Structural build-up model for three-dimensional concrete printing based on kinetics theory

Abstract

The thixotropic structural build-up is crucial in extrusion-based three-dimensional (3D) concrete printing. This paper uses a theoretical model to predict the evolution of static and dynamic yield stress for printed concrete. The model employs a structural kinetics framework to create a time-independent constitutive link between shear stress and shear rate. The model considers flocculation, deflocculation, and chemical hydration to anticipate structural buildability. The reversible and irreversible contributions that occur throughout the build-up, breakdown, and hydration are defined based on the proposed structural parameters. Additionally, detailed parametric studies are conducted to evaluate the impact of model parameters. It is revealed that the proposed model is in good agreement with the experimental results, and it effectively characterizes the structural build-up of 3D printable concrete.

Graphical abstract

Keywords

structural build-up / rheology / thixotropy / 3D printable concrete / kinetics theory / ultra high performance concrete

1 Introduction

Three-dimensional concrete printing (3DCP) is a recently developed technology in civil engineering, offering multiple advantages, such as digital construction, intelligent building solutions, structural optimization, free-form structures, reduced labor costs, and decreased production time [1,2]. Printing normally involves depositing concrete layer upon a previous layer. During printing, once a concrete layer is deposited, supporting the next layer’s weight depends on the hydration process and yield stress development. This is generally achieved by adding suitable mineral admixtures, superplasticizers, viscosity modifying agents, and accelerators [3,4]. Cement-based products undergo structural changes due to hydration and thixotropy over time. Thixotropic behavior is a crucial factor affecting wet concrete pumpability and extrudability, and buildability of 3D printed objects. It generally involves four stages: thixotropic breakdown, dynamic buildup, thixotropic buildup, and static buildup. The first and the third last only seconds, and the other two last for several hundred minutes [5,6]. Different models describe the rheology of fresh cement-based materials on different time scales [7,8]. The constitutive equation and kinetics model used by Ma et al. [9] describe nonlinear increases in static yield stress at rest following varying pre-shear speeds over hundreds of seconds of rest. The static yield stress of cement paste changes bilinearly over time, according to Navarrete et al. [10], Kruger et al. [11], and Perrot et al. [12], simultaneously derives a static yield stress exponential development model for periods of hundreds of minutes.

A critical review of the above mentioned literature shows that the criterion for developing printable concrete is rheology. For modeling the properties of conventional concrete, the ingredients are considered to be in a dry state before mixing, a fluid state during homogeneous mixing, a semi-fluid state during placement and compaction, a semi-solid state during curing, and a solid-state after attaining strength. In the case of 3DCP, the printing mix is assumed to be a fluid during homogeneous mixing and pumping, semi-fluid during extrusion, semi-solid during buildability, and solid after curing or setting. The time-dependent properties of 3DCP, such as thixotropy and chemical hydration, are critical to achieving suitable printing parameters (e.g., open time, interval time, printer traveling speed, and extrusion speed) and acceptable extrudability and buildability. Thixotropy models for cement paste, mortar, and concrete have been developed in the past [13]. In simulations, thixotropy was rarely considered, most likely because more work would have been required, and more challenging model calibration [14]. However, these models must be extended for 3D printable concrete for broad application and acceptance among stakeholders and industries, so this study first presents a background on thixotropy, followed by a critical discussion on developing available thixotropy models and approaches. Based on this discussion, the need to develop the proposed model is highlighted. Subsequently, a new model based on structural kinetics is presented to determine the time-dependent properties of 3DCP. The model is validated by considering experimental observations and followed by parametric studies.

2 Structural kinetics model

The models proposed by Cheng et al. [15,16] are expanded in literature and broadly classified in terms of 1) time-independent relations, 2) indirect kinetics equations, and 3) rheological and structural parameters. Typical time-independent kinetic models are given in Tab.1. The model parameters are yield stress \left({\tau }_{\mathrm{y}}\right), plastic viscosity \left({\mu }_{\mathrm{p}}\right), shear rate \left(\dot{\gamma }\right),and shear thinning or shear thickening intensity. It is reported that the Bingham model needs to be validated for examining mixes with low water-to-binder ratios and is more applicable for instant analysis. The Herschel−Bulkley model has constraints during measurements at low shear rates. The Modified Bingham model is unsuitable for highly shear-thickening cementitious mixes. By contrast, the Casson model is more appropriate for high-viscosity mixtures [17–19].

In the case of indirect kinetic models, a generalized scalar constitutive expression that interrelates \eta ,\dot{\gamma ,} and \lambda, in the form of an equation of state and a rate equation, is given by Eqs. (1) and (2) [15].

(1)\tau =\eta (\lambda ,\dot{\gamma })\dot{\gamma },

(2)\frac{\mathrm{d}\lambda }{\mathrm{d}t}=f\left(\lambda ,\dot{\gamma }\right) .

The time-dependent effects on thixotropy are modeled by considering flocculation in viscosity, concentration, particle shape, and particle size distribution relationships. Flocculation is a phenomenon in which dispersed particles are brought together by van der Waals interactions and electrostatic forces to form clusters. The phenomenon can be examined via a lumped parameter model using a single structural parameter (λ). The rate of change of structure is categorized in terms of breakdown rate (K_2\lambda \dot{\gamma }), build-up rate (K_1(1 -\lambda )) and given by the rate equation (Eq. (3)).

(3)\frac{\mathrm{d}\lambda }{\mathrm{d}t}= K_1(1-\lambda )-K_2\lambda \dot{\gamma }.

The model proposed by Moore [23] has been extended by Worrall and Tuliani [24] by characterizing the behavior of clay suspensions. The kinetic model is developed by considering four parameters: Bingham and true yield stress, shear rate, and differential viscosity. The model has been further improved by Toorman [25] using an iterative method to predict the shear rate precisely. Pinder [26] has suggested that yield value, rate constant and apparent viscosity parameters are enough to characterize the material if exact knowledge about concentration, particle shape, size, and shear rate is available. In the case of viscoelastic materials under constant stress conditions, shear rate is calculated from the product of thixotropy and shear rate retardation values, where the retardation term is calculated based on the Maxwellian approach [27]. To model the nonlinear viscoelastic properties, Yziquel et al. [28] proposed a structural network model featuring three kinetic equations, which are based on 1) the second invariant of the rate-of-strain tensor, 2) polymer theory and the first invariant of the stress tensor, and 3) energy dissipated during microstructure evolution. The model is, however, very complex and needs to be calibrated based on experimental observations. Mujumdar et al. [29] developed a visco-plastic model to incorporate thixotropy. The model can quantify the elastic and viscous contributions from total stress, which arise from intra-floc deformation and inter-floc dissipation, respectively. A summary of the indirect kinetic models can be found in Tab.2.

