Effects of time-varying liquid bridge forces on rheological properties, and resulting extrudability and constructability of three-dimensional printing mortar

Abstract

Extrudability and constructability are two important, yet contradictory issues pertaining to the construction of three-dimensional (3D) printing concrete. Extrudability is easily achieved when 3D printing cement mortar has a high water content and low cohesion, but the printed structure is easily collapsible. However, a 3D printing cement mortar with a low water content and high cohesion has a relatively stable printed structure although the cement mortar might not be extrudable. This study proposes a particle-based method to simulate 3D printing mortar extrusion and construction as an overall planning tool for building design. First, a discrete element model with time-varying liquid bridge forces is developed to investigate the microscopic effects of these forces on global rheological properties. Next, a series of numerical simulations relevant to 3D printable mortar extrudability and constructability are carried out. The study demonstrates that the effects of time-varying liquid bridge forces on rheological properties and the resulting extrudability and constructability of 3D printing mortar are considerable. Furthermore, an optimized region that satisfies both the extrusion and construction requirements is provided for 3D printing industry as a reference.

Graphical abstract

Keywords

particle-based simulation / liquid bridge force / rheological property / 3D printing mortar / extrudability / constructability

1 Introduction

Three-dimensional (3D) printing technology is recognized as a key link in the next industrial revolution. It has been widely used in various fields, such as mechanical manufacturing, aerospace, industrial design, and biomedicine. Many scholars have explored the application of 3D printing technology in the construction industry.

1.1 3D printing technology in construction industry

Recently, 3D printing technology has been widely applied in the construction industry. Vizotto [1] proposed a construction method using free-form components that are suitable for the progressive accumulation and selective solidification of cement materials. Khoshnevis et al. [2–6] invented the contour crafting technique, in which the nozzle movement is controlled by the programming and contouring of objects where printing is conducted by extruding ink materials. Kazemian et al. [7] proposed the contour crafting cable system. In addition, Lim et al. [8] contributed to developments in the construction-scale 3D printing processes. However, the printing speed was relatively low [9]. According to the pre-set printing path, the printing nozzle might continuously print the structural components layer by layer at a certain speed and angle [10]. Le et al. [11] developed a mix design and determined the fresh properties required for high-performance printing concrete, in which adhesives were used to harden the gravel powder layer. Ma et al. [12] reviewed the emerging techniques proposed for construction. Marketing has undergone a revolution in recent years [13]. Additional potentials and challenges related to 3D concrete printing have also been investigated [14,15]. The early age mechanical behavior of 3D printed concrete has been investigated by Bos et al. [16] and Wolfs et al. [17]. A novel technique based on digital data has also been used for fabrication of a concrete shell and concrete bridge [18,19]. Moreover, a 3D printed bridge was demonstrated to have a relatively long life cycle [20].

1.2 Cohesion model of moist bulk media

Krenzer et al. [21] classified the simulation techniques of cement-based media into three groups, namely the friction model of dry bulk media, cohesion model of moist bulk media, and Bingham model of suspension media, as shown in Tab.1. In general, moist bulk media can be analyzed using the ab initio approach for thixotropy, parameter optimization technique, digital data-driven method, and virtual prototyping approach [22–26]. However, the effects of capillary bridge characteristics on global structural behavior should not be ignored [27,28]. Many scholars conducting particle-based viscoelastic fluid simulations using the discrete element method (DEM) demonstrated that the cohesion model was superior to other methods [29–31]. In addition, the capillary bridge properties among cement particles change over time. For example, Akinci et al. [32] proposed a momentum-conserving two-way coupling method of smoothed particle fluids and arbitrary rigid objects based on hydrodynamic forces. Moreover, Ando et al. [33] conducted highly adaptive liquid simulations on tetrahedral meshes. The advantages of the cohesion model have previously been highlighted [34–39]. Therefore, only the cohesion model of moist bulk media is used to investigate the rheological properties and resulting extrudability and constructability of 3D printing mortar in this study.

1.3 Research gap

Extrudability and constructability are two of the most important issues in 3D printing mortar construction. However, these two factors are contradictory in nature. When 3D printed cement mortar has a high water content and low cohesion, the extrudability is easily achieved, but the printed structure collapses easily. In contrast, when the water content of a 3D printed cement mortar is low and the cohesion is high, the printed structure is relatively stable, but the mortar might be difficult to extrude.

