Fractal approach to the fluidity of a cement mortar

Chun-Hui He; Chao Liu

doi:10.1515/nleng-2022-0001

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Fractal approach to the fluidity of a cement mortar

Chun-Hui He and Chao Liu

Published/Copyright: February 9, 2022

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From the journal Nonlinear Engineering Volume 11 Issue 1

Abstract

The fluidity of a cement mortar is a key factor for 3-D printing technology and cement-based materials. This paper introduces the measurement of the fluidity according China’s national standard, and a mathematical model is established to reveal main factors affecting the measure accuracy. The result shows the fluidity reveals mainly the rheological property of the mortar, but it is also affected by other measuring conditions, e.g., the vibration properties of the measuring table.

Keywords: Fractal rheological model; vibrating table; fluidity; fractal derivative; fluidity equation

1 Introduction

The cement mortar’s fluidity is a basic property for various applications in the 3D printing technology [1–3] and the architecture engineering [4–8]. The fluidity plays an important role in the three-dimensional printing technology, Zhang et al. improved the fluidity by adding lubricant and toughening agent in the printing paste [1]; He et al. showed that the fluidity affected the bond stress–slip relationship of 3-D printed subjects [2]; Zuo and Liu revealed the fluidity affected the mechanical and electrical properties of printed composites [3]. Cement is the dominant building material, and its fluidity is a hot topic for fast constructing of a high building with a short period. Feng et al. studied the effect of polycarboxylate superplasticizers on cement mortar’s fluidity [4]; Espinoza-Moreno et al. found that addition of graphite powder in the mortar could change the fluidity [5]; many other searchers also found the addition of superplasticizer or nanoparticles, or inorganic pigments could improve the fluidity [6–10].

The fluidity will greatly affect the printing or building process and the product’s mechanical, thermal, permeable, electronic properties and the vibration strength as well. He et al. found the microstructure of a concrete can be used for vibration attenuation and vibration absorption [11].

The fluidity is generally measured on a jump table with a given frequency and a given amplitude. A fresh mortar with a truncated conical shape is put on the jump table for vibration in a given period, then the final diameter of the expanded mortar is measured to characterize the fluidity. There are many factors affecting the fluidity, mainly, the contents of mortar’s components, the friction between the table and mortar, vibrating properties, and mortar’s rheological property.

2 Rheological modelling

The rheological property is the main factor affecting mortar’s fluidity. The rheological property depends only the mortar fluid’s properties; while the fluidity is affected by many other factors like vibrating parameters and the table’s surface property. There are many rheological models, for examples, the Bingham model [12], The Herschel-Bulkley model [13], Zuo’s fractal model [14, 15].

Bingham model [12]:

(1) τ(t)=τ0+μdεdt

where τ ₀, μ, and ε are, respectively, the yield strength, the viscosity, and the strain.

The Herschel-Bulkley model [13]:

(2) τ(t)=τ0+μ(dεdt)n

where n is an index.

Kelvin model [16]:

(3) τ=τ0+μdudx

The power-law model [17]:

(4) τ=μ(dεdt)n

Fractal Maxwell rheological model [18]:

(5) dudxα=1Edσdxα+1μσ,

Here d/dx^a is the fractal derivative [19, 20]

Fractal rheological model [14, 15]:

(6) τ=τ0+μdudxα

Fractional Bingham model [21]:

(7) τ(t)=τ0+μλαdαεdtα

Here d^α/dt^α is the fractional derivative. There are many fractional modifications of the rheological model, for example, the fractional Kelvin-Voigt model [22]. There are many definitions on fractional derivatives [23–30], and most of which are meaningless and cannot be used for practical applications, He’s fractional derivative [31, 32] and the two-scale fractal derivative [33–35] have physical understanding and have potential applications in studying mortar’s properties. Additionally, according to the dimensional analysis, n in Eqs. (2) and (4) has to equal to one.

3 Theoretical model for mortar’s expansion under vibration

Though the mortar’s fluidity test has been widely used in fabrication of various functional cements, no theoretical analysis was carried out to elucidate the effect of vibrating properties and mortar’s properties on the fluidity. Figure 1 shows the deformation of a mortar in the vibrating table.

Figure 1

Deformation of a mortar in the vibrating table

We assume that the test table’s vibration can be described as:

(8) y(t)=A cos ωt

where ω and A are, respectively, the frequency and amplitude of the vibrating table. Its velocity is:

(9) u=dydt=−Aω sin ωt

The mortar’s kinetic energy is:

(10) E=12mu2=12mω2A2 sin2 ωt

Its average kinetic energy during the quarter cycle is:

(11) E¯=∫0T/4{ 12mω2A2sin2ωt }sinωtdt=∫0π/2{ 12mωA2sin3s }dt=12mωA223π2=mωA2

The mass conservation law of the mortar during the vibration process requires:

(12) 14πd2δ=V0

where d and δ are, respectively, the diameter and the height of the mortar under vibration, V ₀ is the mortar’s volume. Eq. (12) is obtained under the assumption that the solvent involved in the mortar is not evaporated, so the density of the mortar keeps unchanged.