The correlation between structural parameters with viscosity (\eta) or shear stress (\tau) is indicated in Tab.3. The concept is introduced by Moore [23], where the structural parameter, λ, and initial apparent density are interrelated. The model is further expanded by Refs. [24,25] to the observed initial yield stress (τ_i). Later, Mujumdar et al. [29] assumed that the total existing stress depends on the summation of elastic and viscous stresses. The relative effect of the stresses is governed by λ. A similar observation is suggested by Ref. [30]. The model proposes a time-dependent pre-factor and can predict the stress transients occurring due to rapid variations of shear rate. The complexities in the model have been simplified by Refs. [8,31], with fewer parameters.

2.1 Thixotropic models for three-dimensional printable concrete

The thixotropic characteristics are quantified regarding structuration rate (A_{\mathrm{t}\mathrm{h}\mathrm{i}\mathrm{x}}), defined as the slope of the increase of {\tau }_{\mathrm{S},\mathrm{i}} concerning resting time (t_{\mathrm{o}}). The structuration rate is generally considered constant and, typically in thousands of seconds or 10 to 60 min after mixing. However, in the case of 3DP concrete, the rate of evolution of {\tau }_{\mathrm{y}} cannot be considered to be linear due to the faster structural build-up and chemical hydration that are needed to satisfy the requirement of buildability and thus assumed to be exponential [13] (Fig.1).

This exponential increase in the {\tau }_{\mathrm{y}}. can demonstrate the smooth transition from linear growth and predicts better for more than 1 h. However, this assumption leads to incorporation of several other factors, such as hydration and environmental conditions. In addition, Kruger et al. [11] have proposed a bilinear model considering A_{\mathrm{t}\mathrm{h}\mathrm{i}\mathrm{x}} and R_{\mathrm{t}\mathrm{h}\mathrm{i}\mathrm{x}} as described by Eqs. (4) and (5). R_{\mathrm{t}\mathrm{h}\mathrm{i}\mathrm{x}} considers interatomic as well as intermolecular interactions in the microstructure. This bi-linear model considers physical and chemical influences and is a better indicator for modeling the thixotropy of 3D printing concrete.

(4)\tau ={\tau }_{\mathrm{D},\mathrm{i}}+R_{\mathrm{t}\mathrm{h}\mathrm{i}\mathrm{x}}t,\text{if}t\le t_{\mathrm{r}\mathrm{f}},

(5)\tau ={\tau }_{\mathrm{S},\mathrm{i}}+A_{\mathrm{t}\mathrm{h}\mathrm{i}\mathrm{x}}\left(t-t_{\mathrm{r}\mathrm{f}}\right), \text{if}t>t_{\mathrm{r}\mathrm{f}}.

3 Proposed model in this study

Several models have been developed that depend on time to characterize thixotropic materials. Most existing thixotropic models of 3D printing concrete are based on yield stress ({\tau }_{\mathrm{y}}). However, the rheological characterization of concrete with printing ability is much more complex due to the high dependency on various factors such as solification gradient, hydration and structural build-up, and process parameters [32]. Fig.2 schematically presents the underlying phenomenon occurring during concrete printing, stating the differences between the desirable and real yield stress. The development of {\tau }_{\mathrm{y},} depends on the micro-structure of the particle-particle matrix, which can be due to colloidal forces or direct contact. In the resting state, the material displays higher yield stress ({\tau }_{\mathrm{y}}) because the microstructure of the matrix strengthens by colloidal flocculation and cement hydration bonding. During the shear state, the microstructure breaks down, and the bonding network is dilated, reducing {\tau }_{\mathrm{y}}. In the case of 3DPC, the static yield stress is a significant parameter for developing load-bearing ability and shape stability. Pumpability and extrusion are more efficient when the initial yield stress is less. It is noteworthy that thixotropic build-up is a reversible phenomenon, while hydration build-up is an irreversible one.

In Fig.2, the structural build-up during 3DCP is presented. The structural build-up is modeled by considering flocculation, deflocculation, reflocculation, hydration induction period, and concrete set in Zone I, Zone II, Zone III, Zone IV, and Zone V, respectively. It is assumed that during the Zone I stage, the dry materials are mixed with step-wise additions of water and chemical admixtures. Due to the short duration of Zone I, the hydration effects are not prominent on {\tau }_{\mathrm{y}}. In Zone II, the structural buildability evolves linearly with the development of green strength. During this period, the mixed materials undergo flocculation and achieve equilibrium between breaking down and building up thixotropic forces. However, it has been reported that green strength is not competent to withstand a weight of printed layers having a height of more than 150 cm [33]. During open time till Zone II, the constant dynamic yield stress is observed, which increases suddenly during the bonding time.

In Zone III, the material gains strength rapidly to hold the weight of more layers. The concrete printing process during this phase can be increased or adequately controlled to print more layers without defects. Zone IV is the time during which the {\tau }_{\mathrm{y}} grows exponentially with an increase in the rate of rigidification, and the concrete sets during Zone V, with irreversible structural build-up [34]. The actual or real yield stress differs from the desired yield stress. According to various studies reported in the literature, if the material is left to rest, then {\tau }_{\mathrm{y}} quickly increases due to flocculation behavior. If the material is subjected to a constant strain rate, then {\tau }_{\mathrm{y}} decreases over time and tends to become stable. This is aided by de-flocculation during pumping and by the extrusion strain rate. The above phenomenon is modeled in the present study. In other words, the model proposed in this study considers that the thixotropic relation for concrete 3D printing depends on stress, strain rate, and viscosity. The details are as follows.

3.1 Mathematical formulation

The basic assumptions for the development of the model are as follows.

1) The mixing process of the ingredients during printing is uniform and homogeneous. The material is considered to flow from the rest condition. Applying a superplasticizer during printing does not reduce the effect of flocculation. Hysteresis loss is considered negligible during the initial period.

2) A steady pumping rate is required to develop the initial dynamic yield stress. To have dynamic yield stress build-up, the composition of the printable mix should have adequate flocculating particles to aid the development of early static yield stress

3) The printable mix can attain yield strength due to only hydration build-up of printed concrete after the re-flocculation period. The effects of temperature variations during printing on the hydration rate and evolution of C-S-H hydrates are negligible.

