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doi: 10.3934/dcdss.2021161
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## A novel bond stress-slip model for 3-D printed concretes

 1 School of Civil Engineering, Xi'an University of Architecture and Technology, Xi'an, China 2 School of Science, Xi'an University of Architecture and Technology, Xi'an, China 3 Mechanical Engineering Department, Faculty of Engineering, Shahid Chamran University of Ahvaz, Ahvaz 61357-43337, Iran 4 Drilling Center of Excellence and Research Center, Shahid Chamran University of Ahvaz, Ahvaz, Iran

*Corresponding author: chaoliu@xauat.edu.cn (C. Liu) and h.msedighi@scu.ac.ir (H.M. Sedighi)

Received  August 2021 Revised  September 2021 Early access December 2021

This paper considers the 3D printing process as a discontinuous control system and gives a simple and coherent bond stress-slip model for a new and intelligent building 3-D printed concrete. The previous models focused on either the maximal stress or the maximal slip, however, the novel model uses an energy approach by the dimension analysis, so that the main factors affecting the bond stress-slip relationship can be clearly revealed, mainly including the concrete's properties (its porous structure and its strength), the steel bar's properties (its printing direction, its strength, its surface roughness and its geometrical property) and the printing process. It is confirmed that the proposed model, similar to the constitutive relationship in elasticity, plays a key role in concrete mechanics, and it can conveniently explain the observed phenomena from the experiment.

Citation: Chun-Hui He, Shu-Hua Liu, Chao Liu, Hamid Mohammad-Sedighi. A novel bond stress-slip model for 3-D printed concretes. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2021161
##### References:

