface area of the perfectly spherical particles of the same diameter. In a previous work, a linear relationship was established between the packing density of the monofraction particles and the coefficient f [65]:

$$f=-1.57·{\mathrm{\Phi }}_{agg,i}+2.13$$

(3)

3. Materials and Methods

3.1. Raw Materials

Gray Portland cement was used as the binder. The specific surface area was 353.3 ${\mathrm{m}}^2/\mathrm{kg}$. The compressive and flexural strengths, according to EN 196-1 [66], were 56.5 and 8.04 MPa, respectively. The mineralogical composition of the applied cement was determined using an XRD analysis. The measurements were conducted using an ARL Equinox 1000 diffractometer (Thermo Fisher Scientific, Waltham, MA, USA). The mineralogical composition of the cement is shown in

Table 2. Mineralogical composition of Portland cement.

${\mathit{C}}_3\mathit{S}$, %	${\mathit{C}}_2\mathit{S}$, %	${\mathit{C}}_3\mathit{A}$, %	${\mathit{C}}_4\mathit{A}\mathit{F}$, %
63.3	18.0	5.6	9.5

.

Quartz powder containing more than 95% crystalline quartz was used as the inert mineral additive. Condensed silica fume with an amorphous silicon dioxide content of more than 85% was used as the active mineral additive. The main properties of the mineral additives are presented in

Table 3. Properties of powder materials.

Powder	$\mathbf{Density}, \mathbf{k}\mathbf{g}/{\mathbf{m}}^3$	$\mathbf{Fineness}, {\mathbf{m}}^2/\mathbf{k}\mathbf{g}$
Quartz powder	2650	364.0
Silica fume	2200	17,000

.

Two fractions of quartz sand were utilized as the fine aggregate, namely 0.1–0.4 and 0.4–0.8 mm. The main properties of the quartz sand are listed in

Table 4. Properties of aggregates.

Aggregate ID	Fraction	$\mathbf{Density}, \mathbf{k}\mathbf{g}/{\mathbf{m}}^3$	$\mathbf{Bulk} \mathbf{Density}, \mathbf{k}\mathbf{g}/{\mathbf{m}}^3$	Mean Particle Size, mm	Packing Density, -
Q.0.4	0.1–0.4	2637	1403	0.25	0.532
Q.0.8	0.4–0.8	2632	1524	0.60	0.579

.

Brass-coated corrugated steel fiber made of high-carbon steel wire was used. The length and diameter of the fiber were 22 and 0.3 mm, respectively. The tensile strength was equal to 2700 MPa. The tensile strength of the fiber was determined using an electromechanical tensile testing machine, Instron 3382 (Instron Corporation, Norwood, MA, USA). The fiber was clamped on both sides using grippers and tested at a loading speed of 0.4 mm/min. The appearance of the steel fiber is shown in Figure 1.

To ensure the required flowability of the fresh mixtures, a superplasticizer based on polycarboxylate esters, MasterGlenium 115 (BASF, Ludwigshafen, Germany), was used. The superplasticizer had a density of 1.08 $\mathrm{g}/{\mathrm{cm}}^3$ and a water content of 70%.

3.2. Mix Design

In this work, five various types of concrete matrixes were tested, differing in the type and volumetric content of quartz sand, further designated by Arabic numerals from 1 to 5. Composition 1 did not contain any aggregate. Compositions 2 and 3 contained Q.0.4 aggregate in amounts of 0.2 and 0.4 ${\mathrm{m}}^3/{\mathrm{m}}^3$, respectively. Compositions 4 and 5 contained a mixture of two fractions, Q.0.4 and Q.0.8, in a ratio of 30:70 by volume. This ratio was selected based on it achieving the highest packing density of aggregate particles. The optimization of the particle packing density was carried out using the Compressible Packing Model developed by F. de Larrard [67]. The total amount of aggregate in compositions 4 and 5 was also 0.2 and 0.4 ${\mathrm{m}}^3/{\mathrm{m}}^3$. As a result, compositions with parameter values for $S_{agg,tot}$ ranging from 0 to 12.38$·{10}^3$ ${\mathrm{m}}^2$ were obtained. The water/cement ratio, taking into account the water content in the superplasticizer, was constant and amounted to 0.275. The content of the silica fume and quartz powder were 20% by weight of the cement, respectively. The content of the superplasticizer was selected in such a way as to ensure the slump flow on the Hagerman cone in the range of 230–250 mm. Furthermore, five compositions were tested with steel fiber, whose content was taken to be 2% of the total volume of the mixture. The fiber was added to replace part of the cement paste in order to ensure a constant value of $S_{agg,tot}$ compared to the plain concrete matrix. The designation of the compositions with steel fiber is presented in the form “X-F”, where X is the number of the concrete matrix (from 1 to 5), and F is the designation of the presence of fiber. The consumption of components per 1 ${\mathrm{m}}^3$ is presented in

Table 5. Mixture proportions.

Component 1	1	2	3	4	5	1-F	2-F	3-F	4-F	5-F
C, kg	1295	1036	775	1036	775	1277	1014	748	1014	750
W, kg	337	269	199	269	199	332	261	191	261	193
SF, kg	259	207	155	207	155	255	203	150	203	150
QP, kg	259	207	155	207	155	255	203	150	203	150
Q.0.4, kg	-	527	1055	158	316	-	527	1055	158	316
Q0.8, kg	-	-	-	369	738	-	-	-	369	738
SP, kg	32	26	23	26	23	26	25	25	25	23
F, kg	-	-	-	-	-	156	156	156	156	156
$S_{agg,tot}·{10}^3$, ${\mathrm{m}}^2$	0	6.19	12.38	3.53	7.06	0	6.19	12.38	3.53	7.06

¹ C—Portland cement; W—water; SF—silica fume; QP—quartz powder; Q.0.4—quartz sand with a maximum diameter of 0.4 mm; Q.0.8—quartz sand with a maximum diameter of 0.8 mm; SP—superplasticizer; F—steel fiber.

.

3.3. Specimen Preparation and Curing

A Viatto B-30P mixer with a planetary rotation of the blade was used for the preparation of the concrete mixtures. The mixing of the components was carried out in the following sequence: (1) the mixing of the dry components for 2 min at a speed of 105 rpm; (2) the addition of water and 70% of the plasticizing admixture and mixing for 5 min at a speed of 105 rpm; and (3) the addition of the remaining plasticizing admixture and mixing to a homogeneous state at a speed of 408 rpm for 5–8 min.

After molding, the samples were stored in laboratory conditions at +20 °C for 24 h. To avoid moisture evaporation, the exposed surface of the specimens was covered with polyethylene film. After 24 h, the samples were unmolded and stored for the next 24 h under normal conditions. After 48 h from the date of casting, the samples were heat-treated at +80 °C and 100% relative humidity for the next 48 h. After heat treatment, the specimens were stored in the laboratory room until testing.

The mixture setup and the specimens manufacturing process are shown in Figure 2.

3.4. Experimental Methods

3.4.1. Compressive Strength

The compressive strength was determined on cube specimens of 50 × 50 × 50 mm. The specimens were loaded at a rate of 2 MPa/s. The compressive strength of each specimen was determined by the following equation:

$$f_{c,i}=F/A, [\mathrm{MPa}]$$

(4)

where

F is the maximum force, N;

A is the cross-sectional area, ${\mathrm{mm}}^2$.

