The shear strength of the interface between artificial rock and printed concrete at super-early ages

Abstract

3D concrete printing has the potential to replace shotcrete for construction of linings of tunnels in hard rock. The shear strength of the interface between rock and printed concrete is vital, especially at super-early ages. However, traditional methods for testing the shear strength of the interface, e.g., the direct shear test, are time-consuming and result in a high variability for fast-hardening printed concrete. In this paper, a new fast bond shear test is proposed. Each test can be completed in 1 min, with another 2 min for preparing the next test. The influence of the matrix composition, the age of the printed matrices, and the interface roughness of the artificial rock substrate on the shear strength of the interface was experimentally studied. The tests were conducted at the age of the matrices at the 1st, the 4th, the 8th, the 16th, the 32nd, and the 64th min after its final setting. A dimensionless formula was established to calculate the shear strength, accounting for the age of the printed matrices, the interface roughness, and the shear failure modes. It was validated by comparing the calculated results and the experimental results of one group of samples.

Graphical abstract

Keywords

rock tunnel / printed concrete / interface / fast bond shear test / shear strength

1 Introduction

For several decades shotcrete has been applied to construct the primary linings of rock tunnels with many merits, such as high flexibility and no requirement for formwork [1]. Though significant progress has been made in mix proportion [2], spraying process, and equipment [3], rebound of shotcrete cannot be avoided. The rebound contaminates the in situ air quality and decreases the quality of sprayed layer [4,5]. The latter may influence the stability of tunnel linings [6,7]. 3D concrete printing technology provides a formwork-free, high automation degree, and layer-wise method for constructing concrete structures [8]. The first author of this paper was the first to propose the method of using 3D concrete printing technology to construct the linings of hard rock tunnels [9]. Capable of printing concrete precisely, this technology shows the potential to replace shotcrete to build the linings. Study of the bond property of the interface between printed concrete and substrate is a crucial prerequisite of in situ printing of tunnel linings.

Bond strength at the interface between rock and concrete is essential for the tunnel linings [10]. Studies have been conducted, at the Belgium−China Joint Laboratory for Industrialized Construction, to investigate adhesion of the interface between rock and printable concrete. Tao et al. [11] studied the influence of substrate properties, e.g., surface roughness, on adhesion of the interface between artificial rock and printed concrete. Concrete was used as the substrate material to simulate rock, creating ‘artificial rock’ in the investigation. The results show that an increase of the interface roughness can lead to less contact area between artificial rock and printable concrete, compromising the adhesion of the interface. The influence of polymer modification on adhesion was also studied by Tao et al. [12]. Two failure modes were observed, i.e., adhesion failure (failure occurring at the interface) and cohesion failure (failure occurring within the fresh material).

For the printing of tunnel linings, attention should be paid to the evolution of mechanical properties of the hydrating concrete, because the tunnel linings usually restrict stress redistributions in the ground during the hydration of the concrete [13]. At the final setting time, the concrete becomes solid and measurable mechanical properties start to develop [14]. Previous study on early-age mechanical properties of concrete mainly focuses on the age within the first 12 h [15] or within several days [16,17]. This paper focuses on an even earlier stage, i.e., the super-early ages, from the final setting time of the printable concrete until the 64th min after its final setting time. During this stage, the mechanical strength of concrete significantly increases as the microstructure of concrete rapidly changes, i.e., following a time-dependent exponential trend [18]. The fast development of strength, including the compressive strength and bond strength, determines the success of the printing [19]. In this study, the unit of min [20] is used to precisely characterize the time-dependent properties of printable concrete at the super-early ages.

Fast-hardening printable concrete is commonly used to improve buildability, e.g., the capacity of printing a concrete structure of 3-m height in 9 min [21]. Such printable concrete has the potential to replace shotcrete in construction of tunnel linings. However, the fast evolution of concrete strength renders traditional shear strength tests inapplicable. Traditional shear tests, such as direct shear tests between rock and concrete, commonly require 10–15 min to conduct each test. Within these 10–15 min, the mechanical properties of the fast-hardening printable concrete evolve rapidly, resulting in significant test errors with these traditional tests. Although achievements on interface strength [22] and cracking [23,24] of the hardened stage of concrete could be accomplished, the absence of practical test methods resulted in the insufficient understanding on the process of shear strength development of the interface during the super-early ages of printable concrete. A fast method for conducting the shear test on the interface between rock and printable concrete has been urgently needed to capture the shear strength of the interface. In this paper, a fast bond shear test (FBST) is developed. The shear test as such can be performed in 1 min and, as it takes about 2 min to proceed to the next test, thus a complete test cycle can be completed every 3 min. By shortening the test duration, the variability of the test results will be insignificant even for fast-hardening printable concrete. In this method, the time-dependent evolution of shear strength at the interface between artificial rock and printable concrete can be quantitatively studied.

