Nonlinear predictive modeling of building rates incorporating filament compression deformations in 3D printed geopolymer concrete

Abstract

3D printed concrete undergoes compressive deformation when printed fresh, often overlooked by traditional methods, impacting buildability prediction accuracy. In this paper, the buildability prediction model is modified by incorporating the Mohr–Coulomb damage criterion and focusing on the compressive deformation during the printing process. The prediction model combines the following key components: 1) the utilization of bilinear stress−time loading curves to simulate nonlinear stress−time loading curves during the actual printing process; 2) conducting uniaxial unconfined compression tests on cylindrical fresh specimens with different aspect ratios (ranging from 0.25 to 2) to extract the stress–strain response of the material; 3) the refinement of material parameters (including elastic modulus and plastic yield stress) and their variations with aspect ratio derived from the uniaxial unconfined tests. The material experimentation results indicate that the green strength exponentially decreases with increasing aspect ratio, while Young’s modulus exhibits a linear increase with the same parameter. Experimental comparisons were made during hollow drum printing tests using two different printing materials against the Mohr–Coulomb buildability prediction model. The results from these experiments demonstrate the improved accuracy of the new model in predicting failure heights (with relative error rates of 5.4% and 10.5%).

Graphical abstract

Keywords

3D concrete printing / buildability / green strength / Young’s modulus / nonlinear building prediction model

1 Introduction

3D printing technology is currently receiving widespread attention across various fields [1–3]. And one of the 3D concrete printing is an innovative construction technology that utilizes digital techniques and additive manufacturing concepts to rapidly and efficiently construct complex structures and building elements, thereby eliminating the need for traditional formwork [4–6]. This technology offers significant advantages in terms of cost reduction and waste reduction in the construction industry, while also positively impacting the fields of architecture and sustainable development [6]. Currently, the extrusion-based 3D printing technique is most widely adopted, where the printing material is extruded through a print head and deposited layer by layer to assemble structures without the need for formwork. On one hand, the extruded material must maintain its shape under the self-weight and the weight of the overlying layers, avoiding plastic collapse [7]. On the other hand, the overall structure must remain stable and prevent elastic buckling [8]. These characteristics represent the behavior commonly known as “buildability” of 3D printed concrete (3DPC) [9,10].

According to research findings, an increased Young’s modulus has the potential to enhance the local or global stability of printed structures [11]. The static yield stress refers to the minimum stress required to initiate material flow in a quiescent state [12], and a higher static yield stress contributes favorably to achieving improved green strength [13]. The term “higher” is relative and dependent on process parameters such as printer type (e.g., stamping or screw extruder), printing speed, and vertical stacking rate. Optimal buildability performance can be achieved through material formulation optimization, including adjustments in water content and admixtures to attain higher static yield stress and Young’s modulus, combined with process optimization encompassing the mixing process, printing interval, and vertical stacking rate. The performance of 3DPC structures predominantly relies on the buildability characteristics of the printing material. Previous studies have extensively investigated the rheology of freshly printed material, green strength, Young’s modulus, and the evolution of material properties over time in relation to the buildability performance of printing materials [11,14,15]. Consequently, comprehending the characteristics and buildability of printing materials plays a vital role in optimizing the mixture proportions, selecting appropriate processes, and establishing specifications and quality control standards for novel concrete formulations designed for 3D printing [16].

Buildability refers to the maximum height at which a specific material can be successfully printed without experiencing failure in a given cross-sectional geometry. As a result, the buildability of printing materials can be directly assessed through trial printing methodologies [15,17–19]. Nevertheless, accurately predicting the buildability performance of printed structures remains a challenging task. Currently, various models based on rheology [20,21], mechanical properties [22,23], or a combination of both [24] are employed for buildability prediction. Attempts to predict buildability by considering material characteristics, development, and the geometry of the printed component have yielded mixed results. Wolfs et al. [14] established a numerical model based on the Mohr–Coulomb failure criterion to predict the buildability of printing materials. The model demonstrated accurate prediction of the failure mode, with the total predicted number of filaments being 27.5% higher than the experimental value. Tripathi et al. [25] developed a buildability prediction model for different potential failure modes and achieved a relative error prediction value of 15% through experimental investigations. The Mohr–Coulomb model proves effective in accurately estimating the height at which plastic failure occurs. Kruger et al. [20] successfully predicted the plastic failure of printed structures based on the Mohr–Coulomb failure criterion, and experimental studies verified the model’s conservative error to be below 10%. Analysis of relative errors indicated that overly conservative model predictions can be attributed to material defects, stress redistribution of individual filaments under self-weight compression, and geometric defects causing eccentricity under vertical loading [14,20,24,25].

Buildability assessment analysis and prediction models frequently rely on the properties of fresh 3D printed materials. The unconfined uniaxial compression test (UUCT) has been widely employed to evaluate the green strength and Young’s modulus of 3DPC at various ages [14,24,26]. Researchers such as Wolfs et al. [14] and Panda et al. [15] utilized specimens with a diameter (D) of 70 mm and a length (L) of 140 mm, while Kurt et al. [27] opted for dimensions of D = 75 mm and L = 100 mm. Most studies adopted specimens with an L/D ratio of 2, enabling the formation of an oblique shear damage surface [14]. This approach eliminates size effects and ensures a dimensional scale that accurately represents the behavior of concrete during the printing process [23]. However, in 3D printed filamentary layers, the aspect ratio typically remains below 1. Consequently, Kruger et al. [20] proposed a correction for the strength of fresh concrete, as the performance behavior derived from previous studies based on an aspect ratio of 2 does not precisely reflect the force behavior of real 3D printed filaments. Hence, it is imperative to conduct mechanical performance tests on fresh concrete with varying aspect ratios to obtain accurate performance information regarding fresh 3DPC.