As mentioned in the previous section, numerous laboratory research experiments have been conducted to investigate the performance of 3D printable cement-based materials. In addition, numerous simulation techniques for granular flow have been developed. However, studies focused on the 3D printable mortar mechanism from a microscopic point of view are limited. Therefore, the objective of this research study is to develop a particle-based method to simulate mortar extrusion and construction as an overall planning tool for the structural design of 3D printable mortar. The proposed discrete element approach considers the cohesion forcing of moist bulk media. Research questions addressed in this simulation-based study include:

1) How do time-varying liquid bridge forces affect the rheological properties of 3D printable mortar?

2) How do the rheological properties of 3D printable mortar affect overall their extrudability and constructability?

1.4 Organization of this paper

The paper is organized as follows. In Section 2, the numerical method, consisting of the discrete element approach, cohesion model, and simulation algorithm on granular flow, is introduced. In Section 3, the effects of time-varying liquid bridge forces on the rheological properties of 3D printable mortar are investigated using discrete element simulations of the slump, uniaxial compression, and shearing tests. In Section 4, extrudability and constructability analyses and evaluations are conducted. An optimized region that satisfies the requirements for both extrudability and constructability is obtained. Finally, the conclusions drawn from this study are set out in Section 5.

2 Numerical method

This section introduces certain novel methods and concepts of virtual reality aided by intelligent building design. Furthermore, this section presents the basic concept of multiscale 3D printed cement-based materials, the liquid bridge, close cooperation between particle-based animation and 3D printing construction, and relevant algorithms of the overall planning tools.

2.1 Discrete element approach

A 3D printed concrete building should maintain good concrete extrudability and fluidity at the nozzle extrusion stage, maintain concrete constructability with minimal deformation and no damage in the forming and curing stage, and ensure high concrete heterogeneity. These factors reflect the difficulties facing the application of 3D printed concrete in buildings. Therefore, the control of printing material extrudability and workability is a major challenge in the process of constructing 3D printed concrete buildings.

The literature on multiscale analysis proposes the division of cement-based materials into four levels [40], as shown in Fig.1. At the first level, that is, the nanometer scale, the atomic and molecular structure C-S-H gels can be observed. At the second level, the liquid or paste bridge occurring between two sand particles becomes the research focus. At the third level, 3D printed concrete at the millimeter scale is the research focus. At the fourth level, the concrete structural member at the meter scale is considered. The hydration reaction of cement occurring at the first and second levels is one of the key factors affecting the change in the overall structural performance of 3D printing. Two other key factors are the physical process of deflocculation and build-up of shear resistance. The structuration, or indeed, the chemical process behind underpinning thixotropy as an indicator of shear resistance, captures the rate of increased shear resistance in the early stage after the pumping and extrusion agitation of filament stops. This study investigates the effect of time-varying material properties at the first level on the overall behavior at the fourth level.

Using DEM numerical modeling, this study attempts to quantify the mechanism causing the contradiction between extrudability and constructability. However, all numerical methods have limitations, and DEM is no exception. DEM can only consider two factors, namely the particle itself and interaction between the particles. Although the real mechanism is significantly more complicated than the one which the numerical method can simulate, the particle properties are assumed to remain unchanged and that the interaction changes with time, resulting in a series of subsequent reactions. Although numerous interfacial transition elements are available, surface tension is likely one of the easiest factors that can be used to quantify the interaction changes.

The evolution of a “cement paste bridge” from a “liquid bridge” to a “solid bridge” is thought to be one of the key factors affecting the extrudability and constructability of 3D printing mortar. To verify this conjecture, a corresponding parameter was first obtained in this study and then used as a control parameter. Based on the relevant liquid bridge force formula, the parameter selected for this study was the surface tension of the liquid bridge, \gamma. Microscopic and microscopic stochastic evolution models of cement-based materials have been provided in the literature. These models are worthy of further study. However, owing to limited capacity, this study did not apply these models to simulate the stochastic evolution, but rather applied the surface tension of the liquid bridge as the sole fine-scale control variable. In addition, the key parameters of extrudability and constructability of coarse-scale 3D printing mortar, such as the elastic modulus E and plastic yield strength σ_y, have been studied in Ref. [27]. A two-scale numerical simulation method with a particle and cement paste bridge element was used to explore the relationship between the fine- and coarse-scale parameters. Once the relationship between the parameters was established, the shear stress of the liquid bridge was used to change the corresponding elastic modulus and plastic yield strength. Finally, the reliability of the extrudability and constructability of the 3D printing mortar was evaluated according to the changing path.