The structure of this paper is organized as follows. An FBST is proposed in Section 2 for testing the shear strength of the interface between artificial rock substrate and printed concrete at super-early ages. Section 3 is devoted to the design of experiments mainly based on the FBST. The experiments aim to investigate the influence of the mix proportion of the matrices, the age of the printed matrices, and the interface roughness of the artificial rock substrate on the shear strength. In Section 4, the test results of the time-dependent shear strength and the shear failure modes are presented. Based on Section 4, a dimensionless formula for the shear strength at the interfaces is developed and validated in Section 5. Section 6 contains the main conclusions drawn from the present study.

2 Fast bond shear test

2.1 The mechanism and set-up of the FBST

In the FBST, push-shear tests are conducted at the interface between printed concrete discs and artificial rock substrate to measure the time-dependent shear strength of the interface. The mechanism of the FBST is shown in Fig.1. First, fresh cementitious matrices are printed into the cylindrical holes of the artificial rock substrate to form the interface bond. The interface bond evolves as the concrete hydrates. Then, the discs of printed concrete are loaded and pushed downwards at predetermined time instants at super-early ages. The maximum shear force of the interface between the printed concrete disc and the artificialrock substrate is measured and used to calculate the shear strength of the interface. After the loading program, the position of the loading rod is aligned with the next test sample for the next test.

2.2 Preparation of the artificial rock substrate with holes

Ultra-High Performance Concrete (UHPC) is used as the substrate material to simulate hard rock. The compressive strength of UHPC, i.e., over 100 MPa, is similar to that of the hard rock in mountain tunnels [25].

The artificial rock substrates are prefabricated for the FBST. Each substrate contains six holes. The upper part of each hole serves as the contact zone for bonding printed concrete, with a diameter of 50 mm and a depth of 25 mm. The lower part of the hole provides a hollow for the printed concrete to be pushed downwards, with a diameter of 60 mm and a depth of 55 mm. The dimensions of the substrate are shown in Fig.2.

The surface roughness in the upper part of the hole is designed based on two parameters. The root mean square of the surface profile Z_1 and the root mean square of the first derivative of the surface profile Z_2, are selected to quantify the interface roughness. Z_1 is an important parameter for reflecting the height deviation from the mean line through the profile [26]. Z_2 is a useful parameter for prediction of friction [27]. The expressions of Z_1 and Z_2 of the hole are shown as

(1)Z_1=\sqrt{\frac{1}{L}{\int }_0^L{y}^2\mathrm{d}x},

(2)Z_2=\sqrt{\frac{1}{L}{\int }_0^L{\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)}^2\mathrm{d}x},

where x is the vertical distance from the upper surface of the hole (mm), y is the horizontal distance between the surface profile and the mean line of the surface profile (mm), and L equals the depth of the hole (mm).

The artificial rock substrate is molded. The mold consists of 3D-printed tubes and acryl glass plates. The tubes are 3D-printed by a 3D printer iSLA660 using ultraviolet curable resin. The printed tubes serve as the confinement to form the holes in the substrate, while the acryl plates determine the boundary of the substrate. The assembled molds are shown in Fig.3(a).

The molding, demolding, and curing of the substrate are performed in a curing room with a temperature of (20 ± 2) °C and a relative humidity of 50% ± 5%. The UHPC is blended (see Subsection 3.2) and filled into the assembled molds. The molds are disassembled on the 3rd day. The resin tubes are removed from the substrate by breaking the weak connections of the printed tubes. After the demolding, the artificial rock substrates are placed in the curing room until an age of 28 d. The molding and demolding of the substrate are shown in Fig.3.

2.3 Printing concrete into the holes

Before printing, the lower parts of the holes are temporarily filled with foam cylinders, as shown in Fig.4(a). The diameter of the foam cylinders is identical to that of the lower part of the holes, i.e., 60 mm. The length of the foam cylinders is 100 mm, which is longer than the depth of the lower part of the holes so that they can be easily pulled out. A venthole in the center of each foam cylinder is preset. The top of the foam cylinder is covered with Teflon film.

Concrete printing is performed in a room with a temperature of \text{(}\text{20}\text{±}\text{2}\text{)} °C and a relative humidity of \text{50}\mathrm{\%}\text{±}\text{5}\mathrm{\%}. The cementitious matrices (see Subsection 3.1) are mixed (see Subsection 3.2) and filled into a cylindrical printer. The printer has a circular outlet with a diameter of 50 mm. Since the FBST mainly focuses on the interface bond between artificial rock and printed concrete, the layer-wise printing procedure is not used in this test. The printer is aligned with the hole to be printed and descends until the gap between the outlet and the top surface of the substrate is 10 mm. Since the outlet of the printer and the hole have the same diameter, i.e., 50 mm, the matrix is directly printed into the hole from the printer. The printing process is shown in Fig.4(b). When the printing of one hole ends, the printed matrix in the hole is separated from the matrix in the printer by a trowel. Then, the top of the printed sample is finished and the printer is moved to the next hole. The printed samples with finished top surfaces are also shown in Fig.4(b). Before the final setting time of each matrix, the foam cylinders are removed. The preset venthole allows for equalising the air pressure at both sides of the foam cylinder while it is pulled out. Furthermore, the Teflon film reduces the adhesion at the interface between the foam cylinder and the printed concrete. No settlement is observed on the upper surface of the printed samples when the cylinders are removed. The printed matrix–artificial rock substrate composite is placed in the loading position in a universal testing machine, as shown in Fig.4(c).