The proposed evolution of {\tau }_{\mathrm{y}} is given in Fig.3. It incorporates phenomena starting from inter-particle cohesion, flocculation, and cement hydration. The model aims to capture the growth of yield stress of fresh printable concrete concerning time. The dynamic yield stress is lower than static yield stress due to the applying constant strain rate during the pumping and extrusion phase. Once the concrete layers start printing, the printing material’s dynamic yield stress grows rapidly and nonlinearly. The evolution rate depends on the flocculating particle concentration, size distribution and shape, and application of de-flocculation agents. If the gap between flocculation forcing agents and de-flocculation forcing is high, the difference between static and dynamic yield stress will be more significant. Low yield stress in the dynamic state indicates predominant de-flocculation behavior. In Fig.3, t_0, t_{\mathrm{r}\mathrm{f}}, and t_{\mathrm{h}} denote the time during rest state, re-flocculation time and time of hydration during induction period, respectively. During t_0-t_{\mathrm{r}\mathrm{f}}: de-flocculation decreases, and flocculation increases with negligible hydration build-up. At t_{\mathrm{r}\mathrm{f}}: the de-flocculation is negligible, flocculation is approximately complete, and hydration build-up gains momentum. During t_{\mathrm{r}\mathrm{f}}-t_{\mathrm{h}}: hydration build-up is exponential.

3.2 Generalized expression

3DPC in a fresh state is physically semi-solid. Both solid and fluid state behaviors contribute to the shear stress development in this state. In the case of solid state, as per Hooke’s law, the shear stress (τ) is directly proportional to strain (γ), within the elastic limit, and can be expressed by Eq. (6). The proportionality constant is given by shear modulus (G) and Eq. (6) then becomes Eq. (7).

(6)\tau \propto \gamma ,

(7)\tau =G\gamma .

On differentiating Eq. (7) with respect to time t, the interrelation between shear stress rate \left(\dot{\tau }\right), shear modulus rate \left(\dot{G}\right), and strain rate ({\dot{\gamma }}_{\mathrm{s}}) is given by Eq. (8).

(8)\dot{\tau }=\dot{G}{\dot{\gamma }}_{\mathrm{s}}.

In the case of the fluid state, the τ is related to apparent viscosity (η) and by Eq. (9).

(9)\tau =\eta {\dot{\gamma }}_{\mathrm{f}}.

Combination of Eq. (8) and (9) gives Eq. (10).

(10){\dot{\gamma }}_{\mathrm{s}}+{\dot{\gamma }}_{\mathrm{f}}=\frac{\dot{\tau }}{\dot{G}}+\frac{\tau }{\eta }.

The \dot{\tau } of concrete in a solid state under rest conditions depends on several physical and chemical properties of binder and aggregate. The significant parameters are the specific surface area, particle size distribution, hydration constant (k_3), aggregate volume fraction (ϕ), surface texture, and rate of the heat of hydration ({\dot{\alpha }}_{\text{heat}}). The interrelationship between the parameters mentioned above can be given by Eq. (11).

(11)\dot{\tau }=k_3\varphi {\dot{\alpha }}_{\text{heat}}\lambda .

The value of shear stress in the fluid state given in Eq. (10) is obtained from Eq. (12), a generalized expression for the Herschel-Bukley model regarding yield stress \left({\tau }_{\mathrm{y}}\right) and \dot{\gamma }. On further simplification of Eq. (12), it is found that the apparent viscosity is proportional to {\dot{\gamma }}^{n-1} as is given in Eq. (14).

(12)\tau ={\tau }_{\mathrm{y}}+k{\dot{\gamma }}^n,

(13)\frac{\tau }{\dot{\gamma }} =\frac{{\tau }_{\mathrm{y}}}{\dot{\gamma }}+\frac{k{\dot{\gamma }}^n}{\dot{\gamma }},

(14)\eta =k{\dot{\gamma }}^{n-1}.

In general, the {\tau }_{\mathrm{y}} value for the case of non-reactive/ fully stable materials is constant and time-invariant. However, in the case of reactive material showing flocculative behavior, the yield stress is time-dependent [35]. The instantaneous structural level of fresh cementitious binder paste comprises 1) a reversible contribution \left({\lambda }_{\mathrm{r}}\right) induced by colloidal forces and 2) an irreversible contribution \left({\lambda }_{\mathrm{i}\mathrm{r}}\right) obtained by hydrate formation. As a result, the sum of contributions from {\lambda }_{\mathrm{r}} and {\lambda }_{\mathrm{i}\mathrm{r}} yields the rate of the transient structural parameter (λ) [22]. Λ_r is obtained as a difference between flocculation level and deflocculation. The hydration build-up gives {\lambda }_{\mathrm{i}\mathrm{r}}. The structural parameter (λ) is defined as the ratio of yield stress (τ_y) and initial reference static yield stress {\tau }_{\mathrm{S},\mathrm{i}} and represented in Eq. (15):

(15)\frac{{\tau }_{\mathrm{y}}}{{\tau }_{\mathrm{S},\mathrm{i}}}=\lambda ={\lambda }_{\mathrm{r}}+{\lambda }_{\mathrm{i}\mathrm{r}}.

On substituting the value of τ_y in Eq. (12) and considering the flocculation to be over (λ_r = 1), Eq. (16) becomes Eq. (17), which is further simplified to obtain the static yield stress and is expressed in Eq. (18).

(16)\tau =\left({\lambda }_{\mathrm{r}}+{\lambda }_{\mathrm{i}\mathrm{r}}\right){\tau }_{\mathrm{S},\mathrm{i}}+k{\dot{\gamma }}^n,

(17)\tau =\left(1+{\lambda }_{\mathrm{i}\mathrm{r}}\right){\tau }_{\mathrm{S},\mathrm{i}}+k{\dot{\gamma }}^n,

(18){\tau }_y=\left(1+{\lambda }_{\mathrm{i}\mathrm{r}}\right){\tau }_{\mathrm{S},\mathrm{i}}.

Similarly, structuration rate (A_{\text{thix}}) can be written in the form of partial differentiation of {\lambda }_{\mathrm{r}} and {\lambda }_{\mathrm{i}\mathrm{r}} and is represented in Eq. (19). According to the kinetics model, the rate of reversible structuration is expressed as the rate of flocculation (build-up), and the rate of de-flocculation (breakdown). In contrast, the rate of irreversible structuration is expressed in the hydration build-up rate. Based on the assumptions above, Eq. (19) is further simplified into Eq. (20) where F, D, and B denote the flocculation, deflocculation, and hydration build-up, respectively.

