
==== Front
Discov Nano
Discov Nano
Discover Nano
2731-9229
Springer US New York

39256267
4083
10.1186/s11671-024-04083-9
Research
Analytical solution of nano-concrete-epoxy interaction area considering static equilibrium
Haque Md. Foisal mfh.civil@gmail.com

https://ror.org/02m32cr13 grid.443015.7 0000 0001 2222 8047 International University of Business Agriculture and Technology, Dhaka, 1230 Bangladesh
11 9 2024
11 9 2024
12 2024
19 1 1463 7 2024
16 8 2024
© The Author(s) 2024
2024
https://creativecommons.org/licenses/by-nc-nd/4.0/ Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.
This research proposes an analytical solution of the nano-concrete-epoxy interaction area within nano crack region of the reinforced concrete beam by applying Newton’s third law in static equilibrium. For deriving the governing equation, the imaginary beam with free ends (no support) is considered within nano crack region. This imaginary beam is acted along the imaginary line of concrete-epoxy interface. Newton’s third law is applicable for deriving the governing equation because of assuming the absence of frictional and other external forces. The parametric study is performed for implementing the proposed formula of nano interactive area considering variable nano crack depths and thicknesses. The nano interactive area is increased gradually with the increment of depths and thicknesses based on the parametric study because of linear functionality of interactive area and geometry of nano crack region. The maximum interactive area is found to be 314 nm2 at 0.6 ratio of depths and thicknesses of the nano crack. The incremental differences in interactive area between the crack depth or thickness ratios of 0.1 and 0.6 are found to be 25.4% and 1.6% for variations of the crack depth and thickness ratios, respectively. So, the crack depth shows higher impact on the interaction area compared to the thickness of the crack. However, there is a scope for enhancing this research in future by deriving closed-formed analytical formulations to consider appropriate boundary conditions.

Keywords

Analytical solution
Geometry of nano crack region
Nano area at concrete-epoxy interface
Newton’s third law in static equilibrium
Parametric study
issue-copyright-statement© Springer Science+Business Media, LLC, part of Springer Nature 2024
==== Body
pmcIntroduction

In engineering structures, there are two types of crack generally seen such as structural and non-structural, whereas structural cracks are visible in most cases. On the other hand, non-structural cracks are invisible, for example; hair line crack, internal crack of the mortar, internal crack of the paint, etc. The non-structural crack is not influenced to damage the structure, but the invisible structural crack can gradually damage the life period of the structure. These structural invisible cracks are very dangerous for structure so, it is called as “silent killer” of the structure according to the opinion of this research author. The classification of these invisible cracks are flexural, shear, torsion, etc. These cracks are normally seen in the beam, column, and slab. The research on the direct structural invisible crack is very scarce in literature. Some studies [2, 12, 16, 22] were conducted on invisible nano crack in the case of retrofitting of beam by using fiber reinforced polymer (FRP). Also, the radial cracks were considered for the calculation of the stress intensity factor for functional graded cylinder to use weight function method [17]. The gradation curve was prepared for the laboratory compressive test of concrete cube sample contained granite aggregates to investigate the effect of Larestan’s pipeline water [4]. The analytical [18] and finite element [19] methods were used to predict post-buckling behaviors of graphene oxide reinforced concrete members with geometrical imperfection. In addition, the nonlocal strain gradient theory was applied to evaluate transient vibration responses of porous nano beams [20] and shells [3]. For this reason, the present research proposed the analytical solution of invisible nano structural crack of beam by considering epoxy-concrete interaction. The analytical formula of nano-epoxy-concrete interaction area is derived by considering Newton’s third law of motion in static condition.