2.4 Pushing out the printed cylindrical samples

The printed discs of matrices are loaded and pushed downwards from the upper part of the holes after the final setting time of each matrix. First, the axes of the printed concrete disc and of the steel cylindrical loading rod (with a diameter of 50 mm) are aligned, as shown in Fig.5(a). Then, the bottom of the rod is lowered to the top of the disc, as illustrated in Fig.5(b). The upper surface of the matrix is finished (see Subsection 2.3) for achieving uniform contact with the bottom of the steel rod. The displacement control mode is used in the loading procedure with a loading speed of 10 mm/min. The loading procedure ends when the bond between the artificial rock substrate and the printed disc sample fails. The sample is pushed into the lower part of the hole, as illustrated in Fig.5(c). The relative position between the substrate and the steel rod is adjusted for testing the next disc sample, as shown in Fig.5(d). The whole procedure can be accomplished within 3 mins.

2.5 Calculation of the shear strength of the interface

The shear stress is assumed to be uniformly distributed at the interface considering the small thickness of the disc, i.e., 25 mm. The peak value of the loading force is recorded while pushing out the printed disc. The mean shear stress at the interface is calculated according to

(3)\tau =\frac{F_{\mathrm{m}\mathrm{a}\mathrm{x}}}{S},

where \tau represents the shear strength of the interface (Pa), F_{\text{max}} represents the maximum loading force (N), S represents the lateral surface area of a cylinder with a diameter of 0.050 m and a height of 0.025 m.

3 Experimental design

3.1 Mixture proportions

Three matrices were used in this study, labeled as matrix A, B, and C. Matrix A consisted of Portland cement paste serving as the reference mix. Matrix B represented traditional mortar without admixtures. Matrix C was fast-setting printable mortar with admixtures, i.e. superplasticizer, viscosity modifying agent (VMA), and accelerator. Matrix C was modified from a printable concrete mixture proportion from Tao et al. [11]. The accelerator was added to shorten the setting time and to increase the strength development rate at super-early ages [21].

The mixture proportions of the three matrices are shown in Tab.1. Standard Portland cement \text{P∙I 42.5} was provided by China United Cement Corporation. The cement had a specific gravity of \text{3.16} and a specific surface area of \text{340}{\text{mm}}^{\text{2}}/\text{kg} with its chemical compositions shown in Tab.2. China ISO Standard Sand, with a mean diameter of 0.7 mm, was used as the fine aggregate produced by Xiamen ISO Standard Sand Co., Ltd. Commercial liquid polycarboxylate superplasticizer PCA-I and liquid accelerator SBT-N II (no alkali type) were provided by Sobute New Materials Co., Ltd. Hydroxyethyl methyl cellulose (HEMC) with a viscosity of \text{100000 mPa∙s} was used as VMA.

The UHPC with the mixture proportions shown in Tab.3 was provided by Shanghai Ruishitan Environmental Protection Technology Co., Ltd. The premix binder of the UHPC mainly consisted of Portland cement and silica fume. The compressive strength of the UHPC reached 72.3 MPa on the 3rd day, 82.8 MPa on the 7th day, and 126.5 MPa on the 28th day.

3.2 Mixing procedure

The three matrices were mixed by a JJ-5 planetary mortar mixer. A compulsory concrete mixer (speed adjustable) was used to mix the UHPC. All ingredients were stored in a laboratory room with a temperature of (\text{20}\text{±}\text{2}\text{)} °C and a relative humidity of \text{50}\mathrm{\%}\text{±}5\mathrm{\%}.

The mixing steps for matrix A, B, and C included: 1) manually blending the liquid polycarboxylate superplasticizer with water for 10 s and mixing sand with the HEMC for 10 s (applicable only for matrix C); 2) adding water (with superplasticizer) to the cement in the bowl and mixing with the rotational speed of 140 r/min for 30 s; 3) adding sand (with the HEMC applicable for matric C) in the cement paste and mixing with the rotational speed of 140 r/min for another 30 s (not applicable to matrix A); 4) mixing with the rotational speed of 285 r/min for 30 s; 5) halting for 90 s; 6) mixing for 60 s with the rotational speed of 285 r/min (adding accelerator at the 45th second for matrix C).

The mixing steps for the UHPC included: 1) adding sand to the premix binder and mixing with the rotational speed of 60 r/min for 120 s; 2) adding water and the superplasticizer then mixing with the rotational speed of 100 r/min for 300 s; 3) adding the steel fiber and mixing with the rotational speed of 100 r/min for 180 s.

3.3 Experimental program

3.3.1 Fast bond shear test

FBST was designed to study the influence of the composition of the printed matrices, the age of the matrices, and the interface roughness on the interface shear strength at super-early ages. 3 matrices (see Tab.1), 6 time instants, and 4 levels of interface roughness (see Tab.4) were selected. The influence of adding fine aggregates on the shear strength of the interface was studied by comparing the test results of matrix A and matrix B. The influence of adding certain types of admixtures was analyzed by comparing the test results between matrix B and matrix C.