(19)\frac{\mathrm{d}\lambda }{\mathrm{d}t}= \frac{\mathrm{\partial }{\lambda }_{\mathrm{r}}}{\mathrm{d}t}+ \frac{\mathrm{\partial }{\lambda }_{\mathrm{i}\mathrm{r}}}{\mathrm{d}t},

(20)\frac{\mathrm{d}\lambda }{\mathrm{d}t}= \left(\frac{\mathrm{d}F}{\mathrm{d}t}-\frac{\mathrm{d}D}{\mathrm{d}t}\right)+\frac{\mathrm{\partial }B}{\mathrm{d}t}.

In the following, Eqs. (19) and (20) are further expanded.

3.2.1 Case I: Ideal flocculation

The viscosity of a suspension with flocculated cement particles is high but reduces when the particles are deflocculated. Because of flocculation, thixotropy is found in cement paste. At constant shear, its viscosity decreases and then restores once the shear is ceased. Although thixotropy is reversible, irreversible hardening of fresh mixture boosts the viscosity steadily over time and eventually causes concrete to set. The time-dependent property of flocculation is given in Fig.4. It is found that the rate of flocculation is very aggressive initially after mixing, and has an exponential nature. The flocculation behavior can be quantified in terms of the amount of structure that can be reversibly built up and then given by Eq. (21) [36].

(21)\frac{\mathrm{d}F}{\mathrm{d}t}=R_{\mathrm{F}}= K_1\left(1-{\lambda }_{\mathrm{r}}\right),

(22)F={\int }_0^t{R}_{\mathrm{F}}\mathrm{d} t.

The reversible structural parameter at t = 0 and time t = t is designated as λ_r0 and λ_rt, respectively. The rate of flocculation at t = 0 and time t = t, considering structure build-up without de-flocculation is given by Eqs. (23)–(25).

(23){\int }_{{\lambda }_{\mathrm{r}0}}^{{\lambda }_{\mathrm{r}}} \mathrm{d}{\lambda }_{\mathrm{r}}= {\int }_0^t K_1\left(1-{\lambda }_{\mathrm{r}}\right)\mathrm{d}t,

(24){\lambda }_{\mathrm{r}\mathrm{t}}=1-\left(1-{\lambda }_{\mathrm{r}0}\right){\mathrm{e}}^{-K_{1}t},

(25)\frac{\mathrm{d}F}{\mathrm{d}t}=R_{\mathrm{F}}= K_1\cdot \left(1-{\lambda }_{\mathrm{r}0}\right){\mathrm{e}}^{-K_{1}t},

If {\lambda }_{\mathrm{r}0} =0, then

(26)\frac{\mathrm{d}F}{\mathrm{d}t}=R_{\mathrm{F}}= K_1\cdot {\mathrm{e}}^{-k_{1}t}.

If Eq. (26) is solved, considering the scenario of mixes having very fine binders, good particle packing, high rate of inter and intraparticle forces/interactions on an increasing scale of 4 to 1, the flocculation rate can be depicted by Fig.5.

3.2.2 Case II: Ideal deflocculation

The thixotropy of cementitious material depends on the flocculation/de-flocculation property, which in turn depends on the evolution or breaking of C-S-H bridges in the matrix. The C-S-H nucleation is significant for maintaining shape stability and building rate [37,38]. The break-up of flocs or deagglomeration of the particles due to external or internal strain rate, considering flowability and structuration, is characterize de-flocculation of cementitious materials [39]. The determination of the de-flocculation rate indirectly relates to the rate of flocculation [40]. The rate of de-flocculation (R_D) (Eq. (27)) is found to be directly proportional to flowability (β), structuration (λ_r), and strain rate ( \dot{\gamma }) (Eq. (28)) [41]. To simplify the mathematical formulations, the proportionality constant in this study is assumed to be K_2 (Eq. (29)). The flowability of the mix can be controlled by proper selection of water-to-binder ratio, the addition of admixtures, accelerators/hardeners, superplasticizer, fiber dosage, etc. The same effect is considered for three cases and given in Eq. (30).

(27)\frac{\mathrm{d}D}{\mathrm{d}t}=R_{\mathrm{D}},

(28)R_{\mathrm{D}}\propto \beta {\lambda }_{\mathrm{r}}\dot{\gamma },

(29)\frac{\mathrm{d}D}{\mathrm{d}t}=R_{\mathrm{D}}= K_2\beta {\lambda }_{\mathrm{r}}\dot{\gamma },

(30)

In the case of standard concrete, the strain rate of mixes has an inverse relationship with resistance against flowability, and is given by Fig.6. It is assumed that on the release of the material under a free state, the material flow is faster in the initial stages and decays exponentially at a later stage (Fig.6(a)). The rate of flow depends on both external and internal factors. The internal factors are the interatomic force’s ability to rebuild strength, while external factors are the type of load during mixing, pumping, extruding, or casting. However, in the case of 3DCP, the strain rate is maintained constant to enable proper buildability and extrudability (Fig.6(b)).

In the cases shown in Fig.6(a) and Fig.6(b), the strain rates are assumed to be as given by Eqs. (31) and (32), respectively.

(31)\dot{\gamma }\left(t\right)={\dot{\gamma }}_{\mathrm{m}\mathrm{a}\mathrm{x}}\cdot {\mathrm{e}}^{-C_{0}t},

(32)\dot{\gamma }\left(t\right)=K.

By integrating Eq. (29), Eq. (33) is obtained:

(33)D={\int }_0^t K_2\cdot \beta \cdot {\lambda }_{\mathrm{r}}\cdot \dot{\gamma }\mathrm{d}t,

Generalized figures for the rates of de-flocculation under natural and constant strain rate conditions, using Eq. (29), and Eqs. (31)–(33), are shown in Fig.7(a) and Fig.7(b).

3.2.3 Case III: Hydration build-up

The structural build-up consisting of reversible and irreversible processes includes the following processes: 1) colloidal interactions; 2) evolution of C-S-H and ettringite network, and 3) inter-particle interaction. The irreversible build-up can be evaluated by characterizing the processes involved during induction, nucleation, and diffusion. The kinetic law for irreversible build-up is derived from the Arrhenius concept [42]. The rate of hydration is found to be proportional to the heat of hydration (α) and solid volume fraction ϕ [43]. The quantities k₃, α, and ϕ are integrated into Eq. (35) by considering constant K_3. The {\tau }_{\mathrm{S},\mathrm{i}} of the binder materials such as cement (C), silica fume (SF), and fly ash (FA) are dependent on (α) and ϕ (Fig.8). The values of K_3 for mixes, detailed in Tab.4 [44–48], are evaluated by Eq. (38). The values of k₃ are further inversely calculated from Eq. (34). The ranges of k_3 and K_3 are found to be 0.5−3.0 (g·mJ⁻¹) and 1.2- 6.5\times {10}^{-4}\left({\mathrm{s}}^{-1}\right)), respectively. However, during the reversible process, the gain in the hydration build-up is represented by Eq. (34) and additional constant C and expanded in Eqs. (36) and (37). The solution obtained after simplifying Case I (Subsubsection 3.2.1), Case II (Subsubsection 3.2.2) and Case III (Subsubsection 3.2.3) is given in Eqs. (25), (29), and (39), respectively. In Fig.8 the development of static yield stress for different binder materials is given.