2.2 Liquid bridge

One of the typical constraints in a particular assembly is the liquid bridge between two particles. Fig.2 shows the liquid bridge model between particles i and j, where R_i and R_j represent the radius of particles i and j, respectively; R represents the equivalent radius; a represents the distance between two particles; d_sp/sp represents the immersion distance; and θ represents the gas–liquid contact angle of the particle surface. For example, by applying the Gladkyy and Schwarze model [28] for two particles of unequal diameters, the equivalent radius is given as

(1)R=\frac{2R_i{R}_j}{R_i+R_j}.

The immersion distance is given as

(2)d_{\text{sp/sp}}=\frac{a}{2}\left(-1+\sqrt{1+\frac{2V}{\text{π}Ra}}\right),

where V denotes the volume of the liquid bridge. The liquid bridge force is expressed as

(3)F=-\frac{2\text{π}R\gamma \cos \theta }{1+a/2d_{\text{sp/sp}}},

where \gamma denotes the surface tension of the liquid. It might be defined as a stochastic function of position x, namely \gamma(x, \phi).

Jennings et al. [27] developed a simple digital technique for generating nanogranular structures. The simulation starts with an individual particle at the center of the area, and new particles are then randomly generated in contact with the existing particles until the particles reach the edge of the area, as shown in Fig.2. The generated particles consist of LD C-S-H, HD C-S-H, UHD, and clinker phases.

Based on the above multiscale analysis and liquid bridge force theory algorithm, the first research problem can be considered: Which material property dominates both extrudability and constructability? The performance of cement-based materials in 3D printing processes changes with the hydration reaction, which is the key influencing factor affecting the extrudability and constructability of 3D printed materials. However, this change, called the stochastic evolution at the micro scale, might be extremely complex. For easy controllability, the material coefficient that dominates the overall macro behavior of 3D printing must be determined. The liquid bridge force is posited as a good choice, that is, the hydration reaction could be regarded as the solidification of the liquid bridge at the micro scale. Using Eqs. (1)–(3), most of the coefficients are shown to be related to the shape of the liquid bridge, whereas \gamma is the only parameter related to the material properties. In the subsequent parts sections, a series of numerical experiments are used to observe the effects of the change in parameters on the overall behavior of 3D printing.

2.3 Simulation algorithm for granular flow

A 3D printing mortar material is a type of composite material with high heterogeneity and complex particles. For the virtual simulation of particle filling, many different algorithms have been proposed to simulate the complex packing behavior of discrete particle elements. Several particle filling simulation algorithms have been successfully applied in the field of computer-aided design. These simulation methods can be roughly divided into two categories, namely the Eulerian gird approach and Lagrangian particle method. In this study, the Lagrangian particle technique was applied to simulate 3D printing mortar construction at the micrometer scale, as shown in Fig.3.

In a particular system with a set of N particles and scalar constraints, a particle i has a mass m_i, position p_i, and velocity v_i. The position of each particle is iteratively updated by resolving the constraints while the linear and angular momenta of the system is balanced, where p denotes a particle position array and ζ(p) denotes a constraint array.

To find a correction vector Δp, such that ζ( p + Δp) = 0, the constraint is given as

(4)\zeta (p+\mathrm{\Delta }p)\approx \zeta (p)+\mathrm{\nabla }p\zeta (p)\mathrm{\Delta }p=0,

and Δp is restricted in the direction \mathrm{\nabla }p\zeta (p) with a scaling factor ξ to conserve the linear and angular moments.

(5)\mathrm{\Delta }p=\xi \mathrm{\nabla }p\zeta (p),

where the scaling factor ξ_i for particle i is expressed as

(6){\xi }_i=-\frac{{\zeta }_i(p)\text{/}{m}_i}{{\sum }_j{\Vert \mathrm{\nabla }{p}_j{\zeta }_i(p)\Vert }^2/m_j},

and the position correction array is given by

(7)\mathrm{\Delta }{p}_i=-\frac{{\zeta }_i(p)\text{/}{m}_i}{{\sum }_j{\Vert \mathrm{\nabla }{p}_j{\zeta }_i(p)\Vert }^2/m_j}\mathrm{\nabla }{p}_i{\zeta }_i(p).