The compressive strength of each composition was determined as the arithmetic mean of the results of 3 parallel tests.

3.4.2. Fiber Pullout Test

This test was carried out by pulling out a fiber embedded 10 mm into a concrete matrix. For testing, half-dog-bone samples were used, in the center of which a fiber was installed using a polyvinyl chloride membrane. Cyanoacrylate glue was used to attach the fiber to the membrane to avoid displacement during the pouring of the fresh concrete mixture. After the hardening of the samples, the membranes were removed. The test was carried out on an Instron 3382 tensile testing machine. The loading speed was 0.4 mm/min until the displacement value reached 2 mm and 3 mm/min until the fiber was fully extracted. Figure 3 shows the sample manufacturing and testing process.

During the test, a load–fiber slip diagram was recorded, which was used to calculate the bond strength, fiber utilization factor, and fracture energy per unit fiber surface area.

The bond strength is defined as the maximum shear stress that occurs between the fiber and concrete, and is calculated using the following formula:

$${\tau }_f={\displaystyle \frac{F_{max}}{\pi ·d_f·l_e}}, [\mathrm{MPa}]$$

(5)

where

$F_{max}$ is the maximum pullout load, N;

$d_f$ is the fiber diameter, mm;

$l_e$ is the fiber embedment length, mm.

The fiber utilization factor is determined by using the following equation:

$$K_f={\displaystyle \frac{{\sigma }_{f,max}}{{\sigma }_f}},$$

(6)

where

${\sigma }_f$ is the fiber tensile strength, MPa;

${\sigma }_{f,max}$ is the maximum tensile stress in the fiber cross-section, MPa.

The maximum tensile stress in the fiber cross-section is calculated by the following equation:

$${\sigma }_{f,max}={\displaystyle \frac{4·F_{max}}{\pi ·d_f^2}} [\mathrm{MPa}]$$

(7)

The fracture energy per unit area of fiber surface is determined by the following equation:

$$G={\displaystyle \frac{W_p}{\pi ·d_f·l_e}}, [\mathrm{N}/\mathrm{mm}]$$

(8)

where

$W_p$ is the work is required to pull the fiber, $\mathrm{N}·\mathrm{mm}$.

The value of $W_p$ is determined by integrating the load–fiber slip diagram:

$$W_p={\int }_{s=0}^{s=l_e}F\left(s\right)ds, [\mathrm{N}·\mathrm{mm}]$$

(9)

where

$F\left(s\right)$ is the load value at a certain fiber slip, N;

S is the fiber slip, mm.

The values of each parameter were determined based on the results of testing 6 samples.

3.4.3. Critical Stress Intensity Factor

The critical stress intensity factor, $K_{I,c}$, was determined on 40 × 40 × 160 mm beam specimens with an artificial crack, which was forming by cutting the samples with a saw. The cut’s depth was 10 mm. The test scheme is shown in Figure 4. The loading rate of the specimens was 0.1 mm/s. The critical stress intensity factor for each specimen was calculated by Equation (10), according to the Russian standard GOST 29167-2021 [68]:

$$K_{I,c}={\displaystyle \frac{1.5·F·L}{b^{3/2}·t}}·\sqrt{a_0·b}·(1.93-3.07·\lambda +14.53·{\lambda }^2-25.11·{\lambda }^3+25.8·{\lambda }^4, [\mathrm{MPa}·\sqrt{m}]$$

(10)

where

$F$ is the force at cracking, MN;

L is the distance between supports, m;

b, t are the height and width of the cross-section, m;

$a_0$ is the notch depth, m;

$\lambda$ = $a_0/b$.

3.4.4. Modulus of Elasticity

The modulus of elasticity and Poisson’s ratio were determined on 70 × 70 × 280 mm prism specimens according to the Russian standard GOST 24452-80 [69]. The longitudinal and transverse strains were measured using digital sensors mounted on 4 faces of the specimen (Figure 5). A total of 4 sensors were used to measure the longitudinal strain and 2 to measure transverse strain.

The modulus of elasticity and Poisson’s ratio were determined using Equations (11) and (12) at a stress equal to 0.3 of the prismatic strength of the concrete:

$$E_c={\sigma }_c/{\epsilon }_{1,m}, [\mathrm{MPa}]$$

(11)

$${\nu }_c={\epsilon }_{2,m}/{\epsilon }_{1,m},$$

(12)

where

${\sigma }_c$ is the compressive stress equal to 30% of the prismatic compressive strength, MPa;

${\epsilon }_{1,m}$ and ${\epsilon }_{2,m}$ are the average longitudinal and transverse strain at a stress level of 30% of the prismatic compressive strength, -.

The result of the test was the arithmetic mean of 3 parallel measurements.

3.4.5. Drying Shrinkage

The drying shrinkage was determined on 40 × 40 × 160 mm prismatic samples according to the Russian standard GOST 24544-2020 [70]. During the testing process, the samples were stored at a temperature of +20 °C and a relative humidity of 50%. The change in the length of the samples was determined using a CONTROLS 62-L0035/A device (CONTROLS, Milan, Italy) (Figure 6). The initial reading of the device was determined after 12 h of hardening of the samples. Further readings were performed every 24 h until 7 days of hardening, every 7 days until 28 days of hardening, and every 28 days until 110 days of hardening. The shrinkage strain of the sample was determined by the following equation:

$${\epsilon }_{sh}={\displaystyle \frac{l\left(t\right)-l_0}{l_1}},$$

(13)

where

$l\left(t\right)$ is the instrument reading at time t, mm;

$l_0$ is the initial reading after 12 h of hardening, mm;

$l_1$ is the measurement base, mm.

The result of the test was the arithmetic mean of 3 parallel measurements.

3.4.6. Axial Tensile Test

Dog-bone specimens with a cross-sectional dimension of 50 $\times$ 30 mm were used for axial tensile testing. To prevent a bending moment during the loading process, grips with three degrees of freedom were used. The measurement of the elongation of the sample during the test was conducted using digital displacement sensors that were positioned on two opposing front faces of the specimen. The sensors were bonded to the surface using cyanoacrylate glue. The measurement base was 110 mm. The recording frequency of the sensors was 4 Hz. The test was carried out on an Instron 1100 HDX tensile testing machine (Instron Corporation, Norwood, MA, USA). The loading speed was set at 0.4 mm/min until the maximum load value was reached, and then at 3 mm/min until the complete fracture of the sample. In order to prevent the fracture of the sample within the gripping area, WallWrap Tape 535 carbon tapes with a density of 530 $\mathrm{g}/{\mathrm{cm}}^2$ (Chemproduct, Lyubertsy, Russia) and impregnated with FibArm Resin 530+ epoxy resin (Umatex, Moscow, Russia) were adhered to the surface of the front faces. After gluing the carbon tapes, the samples were kept for 7 days at a temperature of +20 °C and a humidity of 50%. The appearance of the samples and the test setup are shown in Figure 7.

Based on the test results from the control samples, stress–strain curves were generated, from which the following parameters were determined: cracking strength (${\sigma }_{cc}$), post-cracking strength (${\sigma }_{pc}$), cracking strain (${\epsilon }_{cc}$), post-cracking strain (${\epsilon }_{pc}$), and fracture energy-related parameters.