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##### References:
Sketch of test specimen (unit: mm)
Cubic specimen
Damaged shape of specimen (The main damage of the specimen is the split failure, and the pull-out failure mainly occurred in the specimens with the 45 printing direction) (a) Herringbone splitting (b) In-line splitting (c) Pull out
Bond stress-slip relationship
Printing direction
]">Figure 7.  Nonlinear bond stress-slip relationship [21]
Main chemical components of cementitious materials wt. %
 Component Na$_{2}$O MgO Al$_{2}$O$_{3}$ SiO$_{2}$ P$_{2}$O$_{5}$ SO$_{3}$ Cl K$_{2}$O FeO$_{3}$ TiO$_{2}$ SrO Cement 0.08 0.65 4.65 20.9 0.12 2.65 0.05 0.87 65.00 3.23 0.22
 Component Na$_{2}$O MgO Al$_{2}$O$_{3}$ SiO$_{2}$ P$_{2}$O$_{5}$ SO$_{3}$ Cl K$_{2}$O FeO$_{3}$ TiO$_{2}$ SrO Cement 0.08 0.65 4.65 20.9 0.12 2.65 0.05 0.87 65.00 3.23 0.22
Mix ratio of 3D printed concrete (wt. %)
 Water-cement ratio cement Early Strength Agent Sand Fly ash Silica Fume Cellulase PVA Water reducing agent 0.3 26.6% 2.66% 66.46% 2.68% 1.34% 0.027% 0.0225% 0.186%
 Water-cement ratio cement Early Strength Agent Sand Fly ash Silica Fume Cellulase PVA Water reducing agent 0.3 26.6% 2.66% 66.46% 2.68% 1.34% 0.027% 0.0225% 0.186%
Mechanical properties of the casted and printed concretes
 Samples Compressive strength Casted concrete 50.2 MPa Printed concrete X-direction 37.7 MPa Y-direction 42.9 MPa Z-direction 40.0 MPa
 Samples Compressive strength Casted concrete 50.2 MPa Printed concrete X-direction 37.7 MPa Y-direction 42.9 MPa Z-direction 40.0 MPa
Mechanical properties of steel bars
 Rebar type Diameter (mm) Yield Strength (MPa) Ultimate strength (MPa) Strain Elastic Modulus (MPa) HRB400 10 330 400 14% 2.00$\times$10$^{5}$
 Rebar type Diameter (mm) Yield Strength (MPa) Ultimate strength (MPa) Strain Elastic Modulus (MPa) HRB400 10 330 400 14% 2.00$\times$10$^{5}$
Pull-out test results of 3D printed concrete
 Samples Rebar type $\tau_\text{max}$ (MPa) $s_\text{max}$ (mm) $\tau_\text{max}s_\text{max}$ (MPa$^{*}$mm) Average (MPa$^{*}$mm) Our group Casted sample HRB400 10.13 0.933 9.4512 9.4512 Parallelly printed samples 8.75 0.93 8.1375 8.1375 Vertically printed samples 8.22 0.99 8.1378 8.1378 Inclined printed samples with 45° 6.15 0.80 4.92 4.92 Ref. [40] Casted samples BFRP Unsmooth bar 26.57 0.54 14.3478 16.7413 28.17 0.59 16.6203 28.74 0.67 19.2558 BFRP Smooth bar 23.94 1.35 19.36953 22.86019 25.71 1.42 23.60083 24.82 1.33 25.61021 Parallelly printed samples BFRP Unsmooth bar 22.81 0.49 11.1769 12.37963 24.27 0.56 13.5912 23.79 0.52 12.3708 BFRP Smooth bar 21.35 1.29 14.4182 16.80665 22.79 1.32 17.94038 21.42 1.46 18.06137 Vertically printed samples BFRP Unsmooth bar 20.48 0.52 10.6496 10.47707 20.65 0.48 9.912 19.41 0.56 10.8696 BFRP Smooth bar 13.51 1.01 10.7561 12.68027 18.84 1.36 13.48032 18.91 1.27 13.80439 Inclined printed samples with 45° BFRP Unsmooth bar 23.59 0.58 13.6822 12.42147 22.58 0.49 11.0642 22.76 0.55 12.518 BFRP Smooth bar 15.46 1.37 18.74461 16.44966 20.07 1.25 13.83025 21.94 1.34 16.77412