Using position-based dynamics, a fast simulation of viscous fluids with elasticity and thermal conductivity was executed. In this study, the Karhunen−Loève stochastic particle element modeling method is proposed to conduct in-depth research on the virtual simulation of the entire 3D printed mortar building construction process. The Karhunen−Loève stochastic chaos expansion has been applied in the field of stochastic numerical theory for many years. However, in this study, the Karhunen−Loève stochastic chaos expansion is combined with the particle element modeling method for the first time and applied to the entire 3D printed mortar building construction process. The most significant feature of this method is the accurate and random generation of a pixel image by combining nano-indentation, scanning electron microscope, and X-ray tomography of a particle element with high heterogeneity material. Thus, the entire 3D printed concrete building construction process might be simulated using this particle element. However, a significant issue is the means by which to generate high heterogeneity as a constraint property.

The conventional constraint control procedures were investigated for generation, modification, and cancellation, as shown in Box 1. Note that the viscous label was denoted for viscous fluid particles and elastic for particles in elastic materials.

Box 2 presents the complete procedure of the steps performed in our proposed method. A particle n has M_n constraints, denoted as ζ_n,1,…,ζ_n,Mn, which are generated, modified, and deleted using Box 1 (line 11). The position correction step was carried out until its iteration reached a specified count. The cohesion constraints were solved with density constraints in parallel to satisfy both types of constraints as far as possible and because the effect of one type of constraint can be ignored if one side has previously been solved.

The granular simulation in this study was primarily implemented using the open-source software package Yade [41], which is a community driven open-source software written in C++ and Python under Linux, and its extensible open-source framework is based on DEMs. The calculations were written in C++ using a flexible object model that allowed independent execution of new algorithms and interfaces. Python was used for fast and concise scene construction, simulation control, and post-processing debugging. This platform is one of the core elements of the overall simulation tool proposed in this paper.

3 Effects of time-varying liquid bridge forces

In this section, a series of numerical experiments relevant to extrudability and constructability that were carried out are described. These numerical experiments include slump, uniaxial compression, and shearing tests for 3D printed mortar samples. The purpose of these simulations is to investigate the effect of the microscopic material parameter \gamma on the global behavior. Thereafter, the relationship between the microscopic material parameter \gamma and overall behavior was used to analyze the extrudability and constructability of the 3D printed cement-based material.

3.1 Slump test

3.1.1 Discrete element simulation

The simulation of the slump test was conducted to determine the influence of microscopic material parameter \gamma on the slump phenomenon. The slump cone mold was 300 mm in height, 100 mm in diameter at the top, and 200 mm in diameter at the base. For the experiment, the cone was filled with 3D printed cement-based material. The slump distance is measured as the displacement between the initial top surface of the concrete and that of the concrete after removing the mold. In general, scholars typically set up and calibrate the material parameters of the DEM model through simple experiments, such as the slump test. For example, Yang et al. [42] used values of 10⁶ for the normal stiffness and viscosity, and values of 10⁵ for the tangential stiffness and viscosity based on slump tests. Subsequently, Mechtcherine et al. [43] proposed the calibration of parameters for contact models based on simple reference experiments that can be processed in the laboratory and modeled though simulation. In addition, Mechtcherine and Shyshko [44] suggested that the relationship between the yield stress and particle bond strength can be obtained using the slump test. Moreover, Jayathilakage et al. [45] proposed that for 3D printed mortar discrete element simulation, the model parameters can be calibrated step by step using the extrusion test.

The simulation of the slump test included the following three steps. First, the boundary wall of the slump cone was established and parameters values were set. Second, the slump cone was filled with the particles; the particles accumulated under the action of gravity. Once filled, the particles outside the cone were deleted, thus retaining only those inside the cone. Finally, the boundary wall of the slump cone was removed. The mortar particles then collapsed under the influence of gravity. Once the mortar height stabilized, the height difference was recorded between the highest point of the sample and the height of the slump cylinder at this time. In most cases, after the simulation time exceeded 20 s, the particles ceased changing; therefore, the slump time was set to 30 s. The time step was set to 1 × 10⁻⁵ s.

The material properties used in the slump test are listed in Tab.2. In this study, the particle density was 2000 kg/m³ because the particle was considered to be an aggregate wrapped in cement mortar. The contact friction angle is 80° because the influence of the rough aggregate surface roughness was considered. The solid–liquid contact angle selected was 45°, in accordance with Ref. [46]. The parameters were calibrated according to the slump test.