The cracking strength, ${\sigma }_{cc}$, is defined as the point at which the stress–strain diagram changes from linear to nonlinear. The tensile strength, or post-cracking strength, ${\sigma }_{pc}$, is defined as the maximum tensile stress that was achieved in a cross-section of the specimen during testing. For the materials in the UHPC class, ${\sigma }_{pc}$ is achieved at the end of the strain-hardening process, which is characterized by the formation of many microcracks along the entire length of the specimen. This is followed by the localization of deformations in one of the previously formed cracks and the beginning of fiber pullout, which is accompanied by a decrease in tensile stress. The post-cracking strain, ${\epsilon }_{pc}$, is defined as the strain corresponding to the maximum tensile stress. Strain hardening during axial tension occurs when the following inequality is satisfied [71]:

$${\sigma }_{pc}\ge {\sigma }_{cc}$$

(14)

If the inequality (14) is not met, only one crack will form in the sample, leading to the pulling out of fibers and the fracture of the material.

The fracture energy of the strain-hardening materials consists of two parts [72]: the energy dissipated per unit volume in the process of strain hardening, $g_{f,a}$, and the amount of energy dissipated to completely separate the UHPFRC into two parts, $G_{f,B}$ (energy per unit area). The value of $g_{f,a}$ is calculated by integrating the stress–strain diagram up to the post-cracking strain value, ${\epsilon }_{pc}$:

$$g_{f,a}={{\displaystyle \int }}_0^{{\epsilon }_{pc}}\sigma \left(\epsilon \right)·d\epsilon [{\displaystyle \frac{\mathrm{kJ}}{{\mathrm{m}}^3}}]$$

(15)

The fracture energy per unit area, $G_{f,B}$, is determined by integrating the stress–crack opening diagram up to the maximum possible crack opening, which is equal to ${\delta }_u={\displaystyle \frac{l_f}{2}}$ mm:

$$G_{f,B}={{\displaystyle \int }}_{\delta }^{{\delta }_u}\sigma \left(\delta \right)·d\delta [{\displaystyle \frac{\mathrm{kJ}}{{\mathrm{m}}^2}}]$$

(16)

The total fracture energy is calculated using the following equation:

$$G_f=G_{f,A}+G_{f,B} [{\displaystyle \frac{\mathrm{kJ}}{{\mathrm{m}}^2}}]$$

(17)

To calculate $G_{f,A}$, it is necessary to know the number of microcracks, $N_{cr}$, formed during the strain-hardening process. In order to determine that, ethyl alcohol was applied to the samples after testing. The cracks, which absorbed more alcohol, dried later than the specimen’s surface, leading to darker visible cracks. The value of $G_{f,A}$ was then determined by the following equation:

$$G_{f,A}=g_{f,a}·{\displaystyle \frac{L_g}{N_{cr}}}, [{\displaystyle \frac{\mathrm{kJ}}{{\mathrm{m}}^2}}]$$

(18)

where

$L_g$ is the gauge length, m.

The typical stress–strain diagram of UHPFRC under axial tension is presented in Figure 8.

In order to compare the performance of the fiber in the composition of the UHPFRC and during the pulling out of an individual fiber, the utilization coefficient of steel fiber was calculated using the following equation [15]:

$${K^{\prime }}_f={\displaystyle \frac{{\sigma }_{pc}}{{\alpha }_0·V_f·{\sigma }_{f,max}}},$$

(19)

where

${\alpha }_0$ is the fiber orientation coefficient, -;

$V_f$ is the fiber volume fraction, -.

If the number of fibers in the fracture plane of the sample is known, the orientation coefficient can be determined using the following equation [73]:

$${\alpha }_0={\displaystyle \frac{A_f·N_f}{b·h·V_f}},$$

(20)

where

$A_f$ is the fiber cross-section area, ${\mathrm{mm}}^2$;

$N_f$ is the number of fibers crossing a plane, -;

b, h are the width and height of the cross-section, mm.

The $N_f$ value was determined by sawing the sample after the tensile test at a distance of 5–10 mm from the main crack. After this, the cross-section was photographed, and the number of fibers was determined from the image using special software [74].

3.4.7. Microscopic Observations

After the pullout test, the surface of the steel fiber was examined using a FEI Quanta 250 scanning electron microscope (FEI Company, Hillsboro, OR, USA)at 5000× magnification.

To count the number of contacts of the aggregate particles with the fiber surface, $N_g$, a LEVENHUK 320 BASE optical microscope was used (Levenhuk Inc., University, Tampa, FL, USA). The number of contacts was determined for the same sample used to determine the fiber orientation coefficient. The following formula was used to determine the $N_g$ value:

$$N_g={\displaystyle \frac{{\sum }_{i=1}^n{N}_{g,i}}{n}}, [\mathrm{EA}]$$

(21)

where

$N_{g,i}$ is the number of contacts with the i-th fiber, EA;

$n$ is the number of individuals measurements (not less than 30).

Figure 9 demonstrates the principle for determining the value of $N_{g,i}$.

4. Results and Discussion

4.1. Fiber Pullout Test

The results of the fiber pullout tests are presented in

Table 6. Average fiber pullout parameters of tested mixtures.

Mix ID	${\mathbf{\tau }}_{\mathit{f}}, \mathbf{M}\mathbf{P}\mathbf{a}$	${\mathit{\sigma }}_{\mathit{f},\mathit{m}\mathit{a}\mathit{x}}, \mathbf{M}\mathbf{P}\mathbf{a}$	${\mathit{K}}_{\mathit{f}}$	${\mathit{W}}_{\mathit{p}}, \mathit{N}\mathit{·}\mathbf{m}\mathbf{m}$	$\mathit{G}, \mathit{N}/\mathbf{m}\mathbf{m}$
1	14.60	1928	0.69	783.3	83.9
2	18.60	2420	0.86	801.9	87.2
3	20.20	2759	0.99	1307.9	135.6
4	16.75	2172	0.78	916.3	100.3
5	18.90	2440	0.87	888.6	98.2

. The average load–fiber slip diagrams are presented in Figure 10.

The pullout curve of corrugated fiber is different from that of a straight or hooked-end fiber, as presented in [75]. There exist three distinct peaks on the diagram, which arise from the fiber’s displacement of approximately 1.5, 5, and 9 mm, irrespective of the composition of the concrete matrix. The increase in the load at the indicated displacement values is attributed to the plastic deformation of the wave bends of the fiber [76], whose location coincides with the abscissa values of the observed peaks.

With an increase in the parameter $S_{agg,tot}$, from 0 to 12.38$·{10}^3$ ${\mathrm{m}}^2$ (compositions 1, 2, and 3), the bond strength increased by 38.4%, from 14.60 to 20.20 MPa. Similar results were obtained for mixtures 4 and 5, which employed a larger particle size aggregate. For these compositions, the bond strength experienced an increase of 29.5%, from 14.60 to 18.90 MPa. The experimental results are in accordance with the data obtained by other authors for straight and hooked-end fibers, when varying the particle size or aggregate content in the composition [53,54,55,56,57]. It is worth noting that composition 2 and 5 have very similar ${\tau }_f$ values, despite having different aggregate contents. This is explained by the almost identical values of the parameter $S_{agg,tot}$, as shown in Figure 11.

The experimental data are satisfactorily described by an exponential function of the form $y=a-b·e^{-c·x}$, with a coefficient of determination equal to 0.99. The nature of the function suggests the presence of an ultimate value of bond strength, which can be achieved by increasing the value of the parameter $S_{agg,tot}$. For the corrugated fiber and matrix composition used in this work, the ultimate value of the bond strength is 22.05 MPa, which is 51% higher compared to the composition without aggregate.