The surface roughness of the hole (\varphi 50\mathrm{m}\mathrm{m}) in the substrate was realized according to sinusoidal function curves. The Z_1 and Z_2 of the sinusoidal function curve are shown as

(4)\begin{array}{rl}{Z}_{1,\mathrm{s}\mathrm{i}\mathrm{n}}=& \sqrt{\frac{1}{L}{\int }_0^L{y}^2\mathrm{d}x}=\sqrt{\frac{1}{L}{\int }_0^L{\left[A\cdot \mathrm{s}\mathrm{i}\mathrm{n}\left(\frac{2\text{π}}{L}x\right)\right]}^2\mathrm{d}x}\\ =& \frac{\sqrt{2}}{2}A\approx 0.707A,\end{array}

(5)\begin{array}{rl}{Z}_{2,\mathrm{s}\mathrm{i}\mathrm{n}}=& \sqrt{\frac{1}{L}{\int }_0^L{\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)}^2\mathrm{d}x}\\ =& \sqrt{\frac{1}{L}{\int }_0^L{\left[A\cdot \frac{2\text{π}}{L}\cdot \mathrm{c}\mathrm{o}\mathrm{s}\left(\frac{2\text{π}}{L}x\right)\right]}^2\mathrm{d}x}=\sqrt{2}\text{π}\frac{A}{L}\approx 4.44\frac{A}{L},\end{array}

where Z_{\text{1,sin}} represents the root mean square of the sinusoidal curve (mm); Z_{2,\mathrm{s}\mathrm{i}\mathrm{n}} represents the root mean square of the first derivative of the sinusoidal curve (dimensionless); A represents the amplitude of the sinusoidal curve (mm); x represents the vertical distance from the upper surface of the hole in the artificial rock substrate (mm); y represents the horizontal distance between the surface profile and the mean line of the surface profile (mm); L represents the wavelength of the sinusoidal curve (mm).

The Z_{\text{1}} of natural rock interfaces ranges between 0.265 mm and 2.113 mm, while the Z_{\text{2}} of natural rock interfaces ranges between 0.127 and 0.615, obtained by statistical analysis of the morphology of natural rock joint samples [28]. The Z_{\text{1}} and Z_{\text{2}} of the surface roughness of the hole were designed to be in the same range as the Z_{\text{1}} and Z_{\text{2}} of natural rock interface. Therefore, the A of the sinusoidal curve ranged from 0.375 to 2.99 mm, and the value of A/L ranged from 0.0286 to 0.139. Within these ranges, 4 classifications of surface roughness were designed, with parameters shown in Tab.4 and pictures in Fig.6.

Six tests were conducted at various time instants for each matrix and roughness classification. The test started after the final setting time of the matrix. Therefore, the age of concrete in the FBST was measured from the final setting time of each matrix. Six test time instants were set: the 1st min, the 4th min, the 8th min, the 16th min, the 32nd min, and the 64th min after the time zero. Notably, performing a cycle of the FBST took about 3 min, consisting of 1 min for the testing and 2 min for preparing the next sample. Thus, a time interval of at least 2 min was required between two tests. From the 4th min, test time instants followed a geometric progression with a common ratio of 2. The test time instants enabled a short-interval measurement of the shear strength of the interface right after the final setting, while the interval between tests gradually increased as the printed matrix hardened.

The test was conducted in a laboratory with a temperature of (\text{20 ± 2}\text{)} °C and a relative humidity of 50\mathrm{\%}\pm 5\mathrm{\%}. Artificial rock substrates were first stored in the room for 24 h. A computer-controlled universal testing machine with a load capacity of 25 kN was used, as shown in Fig.1. The universal testing machine recorded the force for pushing out the samples with a relative error of \text{±}0.5\mathrm{\%}.

The samples were labeled as matrix-roughness-time instant. For instance, A-II-1 indicates that the matrix A bonded to the interface roughness II was tested at the 1st min after the final setting of the matrix A.

3.3.2 Workability tests

The setting time test, mini-slump test, and flow table test were conducted on each matrix to study the workability [29]. The setting time test was performed using a manual Vicat apparatus based on the ISO standard ISO 9597 [30]. The zero time for the matrices was established as the moment when cement and water were brought into contact in the mixing. The initial setting time and the final setting time were recorded with precision to the nearest min. The mini-slump test for the matrices followed the Chinese standard DL/T 5126-2001 [31]. A truncated cone with \varphi 50 mm at the top, \varphi 100 mm at the bottom, and 150 mm height was used. The matrices were poured into the truncated cone and compacted. Then the cone was lifted, and the slump value of each matrix was measured with an accuracy of ± 1 mm. The test was completed within 2 min after the mixing. The flow table test for the matrices was conducted following the Chinese national standard GB/T 2419-2005 [32]. This test was completed within 3 min after the mixing.