In freshly printed 3D concrete, two primary modes of failure can occur: plastic failure and stability failure. The specific type of failure is determined by the geometric shape of the object, loading rate, material properties, and boundary conditions [28]. Plastic failure takes place when the self-weight stress imposed by subsequent layers exceeds the strength of the underlying printed material. Previous studies have suggested that the failure of printed structures is primarily attributed to plastic failure caused by the self-weight of the printing components [11,26,28–31].

During the process of layer stacking in 3D printing, printed filaments frequently undergo vertical compression deformation caused by their own weight [32]. This deformation leads to a gradual separation between the print head and the filament, resulting in a progressive increase in filament extrusion height [25,32]. Unfortunately, the physical phenomena of the filament compression deformation are often disregarded in the buildability prediction models, which is also another key factor affecting the buildability prediction accuracy of printed concrete. Hence, it is essential to investigate the effect of filament compression on buildability prediction.

This study focuses on the buildability analysis of the fresh 3D concrete printing, especially for the failure mode of plastic yielding. First, the Young’s modulus and green strength for different fresh printable mortar mixtures with varying aspect ratios were tested and analyzed. The UUCT was employed to investigate these patterns. Furthermore, the impact of filament compression on buildability was investigated through hollow column structure printing experiments, utilizing the Mohr–Coulomb prediction model. Lastly, a modified prediction model was developed, taking into account filament compression, and its accuracy was also evaluated through experimental validation.

2 Materials and methods

2.1 Materials and mix design

In this study, granulated blast furnace slag (GGBFS) and fly ash (FA) were employed as precursor materials, while powdered sodium silicate powder was used as an activator (with a molarity of 0.9 (nSiO₂/nNa₂O), containing 47.5% SiO₂, 50% Na₂O, and 3% H₂O by mass fraction). Common river sand served as the fine aggregate in the mixture. Sodium gluconate powder and barium chloride powder were incorporated as retarders, with mass fractions of 1% and 0.5%, respectively. A control group consisting of materials with different buildability characteristics was established, and steel fibers (8 mm in length, 0.12 mm in diameter) were added at a volume content of 1%. It should be noted that the 1% volumetric dosage is selected by considering the maximum dosage to ensure that the extrusion process is not blocked during the printing process. The chemical composition of the materials is presented in Tab.1. The proportions of the mixed materials are provided in Tab.2, and all values are expressed as mass ratios of the geopolymer precursors (GGBFS + FA). The values in parentheses denote the corresponding values in kg/m³. The particle size distribution of the materials is illustrated in Fig.1.

2.2 Unconfined uniaxial compression test

The influence of specimen size and shape on the properties of common limestone rock has been examined. In contrast, the Young’s modulus of limestone rocks demonstrates an increment with increasing size [33]. However, limited research has been conducted on the effect of aspect ratio on the early Young’s modulus of mortar in the context of 3D printed materials. UUCTs were performed on columnar mortar specimens to evaluate their Young’s modulus characteristics, as well as their green strength properties. Similar tests have been utilized in previous studies investigating the early development of mortar strength [14,15].

The specimens were fabricated using acrylic circular molds with heights and internal diameters of L = 140 mm, D = 70 mm; L = 70 mm, D = 70 mm; L = 35 mm, D = 70 mm; and L = 17.5 mm, D = 70 mm, respectively, resulting in L/D ratios of 2, 1, 0.5, and 0.25 (Fig.2(a)). For each L/D ratio, four parallel samples were cast in a single mold. Prior to testing, all specimens underwent uniform compaction and were subjected to 5 s of vibration on a shaking table. The molds were then removed after 1 min of resting, and the deformation of the specimens (diameter and height) caused by their self-weight was measured (Fig.2(b)).

The loading initiation time was synchronized with the start of the printing process, specifically after 13 min (as described in Subsection 2.4). This was done to ensure that the plastic mechanical properties of the test specimens aligned with the elastic compression and strength characteristics exhibited by the material during printing. The specimens were subjected to loading at a constant deformation rate, which varied across different studies, ranging from 0.05 [23] and 0.2 [34] to 0.5 mm/s [14,23,24,35] in different studies. Wolfs et al. [14] concluded that the deformation rate should be sufficiently high to avoid the influence of thixotropic buildup. In this study, a loading rate of 0.5 mm/s was selected to enable the completion of each test within 14–150 s, thereby minimizing the impact of thixotropic buildup. The UUCT test was carried out using a universal testing machine with a maximum test force of 30 kN. Compression tests were conducted with vertical strains of up to 40%.

During the tests, three parallel samples were employed for the loading experiments for each aspect ratio. The corresponding vertical displacements and pressures were recorded using the testing machine. After loading, one sample was chosen, and the loading process was halted at four different vertical displacements. The maximum diameter of the sample was then measured using a vernier caliper.

2.3 Rheology testing

If the shear stress induced by the applied load exceeds the static yield shear stress of the material, it will result in damage known as plastic yielding. In the case of 3D printed structures, plastic yielding of the filament layer leads to the plastic collapse of the entire structure. Kruger et al. [36] conducted multiple stress growth tests using a rheometer to determine the time-dependent evolution of the static yield shear strength of the material. Rheological tests in this study were performed using a Brookfield rheometer equipped with a vane probe. The tests were conducted at various rest periods (min): 0, 1.5, 3, 4.5, 6, 9, 12, 15, 20, 25, 30, and 35. Nerella et al. [13] recommended employing a strain-based approach to rheological characterization with the lowest possible shear rate to initiate flow. Furthermore, Panda et al. [15] examined the development of static yield stresses in printable mortars with a maximum aggregate size of 1.2 mm by subjecting them to a constant shear rate of 0.1 s⁻¹ for 60 s using a rotational rheometer equipped with a vane probe. In this study, the same shear rate of 0.1 s⁻¹ for 60 s was employed to determine the static yield stress of the material.