To investigate the effect of microscopic material parameter \gamma on the global behavior, the laboratory experiment was simulated using the particle-based model proposed in this study. Fig.6 presents the results of this numerical experiment. The results indicated that the microscopic material parameter \gamma increases, whereas the slump distance H considerably decreases. Furthermore, the microscopic material property was shown to have a significant influence on the overall behavior of the 3D printed material.

Subsequently, the third research question was investigated: How could a simplified reliability assessment be made using the overall planning technique? To answer this question, the relationship between the micro and macro scales should be established. In this regard, the simulation at the macro scale is based on the representative volume element (RVE). The following Mohr–Coulomb criterion is used in RVE as follows:

(8)\left|\tau \right|\text{=}C-\sigma \tan \varphi ,

where C is the cohesion coefficient, and ϕ is the angle of internal friction. The aim of this simulation is to understand the relationship between \gamma, C, and ϕ. Fig.7 shows that C and ϕ determine the size and shape of the elastic region. An ideal uniaxial plastic model is shown in Fig.8. The transition of material from a liquid to a solid at the macroscopic scale is dependent on three key parameters, namely the elastic modulus E, yield strength σ_y, and cohesion coefficient C. During solidification of the 3D printed cement-based material, all three key parameters gradually increase from almost zero to a certain magnitude. The main key factor influencing the slump test is the yield strength σ_y.

The collapse of the specimen can reasonably be inferred to result from the stress at certain points exceeding the yield stress. In contrast, the collapse of the block stopped because the stress value was just lower than the yield stress. In this study, the yield strength σ_y was assumed to potentially have a positive linear relation to the final height of the test specimen, based on the following approximation:

(9){\sigma }_{\text{y}}\approx \frac{W}{A}=\frac{WH}{V}=dgH,

where W denotes the specimen weight, A denotes the mean area projected on the ground plane, H is the specimen height after removing the slump cone, V is the specimen volume, d is the specimen density, and g is the gravitational acceleration.

3.1.2 Experimental validation

In this study, the slump test was used to calibrate and verify the parameters of the viscoelastic liquid bridge model. Our previous work [47] indicates that the simulation results are in good agreement with the experimental work by Ma et al. [12] with a maximum error of approximately 3.8%. The proposed method was demonstrated to be capable of simulating the 3D printing process for mortar.

Once the proposed method was validated, the relationship between the yield strength σ_y and liquid bridge parameter \gamma was investigated, as shown in Fig.9.

Thus, the yield strength σ_y can be expressed as a function of the liquid bridge parameter \gamma as follows:

(10){\sigma }_{\text{y}}=-36.29{\gamma }^2+827.3\gamma +1667.

Many studies have assumed that the yield strength increases during solidification [10–13]. Once the relationship between the surface tension and yield strength is established, the yield stress can be increased by controlling the corresponding surface tension value in the simulation.

3.2 Uniaxial compression test

The simulation of the uniaxial compression test was conducted to determine the influence of the microscopic material parameter \gamma on uniaxial compression. The material properties used in the uniaxial compression test are listed in Tab.2, where the surface tension force is from 2.0 to 20.0 N/m. In the simulation, a cylindrical container with a diameter of 200 mm and a height of 400 mm was created; then the particles were piled up under gravity and the container was removed after completion, as shown in Fig.10. The final compression rate was 50%. Fig.11 shows the results of the numerical uniaxial compression test. The results indicate that all stress–stain relationships were approximately linear. Thus, the slopes of the straight lines were assumed to be Young’s modulus E, and the relationship between \gamma and E could be determined. Fig.12 reveals the relationship between E and \gamma. Thus, E can be written as

(11)E=-96.27{\gamma }^2+3529\gamma +2699.

Once the relationship between the surface tension and elastic modulus is established, the elastic modulus can be increased by controlling the surface tension value in the simulation.

3.3 Shearing test

The simulation of the shearing test conducted to determine the influence of the microscopic material parameter \gamma on the shearing stress. The material properties used in the shearing test are listed in Tab.2, where the surface tension force is from 2.0 to 10.0 N/m. Fig.13(a) illustrates schematic of the shearing test. Fig.13(b) shows the process of the numerical shearing test. Fig.14 shows the results of the numerical shearing test. The shear stress τ is calculated according to the formula mentioned in Ref. [17]. Thus, The shear stress τ can be written as

(12)\tau =-3.125{\gamma }^2+225\gamma +1662.