The observed increase in strength may be due to the fact that, as the value of $S_{agg,tot}$ increases, the number of aggregate particles in the mixture increases. As a consequence, the probability of their contact with the fiber’s surface also increases. The steel fiber used in this study has a brass coating with a Mohs hardness of 3.5. When a displacement between the fiber and concrete occurs, a particle of quartz sand with a hardness of 7.0 will scratch the brass coating, as a result of which the maximum pullout load will increase due to the increased friction force. In work [54], the authors used zirconium sand with a Mohs hardness of 7.5 instead of quartz, which led to an increase in the bond strength of straight steel fiber of 40%.

The theoretical number of contacts between the aggregate particles and the surface of the steel fiber is determined as follows. It has been suggested that the aggregate particles in the volume of the mixture are located at equal distances from each other, resulting in a grid consisting of square cells. Since the area occupied by the aggregate particles on a plane is equal to its volume fraction, the following equation for calculating the distance between the centers of the aggregate particles on a plane can be used:

$$S_g=\sqrt{{\displaystyle \frac{\pi ·d_g^2}{4·V_{agg}}}}, [\mathrm{mm}]$$

(22)

where

$d_g$ is the mean diameter of the aggregate particles, mm.

Figure 12 presents a schematic representation of a unit cell of the fiber-reinforced concrete from the point of view of the location of the aggregate particles. In the center of the cell, a square zone is formed with the side and diagonal $S_{g,1}$ and $S_{g,2}$, respectively, within which the steel fiber may be located. Based on the specified scheme, the maximum possible number of contacts of the aggregate particles with the surface of the steel fiber, $N_{g,max}$, is determined as follows:

• If $d_f\ge S_{g,2}$, then $N_{g,max}=4$;
• If $S_{g,2}>d_f\ge S_{g,1}$, then $N_{g,max}=2$;
• If $d_f<S_{g,1}$, then $N_{g,max}$ = 1.

The results of the calculation of $N_{g,max}$ for the compositions studied in this work are presented in

Table 7. The experimental and theoretical values for the number of contacts between the aggregate particles and steel fibers.

Mix ID	${\mathit{d}}_{\mathit{g}}$, mm	${\mathit{V}}_{\mathit{a}\mathit{g}\mathit{g}}$$, {\mathbf{m}}^3$	${\mathit{S}}_{\mathit{g}}$, mm	${\mathit{S}}_{\mathit{g},1}$, mm	${\mathit{S}}_{\mathit{g},2}$, mm	${\mathit{N}}_{\mathit{g},\mathit{m}\mathit{a}\mathit{x}}$, EA	${\mathit{N}}_{\mathit{g}}$, EA
1	-	0	0	0	0	0	0
2	0.25	0.2	0.50	0.25	0.45	2.0	2.1 ± 0.9 1
3	0.25	0.4	0.35	0.10	0.24	4.0	3.9 ± 1.2 1
4	0.5	0.2	0.99	0.49	0.90	1.0	1.3 ± 0.9 1
5	0.5	0.4	0.70	0.20	0.49	2.0	2.7 ± 1.1 1

¹ Standard deviation.

. The table also contains the experimental values of the number of contacts of the aggregate with the fiber, $N_g$, which were determined by analyzing the cross-section of the samples after an axial tensile test.

The proposed method for calculating the number of contacts between the aggregate particles and fiber has a good agreement with the experimentally obtained values. This confirms the assumption made regarding an increase in the bond strength due to an increase in the friction force in the contact zone, due to the presence of aggregate particles. It is also worth noting that the relationship shown in Figure 11 is a special case. As follows from the proposed geometric model, when the fiber diameter decreases, to achieve the same increase in bond strength, the value of the parameter $S_{agg,tot}$ should be higher, and vice versa.

To study the extent of surface damage, microscopic examinations were carried out on the surface of the steel fibers before and after they were pulled out of the UHPC. Figure 13a shows a picture of a fiber before pullout testing, on the surface of which small longitudinal scratches can be observed, which were probably formed during the fiber manufacturing process. After pulling the fiber out of the concrete, deeper scratches appear across the entire surface. Moreover, the degree of damage is higher in the compositions containing aggregate (Figure 13c–e) compared to the fiber extracted from composition 1, which does not contain aggregate (Figure 13b). This is due to an increase in the value of the parameter $S_{agg,tot}$, which results in an increase in the number of aggregate particles in the mixture, and a more intense scratching of the surface during the process of pulling out the fiber. Similar photographs were obtained in [54] during the testing of straight fiber. It is also worth noting that with an increase in $S_{agg,tot}$ from 6.19$·{10}^3$ to 12.38$·{10}^3$ ${\mathrm{m}}^2$, the visual appearance of the surface practically does not change (Figure 13d,e). This is associated with the saturation of the contact zone with aggregate particles and corresponds to the nature of the relationship presented in Figure 11.

Figure 14 shows the relationships between the fiber utilization factor, the fracture energy during pulling, and the parameter $S_{agg,tot}$. With an increase in $S_{agg,tot}$ from 0 to 12.38$·{10}^3$ ${\mathrm{m}}^2$, the fiber utilization factor increases from 0.69 to 0.99, indicating a near-full utilization of the fiber’s potential in terms of its strength. A further increase in the value of $S_{agg,tot}$ would likely lead to fiber rupture due to the occurrence of tensile stress in the cross-section greater than the tensile strength of the fiber material. The obtained utilization factor values are in agreement with other types of mechanically deformed steel fibers. For example, for hooked-end and twisted fibers, $K_f$ takes values of 0.87–1.0 and 0.58–0.96, respectively [77,78,79,80]. The utilization factor of straight fiber in UHPC, depending on the matrix composition, takes values in the range of 0.28–0.63, which is more than two times less than for other types of fibers. This is because the deformed fibers resist the pullout force by chemical bonds, frictional shear resistance, and additional mechanical anchorage from their deformation [79]. The corrugated fiber could be extracted from the matrix subsequent to the complete bending of the plastic hinges throughout its entire length. This would lead to a higher average bond strength and utilization factor than that of the straight fiber.

As shown in Figure 14b, the fracture energy increases with an increase in parameter $S_{agg,tot}$ from 83.9 to 135.6 N/mm. The experimental values are also described by an exponential function. However, a significant increase in the fracture energy is observed after the value of $S_{agg,tot}$ exceeds 10$·{10}^3$ ${\mathrm{m}}^2$. The same trend was observed in [81] while testing an inclined straight steel fiber, the surface of which was treated with sandpaper. An inclined steel fiber can be considered as a type of corrugated steel fiber, because during pullout the same mechanism of the plastic deformation of steel occurs. As the sandpaper grit increases from 120 to 800, the pullout work increases from 816.06 to 1109.57 J. The overall trend is exponential and similar to the graph in Figure 14b. As the parameter $S_{agg,tot}$ increases, the number of aggregate particles in contact with the fiber surface also increases, resulting in more scratches on the surface, as shown in Figure 13. This is similar to the sandpaper treatment used in [81].

The proposed mix-design parameter, $S_{agg,tot}$, can be used to optimize the pullout performance of steel fibers considering previously studied factors, such as the water/cement ratio and silica fume content [82,83]. Since the aggregate is the cheapest component of UHPC/UHPFRC, adjusting the value of $S_{agg,tot}$ will not lead to a significant increase in the cost of the mixture [84]. Improving the material properties at the meso level leads to enhanced macroscopic strength properties, as will be shown in the following sections.