(34)\frac{\mathrm{d}{\lambda }_{\mathrm{i}\mathrm{r}}}{\mathrm{d}t}= k_3{\varphi }_0{\dot{\alpha }}_{\text{ind}}\lambda .

(35)\frac{\mathrm{d}{\lambda }_{\mathrm{i}\mathrm{r}}}{\mathrm{d}t}= R_{\mathrm{B}}= K_3\left(C+{\lambda }_{\mathrm{i}\mathrm{r}}\right),

(36){\int }_{{\lambda }_{\mathrm{i}\mathrm{r}\mathrm{r}},t=0}^{{\lambda }_{\mathrm{i}\mathrm{r}\mathrm{t}},t=t}\mathrm{d}{\lambda }_{\mathrm{i}\mathrm{r}}={\int }_0^t{R}_{\mathrm{B}}\mathrm{d} t={\int }_0^t{K}_3\left(C+{\lambda }_{\mathrm{i}\mathrm{r}}\right)\mathrm{d}t,

(37){\lambda }_{\mathrm{i}\mathrm{r}\mathrm{t}}=\left(C+{\lambda }_{\mathrm{i}\mathrm{r}0}\right){\mathrm{e}}^{K_{3}t}-C.

If {\lambda }_{\mathrm{i}\mathrm{r}0}=0, then

(38){\lambda }_{\mathrm{i}\mathrm{r}\mathrm{t}}=C\left({\mathrm{e}}^{K_{3}t}-1\right).

Differentiating Eq. (37), then

(39)\frac{\mathrm{d}B}{\mathrm{d}t}=K_3\left(C+{\lambda }_{\mathrm{i}\mathrm{r}0}\right){\mathrm{e}}^{K_{3}t}.

On further substituting Eqs. (25), (29), and (39) in Eqs. (19) and (20), the generalized expression for structural build-up is then given by Eq. (40).

(40)\begin{array}{rl}\frac{\mathrm{d}\lambda }{\mathrm{d}t}=& \frac{\mathrm{\partial }{\lambda }_{\mathrm{r}}}{\mathrm{d}t}+\frac{\mathrm{\partial }{\lambda }_{\mathrm{i}\mathrm{r}}}{\mathrm{d}t}=\left(\frac{\mathrm{d}F}{\mathrm{d}t}-\frac{\mathrm{d}D}{\mathrm{d}t}\right)+\frac{\mathrm{\partial }B}{\mathrm{d}t}\\ =& \left[K_1\cdot \left(1-{\lambda }_{\mathrm{r}0}\right){\mathrm{e}}^{-K_{1}t}\right]-\left(K_2\beta {\lambda }_{\mathrm{r}}\dot{\gamma }\right)\\ & +\left[K_3\left(C+{\lambda }_{\mathrm{i}\mathrm{r}0}\right){\mathrm{e}}^{K_{3}t}\right].\end{array}

3.3 Structural build-up

The structural build-up consists of dynamic and static phases. During the dynamic phase, the ingredients undergo mixing, pumping, and extruding due to the applied strain rate. It is acknowledged that flocculation and de-flocculation occur with insignificant hydration ({\displaystyle \frac{\mathrm{\partial }B}{\mathrm{d}t}}=0). Once the material is deposited in the mold or printed, the structural build-up is known as static structural build-up [33].

3.3.1 Dynamic structural build-up

Considering negligible hydration during the dynamic structural build-up, Eq. (40) can be expressed as Eq. (41) and simplified in Eq. (42).

(41)\frac{\mathrm{d}\lambda }{\mathrm{d}t}= \left[K_1\cdot \left(1-{\lambda }_{r0}\right){\mathrm{e}}^{-K_{1}t}\right]- \left[K_2\beta {\lambda }_{\mathrm{r}}\dot{\gamma }\right],

(42){\lambda }_{\mathrm{r}\mathrm{t}}=\frac{1-\left(1-{\lambda }_{\mathrm{r}0}\right){\mathrm{e}}^{-K_{1}t}}{1+K_2\beta {\int }_0^t \dot{\gamma }\mathrm{d}t}.

The reversible structuration rate for the natural strain rate case (Fig.6(a)) is obtained by Eq. (43).

(43){\lambda }_{\mathrm{r}\mathrm{t}}=\frac{1-\left(1-{\lambda }_{\mathrm{r}0}\right){\mathrm{e}}^{-K_{1}t}}{1+{\displaystyle \frac{K_2\beta {\dot{\gamma }}_{\mathrm{m}\mathrm{a}\mathrm{x}}}{C_0}}\left(1-{\mathrm{e}}^{-C_{0}t}\right)} .

At the maximum and minimum structural build-up (Eq. (41)), {\displaystyle \frac{\mathrm{d}\lambda }{\mathrm{d}t}}=0 and {\lambda }_{\mathrm{r}}= {\lambda }_{\mathrm{e}\mathrm{q}}, respectively.

(44)K_1\cdot \left(1-{\lambda }_{\mathrm{r}0}\right)\cdot {\mathrm{e}}^{-k_{1}t}-K_2\cdot \beta \cdot {\lambda }_{\mathrm{e}\mathrm{q}}\cdot {\dot{\gamma }}_{\mathrm{m}\mathrm{a}\mathrm{x}}\cdot {\mathrm{e}}^{-C_{0}t}=0.

The equilibrium structuration (λ_eq) is obtained by Eqs. (45) and (46).

(45){\lambda }_{\mathrm{e}\mathrm{q}}=\frac{K_1\cdot \left(1-{\lambda }_{\mathrm{r}0}\right)}{K_2\cdot \beta \cdot {\dot{\gamma }}_{\mathrm{m}\mathrm{a}\mathrm{x}}}\cdot {\mathrm{e}}^{t\left(C_0-K_1\right)},

at C_0= K_1

(46){\lambda }_{\mathrm{e}\mathrm{q}}=\frac{K_1\cdot \left(1-{\lambda }_{\mathrm{r}0}\right)}{K_2\cdot \beta \cdot {\dot{\gamma }}_{\mathrm{m}\mathrm{a}\mathrm{x}}}.