It is worth noting that a total of 60 L of 3DPC were prepared, with 10 L reserved for rheological testing and the remaining 50 L used for 3D printing. During the rheological tests, the 3DPC material was placed in 500 mL beakers, and the rheological parameters were measured at a single time point in each beaker. The rheological testing commenced simultaneously with the printing process to ensure accurate characterization of the material properties specific to the 3D printed material.

2.4 Printing procedure

Fig.3 illustrates the procedure for preparing the printing material. Initially, all the components were dry mixed to achieve homogeneity. Subsequently, water was added to the mixture and further homogenized through wet mixing. The resulting print material was then loaded into the printer hopper and used for printing. The entire preparation process of the printing material strictly adhered to the specified time points (in minutes). This meticulous approach was employed to ensure that subsequent tests could accurately capture the relevant characteristics of the printing material parameters. Throughout the printing process, the ambient temperature and relative humidity were controlled at (20 ± 5) °C and 60% ± 10%, respectively. This stringent control was implemented to prevent any influence of environmental temperature and humidity on the density of the printed material.

The 3D printing process was conducted using a three-axis gantry printer as depicted in Fig.4. The printing platform consisted of a 3 m × 3 m steel plate. Control of the print head’s movements in the X, Y, and Z axes was achieved by the control system, enabling precise control of the printing parameters outlined in Tab.3. The discharge speed of the printed material was regulated by adjusting the extruded blades’ speed within the print head’s hopper. The print head itself featured a circular nozzle with a diameter of 20 mm. It should be noted that different screw rotation parameters were employed to ensure the consistent size of the printed filament for V0 and V1 ratios.

To assess the buildability of the printed structure, the occurrence of plastic yielding damage in the bottom filament print layer was examined. This was achieved by printing a circular hollow column with a diameter of 250 mm until failure, as shown in Fig.5. This test approach has been employed in previous studies [20,25]. Additionally, the printing process was recorded using a camera to capture the evolution of the printed layers and the moment of failure. Specifically, a printing speed of 150 mm/s for a 250 mm diameter printed hollow column corresponds to a building speed of 5.23 s per layer.

3 Results and discussion

3.1 Green strength and Young’s modulus of the fresh material

3.1.1 Correction of cross-sectional area in green strength calculation

Contrary to the stress calculation method employed for fully hardened specimens, the initial height (L₀) and initial diameter (D₀) of the prepared specimens undergo changes due to the influence of self-weight before subjecting them to the UUCT test. L₀ and D₀ are crucial parameters that govern the stress–strain behavior during subsequent loading of the fresh state specimens. Furthermore, throughout the loading process, the specimen’s diameter (D_i) changes along with its vertical height (L_i), thus significantly affecting the true stress within the cross-section. Therefore, it is imperative to consider both the changes in diameter and vertical height when calculating the true stress. In Subsection 2.2, electronic vernier calipers were utilized in conjunction with vertical load–displacement data to determine the specimen’s diameter (D_i) at a vertical strain of ε = (L₀ − L_i)/L₀.

Tab.4 demonstrates that as the L/D ratio of the specimen decreases (with a constant diameter and reduced height), the influence of self-weight on the initial height diminishes. Additionally, it can be observed that the specimen V1, which contains steel fibers, exhibits a lower rate of diameter increase with strain during loading compared to the specimen V0 without steel fibers. This phenomenon can be attributed to the enhanced viscous and internal frictional resistance of the mixture due to the presence of steel fibers [37–39], as indicated by the slope of the fitted curve depicting the diameter-strain relationship (Fig.6). Reference [40] also mentions that the addition of steel fibers can improve the shape retention capability of 3D printed materials. Moreover, as the aspect ratio decreases, the rate of increase in specimen diameter reduces with strain. In other words, for the same level of strain, a smaller aspect ratio leads to a slower increase in specimen diameter. This observation is clearly evident from the change in slope of the fitted curve (0.5–0.2). This phenomenon can be primarily attributed to the increased restraining effect exerted by the contact surface between the specimen and the loading plate as the aspect ratio decreases.

3.1.2 Green strength and Young’s modulus

The damage modes of fresh specimens with varying aspect ratios are presented in Tab.5. It is noteworthy that the fresh specimens exhibit a consistent damage pattern. The failure mode of the material resembles the barrel-shaped failure mode observed in plastic materials, wherein the cross-sectional area increases with vertical deformation. The material displays typical plastic and deformable behavior without the formation of a distinct shear damage surface. However, the specimen with an L/D ratio of 2 shows significant shear cracks. This occurrence can be primarily attributed to the friction between the loading plate and the specimen, which restricts the lateral deformation of the specimen during the loading process. Consequently, certain regions within the specimen experience a multi-axial stress state. In contrast, the remaining specimens, which did not exhibit shear cracking, undergo a multiaxial stress state in their cracked regions under the influence of vertical pressure [20]. Additionally, the lateral deformation resulting from vertical compression, along with an increase in tensile stress within the specimens, leads to the formation of vertical cracks (for specimens with L/D ratios of 1, 0.5, and 0.25) and shear cracks (for specimens with an L/D ratio of 2) when the particle cohesion is exceeded. This phenomenon has been observed in fresh 3DPC materials [41,42] as well as in cohesive soils during tests [43–45].