The energy conservation law reads:

(13) 4nE¯+mgh=Wf+Wμ

where 4̅ is the kinetic energy in a vibrating cycle, n is the total vibrating times in the given period, m is the mortar’s mass, h is given as:

(14) h=12(h0−δ)

h ₀ is the original height of the mortar before vibration. W_f and W_μ are, respectively, dissipation works due to the friction and viscosity:

(15) Wf=∫02π∫d0/2d/2f drdθ

(16) Wμ=∫02π∫d0/2d/2τ drdθ

For simplicity, we adopt the coulomb friction for the friction between the mortar and the table, it can be expressed as:

(17) f=kmg

where k is the friction coefficient. So, the friction dissipation work is:

(18) Wf=πk(d−d0)mg

According to Zuo’s fractal rheological model [14], we have:

(19) τ=τ0exp(−ζωα)

where τ₀ is the viscous force when ω = 0, ζ is Zou’s rheological coefficient, α is the mortar’s fractal-related parameter. After a simple calculation, Eq. (19) becomes:

(20) Wμ=πτ0(d−d0)exp(−ζωα)

Now Eq. (13) becomes:

(21) 23πnmωA2+12mg(h0−δ)=πk(d−d0)mg+πτ0(d−d0)exp(−ζωα)

In view of Eq. (12), the fluidity of a cement mortar can be expressed as:

(22) 23πnmωA2+12mg(h0−4V0πd2)=πk(d−d0)mg+πτ0(d−d0)exp(−ζωα)

Eq. (22) reveals that the fluidity depends upon the vibration property, mortar’s geometry, friction, and rheological property. In Eq. (22) Zou’s rheological coefficient is determined experimentally, that makes Eq. (22) inconvenient for practical applications. Here is suggested another approach to find the fluidity equation by the fractal modification of the rheological model.

4 Fractal fluidity model

Fractal theory is a useful tool to analysis of various functional concretes, Tang et al. gave a complete review on fractal approach to cement-based materials [36], and it is Yu-ting Zuo who was the first scientist to apply the fractal theory to study the fluidity, and a fractal rheological model was successfully suggested [5, 6]. Here we apply the basic ideas of the two-scale fractal theory [34] to study the main factors affecting the expanded mortar’s diameter.

The average expanding velocity per cycle of the mortar vibrating in the horizontal direction is:

(23) v=d/2−d0/2nT

Where:

(24) T=2πω

In Eq. (23), we assume that the mortar expands (d/2 – d ₀/2)/n each cycle. Eq. (23) becomes:

(25) v=(d−d0)ωπn=2(r−r0)ωπn

According to the definition of fractal derivative, the radial viscous force can be calculated as:

(26) τr=μdvdra=μΓ(1+α)limr−r0→Δrv(r)−v(r0)(r−r0)α

The fractal derivative has the following properties [35]:

(27) dvdra={v,α=0dnvdrn,α=n;(n=1,2,3,⋯)

and

(28) ddtαtm=Γ(1+m)Γ(1+α−N)Γ(1+m−N)tm−α,α>N

Eq. (26) can be approximated as:

(29) τr=μdvdrα≈μΓ(1+α)2ωπn(r−r0)α−1

Similarly, the axial viscous force is:

(30) τy=μdudyα≈μΓ(1+α)Aω4n(h0/2−h/2)α

The dissipation works in radial and axial directions are, respectively, as:

(31) Wr=∫02π∫r0rτrdrdθ=2μωΓ(1+α)π(2−α)n(r−r0)2−α

(32) Wy=∫02π∫h/2h0/2τydrdy=2πμΓ(1+α)Aω4n(h0/2−h/2)α−1

So, Eq. (23) can be updated as:

(33) 23πnmωA2+12mg(h0−4V0πd2)=πk(d−d0)mg+2μωΓ(1+α)π(2−α)n(r−r0)2−α+2πμΓ(1+α)Aω4n(h0/2−h/2)α−1

where h can be calculated as:

(34) h=12(h0−4V0πd2)

Finally, we obtain the following fluidity equation:

(35) 23πnmωA2+12mg(h0−4V0πd2)=πk(d−d0)mg+2μωΓ(1+α)π(2−α)n(d2−d02)2−α+2πμΓ(1+α)Aω4n[12h0−14(h0−4V0πd2)]α−1

For given test conditions, all parameters except α in Eq. (33) can be determined experimentally, so the fluidity experiment can be used to determine the value of α in Zuo’s fractal rheological model [15].

5 Conclusion

This paper studies the experiment for measuring the fluidity of a cement mortar, the theoretical analysis shows that the measured fluidity depends upon not only the mortar’s rheological property, but also measuring conditions. When all measuring conditions are fixed, the fluidity experiment can be used for determination of the fractal dimension of α in Zuo’s fractal rheological model [15], an inexplicit formulation for calculation of the fractal order is given Eq. (35), it can be solved by Newton’s iteration method or Chun-Hui He’s iteration algorithm.

Acknowledgement

The authors would like to acknowledge the National Key Research and Development Program of China (2019YFC1907105), National Natural Science Foundation of China (51878546, 52178251), the Excellent Youth Science Foundation Project of Shaanxi Province (2020JC-46), the Innovative Talent Promotion Plan of Shaanxi Province (2018KJXX-056), the Key Research and Development Projects of Shaanxi Province (2018ZDCXL-SF-03-03-02), the Science and Technology Innovation Base of Shaanxi Province (2017KTPT-19) and Key Research and Development Project of Shaanxi Province (No. 2020SF-367) for financial support.

Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.

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Received: 2021-11-05

Accepted: 2021-12-11

Published Online: 2022-02-09

Published in Print: 2022-01-01

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/nleng-2022-0001

Keywords for this article

Fractal rheological model; vibrating table; fluidity; fractal derivative; fluidity equation

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