Similarly, the reversible structuration rate for the constant strain rate case (Fig.6(b)) is obtained by Eq. (47).

(47){\lambda }_{\mathrm{r}\mathrm{t}}=\frac{1-\left(1-{\lambda }_{\mathrm{r}0}\right){\mathrm{e}}^{-K_{1}t}}{1+K_2\beta Kt}.

In the case of constant strain rate, during the maximum and minimum structural build-up (Eq. (41)) {\displaystyle \frac{\mathrm{d}\lambda }{\mathrm{d}t}}=0 and {\lambda }_{\mathrm{r}}= {\lambda }_{\mathrm{e}\mathrm{q}}, respectively, so that, Eqs. (48) and (49).

(48)K_1\cdot \left(1-{\lambda }_{\mathrm{r}0}\cdot {\mathrm{e}}^{-k_{1}t}\right)- K_2\cdot \beta \cdot {\lambda }_{\mathrm{e}\mathrm{q}}\cdot K= 0,

(49){\lambda }_{\mathrm{e}\mathrm{q}}=\frac{K_1\cdot \left(1-{\lambda }_{\mathrm{r}0}\right)}{K_2\cdot \beta \cdot K}\cdot {\mathrm{e}}^{-K_{1}t}.

Finally, the yield stress (τ_y) in this case is obtained by the product of initial static yield stress and structuration and found in Eq. (50) \left(\lambda ={\lambda }_{\mathrm{r}\mathrm{t}}\right).

(50){\tau }_{\mathrm{y}}= {\tau }_{\mathrm{S},\mathrm{i}}{\lambda }_{\mathrm{r}\mathrm{t}}.

3.3.2 Static structural build-up

It is assumed that the flocculation is complete in this stage, and the build-up contribution is largely due to hydration and is termed irreversible structuration \left({\lambda }_{\mathrm{i}\mathrm{r}\mathrm{t}}\right), i.e., {\lambda }_{\mathrm{r}}= 1. Hence Eq. (15) can be written as Eq. (51).

(51){\tau }_{\mathrm{y}}= \left(1+{\lambda }_{\mathrm{i}\mathrm{r}\mathrm{t}}\right)\cdot {\tau }_{\mathrm{S},\mathrm{i}},

Combining Eq. (37) with Eqs. (51) and (52) is obtained.

(52){\tau }_{\mathrm{y}}=\left(1+\left(C+{\lambda }_{\mathrm{i}\mathrm{r}0}\right)e^{K_{3}t}-C\right)\cdot {\tau }_{\mathrm{S},\mathrm{i}},

At t = 0, Eq. (52) is simplified as Eq. (53). Similarly, the slope of Eq. (53) at t = 0 gives A_thix as is shown by Eq. (54):

(53){\tau }_{\mathrm{y}}= \left(1+C{\mathrm{e}}^{K_{3}t}-C\right)\cdot {\tau }_{\mathrm{S},\mathrm{i}},

(54)\frac{\mathrm{d}{\tau }_{\mathrm{y}}}{\mathrm{d}t}=C\cdot K_3{\tau }_{\mathrm{S},\mathrm{i}}=A_{\mathrm{t}\mathrm{h}\mathrm{i}\mathrm{x}},

Hence, if K_3= 1/ c_{3,} then C·{\tau }_{\mathrm{S},\mathrm{i}} becomes A_{\mathrm{t}\mathrm{h}\mathrm{i}\mathrm{x}}\cdot c_3, then Eq. (53) can be expressed as Eq. (55).

(55){\tau }_{\mathrm{y}}= A_{\mathrm{t}\mathrm{h}\mathrm{i}\mathrm{x}}\cdot c_3\cdot \left({\mathrm{e}}^{\frac{t}{c_3}}-1\right)+{\tau }_{\mathrm{S},\mathrm{i}},

The derived Eq. (55) is in accord with the model proposed by Perrot et al. [13]. On rearranging Eq. (53), Eq. (56) is obtained. On expanding the exponential term considering , Eq. (56) becomes Eq. (57).

(56){\tau }_{\mathrm{y}}= (1+C\left({\mathrm{e}}^{K_{3}t}-1\right)\cdot {\tau }_{\mathrm{S},\mathrm{i}},

(57){\tau }_{\mathrm{y}}= (1+C\left(1+K_{3}t+\frac{K_3{t}^2}{2!}+\frac{K_3{t}^3}{3!}+\cdots -1\right)\cdot {\tau }_{\mathrm{S},\mathrm{i}},

On neglecting the later terms in the series, Eq. (57) can be written as Eq. (58).

(58){\tau }_{\mathrm{y}}= \left(1+C{K}_{3}t\right)\cdot {\tau }_{\mathrm{S},\mathrm{i}},

As discussed above, C\cdot {\tau }_{\mathrm{S},\mathrm{i}}\cdot K_3 can be written as A_{\mathrm{t}\mathrm{h}\mathrm{i}\mathrm{x}}. The derived Eq. (59) corresponds with the model proposed by Roussel [31].

(59){\tau }_{\mathrm{y}}= {\tau }_{\mathrm{S},\mathrm{i}}+A_{\mathrm{t}\mathrm{h}\mathrm{i}\mathrm{x}}\cdot t.

3.3.3 Structural build-up during pumping and extrusion

Case I: Pumping stage

At this stage, a constant strain rate exists in the material. If the pumping time is assumed as t, Eq. (48) can be re-presented as Eq. (60).

(60){\lambda }_{\mathrm{r}}= {\int }_0^t K_1\left(1-{\lambda }_{\mathrm{r}0}\right){\mathrm{e}}^{-K_{1}t}\mathrm{d}t- {\int }_0^t K_2\cdot \beta \cdot {\lambda }_{\mathrm{r}}K\mathrm{d}t.

For the time ranging between 0−t_p, the structural build-up at a constant strain rate is given by Eq. (61):

(61){\lambda }_{\mathrm{r}\mathrm{t}}=\frac{1-\left(1-{\lambda }_{\mathrm{r}0}\right){\mathrm{e}}^{-K_{1}t}}{1+K_2\beta Kt}.