4.2. Compressive Strength

The results of the compressive strength determination of the compositions with and without steel fiber are presented in Figure 15.

When the parameter $S_{agg,tot}$ is changed from 0 to 12.38$·{10}^3$ ${\mathrm{m}}^2$, the compressive strength of the UHPFRC increases from 158.7 to 178.7 MPa. The relationship between the strength of the UHPFRC and the parameter $S_{agg,tot}$ is linear. This is comparable to the results reported in [58], which showed that an increase in the aggregate/binder ratio from 0.16 to 0.80 exhibited an increase in the compressive strength of the UHPFRC from 139.0 to 160.7 MPa.

In the compositions without fiber, the compressive strength changes according to a nonlinear law with an increase in parameter $S_{agg,tot}$. When $S_{agg,tot}$ changes from 0 to approximately 7.0$·{10}^3$ ${\mathrm{m}}^2$, the compressive strength decreases from 158.9 to 149.2. After that, an increase in strength is observed. An increase in aggregate content to 0.2 ${\mathrm{m}}^3$ results in a slight decrease in strength, although this effect is less pronounced for compositions 2 and 3 (Figure 15b). This may be due to the smaller particle size and higher homogeneity of the material structure. Such relationships are typical not only for UHPC, but also for HPC and ordinary concrete. In [85], for example, it was found that by increasing the amount of aggregate in the HPC composition from 0 to 0.3 ${\mathrm{m}}^3$, the compressive strength decreased from 91.9 to 74.7 MPa. When the aggregate content was further increased to 0.45 and 0.6 ${\mathrm{m}}^3$, the compressive strength was 76.6 and 74.7 MPa, respectively. The strength of ordinary concrete decreases when the aggregate content reaches 0.4 ${\mathrm{m}}^3$, after which hardening is observed [86].

The different behavior of the UHPC and UHPFRC can be explained using the so-called “wing-crack model”, which was presented in [87] and used to predict the compressive strength of fiber-reinforced composites [88]:

$$f_c={\displaystyle \frac{f_t·\sqrt{\frac{\pi ·a_0·b}{c^2}}+\sqrt{2}·{\tau }_B}{1-\mu }},$$

(23)

where

$f_t$ is the tensile strength of the concrete, MPa;

$a_0$, b, c are the geometric parameters of the crack (see Figure 15b);

${\tau }_B$ is the crack-sliding resistance due to fiber bridging;

$\mu$ is the coefficient of friction between two crack faces.

According to the model, the compression failure of concrete is caused by the beginning of the growth of the most dangerous inclined cracks located at an angle of 45° to the axis of action of the compressive load. When an inclined crack shears, vertical cracks of mode I are formed (Figure 16a,b). The fracture of the material occurs due to the growth and coalescence of the vertical cracks. The expression located beneath the root of the numerator in Equation (23) determines the dimensions of the initial crack and the distance between them. Work [88] proposed equations for calculating the values of $a_0$, b, and c in Equation (23), which depend on the size of the aggregate and its volumetric concentration. It is assumed that the initial cracks form in the contact zone between the aggregate and the cement paste due to shrinkage. By increasing the proportion of aggregate in the volume of the mixture, the distance between the cracks decreases, which leads to their coalescence under a lower compressive stress. The growth of the vertical cracks is controlled by the axial tensile strength of the concrete. As shown in [89], a decrease in the particle size and an increase in the volume fraction of the aggregate leads to an increase in the axial tensile strength of concrete, due to the effect of crack deflection or crack shielding. As a result, with an increase in the volume fraction of the aggregate in the concrete composition, two competing processes take place: strengthening and destructive. The graph in Figure 16c shows how the values of $f_t$ and the expression $\sqrt{{\displaystyle \frac{\pi ·a_0·b}{c^2}}}$ change. The resulting curve in Figure 16d is a concave parabola, which corresponds to the experimental values presented in Figure 15b.

When fiber is incorporated into the UHPC composition, the relationship $f_c=f\left(V_{agg}\right)$ transitions to a linear form, as depicted in Figure 15a. The explanation for this follows from a further analysis of Equation (23). The variable ${\tau }_B$ reflects the increase in the sliding resistance of the two faces of an inclined crack due to the presence of fiber. According to [87], the value of ${\tau }_B$ depends on the volume fraction, geometric dimensions, and bond strength of the fiber to the concrete, ${\tau }_f$. As demonstrated in Section 4.1, the bond strength of corrugated fibers increases with increasing increments of the aggregate volume in the composition. It appears that this factor has the greatest impact on the compressive strength, which results in a proportional increase in this characteristic.

The literature currently presents several models for predicting the compressive strength of UHPC/UHPFRC, taking into account the degree of hydration of the cement in the presence of mineral additives, as well as the packing density of the binder particles [90,91]. As shown in Section 4.2, in order to achieve the required compressive strength, it is also necessary to consider the properties of the aggregate through the parameter $S_{agg,tot}$. The impact of this parameter on the compressive strength is of the same order of magnitude as the influence of the steel fiber volume fraction [92].

4.3. Axial Tensile Test

The results of the axial tensile tests are presented in

Table 8. Average tensile properties of tested UHPFRCs.

Mix ID	${\mathit{\sigma }}_{\mathit{c}\mathit{c}}, \mathbf{M}\mathbf{P}\mathbf{a}$	${\mathit{\epsilon }}_{\mathit{c}\mathit{c}}, \%$	${\mathit{\sigma }}_{\mathit{p}\mathit{c}}, \mathbf{M}\mathbf{P}\mathbf{a}$	${\mathit{\epsilon }}_{\mathit{p}\mathit{c}}, \%$	${\mathit{\alpha }}_0, \mathbf{M}\mathbf{P}\mathbf{a}$	${\mathit{\sigma }}_{\mathit{f},\mathit{m}\mathit{a}\mathit{x}}, \mathbf{M}\mathbf{P}\mathbf{a}$	${{\mathit{K}}^{\prime }}_{\mathit{f}}$
1-F	4.55	0.046	7.53	1.51	0.57	658	0.24
2-F	6.71	0.030	10.54	0.72	0.60	882	0.32
3-F	7.95	0.025	11.13	0.58	0.64	876	0.31
4-F	5.13	0.031	9.56	0.77	0.63	761	0.27
5-F	7.50	0.027	10.76	0.46	0.68	825	0.29

. Figure 17 shows the averaged tensile stress–strain diagrams. The averaged tensile stress–crack opening diagrams are presented in Figure 18.

The behavior of all the tested compositions under axial tension is typical for strain-hardening fiber-reinforced composites. After the formation of the first crack at a stress equal to ${\sigma }_{cc}$, an increase in the tensile stress is observed. The strain-hardening ratio, which is defined as the ratio of the maximum stress to the stress at the formation of the first crack, is in the range of 1.40–1.65, which is typical for this class of materials [14,15,24].

Increasing the parameter $S_{agg,tot}$ changes the position of the key points necessary to describe the tensile behavior of the UHPFRC. The cracking strength increases by 74%, from 4.55 to 7.95 MPa, as $S_{agg,tot}$ changes from 0 to 12.38$·{10}^3$ ${\mathrm{m}}^2$. According to [92], the cracking strength of fiber-reinforced concrete can be determined based on the following expression:

$${\sigma }_{cc}=f_t·\left(1+{\alpha }_0·V_f·{\displaystyle \frac{E_f}{E_c}}, [\mathrm{MPa}]\right)$$

(24)

where

$E_f$, $E_c$ are the modulus of elasticity of fiber and concrete, respectively, GPa.