3.3.3 Unconfined uniaxial compression test

The unconfined uniaxial compression test (UUCT) was conducted on each matrix to study the time-dependent evolution of the compressive strength at super-early ages. The test was conducted in the laboratory at a temperature of (\text{20}\text{±}\text{2}\text{)} °C and a relative humidity of \text{50}\mathrm{\%}\text{±}5\mathrm{\%}. Cylindrical test samples of the matrices with dimensions of \varphi \text{50}\text{mm}\text{× 100}\text{mm} were prepared by molding. The molds for cylindrical samples were designed, particularly, for fast demolding. The mold comprised of three parts, i.e., one bottom plate and two half hollow cylinders. The two half hollow cylinders could be attached to the bottom plate. Each of them had a handle on its outer surface. The inner surfaces of the mold were coated with Teflon, minimizing the friction and adhesion during demolding. Demolding was accomplished by sequentially lifting and removing the two half hollow cylinders from the bottom plate. This process typically took less than 0.5 min. Afterward, it took an additional 0.5 min to place the sample in the testing machine. Therefore, the first test of UUCT could be conducted at the 1st min after the final setting time for each matrix. The above demolding process was started at the final setting time of each matrix (see Tab.5). The universal testing machine (see Subsubsection 3.3.1) was used for loading. The displacement rate was controlled to 10 mm/min. The time zero and the time instants for conducting the UUCT were the same as those for the FBST (see Subsubsection 3.3.1).

3.4 Workability

The results of the workability test are shown in Tab.5. As the reference matrix, matrix A shows the longest initial setting time (255 min) and final setting time (310 min). The high flowability of matrix A rendered the mini-slump test and flow table test unavailable. Matrix B, with the addition of fine aggregates, exhibited shortened setting times. The initial setting time and final setting time were about 70 min earlier than those of matrix A. The addition of admixtures, i.e. superplasticizer, HEMC, and accelerator, significantly reduced the initial setting time and final setting time of matrix C to 12 and 19 min, respectively.

3.5 Compressive strength of the matrices at super-early ages

The time-dependent compressive strengths of the three matrices are presented in Fig.7. The X-axis represents the time after the final setting of each matrix (time zero). The final setting time of each matrix is shown in Tab.5. The trends of the compressive strength development of matrix A and matrix B were basically identical, e.g., around 170 kPa in the 1st min and about 310 kPa in the 64th min. The addition of admixtures, especially the accelerator, accelerated the strength development of matrix C. The accelerator was composed of aluminosilicates and organic polymer, which caused the rapid generation of ettringite in a few mins after the mixing [33]. Rapid setting (see Subsection 3.4) and fast hardening were observed for matrix C. The compressive strength of matrix C surpassed those of matrix A and matrix B from the 4th min until the 64th min after the time zero. After the 32nd min, the rapid growth of C-S-H gel structure in matrix A and matrix B resulted in a steeper increase of the compressive strength [34].

4 Results

4.1 Interface shear strength development

The shear strength of the interface between artificial rock and the printed concrete is shown in Fig.8. The test results for matrix A, B, and C are shown separately. Each line diagram contains the shear strength results for 4 levels of the interface roughness. The X-axis represents the age after the final setting (time zero), which uses a logarithmic scale. The Y-axis is the shear strength of the interface using the linear scale.

The shear strength of the interface continuously increased as the concrete aged, as observed for all matrices. The characteristic of the shear strength of the interface varied for each matrix. For matrix A, the shear strength evolution of the interface for roughness classification II, III, and IV was similar, as shown in Fig.8(a). The shear strength of the abovementioned roughness classifications was more than twice the shear strength of roughness classification I.

For matrix B and C, stepwise increments in the shear strength of the interface were found as the interface roughness increased, e.g., from classification I to classification IV. From roughness I to roughness III, distinctive increments in shear strength were observed for samples of the same age. However, when the roughness increased from classification III to IV, the growth in the shear strength slowed down.

4.2 Interface shear failure mode

Three types of shear failure modes were observed in the FBST for the matrices. Type 1 shear failure mode was an interface shear failure between the substrate and the printed matrix, with the substrate distinctively observed at the failure interface. Type 2 shear failure mode was a mixed shear failure mode. In type 2 shear failure mode, both substrate-matrix interface shear failure and matrix-matrix shear failure were observed at the failure interface. In the matrix-matrix shear failure plane, coarse sand particles were rarely observed. Type 3 shear failure mode was a shear failure in the bulk material of the printed matrices.

Type 1 and type 3 shear failure modes were observed for matrix A, as shown in Fig.9. The red dashed lines in the illustration part indicate the interface shear failure position. The classification of shear failure modes is based on the shear failure characteristics of samples of matrix A in the FBST, with photos shown in Table A.1 in Electronic Supplementary Material.

For matrix B and matrix C, all shear failure modes, i.e. type 1, type 2, and type 3, were observed, as shown in Fig.10 and Fig.11. The appearance of coarse particles in the failure surface in type 3 failure mode was observed, and this varied significantly from their appearance in interface of type 2 shear failure mode. The pictures of the shear failures of each sample in the FBST are shown in the Appendix in Electronic Supplementary materials, from Tables A.2 and A.3 in Electronic Supplementary Material.