For time t_p, the structural build-up at a constant strain rate is given by Eq. (62):

(62){\lambda }_{\mathrm{r}{\mathrm{t}}_1}={\lambda }_{\mathrm{D}}=\frac{1-\left(1-{\lambda }_{\mathrm{r}0}\right){\mathrm{e}}^{-K_1{t}_1}}{1+K_2\beta K{t}_1},

(63){\tau }_{\mathrm{y}}= {\tau }_{\mathrm{S},\mathrm{i}}\cdot {\lambda }_{\mathrm{D}},

(64){\tau }_{\mathrm{y}}= {\tau }_{\mathrm{S},\mathrm{i}}\left[\frac{1-\left(1-{\lambda }_{\mathrm{r}0}\right){\mathrm{e}}^{-K_1{t}_1}}{1+K_2\beta K{t}_1}\right].

Case II: After deposition

The material is subject to natural strain and build-up during this stage. Build-up time after deposition is labeled as t₀, the structural build-up after deposit can be expanded from Eq. (40) and written as Eqs. (65) and (66).

(65)\begin{array}{rl}\lambda =& {\int }_0^{t_0} K_1\cdot \left(1-{\lambda }_{\mathrm{D}}\right){\mathrm{e}}^{-k_{1}t}\mathrm{d}t-{\int }_0^{t_0}{K}_2\beta {\lambda }_{\mathrm{r}}K{\mathrm{e}}^{-C_{0}t}\mathrm{d}t\\ & +{\int }_0^{t_0}{K}_3\left(C+{\lambda }_{\mathrm{i}\mathrm{r}0}\right){\mathrm{e}}^{K_{3}t}\mathrm{d}t,\end{array}

(66){\lambda }_{{\mathrm{t}}_0}=\frac{1-\left(1-{\lambda }_{\mathrm{D}}\right){\mathrm{e}}^{-K_1{t}_0}}{1+{\displaystyle \frac{K_2\beta K}{C_0}}\left(1-{\mathrm{e}}^{-C_0{t}_0}\right)}+\left(C+{\lambda }_{\mathrm{i}\mathrm{r}0}\right){\mathrm{e}}^{K_3{t}_0}-C.

If λ_ir0 = 0 then Eq. (66) can be deduced to Eq. (67):

(67){\lambda }_{{\mathrm{t}}_0}=\frac{1-\left(1-{\lambda }_{\mathrm{D}}\right){\mathrm{e}}^{-K_1{t}_0}}{1+{\displaystyle \frac{K_2\beta K}{C_0}}\left(1-{\mathrm{e}}^{-C_0{t}_0}\right)}+C\left({\mathrm{e}}^{K_3{t}_0}-1\right),

(68){\tau }_{\mathrm{y}}= {\tau }_{\mathrm{S},\mathrm{i}}\cdot {\lambda }_{{\mathrm{t}}_0}.

Hence, the generalized expression for the yield stress τ_y during the time t₀ is given by Eq. (69):

(69){\tau }_{\mathrm{y}}= {\tau }_{\mathrm{S},\mathrm{i}}\left[\frac{1-\left(1-{\lambda }_{\mathrm{D}}\right){\mathrm{e}}^{-K_1{t}_0}}{1+{\displaystyle \frac{K_2\beta K}{C_0}}\left(1-{\mathrm{e}}^{-C_0{t}_0}\right)}+C\left({\mathrm{e}}^{K_3{t}_0}-1\right)\right].

4 Experimental investigation

The current study uses ultra-high-performance concrete (UHPC) 3D printable mixes developed in the previous studies [4]. The mix consists of ordinary Portland cement of Grade 53 satisfying the specifications outlined in IS 12269:2013, Class F FA, conforming to IS 3812:2013, un-densified micro silica, complying with IS 15388:2003 and using limestone powder as binders. The water cement ratio of the mix is 0.23. To enhance the workability of the fresh 3D printable mix, a high-range water-reducing admixture (HWRA) is utilized at a dosage of 0.8% by cement mass. Grade I sand, with particle sizes ranging from 1 to 2 mm and fineness modulus ranging from 2.3 to 3.1, and Grade II sand, with particle sizes ranging from 0.5 to 1 mm and fineness modulus ranging from 2.7 to 3.1 are used as fine aggregate. Straight steel fibers with an aspect ratio of 62 are incorporated as 0 and 1% by weight of binder, as shown in Tab.5. The materials are mixed homogeneously and then subjected to the rheology test using a rotational rheometer with cylindrical cup and vane geometry as shown in Fig.10(a). A pre-shear of 100 s⁻¹ for 30 s is applied initially to eliminate any yield stress buildup, followed by a constant shear of 0.5 s⁻¹ for 60 s. The data are collected for every one sec for the first 40 s to evaluate the yield stress during pumping and then for every 5 min up to 35 min for evaluating the hydration build-up. A typical 3D printed UHPC specimen (U_0) is shown in Fig.10(b).

5 Validation and parametric studies

Wang et al. [39] investigated the variation of shear stress at different rest times. In the current study, one typical experimental shear stress plot with a resting time of 40 min is considered with the proposed model detailed in Section 3. From the model, initial static yield stress, flocculation, deflocculation, and irreversible build-up parametric constants are obtained. The proposed model follows a similar trend to that of the experiments reported by Wang et al. [39] (Fig.11). Further, the model is evaluated to predict the development of static yield stress at resting times of 10, 20, 40, 60, 80, 100, and 130 min. The rate of evolution of static yield stress is found to be exponential (Fig.12) and perfectly matches the experimental observations. The obtained kinetic parameters and model comparison are given in Fig.12.

The time dependence of the rheology of fresh cement-based materials is explained by multiple theories at different time scales. The models assume a nonlinear rise in static yield stress when the rest time is in hundreds of seconds. However, any previous time-dependence model relating to rheological characterization considering wet mixing, rest duration, temperature effect, and hydration for C3DP is limited, and this study makes new proposals.

Further, the variation of the yield stress over time for U_0, U_0.5, and U_1 sample are presented in Fig.13, which matches with the proposed hypothesis as featured in Fig.2.

According to the proposed model, the samples experience two stages: a pumping and deposition stage in the first 40 s, where the yield stress shows an initial value, a maximum, and a steady-state, and a re-flocculation and hydration stage from 40 s to 35 min, where the yield stress rises gradually. The stress growth tests show that static and dynamic yield stresses during C3DP are different. The prediction of yield stress using Eq. (64) for the pumping stage is superimposed over experimental data in Fig.14. The prediction of yield stress using Eq. (69) after deposition of material is superimposed over experimental data in Fig.15. The experimental data points are plot in Fig.14 and Fig.15 using blue, red and black markers for mixes U_0, U_0.5, and U_1, respectively. The model satisfactorily predicts the experimental observations. The model results are further fit using the least squares method. The obtained parameters from the fit and adjusted R-square values are shown in Tab.6. The results indicate that the model can capture the rheological behavior of UHPC mixes with different fiber dosages, as the adjusted R-square values are close to 1. The parameters reflect the effects of fiber addition on the dynamic structuration, static yield stress, flocculation, de-flocculation, and irreversible build-up of UHPC.