From an analysis of Equation (24), it is evident that the cracking strength is determined primarily by the strength of the concrete matrix. A fiber with a volumetric content of 2% contributes about 5% to the strength. As previously stated, the tensile strength of concrete is contingent upon the volumetric content and particle size of the aggregate, which are taken into account by the parameter $S_{agg,tot}$ employed in this study. The greatest influence on strength is attributed to the volume concentration of the aggregate. For mixtures with a maximum grain size of 0.4 mm (compositions 2-F and 3-F), an increase in the volume fraction of the aggregate to 0.4 ${\mathrm{m}}^3$ results in a 74% increase in ${\sigma }_{cc}$, whereas, for compositions with a maximum particle size of 0.8 mm (compositions 4-F and 5-F), the increase is 64%. For mixtures with equal amounts of aggregate, it is observed that when the mean aggregate diameter decreases, the strength increases by 6–31%.

Such results are typical for concrete. In [86], an increase in the aggregate concentration from 0 to 80% led to an increase in the tensile strength from 2.3 to 2.78 MPa. In [93], an increase in the aggregate concentration from 19 to 50% also led to an increase in the splitting tensile strength from 3.80 to 4.58 MPa. According to [94], an increase in the maximum particle size resulted in a decrease in the splitting tensile strength of UHPC, which is also in agreement with the findings presented in this paper.

It was also found that the value of ${\sigma }_{pc}$ increased by 52% when the value of the parameter $S_{agg,tot}$ was increased from 0 to 12.38$·{10}^3$ ${\mathrm{m}}^2$. The maximum tensile stress that strain-hardening fiber-reinforced concrete can withstand after the formation of a crack is determined by the bridging stress of a group of fibers crossing the crack. According to [15], the analytical expression for ${\sigma }_{pc}$ is written as follows:

$${\sigma }_{pc}=\lambda ·{\tau }_f·V_f·{\displaystyle \frac{l_f}{d_f}}, [\mathrm{MPa}]$$

(25)

where

$\lambda$ is the factor accounting for fiber orientation, the mean embedment length, and the group effect during the fiber pullout.

In works [95,96], which considered UHPFRC with straight fiber, the variable $\lambda$ is defined as the product of two coefficients that take into account the orientation of the fiber and the efficiency of the fiber when it is oriented at different angles to the fracture plane ($\lambda ={\alpha }_0·{\alpha }_1$). For UHPFRC with deformed fibers (hooked-end, twisted, and corrugated), it is also important to consider the group effect during the pullout of such fibers.

For the tested compositions, the orientation coefficient, volume fraction, and geometric parameter of the fiber are constant values. Therefore, the growth in the value of ${\sigma }_{pc}$ can solely be attributed to an enhancement in the bond strength between the fiber and the concrete matrix, as depicted in Figure 11. Such a correlation between the fiber bond strength and post-cracking strength of the UHPFRC was also observed in [97].

Figure 19 illustrates the relationship between the strength characteristics of the UHPFRC and the parameter $S_{agg,tot}$.

Despite the increase in the tensile strength of the UHPFRC, the post-cracking strain decreases with an increasing value of $S_{agg,tot}$. The composition 1-F, which does not contain aggregate, yields the highest post-cracking strain. The post-cracking strain decreases with an increasing concentration of aggregate in the composition. This effect is explained by the fact that during the process of strain hardening, evenly distributed multiple cracks are formed. The mechanism of multiple cracking in strain-hardening cementitious composites is explained as follows in [98]: after the formation of a crack in the matrix, the load is assumed by a group of fibers crossing the fracture plane. If the applied load is not sufficient to pull out a group of fibers, the load is transferred back to the concrete matrix, which leads to the formation of a new crack. This process continues until the applied load leads to the pulling out of fibers in one of the previously formed cracks, which marks the transition of the material to the strain-softening stage. The decrease in the value of ${\epsilon }_{pc}$ observed in this study with an increasing value of $S_{agg,tot}$ is associated with the strengthening of the concrete matrix and the formation of fewer cracks (Figure 20a). In fiber-reinforced composites, commonly referred to as ECCs (Engineered Cementitious Composites), the aggregate content is frequently restricted to a range of 400–600 kg/m3 (0.15–0.22 ${\mathrm{m}}^3$), in order to enhance the difference between ${\sigma }_{cc}$ and ${\sigma }_{pc}$. This results in a significant increase in the post-cracking strain and energy absorption capacity [99,100]. In [101], numerical experiments were carried out, which showed that an increase in the strength of the concrete matrix led to a decrease in the post-cracking strain of the UHPFRC.

The graph in Figure 20b shows the relationship between ${\epsilon }_{pc}$ and the number of cracks, $N_{cr}$. From this figure, it is evident that there is a direct relationship between the two quantities, which confirms the mechanism described above.

Figure 21a shows the comparison between the fiber utilization factor during the pullout test and the composite’s axial tension. Even though the utilization factor tends to increase with an increasing parameter $S_{agg,tot}$, the absolute values are more than 2.5 times higher when a single fiber is being pulled out. The reason for this is the concentration of stress near the bends of a fiber when it is pulled out. In the case of the simultaneous pulling out of a group of fibers located next to each other, the stress fields are superimposed, which leads to the destruction of the concrete matrix. This effect is especially noticeable when deformed fibers are used.

The degree of destruction of the concrete matrix can be assessed by evaluating the parameter $\lambda$ from Equation (25), which encompasses the fiber group effect.

Table 9. Values of $\lambda$ for UHPFRCs with different fiber types.

Fiber Type	Mix ID	$\mathit{\lambda }$, -	Average Value	Reference
Corrugated	1-F	0.35	0.38	Present study
	2-F	0.39
	3-F	0.38
	4-F	0.39
	5-F	0.39
Straight	UHP-FRC-2.0	0.94	0.96	[15]
	U-S-2.0	0.97		[14]
	SL-2.0	0.96		[102]
Hooked-end	UHP-FRC-2.0	0.21	0.33	[15]
	U-H-2.0	0.53		[14]
	HL-2.0	0.26		[102]
Twisted	UHP-FRC-${\mathrm{T}}_1$-2.0	0.26	0.43	[15]
	U-T-2.0	0.59		[14]

presents the values of this parameter for compositions with different types of fiber. Since λ depends on the volume concentration of the fiber, the table shows values only for compositions with $V_f=2\%$.

The highest value of the coefficient $\lambda$ corresponds to compositions with straight fiber. This suggests that when straight fiber is pulled out, there is practically no damage to the matrix, and all the deformations are localized inside the channel [54]. The lowest value of $\lambda$ was obtained for the hooked-end fiber, which indicates the significant destruction of the concrete. Corrugated fiber yielded an intermediate value between the twisted and hooked-end fiber.

Figure 21b compares the appearances of the fibers after the pullout tests with those after an axial tensile test of the UHPFRC. In the first case, significant plastic deformation of the fibers are observed. In the second case, the fibers retains their original shape, which indicates their premature pulling out of the concrete. The results of the macroscopic observations correlate with the values of the fiber utilization factors in Figure 21a.