Nano crack can gradually decreases the service period of structures because of invisibility hampering the protection technique. The remedial measure can possible when it forms a visible crack. People are not known about the nano crack until leakage of water through the structural members especially slab and beam. When people find the main reason of leakage problem, by this time, nano cracks are converted to the micro or millimeter level cracks. The epoxy is used for repairing the micro or millimeter level cracks. An interface is developed, when epoxy is inserted into the crack region. This interface is known as the epoxy-silica interface, which was developed in concrete construction according to the previous studies [14, 25]. Previously,nano composites of Aluminium Oxide (Al2O3) [23, 24, 26], Titanium Oxide (TiO2) [1], and Calcium Carbonate (CaCO3) [5] of Achatina fulica snail shell were used as an epoxy. Another interface is generated during inserting epoxy in the crack, which is known as concrete-epoxy interface. The experimental test with analysis of concrete-epoxy interface was performed by considering the concept of fracture mechanics [12]. The fracture mechanics was used to evaluate crack propagation of nano-multilayered [6] and gradient effect of nano-structured [15] materials. A nano-scale experimental study was conducted to evaluate the interfacial fracture energy of epoxy-based polymer considering debonding mechanism [13]. Various nano sizes of silica were used for the epoxy-silica interface in concrete construction such as 2, 4, and 8 nm [14]. The dimension of each non-structural nano specimen was used to be 16 × 16 × 1.4 nm for studying fatigue behaviour by using atomistic modeling of molecular dynamics [11]. Most of studies described behaviors of nano components of epoxy-material interface based on the above-mentioned literature. So, the main novelty of this research is the direct dealing of nano cracks that are invisible to people without using SEM (Scanning Electron Microscopy). Previous studies [1, 13, 21, 25] used SEM for evaluating behaviors of nano-materials.

The main goal of the present research is the derivation of governing equation of normalized nano-concrete-epoxy interaction area in static condition by using Newton’s third law of motion. The proposed governing equation is automatically validated, because it is matched with the general phenomenon (e.g. applying opposite force to structures by soil, applying opposite force to book by table, applying opposite force to human by ground, etc.). In addition, it is worldwide proved that Newton’s third law is valid for statically equilibrium structures within gravitational field according to the classical physics. Finally, parametric study is performed by varying thickness and depth of the nano crack to evaluate the variations of normalized nano-concrete-epoxy interaction area.

Analytical solution for normalized interactive area of nano crack region

The nano crack of the beam is started from bottom presented in Fig. 1a. Usually, the size of crack shows the irregular pattern [2, 11, 16, 22] started from the bottom of the beam. An irregular crack pattern of this research is depicted in Fig. 1b. The irregularity is formed due to lack of surface smoothness around the crack region. The surfaces of all cracks in practically are not smooth because of non-uniform shear flow and moment distribution, impact of material nonlinearity, etc. For the analytical solution, the smooth surface of crack is considered for deriving the governing equation. This smooth surface consideration of the crack pattern is defined as the regular crack. Details geometry of simplified regular crack pattern are shown in Fig. 1c. This simplified shape of crack is similar to the previous studies [6, 12, 16] crack patterns. This crack is sub-divided into two regular shapes such as trapezium and triangle for deriving the analytical formulations to obtain solution. An epoxy is injected inside the crack by using SEM because of difficult to visibility of nano scale cracks. A nano diameter needle is attached with the SEM body for injecting epoxy. Also, the nano crack region of concrete sample is free from liquid or fluid before injected. It means that this region must be vacuumed before injected epoxy by using SEM. Only one sided concrete is considered at interface for derivations of analytical formulations. The loads of epoxy and concrete are considered to be the same at the interface because of static equilibrium of bonding between concrete and epoxy. Although, bonding strenghts of two materials are varied in reality. In this research, the static equilibrium is considered to apply Newton’s third law at the interface by taking impacts of equal and opposite directional attractive (i.e. bonding) forces of epoxy and concrete, whereas bonding strenghts are avoided from the solution. Newton’s third law (i.e. each applied force has an equal and opposite directional resisting force to maintain the static equilibrium) is applicable to solve this type of loading condition presented in Fig. 2. For the analytical solution, the imaginary beam presented in Fig. 2 is free (i.e. no support), because starting (i.e. B0) and ending (i.e. E0) points of this beam are stayed within nano concrete-epoxy crack zone depicted in Fig. 1c. Due to this reason, Newton’s third law is used as an alternative of closed-formed solution. Only geometry of nano crack region is considered to obtain governing equation of normalized interactive area. The stresses and strains distributions of epoxy and concrete are not considered in this research because of avoiding complexity of deriving formulations. Also, boundary effect is not required in this research because of free beam to obey Newton’s third law in static equilibrium. Recently, analytical formulations were derived by considering the closed-formed solutions with appropriate boundary conditions in cases of tunnel-soil-pile [10] and tunnel-soil-tunnel [8] interactions. The area of the trapezium and triangle shown in Fig. 1c are expressed by Eqs. (1) and (2), respectively. The area of trapezium is the product of half of summation of two thicknesses (i.e. x1 and x2) and upper depth (i.e. z1) of nano crack. Also, the area of triangle is the product of half of lower depth (i.e. z2) and intermediate thickness (i.e. x2) of the nano crack. The uniformly distributed loads of epoxy and concrete are shown in Eqs. (3) and (4), respectively. The loads of epoxy and concrete are the function of unit weight, height and depth of crack, and interactive area. The depth of nano crack must be equal or less than the width of the beam. For free beam, the loads of concrete and epoxy are equal according to Newton’s third law in static equilibrium. The interactive area of Eq. (5) is obtained by yielding of Eqs. (3) and (4). The normalized interactive area of Eq. (6) is found after simplifying Eq. (5), which is the summation of areas of trapezium and triangle in the nano crack region depicted in Fig. 1c. The normalized nano-concrete-epoxy interaction area expressed by Eq. (6) is free from material properties. Only geometry of nano crack region influences the nano interaction area.Fig. 1 Details of idealized nano-concrete-epoxy interaction area of beam