The equations for the initial height (H₀) and diameter (D_i) of the specimen with strain (ε) (Fig.6) are employed in conjunction with the load–displacement curve data obtained through the UUCT in Subsection 2.2 and processed. This enables the generation of a compressive stress–strain curve. A stress–strain curve is a curve drawn by testing the stress–strain relationship of three specimens under the same stress conditions and taking the average of the stress–strain curves of each sample. Fig.7 presents the stress–strain curves for fresh specimens with varying aspect ratios, providing insights into the material’s mechanical behavior under compressive forces and illustrating the variation in green strength among specimens with different aspect ratios. Notably, the stress–strain curve of fresh mortar exhibits significant changes as the aspect ratio of the specimens decreases (Fig.7). For specimens with an aspect ratio of 2, the vertical stress increases with increasing vertical displacement. When the vertical strain reaches 25%, the stress–strain curve exhibits a distinct peak point. Subsequently, the vertical stress begins to decrease with further vertical displacement. As the specimen aspect ratio decreases from 1 to 0.25, the vertical stresses in all specimens increase with increasing vertical displacement, without any peak stresses or descending branches observed. When the vertical strain reaches 40%, extensive vertical cracks appear throughout the specimens. At this stage, the obtained strength no longer represents the true green strength. Therefore, the peak strength is considered for specimens with an aspect ratio of 2, while for specimens with aspect ratios ranging from 1 to 0.25, the stress at 35% strain is taken as the maximum stress. Since the specimen with an L/D ratio of 0.25 is more affected by the restraining effect of the loading surface during compression, its stress–strain curve does not show a peak point. This does not mean that the specimen is not damaged. In certain studies [41,42], the stresses at 30% or 33% strain have been adopted as the maximum stresses without peak stresses for calculation purposes. This choice is primarily influenced by the test duration and the damage pattern observed in the specimens. The Young’s modulus is calculated based on a 5% strain [14,41].

Fig.8 illustrates the increasing pattern of maximum stresses for specimens V0 and V1 as the L/D ratio decreases. During specimen loading, the frictional effect between the specimen and the test machine platen restricts the lateral deformation of the specimen, resulting in regions inside the specimen experiencing a multi-axial stress state. In the actual 3D printing process, the filament undergoes both vertical and lateral deformation due to the self-weight pressure exerted by the overlying filaments. However, the lateral deformation of the lower filament is constrained by either the printing platform or the upper overlying filament. As the L/D ratio decreases, the material’s shear resistance increases due to the greater friction angle resulting from the lateral restraint provided by the printing platform or the contact between upper and lower filaments [20]. Consequently, specimens with smaller aspect ratios exhibit higher uniaxial compression strength. Furthermore, the addition of steel fibers enhances the peak load capacity of the mixture.

To analyze the strengths of specimens with different L/D ratios in a comparative manner, the strengths were normalized based on the specimens with an L/D ratio of 2, as shown in Fig.8(a). The normalized strengths exhibit an exponential variation pattern as the L/D ratio decreases. This finding aligns with the strength correction law proposed in Ref. [20] for fresh materials applicable to 3D printing. Equations (1) and (2) depict the variation of the strength correction factor (F_AR) with the development of the specimen’s L/D ratio (x) for specimens V0 and V1, respectively.

(1)F_{\text{AR}}(x)=0.72{\text{e}}^{\text{(}-x/2.23\text{)}}+0.71,

(2)F_{\text{AR}}\text{(}x\text{)}=0.60{\text{e}}^{\text{(}-x/0.61\text{)}}+0.99.

Fig.8(b) shows the evolution of Young’s modulus for the specimens in relation to the L/D ratio. It is evident that the Young’s modulus increases as the L/D ratio increases. Moreover, the inclusion of steel fibers results in a faster increase in Young’s modulus for specimen V1 compared to V0. Equations (3) and (4) represent linear fits of the Young’s modulus for specimens V0 and V1, respectively, with respect to the variation of the aspect ratio (x).

(3)E\text{(}x\text{)}=-7.87+32.84x,

(4)E\text{(}x\text{)}=-20.88+90.16x.

3.2 Static yield stress

The static yield stresses of the printed materials V0 and V1 were determined through rheological tests, and their evolution with time was characterized by a bilinear model, as shown in Fig.9. This model, proposed by Kruger et al. [20], is consistent with previous studies on silicate cements [46–48], and it describes the strength of the material in its fresh state. Specifically, it represents the shear capacity as a function of time after deposition, tailored specifically for 3D printed cementitious materials. Tab.6 presents the linearly fitted equations that describe the growth of the static yield stress over time for both stage 1 re-flocculation and stage 2 structuring. Thus, the operator governing the development of static yield stress in both stages can be uniformly expressed as Eq. (5).

(5){\tau }_{\text{s}}={\tau }_{\text{s,0}}+A_{\text{thix}}t,

where τ_s: static yield stress of the material at time t (Pa); τ_s,0: initial static yield stress of the material (Pa) (static yield stress of the material at t = 0); A_thix: the structural rate of the material (Pa/min); t: time since cessation of agitation (min).

Fig.9 illustrates that the inclusion of steel fibers enhances the static yield stress (τ_s,0) of the printed material, as evidenced by the larger intercept of the fitted equation in stage 1. This can be attributed to the increased frictional resistance between the steel fibers and the gel-like matrix of the material, leading to an elevated τ_s,0 [49]. Furthermore, the specimens containing steel fibers exhibit higher A_thix values. This can be attributed to the fact that the incorporation of steel fibers reduces the interparticle distance within the mixture, thereby facilitating the microstructural evolution in stages 1 and 2 [50,51]. Such improvements are beneficial for enhancing the buildability of the printed material.