The results of the determination of the fracture energy-related parameters for the studied compositions are presented in

Table 10. Average fracture energy-related parameters of tested UHPFRCs.

Mix ID	${\mathit{g}}_{\mathit{f},\mathit{A}}, \mathbf{k}\mathbf{J}/{\mathbf{m}}^3$	${\mathit{N}}_{\mathit{c}\mathit{r}}, \mathit{E}\mathit{A}$	${\mathit{G}}_{\mathit{f},\mathit{A}}, \mathbf{k}\mathbf{J}/{\mathbf{m}}^2$	${\mathit{G}}_{\mathit{f},\mathit{B}}, \mathbf{k}\mathbf{J}/{\mathbf{m}}^2$	${\mathit{G}}_{\mathit{f}}, \mathbf{k}\mathbf{J}/{\mathbf{m}}^2$
1-F	120.4	31.5	0.42	21.7	22.1
2-F	65.8	24.2	0.30	25.8	26.1
3-F	55.8	17.6	0.36	35.7	36.1
4-F	63.6	22.7	0.32	21.3	21.7
5-F	41.5	16.1	0.28	25.1	25.1

.

The fracture energy during strain hardening, $g_{f,A}$, decreased with an increasing parameter $S_{agg,tot}$, which was associated with a decrease in the number of microcracks formed, and a decrease in the post-cracking strain, as shown in Figure 18a. The relationship between $g_{f,A}$ and the post-cracking strain value was also observed in [25,28]. An increase in $S_{agg,tot}$ from 0 to 12.38$·{10}^3$ ${\mathrm{m}}^2$ led to a decrease in $g_{f,A}$ of 116%, from 120.4 to 55.8 $\mathrm{kJ}/{\mathrm{m}}^3$. At the same time, the fracture energy per unit area, $G_{f,B}$, increased by 65%, from 21.7 to 35.7 $\mathrm{kJ}/{\mathrm{m}}^2$. This was due to an increase in the maximum stress ${\sigma }_{pc}$ and, accordingly, the area under the stress–crack opening diagram. The correlation between $G_{f,B}$ and ${\sigma }_{pc}$ was also noted by other researchers for UHPFRCs with different fiber types [72,103].

It is important to note that changing the parameter $S_{agg,tot}$ leads to the same changes in the properties of the material both at the meso level (when testing a single fiber) and at the macro level (when testing a composite), in terms of the strength and fracture energy. Figure 22 shows how the relative values of strength and total fracture energy change when testing a material at different scale levels. The vertical axis indicates the value of the “relative property”, which is defined as the ratio of the strength/fracture energy of the composition with aggregate to the strength/fracture energy of the composition without aggregate.

From the graphs presented in Figure 22, it is evident that the relationships reflect each other both qualitatively and quantitatively. This allows for the results from a single fiber pullout to be used to predict the behavior of the composite. The slightly higher relative axial tensile strength of the UHPFRC is associated with a slight increase in the fiber orientation coefficient with an increasing parameter $S_{agg,tot}$ (see Table 8).

In general, a particular strength and fracture energy can be achieved mainly by changing the parameters of the fiber reinforcement [14,104]. Adjusting the parameter $S_{agg,tot}$ is also an effective way to change the tensile properties of UHPFRCs, which must be taken into account during the design of the mixture. A more accurate accounting of all the input factors that affect the final properties of the material will make it possible to obtain more economical mixtures and expand the scope of application of this class of material.

4.4. Critical Stress Intensity Factor

The critical stress intensity factor also depends on the parameter $S_{agg,tot}$. As shown in Figure 23, as $S_{agg,tot}$ increases from 0 to 12.38$·{10}^3$ ${\mathrm{m}}^2$, the value of $K_{I,c}$ increases exponentially. The value changes from 0.447 to 0.932 $\mathrm{MPa}·\sqrt{\mathrm{m}}$ for compositions without fiber, and from 3.160 to 4.154 $\mathrm{MPa}·\sqrt{\mathrm{m}}$ for compositions with corrugated steel fiber. In [85], an increase in the aggregate volume content from 0 to 60% resulted in an increase in $K_{I,c}$ from 0.440 to 0.883 $\mathrm{MPa}·\sqrt{\mathrm{m}}$. This improvement is within the same order of magnitude compared to that of the plain UHPC tested in this study.

The observed increase in $K_{I,c}$ is explained by a modification of the fracture toughness of the cement paste, due to the various mechanisms of interaction between a crack and the inclusions that occur when aggregate is introduced into the composition: crack deflection, microcrack shielding, and crack trapping [105]. Compared to conventional concrete, UHPC exhibits a dense contact zone owing to its low water/cement ratio and high silica fume content, which binds large portlandite crystals into smaller-sized CSH. For example, the porosity of the contact zone of UHPC at the age of 28 days is about 2.5% [106], while the porosity of concrete with W/B = 0.4 can reach 36% [107]. Based on this, it can be assumed that the increase in $K_{I,c}$ in the UHPC is explained by only two mechanisms, crack deflection and crack trapping.

The critical stress intensity factor for UHPFRC was previously determined with 15 mm corrugated fiber [108]. With an aggregate concentration of 0.37 ${\mathrm{m}}^3$ and a matrix strength of 152.3 MPa, the $K_{I,c}$ value is 3.122 $\mathrm{MPa}·\sqrt{\mathrm{m}}$, which is 26% lower compared to composition F-5. This implies that increasing the length of the fiber also increases the toughness of the composite.

4.5. Modulus of Elasticity and Poisson’s Coefficient

The elastic modulus and Poisson’s ratio are independent of the particle size of the aggregate, and are solely determined by its volumetric concentration. As shown in Figure 24a,b, changing the size of the aggregate particles leads to a slight change in the deformation characteristics, which is within the measurement error range. Other researchers have reached similar conclusions based on the results of testing ordinary and self-compacting concrete [109,110].

A linear correlation is established between the modulus of elasticity and the volumetric concentration of the aggregate in both the UHPC and UHPFRC mixtures. When steel fiber is introduced into the composition, a constant shift of the approximation line is observed, while the angle of inclination remains the same (Figure 24c). The results obtained are consistent with the well-known rule of mixtures, which is written in the following form:

$$E_c=E_m·\left(1-V_{agg}\right)+V_{agg}·E_{agg},$$

(26)

where

$E_c$, $E_m$, and $E_{agg}$ are the modulus of elasticity of a composite, matrix (cement paste), and aggregate.

The diagram in Figure 25 compares the results from the experiments with the predictions made using Equation (26). There is a good agreement between the data, with a correlation coefficient of 0.98. This allows for Equation (26) to be used to predict the elastic modulus of UHPC and UHPFRC. It is worth noting, however, that this approach is highly simplified and necessitates the experimental determination of the elastic modulus of the cement paste, which may not always be feasible or appropriate. Works [111,112] presented more refined models that take into account the composition of the cement paste and its porosity, and are based on the multi-level Mori–Tanaka homogenization scheme [113].

A decrease in Poisson’s ratio is observed with an increasing volumetric concentration of the aggregate. The experimental data confirm the theoretical models presented in [114]. The authors proposed several different analytical expressions for determining the Poisson’s ratio of concrete, based on different geometric representations of the structure of the composite and the properties of each of the individual components. The resulting curve has the form of a concave parabola, whose extremum is observed at an aggregate concentration of 0.5–0.8 ${\mathrm{m}}^3$, depending on its Poisson’s ratio.