5 Discussion

5.1 The influencing factors on interface shear strength

The results indicate that the composition of printed matrices, the age of matrices, and the interface roughness influence the interface shear strength and the failure modes at super-early ages.

The composition of the matrices influences the interface shear strength and shear failure modes. When fine aggregates are added to matrix B, the interface shear strength of matrix B increases compared with that of matrix A. The shear failure surface of matrix A and matrix B varies significantly, which is obvious for roughness classification III and IV. For matrix A, the shear failure surface is flat, while for matrix B, there are coarse sand particles enclosed by cement observed in the shear failure surface, as shown in the enhanced photos in Fig.10. The existence of the cement-enclosed fine aggregates corresponds to the increase of the interface shear strength in matrix B, as shown in Fig.8(b). Adding admixtures, i.e., superplasticizer, HEMC, and accelerator, further increases the shear strength development rate of the interface at super-early ages. The dominant effect of the accelerator is observed from the experimental results of matrix C. For example, the final setting time t_{\mathrm{f}} reduces from 240 to 19 min, compared to that of matrix B. Moreover, the shear strength of matrix C is the highest of those of the three matrices after the 4th min. The admixtures used in this study (see Subsection 3.1) do not change the shear failure modes, as underlined by the similarity between the shear failure modes for matrix C and matrix B.

The age of the matrices contributes to the development of the interface shear strength at super-early ages. During super-early ages, the time-dependent increments of compressive strength and shear strength of printable concrete have been reported [20]. In this study, the shear strength of all interfaces continuously increases within 64 min after the final setting, as shown in Fig.8. Furthermore, the shear failure modes at the failure surface are independent of the age of the matrices, as shown in Tables A.1 and A.3 in the Appendix in Electronic Supplementary materials. Each test group (with a specific composition and certain interface roughness) demonstrates the same shear failure mode, not changing with the aging of the matrices at super-early ages.

The interface roughness influences both the interface shear strength and the shear failure modes. While there are limited results in the literature on the effect of interface roughness on the interface bond strength at super-early ages, the quantitative parameter of the interface {\text{Z}}_{\text{1}} [21] is found to influence interface adhesion.

In the case of interfacial roughness classification I, all samples fail in type 1 shear failure mode. The interface shear strength in this case is the lowest one among all roughness classifications. Considering the interface is smooth in classification I, the shear strength is mainly contributed by the adhesive bond between the substrate and the matrix at the interface. However, when the interface roughness increases, matrices A, B, and C should be considered separately.

For matrix A, the increase of roughness from classification II to IV leads to only type 3 failure mode with nearly the same shear strength, as shown in Fig.8(a) and Fig.9. Since no fine aggregate exists in matrix A, cement particles and hydration products, e.g., C-H-S gel, are the primary solid substance in matrix A during the test. Since the sizes of the particles of the solid substances are much smaller than the size of surface asperity in the case of interface roughness II, III and IV, the increase of the surface roughness results in a smooth shear surface inside the bulk material of matrix A.

For matrices B and C, the increase of interface roughness leads to type 2 and type 3 failure modes with stepwise incremental shear strength, as shown in Fig.8, Fig.10, and Fig.11. In type 2 shear failure mode, there is no interlocking of coarse sand particles within the shear failure surface. The mean particle size of the fine aggregates in matrices B and C, i.e., 0.7 mm, is 3.5 times larger than the maximum peak-to-valley height of the surface of classification II, i.e., 0.2 mm. The valley of the surface is too shallow to interlock the coarse sand particles when the shear failure occurs. As shown for type 2 shear failure mode in Fig.10 and Fig.11, no coarse sand is interlocked between the shear failure surface and the substrate. Type 3 failure mode is observed for roughness classification III and IV. The maximum peak-to-valley height of the surface for classification III is 0.8 mm, and for classification IV is 3.2 mm. Both heights are larger than the mean particle size of the fine aggregates, i.e., 0.7 mm. Meanwhile, the root mean square of the first derivative of the interface {\text{Z}}_{\text{2}} continuously increases as the roughness classification increases. Both factors contribute to a stronger interlock effect on the coarse sand. As shown in Fig.10 and Fig.11, coarse sand particles enclosed in cement are observed in type 3 shear failure mode. The failure mode transition accompanies the increase in the shear strength of the interface.

As shown in Eq. (6), six variables are selected to quantify the influence of the composition of the matrices, the age of the matrices, and the interface roughness, on the shear strength of the interface. In the equation, the time-dependent shear strength of the interface is represented by \tau (kPa). Three quantified variables are related to the composition of the matrices. The first variable is the shear strength of the smooth interface (roughness classification I) at the age of 1 min after the final setting, namely, {\tau }_0 (kPa). It is determined by the composition of the printed matrix and can be used as the reference value for the further strength development of each matrix. The second variable is the interval between initial setting time and final setting time, namely, t_{\mathrm{f}}-t_{\mathrm{i}} (min). It is also determined by the composition of the matrix and can quantify the setting process of each matrix. The third variable is the mean particle size of the fine aggregates, namely, D_{50} (mm). The age of a matrix after the final setting, namely, t^{\prime } (min), is selected to quantify the influence of the age of the matrices. Z_{\text{1}} (mm) and Z_{\text{2}} (dimensionless) are selected to quantify the influence of the interface roughness.