4 Extrudability and constructability of 3D printing mortar

We previously discussed the influence of the micro scale liquid bridge parameter on the yield strength, elastic modulus, and shear strength. The increasing speed at which the yield strength is achieved in the curing process of cement-based materials significantly affects the extrudability and constructability of 3D printed materials. In this section, the multiscale simulation method was used for different rates of yield strength and critical strength to evaluate the critical performances of extrusion and construction of 3D printed materials among randomly selected samples in a specific range. The principles of elastic buckling and plastic collapse were investigated by Suiker et al. [48], and the critical failure stress approach developed by Perrot et al. [49] have been extended to estimate the critical performances of extrusion and construction of 3D printed cementitious materials.

The cement-based material construction is based on complicated and intricate interactions, including flocculation and solidification, owing to hydration at the contact nodes between the cement particles. According to Roussel et al. [50], flocculation is completed in 1 s, followed by the generation of C-S-H bridges between cement particles for approximately 40–60 min at the imaginary contact points. The rate of C-S-H bridge creation is assumed to be constant because the heat of hydration is constant during the interaction period. Therefore, the increase in yield stress with time (or surface tension of the C-S-H bridges) is deduced to be linear for 40–60 min. Subsequent to this linear increase period, which lasts up to 60 min, the rate of yield stress increases. Thus, the following critical performance analyses are based on the hypothesis that the rate of surface tension increase is linear in the first 40 min.

4.1 3D printing mortar extrusion simulation

In this section, the multiscale simulation method is used to simulate the extrudability of 3D printed materials using the standard open-up testing approach. In the 3D concrete printing process, the material properties of the mortar are time-varying, and the yielding point of the mortar increases. If the mortar is printed too slowly, the material might solidify in the nozzle and thus not be extruded. According to previous research, the solidification process of mortar can be simply simulated by using the surface tension force.

For simplification, the surface tension force was assumed to be proportional to time. The initial surface tension force value was assumed to be 2.0. Five different solidifying rate were chosen, namely \gamma =4/5t+2, \gamma =8/15t+2, \gamma =2/5t+2, \gamma =4/15t+2, and \gamma =1/5t+2. It is worth noting that the hardening rate in the simulation was significantly high. The computational cost of the simulation was significantly high when the real hardening rate was adopted. Tab.3 lists printing parameters used in the open-up test. The material properties are the same as Tab.2. The extrusion performance was investigated by printing long strips.

Fig.15 shows the simulation results at five different hardening rates. When the height of the printed strip changed significantly or the strip broke, extrusion was considered to have failed. The blue vertical line in Fig.15 marks the position at which extrusion failure occurred, and the value at that time was defined as the critical value.

Fig.16 shows five different curves with different hardening rates. The critical values of extrusion failure, 3.9, 3.8, 3.6, 3.4, and 3.3, are marked on the figure. The lower the hardening rate of mortar, the smaller the critical value. The critical value is approximately connected by a curve and can be divided into two areas. The area below the curve is considered to be the printable area where the extrusion performance of mortar is good, and the printed mortar strip is continuous and complete. The area above the curve is considered the non-printable area, where the extrusion performance of mortar is poor.

4.2 3D printing mortar construction simulation

During the construction of 3D printing mortar, the structure may undergo failure owing to plastic deformation and elastic buckling, thus constructability must be considered. At present, there are no unified constructability evaluation criteria. Certain scholars use the maximum number of layers that are printed and stacked without collapse within a given time as the evaluation index, while other use the deformation after printing a fixed number of layers. This study simulated the constructability by printing multi-layer mortar and investigated the constructability of 3D printed mortar through the deformation of the cross section of the mortar layer.

In this study, the screw extrusion nozzle was used, as shown in Fig.17. The diameter of the nozzle was 80 mm. The horizontal movement speed of the nozzle was 500 mm/s. The rotation speed of the screw was 15 rad/s. The printing height was 80 mm. The material parameters of the particles are listed in Tab.2, where the surface tension force is 4.5 N/m. The parameters remained unchanged during the printing.

In this study, a 7-layer hollow square column was continuously printed. Starting from the origin, the nozzle moved 500 mm in the positive direction of the Y axis, 250 mm in the positive direction of the X axis, 500 mm in the negative direction of the Y axis, and 250 mm in the negative direction of the X axis. After returning to the original point, the nozzle was raised vertically (positive direction of the Z axis) by 80 mm. The cycle was completed with a total of 7 layers.