In order to achieve a high elastic modulus, it is necessary to ensure the high volume concentration of the aggregate in the composition. Without reducing the workability of the mixture in a fresh state, this can be achieved by optimizing the packing density of the aggregate particles by adding larger fractions. In this case, compositions with an elastic modulus in the range of 50–55 GPa can be obtained [112]. However, the strength characteristics of materials with an increasing aggregate particle size can be reduced, due to the decrease in the value of the parameter $S_{agg,tot}$, as shown in Section 4.1, Section 4.2, Section 4.3 and Section 4.4.

4.6. Drying Shrinkage

Figure 26 shows the results of the measurements of the drying shrinkage deformation for all the tested compositions.

The experimental data are approximated by the following equation:

$${\epsilon }_{sh}={\epsilon }_{sh,u}·\left(1-e^{-a·T^b}\right),$$

(27)

where

${\epsilon }_{sh,u}$ is the ultimate shrinkage strain;

T is the time, days;

a, b are the empirical coefficients.

Table 11. Coefficient values in Equation (27).

Mix Type	Mix ID	${\mathit{\epsilon }}_{\mathit{s}\mathit{h},\mathit{u}}$	$\mathit{a}, 1/\mathit{d}\mathit{a}\mathit{y}\mathit{s}$	$\mathit{b}$
UHPC	1	2197	0.206	0.70
	2	1426	0.216	0.74
	3	1117	0.224	0.82
	4	1543	0.215	0.71
	5	1174	0.215	0.70
	Average	-	0.215	0.73
UHPFRC	1-F	1743	0.221	0.67
	2-F	1237	0.256	0.75
	3-F	882	0.246	0.90
	4-F	1287	0.270	0.77
	5-F	921	0.221	0.75
	Average	-	0.243	0.77

presents the values of the coefficients in Equation (27), determined using the least squares method.

The size of the aggregate particles does not result in a significant change in the shrinkage deformation of UHPC or UHPFRC. The ultimate shrinkage strain, ${\epsilon }_{sh,u}$, is solely determined by the volumetric content of the aggregate, irrespective of the presence or absence of fibers in the composition (see Table 11). This phenomenon can be attributed to the fact that as the volume of aggregate increases, the proportion of cement paste, which is the main contributor to shrinkage, decreases. The driving force of shrinkage in HPC and UHPC is the emergence of capillary pressure when the relative humidity inside the material decreases (the so-called autogenous shrinkage) [115], as well as the evaporation of moisture that has not yet entered into hydration with the cement. Given the predominant closed nature of UHPC porosity, it can be inferred that the primary cause of shrinkage deformation is attributed to autogenous shrinkage.

The experimental results correspond to the data obtained by other authors for fine-grained concrete with W/C = 0.30, 0.35, and 0.50 [116,117]. Figure 27 shows the relationship between the relative ultimate shrinkage strain and aggregate volumetric content. It can be seen that there is a general tendency towards a decrease in the ultimate shrinkage strain, regardless of the composition of the cement paste.

Paper [118] presents an analytical model for predicting the shrinkage strain of fiber-reinforced concrete. It is postulated that, at a constant volumetric concentration of fiber, an increase in the bond strength leads to a decrease in the shrinkage strain. In Section 4.1, it is shown that reducing the particle size leads to an increase in the bond strength of steel fiber, which would reduce the shrinkage of UHPFRC. The experimental data indicate that there is no relationship between the bond strength and shrinkage, which is contrary to the findings of work [118].

We found that The introduction of steel fibers into matrices with different aggregate concentrations resulted in unequal reductions in the ultimate shrinkage strain. The greatest impact was observed when reinforcing the cement paste, resulting in a reduction of 454 units, from 2197 to 1743. The ultimate shrinkage strain decreased by 235–253 units when reinforcing the matrix with an aggregate concentration of 0.4 ${\mathrm{m}}^3$. This effect can be explained by a change in the ratio between the elastic modulus of the fiber and the matrix. In work [119], it was shown theoretically that an increase in the elastic modulus of the fiber (with a constant modulus of elasticity of the matrix) leads to a decrease in the ultimate shrinkage strain. The same was shown experimentally in [120] by adjusting the elastic modulus of the aggregate. The graph in Figure 28 shows the relationship between the ultimate shrinkage strain reduction and the ratio $E_f/E_m$.

Shrinkage of UHPC and UHPFRC is a significant problem. In restrained conditions, shrinkage generates high tensile stresses that can lead to early cracking and reduced durability of a structure [121]. As with the elastic modulus, shrinkage strains can be effectively reduced by increasing the aggregate concentration. Increasing the volumetric aggregate content leads to the deterioration of the workability of the mixture, due to a decrease in the paste film thickness around the aggregate particles [122]. To maintain a constant paste layer thickness with an increase in the aggregate content of the composition, it is necessary to increase the particle packing density, for example, by adding larger fractions. This leads to a decrease in the $S_{agg,tot}$ value and a decrease in the strength properties. Therefore, when designing a UHPC/UHPFRC composition, it is essential to determine which material properties are of higher priority.

5. Conclusions

This study presents a new mix-design parameter, $S_{agg,tot}$, which reflects the total surface area of the aggregate, and is calculated using the volumetric concentration of the aggregate in the composition and its specific surface area. An extensive experimental program was conducted in which five UHPC and five UHPFRC compositions with different $S_{agg,tot}$ values were tested. Based on the results of the research, the following main conclusions can be drawn:

• The bond strength of corrugated steel fiber to UHPC increases asymptotically with an increase in the value of the parameter $S_{agg,tot}$, which is associated with an increase in the number of contacts between the particles and the surface of the fiber. As a result, the friction force when the fiber is pulled out increases. A geometric model is proposed for calculating the number of contacts between the particles and the surface of the fiber, which is confirmed by the experimental results.
• The compressive strength of UHPFRC increases linearly with an increasing parameter $S_{agg,tot}$, and there is a slight decrease in strength when there is no fiber in the composition. It is found that the experimental data are well described by the “wing crack model”.
• The cracking and post-cracking tensile strength of UHPFRC exhibit the same functional dependence on the parameter $S_{agg,tot}$ as the bond strength of the fiber. The fracture energy during strain hardening, $g_{f,a}$, decreases with an increasing parameter $S_{agg,tot}$, which is explained by the strengthening of the concrete matrix and the formation of fewer cracks in the tensile zone. The total fracture energy, $G_f$, on the contrary, increases, which is associated with an increase in peak stress.
• The critical stress intensity factor increases with an increasing $S_{agg,tot}$, which is associated with various mechanisms for increasing the fracture toughness (deflection and crack trapping by aggregate particles).
• The modulus of elasticity and Poisson’s ratio are independent of particle size and are determined solely by the volumetric concentration of the aggregate.
• The drying shrinkage also depends only on the volumetric concentration of the aggregate. When steel fiber is introduced into a composition, the reduction in shrinkage depends on the ratio of the elastic modulus of the fiber and the matrix.

An improvement in the properties of UHPFRC with an increase in the value of the parameter $S_{agg,tot}$ is observed only when the fiber is activated, which occurs when cracks form and open under mechanical action on the material. If there is no relative displacement of the fiber and the concrete matrix, the properties are determined solely by the volumetric content of the aggregate (shrinkage, modulus of elasticity, and Poisson’s ratio).