3.3 Buildability prediction and testing

3.3.1 Buildability prediction

To ensure the precision of buildability assessment, Kruger et al. [20] introduce the utilization of shear stress rate and the Mohr–Coulomb failure criterion. They combine the Tresca and Rankine limit functions to formulate the assessment equation as shown in Eq. (6):

(6)\tau =\frac{{\sigma }_1}{2F_{\text{AR}}},

where F_AR is the strength correction factor that takes into account the limitations due to the smaller aspect ratio; σ₁ is the positive stress, the self-weight stress of the overlying filament ({\sigma }_1=\sum _{i=1}^n{\sigma }_i).

The self-weight stress of a single filament is shown in Eq. (7):

(7){\sigma }_i=\frac{F}{A}=\frac{G}{A}=\frac{\rho \times g\times h_0\times b\times l}{b\times l}=\rho g{h}_0,

where h₀ is the height of the filament (mm); ρ is the density of the material (g/cm³); g is the gravitational constant, taken as 9.81 m/s²; b is the width of the filament (mm); l is the constant path length per filament layer (mm).

The effectiveness of the buildability predictions given by Eq. (6) for V0 and V1 was validated. To begin with, the strength correction factors (F_AR) for V0 and V1 were computed using Eqs. (1) and (2) correspondingly. Additionally, Kruger et al. [20] proposed a strength correction factor for 3DPC in its fresh state when the material properties are unknown. The strength adjustment factors utilized for buildability predictions are summarized in Tab.7.

By employing the prediction Eq. (6) and considering the material strength curves derived from the rheological tests in Subsection 3.2, the predicted failures in 3D printing are illustrated in Fig.10.

3.3.2 Buildability testing

The process of column failure for V0 and V1 is depicted in Fig.11. It is evident that the bottom layer of the column experienced plastic yielding. Subsequently, the layers underwent plastic flow and were unable to withstand the self-weight of the subsequent printed layers, resulting in structural plastic yielding and the collapse of the column. It is worth noting that the column maintained considerable rigidity throughout the failure process and showed no indications of overall buckling. Minor cracks were observed in the yielding layer (Fig.11(a)), indicating that the static yield stress of the material was surpassed.

V0 experienced plastic collapse at the 19th layer, with a total printing time of 1 min and 7 s. V1 underwent plastic collapse at the 37th layer and required 3 min and 2 s for printing. This implies that the Young’s modulus and density of the printed material are unaffected by time. The results of the experimental validation are summarized in Tab.8. It can be observed that regardless of the employed correction factor, the prediction results are unsatisfactory. The relative error rates compared to the experimental results ranged from 42.1% to 146%.

3.3.3 Analysis of relative error

The building rate, which depends on print-specific parameters and geometry, refers to the rate at which a 3D printed structure is constructed. It is significant because it characterizes the rate of stress increment in the bottom printed layer. Fig.12(a) illustrates a typical 3D printing rate for an object with uniform geometry and printing speed. In this case, a filament layer with constant stress increment is deposited sequentially, leading to a step-function behavior as shown. However, for simplicity and in the context of buildability assessment, the linearized building rate is commonly utilized instead of the step function, as depicted in Fig.12(b). It is important to note that the actual printing process involves changes in the height of the printed material due to filament compression, which deviates from the assumption of uniform filament geometry made in the buildability analysis. Consequently, the original linearized building rate transforms into a nonlinearized building rate. When linear building rates are used in the prediction model, the number of failed layers obtained is significantly higher than the actual test values, resulting in a relative error rate exceeding 42%. In addition, since the number of layers printed on V1 is higher than that of V0, the cumulative error of stacking according to the actual height will be larger. So, the relative error of V1 is greater than V0. Therefore, it is essential to consider the influence of this nonlinear building rate caused by the compression deformation of the printed filament in buildability predictions.

4 Nonlinear building rate prediction simplified model

4.1 Calculation method of nonlinear building rate

The fresh concrete is deposited in layers through the extrusion nozzle at the designated lifting height of the print head. Throughout the printing process, the lower filament experiences the self-weight pressure exerted by the upper printed filament. Simultaneously, a printing strategy employing the print head to compress the print filament and achieve the desired filament size is employed [20,24,25,32,42]. This compression leads to vertical deformation of the filament. The height of the stacked filament in the lower layer undergoes continuous changes. To describe the variation in filament height due to vertical self-weight stress and compression, we define h_i(n) (i = 1,2,3,...,n; n = 1,2,3,...,n) where i represents the number of the printed filament, labeled from 1 to n in the order of deposition, and n denotes the total number of filament layers. h_i(n) is used to represent the height of the filament as the number of deposited layers increases. When i = n, it represents the height of a single filament at the moment of extrusion from the nozzle, denoted as h₁(1),h₂(2),...,h_n(n). Additionally, it signifies the height at which the lower filament begins to act. Specifically, h₁(1) = h₀, where h₀ corresponds to the design height of the filament and the lifting height of the print head.

As shown in Fig.13(a), the printed filaments range from 1 to n₀ layers, with a designated filament height of h₀. For the first layer printed on the platform, the vertical lift of the print head h₀, ensures that the height of the first layer matches the design height. When printing the second layer, the print head maintains a constant vertical lift of h₀. However, due to the vertical deformation caused by the self-weight stress of the first layer (Fig.13(b), where b represents the filament width, l represents the constant path length per filament layer, δh_i represents the elastic compression value of the filament, and F represents the vertical pressure on the filament), the actual height of the second layer, h₂(2), becomes greater than h₁(1) = h₀. Printing of the third layer commences with the print head maintaining the vertical lift of h₀. The first layer experiences the self-weight stress exerted by the second layer, resulting in vertical compression deformation and a height denoted as h₁(2). Simultaneously, the second layer undergoes vertical deformation due to its own gravity. Consequently, the actual height of the third layer of filament, h₃(3), exceeds h₂(2). This pattern continues, and when the cumulative vertical elastic compression height of each layer from the first layer to the n₀th layer reaches Δh_c during printing to the n₀th layer (Eq. (8), Fig.14), the filament will no longer be influenced by the downward extrusion force of the print head. The height of the filament at the moment of extrusion will remain unchanged, represented as h_{n0 + 1}(n₀ + 1) = h_{n0 + 2}(n₀ + 2) = h_f (Fig.14).