Fig. 2 Loadings of imaginary beam of nano-concrete-epoxy interaction

1 Δ1=12x1+x2z1

2 Δ2=12x2z2

3 pE=γEy1Δ1+Δ2z1+z2

4 pC=γCy1Accz1+z2

5 Acc=γE2γCx1+x2z1+x2z2

6 NA=12x1+x2z1+x2z2

Parametric study for application of proposed analytical formulations

Parametric study is an important technique for the application of the proposed analytical formulations. Previously [7–10], the parametric study was performed for implementing analytical formulations. For this reason, present research conducts the parametric study by considering variable nano crack depths and thicknesses to evaluate the nano-concrete-epoxy interaction area. Details of variable parameters are addressed in Table 1. The nature of normalized nano-concrete-epoxy interaction area of Eq. (6) is linear so, it is linearly varied with variations of thicknesses and depths of nano crack. Figure 3a–f present the variations of normalized interaction areas with various crack thicknesses. The nano interactive area is increased gradually with the increment of the crack thickness because of linear functionality of area and thickness. Also, it increases the variations of thickness and depth ratios because of the linear functionality with the interactive area. The crack thickness variations are addressed for fixed depth of the trapezoidal part of the nano crack because previous study [11] used this depth. The maximum value of interactive area is found to be 314 nm2 as shown in Fig. 3f in cases of the maximum thickness and depth ratios. The minimum difference in results for maximum crack thickness between thickness ratios of 0.5 and 0.6 is found to be 7.1% at a specific depth ratio of 0.1. Also, it is obtained to be 8.9% in the case of the depth ratio of 0.6. So, the incremental difference in interactive area between depth ratios of 0.1 and 0.6 is 25.4%. Similarly, the variations of normalized interactive area are shown in Fig. 4a–f with variable crack depths, depth and thickness ratios, and a fixed thickness of 8 nm. The fixed thickness is considered based on the previous study [14]. In this case, the interactive area is gradually increased with the increment of the crack depths, and depth and thickness ratios. The maximum interactive area is found to be 77 nm2 as shown in Fig. 4f at depth and thickness ratios of 0.6 at a specific crack depth of 20 nm. The minimum and maximum differences in results between crack depth ratios of 0.5 and 0.6 are obtained to be 6.6% for 0.1 and 6.7% for 0.6 thickness ratios, respectively, at a specific crack depth of 20 nm. So, the incremental minimum difference in interactive area between crack thickness ratios of 0.1 and 0.6 is 1.6%. The main reason of the incremental normalized interactive area with the variations of geometric parameters in the crack region is the linear functionality of area and parameters according to Eq. (6). Therefore, impact of depth variations on the nano-concrete-epoxy interactive area is higher compared to the thickness variations based on the application results of proposed governing equation.Table 1 Details of variable nano parameters