 Samples Rebar type $\tau_\text{max}$ (MPa) $s_\text{max}$ (mm) $\tau_\text{max}s_\text{max}$ (MPa$^{*}$mm) Average (MPa$^{*}$mm) Our group Casted sample HRB400 10.13 0.933 9.4512 9.4512 Parallelly printed samples 8.75 0.93 8.1375 8.1375 Vertically printed samples 8.22 0.99 8.1378 8.1378 Inclined printed samples with 45° 6.15 0.80 4.92 4.92 Ref. [40] Casted samples BFRP Unsmooth bar 26.57 0.54 14.3478 16.7413 28.17 0.59 16.6203 28.74 0.67 19.2558 BFRP Smooth bar 23.94 1.35 19.36953 22.86019 25.71 1.42 23.60083 24.82 1.33 25.61021 Parallelly printed samples BFRP Unsmooth bar 22.81 0.49 11.1769 12.37963 24.27 0.56 13.5912 23.79 0.52 12.3708 BFRP Smooth bar 21.35 1.29 14.4182 16.80665 22.79 1.32 17.94038 21.42 1.46 18.06137 Vertically printed samples BFRP Unsmooth bar 20.48 0.52 10.6496 10.47707 20.65 0.48 9.912 19.41 0.56 10.8696 BFRP Smooth bar 13.51 1.01 10.7561 12.68027 18.84 1.36 13.48032 18.91 1.27 13.80439 Inclined printed samples with 45° BFRP Unsmooth bar 23.59 0.58 13.6822 12.42147 22.58 0.49 11.0642 22.76 0.55 12.518 BFRP Smooth bar 15.46 1.37 18.74461 16.44966 20.07 1.25 13.83025 21.94 1.34 16.77412
Effects of bars property and concretes property on the bond stress-slip relationship
 Samples D mm $\tau_\text{max}$ MPa $s_\text{max}$ mm $\tau_\text{max}s_\text{max}$ MPa.mm L mm DL mm$^{2}$ $\tau_\text{concrete}$ MPa $\tau_\text{max}s_\text{max}$/ ($\tau_\text{concrete} L$) Casted 10.0 10.13 0.93 9.4512 60 600 50.2 0.003137848 Parallelly 8.75 0.93 8.1375 42.9 0.003161421 Vertically 8.22 0.99 8.1378 37.7 0.003597612 Ref.[45] 9.6 15.88 2.23 35.4124 48 461 37.7 0.019569187 10.4 18.56 1.68 31.1808 54 557 0.015444947 Ref.[46] 8.0 19.49 1.50 29.2350 40 320 36.3 0.020139846 12.0 17.75 3.14 55.7350 60 720 0.025597042 8.0 21.09 1.77 37.3293 20 160 0.051431937 12.0 20.27 2.30 46.6210 30 360 0.042822632 Ref.[28] 16.0 16.42 3.72 61.0517 80 1280 35.0 0.021804189 16.0 18.06 2.15 38.7929 80 1280 35.0 0.013854600 16.0 11.11 3.47 38.5517 80 1280 42.5 0.011338735 16.0 14.81 4.92 72.8652 80 1280 55.5 0.016411081 16.0 17.36 2.67 46.3512 80 1280 60.9 0.009513793 Ref.[13] 12.0 16.48 1.08 17.7984 60 720 30.0 0.009888000 12.0 20.43 1.50 30.6450 60 720 0.017025000 8.0 19.03 0.32 6.08960 40 320 0.005074667 Ref.[29] 10.0 20.00 0.91 18.2000 7.8 78.3 30 0.077479779 8.0 50.00 1.12 56.0000 19.5 156 20 0.143589744 12.0 90.00 1.90 171.000 16.45 197.4 40 0.259878419 Ref.[26] 10.0 13.52 3.20 43.2640 50 500 35.3 0.024512181 10.0 16.33 2.72 44.4176 100 1000 0.012582890 10.0 14.58 0.65 9.47700 150 1500 0.001789802
 Samples D mm $\tau_\text{max}$ MPa $s_\text{max}$ mm $\tau_\text{max}s_\text{max}$ MPa.mm L mm DL mm$^{2}$ $\tau_\text{concrete}$ MPa $\tau_\text{max}s_\text{max}$/ ($\tau_\text{concrete} L$) Casted 10.0 10.13 0.93 9.4512 60 600 50.2 0.003137848 Parallelly 8.75 0.93 8.1375 42.9 0.003161421 Vertically 8.22 0.99 8.1378 37.7 0.003597612 Ref.[45] 9.6 15.88 2.23 35.4124 48 461 37.7 0.019569187 10.4 18.56 1.68 31.1808 54 557 0.015444947 Ref.[46] 8.0 19.49 1.50 29.2350 40 320 36.3 0.020139846 12.0 17.75 3.14 55.7350 60 720 0.025597042 8.0 21.09 1.77 37.3293 20 160 0.051431937 12.0 20.27 2.30 46.6210 30 360 0.042822632 Ref.[28] 16.0 16.42 3.72 61.0517 80 1280 35.0 0.021804189 16.0 18.06 2.15 38.7929 80 1280 35.0 0.013854600 16.0 11.11 3.47 38.5517 80 1280 42.5 0.011338735 16.0 14.81 4.92 72.8652 80 1280 55.5 0.016411081 16.0 17.36 2.67 46.3512 80 1280 60.9 0.009513793 Ref.[13] 12.0 16.48 1.08 17.7984 60 720 30.0 0.009888000 12.0 20.43 1.50 30.6450 60 720 0.017025000 8.0 19.03 0.32 6.08960 40 320 0.005074667 Ref.[29] 10.0 20.00 0.91 18.2000 7.8 78.3 30 0.077479779 8.0 50.00 1.12 56.0000 19.5 156 20 0.143589744 12.0 90.00 1.90 171.000 16.45 197.4 40 0.259878419 Ref.[26] 10.0 13.52 3.20 43.2640 50 500 35.3 0.024512181 10.0 16.33 2.72 44.4176 100 1000 0.012582890 10.0 14.58 0.65 9.47700 150 1500 0.001789802
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