(8)\mathrm{\Delta }{\mathit{h}}_{\mathrm{c}}={\mathit{h}}_{\mathrm{f}}-{\mathit{h}}_0,

where h₀ is filament by the print head extrusion print height (mm) (filament design height determined by the printing parameters); h_f is filament maximum print height (mm) (not subject to the extrusion of the print head, the instantaneous extrusion height of the extruded filament, for the circular print head in accordance with the print head diameter calculation); Δh_c is critical height difference (mm), the height of the print filament for the first time when it is not subject to the role of the print head extrusion, the difference between h_f and h₀.

The stress σ_i acting on the lower filament, considering only the compressive deformation caused by the self-weight stress of the strip and the effect of the nozzle on filament extrusion height, is presented in Eq. (7). It is important to note that once the upper printed filament is extruded and comes into contact with the lower filament, the lower filament experiences self-weight stress from the just-extruded upper filament. Therefore, in Eq. (7), h₀ is replaced by the filament height h_i extruded from the nozzle, i.e., h_i = h_i(n) (i = n, n = 1,2,3,...,n). The extrusion height of the filament from layer 1 to n₀ satisfies h_n0(n₀) > h_n0−1(n₀−1) > ··· > h₃(3) > h₂(2) > h₁(1) = h₀.

The filament extrusion height exhibits nonlinearity from layer 1 to n₀, which implies that the stress σ_i acting on the lower filament layer increases nonlinearly from layer 1 to n₀. Starting from layer n₀, the filament extrusion height remains constant at h_f due to the removal of vertical restrictions (downward extrusion) from the extrusion nozzle. Consequently, the stress σ_i acting on the lower filament layer by subsequently printed filaments from layer n₀ increases linearly. Thus, n₀ is defined as the critical layer.

In the actual printing process, there exists a nonlinear building rate for layers 1 to n₀, followed by a linear building rate after n₀. Both the growth pattern and magnitude of the building rate differ from the linearized building rate described in existing prediction models [20,24,25] (e.g., Fig.15). Additionally, it is evident that at the beginning of printing (1 to n₀ layers), there is little disparity between the linearized building rate and the actual nonlinear building rate. However, as the number of printed layers increases (n > n₀), the difference between the linearized building rate and the nonlinear building rate becomes more pronounced. Accurately calculating the change in extrusion height of subsequently deposited filaments due to self-weight stress is a laborious task. To simplify the process, it can be assumed that the printed filament does not exceed the critical layer n₀ until the extrusion height of the printed layer remains consistent with the design height h₀. After surpassing h₀, the extrusion height of the printed layer becomes h_f. Fig.16 illustrates the simplified building rate, while Fig.15 represents the nonlinear building rate used for model simplification.

4.2 Calculation of critical layer n₀

The crucial parameter in the nonlinear building rate model is denoted as n₀. To determine n₀, Eqs. (9) and (10) provide the self-weight stress exerted by the overlying layers and the corresponding elastic compressive deformation δh_i for each layer ranging from 1 to n₀−1, respectively. The deformation of the filament under its own gravity is disregarded.

(9){\sigma }_i=\{\begin{array}{cc}{n}_0-1\text{layer}:& {\sigma }_{n_{\text{0}}-1}=1\cdot h_0\rho g,\\ n_0-2\text{layer}:& {\sigma }_{n_{\text{0}}-2}=2\cdot h_0\rho g,\\ n_0-3\text{layer}:& {\sigma }_{n_{\text{0}}-3}=3\cdot h_0\rho g,\\ & \cdots \\ \text{l layer}:& {\sigma }_1=\left(n_0-1\right)\cdot h_0\rho g,\end{array}

(10)\delta h_i=\{\begin{array}{cc}{n}_0-1\text{layer}:& \delta h_{n_0-1}={\displaystyle \frac{{\sigma }_{{\text{n}}_{\text{0}}\text{-1}}}{E}}{h}_0,\\ n_0-2\text{layer}:& \delta h_{n_0-2}={\displaystyle \frac{{\sigma }_{{\text{n}}_{\text{0}}\text{-2}}}{E}}{h}_0,\\ n_0-3\text{layer}:& \delta h_{n_0-3}={\displaystyle \frac{{\sigma }_{{\text{n}}_{\text{0}}\text{-3}}}{E}}{h}_0,\\ \dots \\ \text{1 layer}:& \delta h_1={\displaystyle \frac{{\sigma }_1}{E}}{h}_0,\end{array}

where E is the Young’s modulus of the fresh state corresponding to the size of the filament (Pa).

The cumulative elastic deformation Δh is calculated by summing up the elastic compressive deformations of each layer from 1 to n₀ layers, as described in Eq. (11). By substituting the specific expression of compressive deformation from Eq. (10) into Eq. (11), Eq. (12) is obtained, which is then simplified to Eq. (13). The occurrence of the n₀ layer is indicated when the cumulative elastic compression surpasses Δh_c. To establish the relationship between n₀ and Δh_c, Eq. (14) is derived, and solving it leads to Eq. (15).