(6)\tau =f\left({\tau }_0{,t_{\mathrm{f}}-t_{\mathrm{i}},D}_{50},t^{\prime },Z_1,Z_2\right).

5.2 A dimensionless formula for the shear strength at the interface

A dimensionless formula for the shear strength of the interface can be deduced based on the test results. Five essential points are considered to establish the dimensionless formula in Eq. (7).

The first is the nondimensionalization of the shear strength. The shear strength \tau is divided by the reference value {\tau }_0 to obtain a nondimensional number of the shear strength. The value of \tau /{\tau }_0 reflects the development of shear strength for each matrix based on the reference value {\tau }_0.

The second is the relation between the shear strength and the age of the matrices. Based on the test results of Fig.8, the \tau -t^{\prime } curves follow the trend of a power law. This trend is consistent with Lee and Hover’s research that the time-dependent evolution of the yield stress and penetration resistance of cementitious materials around the final setting follows a power function [35].

The third is that the shear failure modes are independent of the age of the matrices at super-early ages. The shear failure mode is determined by the composition of the matrix and the interface roughness classification, as shown in the photos of shear failure modes in Tables A.1−A.3 in Electronic Supplementary Materialss.

The fourth is that the shear failure modes are closely connected with the shear strength of the interface. When the age of the matrix remains the same, the transition of shear failure mode from type 1 to type 3 corresponds with the increment of the shear strength. For all matrices, the lowest shear strength of the interface is found when the shear failure occurs at the interface between the substrate and the matrix. The highest shear strength of the interface appears when the shear failure occurs in the bulk material of each matrix. In the middle lies the shear strength accompanying type 2 shear failure mode, i.e., substrate-matrix and matrix-matrix mixed shear failure, with only small size of fine aggregates interlocked in the surface.

The fifth is that the transition of shear failure mode is determined by the composition of the matrices and the interface roughness. The ratio Z_{\text{1}}{Z}_{\text{2}}/D_{\text{50}} can indicate the transition of the shear failure modes. In this study, when Z_{\text{1}}{Z}_{\text{2}}/D_{\text{50}} equals 0, type 1 shear failure occurs. With the increment of the value of Z_{\text{1}}{Z}_{\text{2}}/D_{\text{50}}, the failure mode transitions toward type 3 shear failure mode.

(7)\frac{\tau }{{\tau }_0}={\left({\displaystyle \frac{t^{\prime }}{t_{\mathrm{f}}-t_{\mathrm{i}}}}+1\right)}^a\cdot \left({\displaystyle \frac{1}{1+c\cdot {\displaystyle \frac{Z_1{Z}_2}{D_{50}}}}}+b\cdot {\displaystyle \frac{c\cdot {\displaystyle \frac{Z_1{Z}_2}{D_{50}}}}{1+c\cdot {\displaystyle \frac{Z_1{Z}_2}{D_{50}}}}}\right),

where a reflects the time-dependent increment rate of the matrix, b is the ratio between the shear strength in the bulk material and the shear strength at the interface, and c is a parameter in connection with the failure mode transition.

The dimensionless formula of shear strength is based on the reference value {\tau }_0. Two parts contribute to the time-dependent shear strength of the interface. The first part is the time-dependent hardening in connection with the age and the composition of the matrix. The time-dependent shear strength follows a power function. The exponent a and (t_{\mathrm{f}}-t_{\mathrm{i}}) rely on the composition of the matrix. The increment of the age of the matrix t^{\prime } ({\text{1}\text{min}⩽t}^{\prime }⩽\text{64 min}) contributes to the shear strength based on the power function. The second part is the shear failure mode determined by the interface roughness and the composition of the matrix. It can describe the transition of shear failure mode from type 1 to type 3. When Z_{\text{1}}{Z}_{\text{2}}/D_{\text{50}} approaches 0, Eq. (7) approaches

(8)\frac{\tau }{{\tau }_0}={\left(\frac{t^{\prime }}{t_{\mathrm{f}}-t_{\mathrm{i}}}+1\right)}^a.

Equation (8) describes the time-dependent evolution of the shear strength when the shear failure occurs at the interface. When Z_{\text{1}}{Z}_{\text{2}}/D_{\text{50}} is large enough, Eq. (7) changes toward

(9)\frac{\tau }{{\tau }_0}={\left(\frac{t^{\prime }}{t_{\mathrm{f}}-t_{\mathrm{i}}}+1\right)}^a\cdot b.

According to the physical meaning of b, Eq. (9) describes the time-dependent evolution of the shear strength when the shear failure occurs in the bulk material. The increase of Z_{\text{1}}{Z}_{\text{2}}/D_{\text{50}} from 0 can be used to describe the transition of the shear failure mode from type 1 to type 3.