Recently, Nerella et al. [41] have proposed a strain-based approach for examining the structural build-up. The experiment observations during C3DP for constant strain conditions of 0.08 s⁻¹ for ages up to 150 min are extracted to predict the static yield stress evolution. The model is also compared with the popular Roussel model [31] in Fig.16. The proposed model includes the structural kinetic parameters such as initial static yield stress (5.08 Pa), flocculation rate (1.88 × 10⁻³ s⁻¹), de-flocculation rate (4.03 × 10⁻³ s⁻¹), rate of irreversible hydration build-up (2.28 × 10⁻⁴ s⁻¹), initial reversible structuration value (0.43), and irreversible build-up constant (1.126). The proposed models satisfactorily predict the static yield stress evolution of cement-based materials satisfactorily.

It is important to note that a number of factors (collectively denoted β), such as water-to-binder ratio, clinker’s characteristics, the distribution of particle sizes, the shape and roughness of the particles, superplasticizers, viscosity-modifying agents, aggregate and fiber sizes, temperature, and others, impact the thixotropic structural build-up. To examine the effect of the factors above, a parametric analysis is conducted to investigate the impact of model variables. The parameters are given in Tab.7.

Three types of scenarios are considered, which are 1) dynamic yield stress under a natural strain rate; 2) dynamic yield stress under a constant strain rate, and 3) a combination of dynamic and static. For Case I, β is varied between 0.5, 1, 2, 3, 4 at strain rates of a) 10 s⁻¹ and b) 100 s⁻¹. For Case II, a) β = 0.5, 1, 2, 3, 4 at strain rate of 10 s⁻¹, and b) β = 1 at strain rates of 10–40 s⁻¹. The input parameters in the table are fixed because the maximum shear rates during mixing, pumping, and casting lie in the range of 10–60, 20–40, and 10 s⁻¹ [54], respectively. Fig.17(a)–Fig.17(d) represent the flocculation and de-flocculation process, a physical reaction primarily occurring during the first few hundred seconds after removing external force. It reveals how the shear stress rebuilds once agitation stops and transitions from the dynamic yield stress to the static yield stress. This process is crucial to 3D printing because it helps the material retain its original form after deposition and when it is sheared after pumping and re-flocculates to its static yield stress. Fig.17(a)–Fig.17(d) show that the de-flocculation rate decays parabolically after a few seconds, as commonly observed in Ref. [5]. For Case I given in Tab.7, the effect of the strain rate and material influencing factors are studied. It is observed that for the same mix properties, if the strain rate is reduced from 100 s⁻¹ (Fig.17(a)) to 10 s⁻¹ (Fig.17(c)), the rate of de-flocculation decreases, and the yield stress of the corresponding mix increases proportionately. If the strain rate is increased 10-fold, the observed yield stress in the material increases approximately by a factor of one and half, while the rate of de-flocculation decreases by a factor of 2.5–3. This, however, also depends on the rheological protocols, testing age, and type of rheometer. Our recent studies observed a similar effect [52,53].

Fig.17(a) and Fig.17(c) additionally depict that as the material parameter (β) increases from 0.5 to 4, the peak value for the de-flocculation curves increases. Fig.17(b) and Fig.17(d) show the yield stress evolution. It means that if that material becomes more flowable or, in other words, non-printable, the yield stress decreases. For the strain rate of 100 s⁻¹, yield stress values are higher than the strain rate of 10 s⁻¹. The case II, given in Tab.7 examines the effect of the constant strain rate observed during pumping. The parametric evaluation is conducted by increasing the flowability value (β) on a scale from 1 to 4. It is observed that if the material becomes stiffer (and β value decreases from 4 to 0.5), the rate of de-flocculation decreases (Fig.18(a)), and the yield stress increases (Fig.18(b)). Hence, during pumping, it is essential to ascertain the required rheology for printing. For the case III, given in Tab.7, the value of the constant strain rate is slowly increased from 10–40 s⁻¹. This is similar to the scenario in which the pumping extrusion rate is increased to increase the material deposition during printing. In such a scenario, it is observed that the de-flocculation rate will increase (Fig.18(c)) and yield stress will reduce (Fig.18(d)). Hence increase in the material pumping rate lowers the yield stress and eases the pumping.

6 Conclusions

This study proposes a thixotropic model describing static yield stress development of printable concrete over time. The developed model is validated based on experimental observations as well as from the literature and extended with parametric comments. The main conclusions of the study are as follows.

1) The proposed model accounts for various structural kinetic parameters such as initial static yield stress, flocculation rate, deflocculation rate, rate of irreversible hydration build-up, initial reversible structuration value, and irreversible build-up constant. This model effectively predicts yield stress evolution in 3D printable concrete at rest, during pumping, and in the deposition stage. From the stress growth tests on 3D printable UHPC, the mixes’ static and dynamic yield stresses are examined. The proposed model captures the rheological behavior of UHPC mixes with different fiber dosages and reflects the effects of fiber addition on the dynamic structuration, static yield stress, flocculation, de-flocculation, and irreversible build-up of UHPC.

2) Based on the parametric studies, it is observed that the increase in flocculation constant (K₁) increases filament shape stability and buildability at the expense of lower inter-layer bond strength. The deflocculation structural parametric constant (K₂) is affected by mixer machine characteristics (such as blade size, drum size, shape, rotational speed, etc.). Hence, using different types of mixer machines for the same mix design can further affect yield stress evolution with noticeable differences over time. The parametric hydration constant (K₃) influences the structuration rate (A_thix). The initial solid volume fraction (ϕ_i) and the rate of the heat of hydration during the induction phase (α_ind) have significant effects on K₃. The water−cement ratio is inversely related to the value of K₃. Printable mixtures with a low water-cement ratio, large solid volume fractions with silica content, or other additives, such as nanomaterials, will have a high K₃ value.

3) The water-binder ratio and superplasticizer dosage determine the flowability parameter (β). The printable concrete’s flowability parameter must stay within a particular range, otherwise the material will be too flowable or too stiff, affecting extrudability, buildability, and pumpability. The flowability parameter has a direct relationship with flow value. The strain rate parameter \left(\dot{\gamma }\right) can consider either natural or external strain rate. Natural strain rate is related to material flowability and has an exponential trend. The external strain rate has a minor impact on cement hydration and primarily affects the flocculating pattern.

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