Fig.17 shows the printing process and results. The deformation of the lower printing layer increased significantly with an increase in the number of layers. Because the lifting height of the nozzle after printing each layer had a fixed value of 80 mm, the distance between the top of the printing strip and nozzle mouth increased with the number of layers. An increase in the number of layers causes the shape of the cross section changed unevenly along the axial direction owing to the irregular deformation of the long side of the printing. The deformation of the four corners of the square column was less than that of the other positions because the deformation of the corners is limited by the sides.

To more clearly demonstrate the change in cross section of each layer during printing, the cross section of the column was visually represented. Fig.18 shows the cross-sectional shape change of the two long sides after each layer was printed, where R_h is the ratio of the surface maximum height to the nozzle height and R_hw is the ratio of the maximum height to the maximum width of the cross section.

After printing the 1st layer, the maximum height of the cross section was approximately 70 mm, R_h was approximately 87.5%, the maximum width was approximately 115 mm, with a R_hw of approximately 61%. The shape was roughly that of a water droplet. After printing the 2nd layer, the 1st layer was greatly deformed, and the deformed height was 49 mm. The height change rate was 30%. The height of the 2nd layer was 89 mm, and the shape of the section was almost circular. The total height of the wall was 138 mm, with a R_h of approximately 86.2%. The maximum width was approximately 125 mm, with a R_hw of approximately 110%. Fig.19(a) shows the curve of section height and number of layers. Fig.19(b) shows the relationship between the maximum cross section width and the number of layers.

Fig.19 shows that the maximum height and width of the cross section of the printed layer are basically proportional to the number of layers when 1−5 layers are considered. The average height of each layer was approximately 65 mm and the average width was approximately 8 mm. The structure was not subject to plastic deformation. Compared with the 5 printed layers, after the 6th layer was printed, the height increased by only 36 mm, the width increased by 23 mm, and the height–width ratio R_hw decreased. The structure had therefore entered the plastic region and undergone great deformation. After printing the 7th layer, the height–width ratio of the cross section continued to decrease. The ratio of the total cross section height to the nozzle height R_h reduced to 72.8% at this time.

Mortar hardening was not considered when printing. Each printed layer was equivalent to the increase in the uniformly distributed loading applied. In the 1st 5 layers, the deformation increased linearly with the increase in loading. For the 6th layer, the load increment was certain, but significant deformation was observed, which is regarded as plastic failure. Therefore, in this case, the maximum number of layers without damage (5 layers) is deemed to be the critical index of constructability of cement-based materials.

The main factor affecting the constructability of 3D printed mortar is the hardening rate of mortar. In the printing process, printed mortar solidifies with time, and the strength and stiffness increases. In this study, the lower and upper limits of constructability are suggested as follows. For the lower limit of constructability, the printed mortar does not harden over time, or the hardening speed is very slow. For the upper limit of constructability, the printed mortar has very high strength, no deformation, or a rapid hardening speed. In the real world, the mortar property is found between these two limits.

The lower and upper limits of constructability were simulated in this study. The lower limit of constructability was considered in the first case, and the upper limit of constructability in the second case. A nozzle similar to that used in the previous section was used in this simulation. Side plates and top scrapers were added underneath the nozzle. The diameter of the nozzle and spacing between the side plates were 120 mm, and the height of the scraper from the bottom was 75 mm. The cross-sectional shape of the long mortar strip was close to that of a rectangle. The printing speed was 300 mm/s, and the printing length was 1250 mm. The constructability was evaluated by printing three layers back and forth. Tab.2 lists material parameters of the particles, where the surface tension force was 2.0 N/m, which corresponded to a mortar slump of approximately 156 mm.

Fig.20 shows the cross-sectional projection and side view after printing each layer for both the lower and upper limits of constructability. After the first layer was printed, almost identical rectangular cross section shapes of 120 mm × 75 mm were obtained for both cases. After printing the second layer, a significant difference between the two cases was observed. For the lower limit, the bottom mortar was subjected to the loading of the upper mortar owing to graving, and the width increased by 57.5%, the overall height was approximately 145 mm, and deformation mainly developed in the direction of the width. After printing the third layer, compared to the upper limit, the printed shape of the lower limit was greatly deformed. The overall height was approximately 180 mm, 20% smaller than that of the upper limit. Furthermore, the maximum width was approximately 270 mm, 260% greater than that of the upper limit.