The solid substance of matrix A mainly consists of cement particles and C-H-S gel; the former are much smaller than the mean particle size of fine aggregates, D_{\text{50}}. When the value Z_{\text{1}}\cdot Z_{\text{2}} approaches 0, type 1 shear failure mode appears, and Eq. (8) can be used. When the value of Z_{\text{1}}\cdot Z_{\text{2}} increases, the shear failure mode transitions directly to type 3 shear failure mode, and Eq. (9) can be used.

The results of the shear strength of roughness I, II, and IV are used to fit the values of a, b, and c. The fitting follows 3 steps. First, the results of roughness I are used to fit the value of a according to Eq. (8). For roughness I, the value of Z_{\text{1}}\cdot Z_{\text{2}} approaches 0, and type 1 failure mode appears for all matrices. Based on the value of a, the shear strength of the interface roughness IV is used to fit the value of b according to Eq. (9). For roughness IV, the shear failure in the bulk materials of the matrices dominates the shear failure. The shear failure strength of roughness IV is considered to be the shear strength in the bulk material. The third step is to fit c using the test results of roughness II following Eq. (7). This step is applicable for matrix B and C, where type 2 failure mode is observed. The least squares method is used to fit the values of a, b, and c following the abovementioned 3 steps. The parameters a, b, and c of each matrix are fitted and shown in Tab.6.

The value of a of matrix B is higher than that of matrix A. It indicates that adding fine aggregates accelerates the shear strength development of the interface at super-early ages. The intervals {(t}_{\mathrm{f}}-t_{\mathrm{i}}) for matrix A and matrix B are similar, at around 60 min. With the shortest interval (t_{\mathrm{f}}-t_{\mathrm{i}}), i.e., only 7 min, matrix C shows the fastest shear strength development, though the value of a of matrix C is the smallest one. Adding fine aggregates increases the ratio of the shear strength in the bulk material to the shear strength at the interface by around 50%, as shown in the value of b. Adding admixtures, i.e., superplasticizer, HEMC, and accelerator, does not influence the value of b. Since no type 2 failure mode occurs in matrix A, the value of c is not determined. The value of c is similar for matrix B and matrix C, indicating a similar transition of the shear failure modes from type 1 to type 3. The fitted curves with the coefficient of determination (R^2) are shown in Fig.12 based on the fitted parameters in Tab.6. In Fig.12, the {\tau }_{\mathrm{A}}, {\tau }_{\mathrm{B}}, and {\tau }_{\mathrm{C}}, respectively, represent the time-dependent shear strength at the interface of matrix A, B, and C. The {\tau }_{0,\mathrm{A}}, {\tau }_{0,\mathrm{B}}, and {\tau }_{0,\mathrm{C}} stand for the reference value {\tau }_0 of matrix A, B, and C.

5.3 Validation of the model

The proposed model is validated by comparing the calculated time-dependent shear strength with the experimental results of the interface roughness III. The experimental results of the interface roughness III are reserved for validation, which was not used for fitting the model parameters in Tab.6. As shown in Fig.13, the dashed curves are the fitted results of the shear strength of the interface roughness III. The fitted results are obtained based on the proposed formulas. For matrix A, Eq. (9) is used to calculate the curve. The reason for selecting Eq. (9) is based on the shear failure type of roughness III for matrix A, i.e., type 3 shear failure mode. Detailed discussion on the shear failure modes for matrix A can be found in Subsections 5.1 and 5.2. For matrices B and C, Eq. (7) is used to calculate the curves. Equation (7) describes the shear strength and the transition of the shear failure modes, which is applicable to the interface roughness III of matrix B and matrix C. The experimental results of interface roughness III are represented by the points in Fig.13. The coefficient of determination (R²) for the validation is shown in Fig.13. The highest value for R² is 0.98, and the lowest value is 0.95, indicating a satisfying correlation between the proposed model and the experimental results.

6 Summary and conclusions

This paper investigates the shear strength of the interface between artificial rock and printed concrete at super-early ages by the FBST. The following conclusions are drawn.

1) The FBST was established. The method shortens the time interval between successive test samples to 3 min. In the 3 min, the whole testing process for one sample and the preparation for testing the next one can be accomplished. The method is applicable for testing the time-dependent shear strength at the interface between fast-hardening concrete and artificial rock substrate at super-early ages.

2) A dimensionless relation was established relating the shear strength of the interface, the age of the printed matrices, and the interface roughness. A formula was deduced based on the shear strength of the interface and the failure mode in the FBST. The formula considers the time-dependent evolution of the matrices and the transition of failure modes with shear strength. The transition of the failure modes from the interface shear failure to the shear failure in the bulk material of the matrices was considered in the formula. The formula was validated by comparing the experimental results and the fitted results from the model. The formula can be used to calculate the time-dependent shear strength of the interface at super-early ages.

3) The FBST developed in this study is suitable for testing fast-setting and fast-hardening matrices without coarse aggregates, i.e., fast-setting mortar. The diameter of holes in the artificial rock substrate is 50 mm. Though far larger than the maximum diameter of sand particles, the diameter of the holes can be similar to the diameter of coarse aggregates in concrete. For testing concrete with coarse aggregates, a larger diameter of the hole should be used in the FBST.

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