(11)\text{Δ}h=\sum _{i=1}^{n_0-1}\delta h_i=\delta h_1+\delta h_2+\delta h_3\cdots +\delta h_{n_{0-1}},

(12)\begin{array}{rl}\mathrm{\Delta }h=& \sum _{i=1}^{n-1}\mathrm{\Delta }{h}_i=\frac{h_0\rho g}{E}{h}_0+\frac{2h_0\rho g}{E}{h}_0+\frac{3h_0\rho g}{E}{h}_0+\cdots \\ & +\frac{(n-1)h_0\rho g}{E}{h}_0,\end{array}

(13)\mathrm{\Delta }h=\sum _{i=1}^{n-1}\mathrm{\Delta }{h}_i=\frac{\rho g{h}_0^2}{E}\left[\frac{\left(n_0-1\right)n_0}{2}\right],

(14)\text{Δ}h=\sum _{i=1}^{n-1}\text{Δ}{h}_i=\frac{\rho g{h}_0^2}{E}\left[\frac{\left(n_0-1\right)n_0}{2}\right]\ge \text{Δ}{h}_{\text{c}},

(15)n_0⩾\frac{1+\sqrt{1+{\displaystyle \frac{8\mathrm{\Delta }{h}_{\text{c}}E}{\rho g{h}_0^2}}}}{2},n_0\in R.

The aforementioned formula for calculating the critical layer n₀ is based on the following assumptions.

1) The parameter “σ_i” representing stress redistribution in the filament deposition process is assumed to be constant in the actual calculation, while it may vary in reality. This assumption introduces conservatism in the calculated value.

2) The extrusion effect of the nozzle on the filament is disregarded, and only the influence of the nozzle on the extrusion height of the filament is considered. The nozzle restricts the extrusion height of the filament, and as the number of filament deposition layers increases, the extrusion effect of the nozzle decreases until it disappears at the n₀ layer. When n₀ is small, this extrusion effect can be considered negligible. Furthermore, neglecting the squeezing effect of the nozzle on the filament leads to an overestimation of the calculated value of n₀ compared to the actual value.

4.3 Nonlinear building rate simplified model

The nonlinear building rate resulting from the vertical elastic compressive deformation of the filament under the self-weight stress of the overlying filament during the printing process is taken into account. By combining Eqs. (6) and (15), the expression for the nonlinear building rate, Eq. (16), is derived.

(16)\tau =\frac{{\displaystyle \sum _{i=1}^{n_0}i\rho \text{g}{h}_0+{\displaystyle \sum _{i=n_0+1}^{n}i\rho \text{g}{h}_{\text{f}}}}}{2F_{\text{AR}}},

where due to the nature of the step function, n denotes the total number of layers as an integer.

However, establishing the relationship between the step function representing the number of printed filament layers (stress growth) and the material strength can be a complex task. To simplify this process, a continuous shear stress expression, Eq. (17), is proposed by linearization to describe the nonlinear building rate during printing.

(17)\tau =\frac{{\displaystyle \underset{0}{\overset{n_0}{\int }}\left(\rho \times \text{g}\times h_0\times i\right)\text{d}i+{\displaystyle \underset{n_0+1}{\overset{n}{\int }}\left(\rho \times \text{g}\times h_{\text{f}}\times i\right)\text{d}i}}}{2F_{\text{AR}}}.

4.4 Validation of the critical layer n₀

Equation (15) for computing the critical layer n₀ is provided. The parameters of the printed filament, such as h₀ = 12 mm and b = 30 mm, are presented in Tab.3. By substituting x = 0.4 into Eqs. (3) and (4), the Young’s modulus of the corresponding filament can be determined. The print head used in the process is a circular nozzle with a diameter of 20 mm, and the maximum filament printing height h_f is set to 20 mm. The filament is compressed by the print head to achieve a filament height of h₀ = 12 mm. The critical height difference Δh_c is calculated as 8 mm using Eq. (8). By substituting the relevant parameters into Eq. (15), the critical layer n₀ is obtained as shown in Tab.9.

Based on the results obtained from Eq. (15) and considering the physical interpretation of the number of printed layers as an integer, it is found that for V0, loading occurs in layers 1 to 6 with a height of h₀ = 12 mm, and from the 7th layer onwards, the height is set to h_f = 20 mm. This implies that the change in loading occurs at the 7th layer. Similarly, for V1, loading is done according to h_f = 20 mm starting from the 11th layer.

To verify the accuracy of the theoretical calculations, a filament hollow cylinder printing test was conducted in Subsection 2.4. It is important to note that the data for the n₀-layer verification test came from the same test as described in Subsubsection 3.3.2. Based on the theoretical assumptions used to calculate the n₀-layer in Subsection 4.2, the extrusion height of the printed filament remains constant when the difference between the design height of the printed layer and the height of the lower printed layer reaches Δh_c. To capture the instantaneous change in extrusion height of the printed filament, a digital camera was employed in Subsection 2.4 to record the layer changes during printing. The captured images were processed and normalized using the outer profile of the print head (represented by the black line in Fig.17) to facilitate comparison of print information across different images. The height of the print head from the bottom to the lower print strip was measured and normalized with respect to the measured height (V0 is based on layer 1). For V1, which takes into account a calculated value of 10 for n₀, the change in height for layers 8 to 13 was considered to clearly visualize the variation in print layer height. Normalization was performed based on layer 8 for this case.

Fig.18 illustrates the normalized height variation of the print head bottom from the printed layer after image processing. It is evident that for V0, the height remains constant starting from layer n = 6, indicating a test value of n₀ = 5. Moreover, it is apparent that at n = 6 layers, the printed filament is no longer influenced by the extrusion from the print nozzle. Similarly, for V1, the test value is n₀ = 10.