Without considering mortar hardening over time, the fluidity of the mortar was too strong, resulting in poor constructability of multi-layer printed material. Improving the constructability of mortar resulted in a reduction of the mortar extrusion quality. Mortar exhibited strong fluidity during extrusion and could be rapidly solidified after extrusion. Thus, the requirements of both extrusion and constructability were fulfilled.

5 Conclusions

The objective of this research was to develop a particle-based method to simulate mortar extrusion and construction to investigate the 3D printable mortar mechanism from a microscopic point of view. This study investigated two research questions. (1) How do time-varying liquid bridge forces affect the rheological properties of 3D printable mortar? (2) How do rheological properties of 3D printable mortar affect the overall extrudability and constructability? The final conclusions drawn to answer these two questions are as follows.

For the moist bulk 3D printed cement-based material, the random solidification of the cement slurry bridge significantly affects the extrudability and workability of the entire 3D printed concrete structure. The liquid bridge parameter \gamma might affect both extrudability and constructability. A series of numerical tests, including the slump, uniaxial compression, and shearing tests, were carried out on 3D printed cement granular materials to approximate the relationship between the liquid bridge parameter \gamma, yielding stress σ_y, Young’s modulus E, and shear strength τ. A straightforward performance evaluation criteria was developed based on the liquid bridge parameter \gamma.

The increasing rate of the liquid bridge surface tension \gamma and 3D printing speed were demonstrated to be two crucial factors for judging extrudability and buildability of moist bulk cement-based materials. The analytical diagram in Fig.16 could provide a good reference to resolve the contradictory problem between extrudability and constructability in 3D printing mortar construction. This diagram clearly indicates which areas are safe, which areas require caution, and which areas are dangerous when constructing 3D printed mortar at a given printing speed. Although there are numerous topics for further study, the proposed analytical method is demonstrated to be feasible. The proposed method is approximately consistent with real-life situations and might be valuable and useful in the 3D printing industry.

Although only the surface tension evolution was considered, other characterizations may be considered to estimate extrudability and constructability. Other researchers have tested the time-evolution of the mechanical properties of fresh 3D printed mortar that relies on rheometer testing. This study ignores the potential effects of thixotropy, flocculation, and the development of chemical hardening products because this research attempts to quantify the most fundamental mechanism of the phenomena using the lowest order factor, that is, the liquid bridge surface tension. To quantify thixotropy and flocculation with other higher order factors is a significant topic for future research. In addition, another limitation of this research is that the complicated evolution of chemical hardening products in the liquid bridge is ignored. For the sake of simplification, the standard liquid bridge model was used to roughly approximate the mechanical change in 3D printing global behavior. Thus, the stochastic evolution of chemical hardening products in the liquid bridge will be a significant topic for research in the future.

Furthermore, the DEM proposed in this paper has one defect. That is, particles of different sizes cannot be considered. In other words, the influence of the particle grading cannot be analyzed by the model proposed in this paper. Thus, one of the future follow-up investigations of this study is to develop a DEM in which the particle grading is considered. In addition, a further shortcoming of this study is that the effect of water evaporation on the overall 3D printed structure was not considered. Thus, one of the future follow-up investigations of this study is to carefully investigate the influence of water evaporation on particle adhesion during cement hydration.

The early strength of cement has an important impact on 3D printing extrusion behavior. The discrete element numerical simulation method proposed in this paper has certain advantages in this respect. Cement strength is an important index for evaluating the quality of cement and provides a basis for dividing the strength grades of cement. Cement strength refers to the ability of a cement mortar-hardened specimen to withstand damage from external forces, and is one of the most important physical and mechanical properties of cement. According to the different forms of stress, cement strength is generally divided into compressive strength, flexural strength, and tensile strength. The maximum stress of a cement mortar-hardened specimen under compression failure is called the compressive strength of cement. The maximum stress of the cement mortar-hardened specimen under bending failure is called the flexural strength of the cement. The maximum stress of the cement mortar-hardened specimen under tensile failure is called the tensile strength of the cement. The method proposed in this paper can input microscopic parameters in the simulation and record the gradual increase in strength of the liquid bridge force during the cement hardening process in detail. Therefore, it is greatly advantageous for the study of the influence of the early strength of cement on 3D printing extrusion behavior.

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