Comparing the theoretical values (n₀ = 6 for V0; n₀ = 10 for V1) with the experimental results, Eq. (15) provides a more accurate calculation of the critical layer. However, for V0, there is a test value of n₀ = 5, which is lower than the theoretical value of n₀ = 6. This discrepancy can be attributed to the omission of print nozzle extrusion in the assumptions 2) of Eq. (15). Consequently, the theoretical calculation yields a higher value than the experimental result.

4.5 Nonlinear building rate prediction model validation

This section presents the validation of the nonlinear building rate prediction model (Eq. (17)) as discussed in Subsection 4.3. The experimental validation results are summarized in Tab.10. It can be observed that the utilization of the nonlinear building rate calculation method, in conjunction with the strength correction factors proposed in this study, yielded relatively conservative errors of 10.5% and 5.4% for V0 and V1, respectively. These results demonstrate the acceptability of the experimental validation process. Furthermore, employing the strength correction factor F_AR from Ref. [20] also produced prediction results with absolute error rates of 5.2% and 16.2%. This indicates that the correction curve presented by Kruger et al. [20] can yield satisfactory prediction results even when the material’s strength correction factor is unknown.

Moreover, the model’s prediction tends to be conservative, primarily due to the coordinated lateral deformation experienced by the material between the lower and upper layers of the filament under vertical pressure during plastic yielding. This phenomenon leads to a redistribution of stresses exerted on the lower layer of the filament, consequently resulting in a conservative prediction. This aspect is also addressed in the assumptions 1) of Eq. (15).

The model (Eq. (17)) effectively predicts the nonlinear building rate phenomenon resulting from filament compression during the printing process. In contrast, the relative error rates obtained using linear building rate predictions (Tab.8) ranged from 42.1% to 146%. This emphasizes the significance of accounting for the nonlinear build rate of the printed material due to the compression of the printed layers. The calculation method of F_AR in Ref. [20] is given based on a large number of experimental statistics. And F_AR in this study may need a large number of experimental statistics to improve the reliability. This is also responsible for the smaller error of the model reported V0 sample in Ref. [20]. However, in terms of the number of relative print error layers predicted, they are 2 and 1, respectively. The advantage of the model in this paper is that based on Ref. [20], a nonlinear calculation method is given, which can effectively improve the accuracy of prediction. However, the calculation of F_AR can still refer to the method of Ref. [20]. The model plays a crucial role in enhancing control over material optimization design and the printing process. It is anticipated that the proposed method can serve as a reliable approach for the analysis and evaluation of the buildable properties of 3D printed structures or printed materials.

5 Conclusions

In this study, the investigation focused on the variations in green strength and Young’s modulus of freshly printed specimens with respect to the aspect ratio using the UUCT method. The buildability of the fresh printed material was also assessed based on the Mohr–Coulomb buildability prediction model. Additionally, a nonlinear building rate prediction model was proposed, and the key parameters and validity of the model were verified. The following conclusions can be drawn.

1) The elastic compression of printed layers caused by the self-weight of the overlying layers significantly affects the building rate. This aspect must not be overlooked in quantitative analyses of buildability prediction. When the linear building rate analysis neglects the influence of compression on the building rate, the relative error rates compared to actual printing range from 42.1% to 146%. In contrast, the relative error rates of the prediction models, which incorporate nonlinear building rate calculations, range from 5.4% to 10.5%.

2) The proposed prediction model effectively captures the failure of freshly printed 3D components due to plastic flow induced by the weight of the upper layers. By considering the nonlinear building rate resulting from the compression of printed layers and the strength correction of the fresh material, the model achieves reasonably accurate predictions. It should be noted that the prediction model represents a lower bound, as it conservatively predicts the number of printed layers that can resist plasticity-induced collapse by neglecting stress redistribution in the lower layers. This was confirmed by the collapse of the V0 and V1 mixtures after printing the 17th and 35th layers, respectively, while plastic yielding occurred after depositing the 19th and 37th layers, respectively, as observed in the validation 3D printing process.

3) The green strength of the fresh printed specimens shows an exponential variation relationship with respect to the aspect ratio. This behavior is primarily attributed to the multi-axial stress state within the specimens due to the constraint effect of the test platform on the contact surface. The constraining effect of the print platform on the printed filament between layers necessitates the correction of the filament’s strength. It should be point out that the accurate adjustment factor should be determined based on the actual material used. However, in the case of unknown material correction factors, Kruger’s fresh material strength correction factor still can be applied. This is evident from the relative error rates of 5.2% to 16.2% observed in the buildability prediction results using the model with Kruger’s strength correction factor.

4) The Young’s modulus of the fresh printed specimens exhibits a linear increase with increasing aspect ratio. Since the total printing time in this work is less than 4 min, the time-dependent change in Young’s modulus is not considered. To accurately capture the elastic compression deformation of the printed structure, it is essential to consider the Young's modulus of the printed material at the designed filament aspect ratio.

This analytical model corrects the buildability problem without considering compression and helps to assess the accuracy of the buildability properties of the printed material. For actual printing in the field it is still necessary to consider its structural form. However, the methodology presented in this paper for calculating the compression critical layers is still applicable to actual printing scenarios. In large-scale printing scenarios, the duration of the printing process tends to increase, highlighting the significance of considering the time-dependent characteristics of the mechanical properties of the printing material in its plastic state. In this regard, the integration of the proposed nonlinear building rate prediction model, which incorporates the time-dependent mechanical properties of 3D printing materials, holds potential for enhancing the adoption of 3DPC buildability prediction in large-scale printing scenarios. Further investigation and research efforts will be dedicated to exploring this area.

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