Parameters	Values	Units	References (if any)	
x1	0.01, 0.1, 1, 2, 4, 6, 8, 10, 12, 14, 16, 18, and 20	nm	2, 4, 8 nm, [14]	
x2x1	0.1, 0.2, 0.3, 0.4, 0.5, and 0.6	–	/	
z1	0.01, 0.1, 1, 2, 4, 6, 8, 10, 12, 14, 16, 18, and 20	nm	16 nm, [11]	
z2z1	0.1, 0.2, 0.3, 0.4, 0.5, and 0.6	–	/	
“–” = No unit; “/” = Reference is not available

Fig. 3 Normalized nano interactive area variations with variable crack thicknesses

Fig. 4 Normalized nano interactive area variations with variable crack depths

Conclusions

This research proposed an analytical governing equation to calculate the nano-concrete-epoxy interaction area by considering Newton’s third law in static equilibrium. For deriving the governing equation, the nano crack region is tightened by epoxy with concrete surface. Also, the frictional and other external forces are avoided for deriving formulations. So, Newton’s third law in static equilibrium is applied for imaginary beam at epoxy-concrete interface within the nano crack region. Two opposite directional loadings are acted on the imaginary beam at interface such as epoxy and one-sided concrete self-weights. The parametric study is performed for implementing the proposed governing equation of nano-concrete-epoxy interaction area. According to the parametric study, the nano interactive area is increased gradually with the increment of depths and thicknesses of nano crack because of linear functionality of interactive area and geometry of nano crack. The maximum interactive area is obtained to be 314 nm2 at the same ratio of depths and thicknesses of 0.6 with the maximum depth of nano crack of 16 nm. The minimum and maximum differences in interactive area for the maximum crack thickness between thickness ratios of 0.5 and 0.6 are found to be 7.1% for 0.1 and 8.9% for 0.6 depth ratios, respectively. These differences are obtained to be 6.6% and 6.7% in cases of thickness ratios of 0.1 and 0.6, respectively, to consider the maximum crack depth. The incremental differences in interactive area are found to be 25.4% and 1.6% in cases of variations of crack depth and thickness ratios, respectively. So, it can be said that the crack depth impact on the interaction area is higher compared to the thickness of crack. Therefore, there is a further scope to enhance this research by performing closed-formed analytical solution to consider appropriate boundary conditions.

List of symbols

Δ1 Area of trapezium ABCD (nm2)

Δ2 Area of triangle DCE by assuming 90° angle of C (nm2)

x1 Top thickness of the crack along the length of the beam (nm)

x2 Intermediate thickness of the crack along the length of the beam (nm)

z1 Upper depth of the crack along the width of the beam (nm)

z2 Lower depth of the crack along the width of the beam (nm)

γE Unit weight of epoxy (N/nm3)

y1 Height of crack along the height of beam (nm)

pE Uniformly distributed load intensity of epoxy on the interaction line (N/nm)

γC Unit weight of concrete (N/nm3)

Acc Interactive area (nm2)

pC Uniformly distributed load intensity of concrete on the interaction line (N/nm)

NA Normalized interactive area (nm2)

Author contributions

Md. Foisal Haque: Conceptualization, Deriving Formulations, Data Analysis, Writing, Reviewing, Editing.

Data availability

Data are not available.

Declarations

Competing interests

The authors declare no competing interests.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
==== Refs
References

1. Al-Ajaj I Abd M Jaffer H Mechanical properties of micro and nano TiO2/epoxy composites Int J Min Metall Mech Eng 2013 1 2 93 97
Al-Ajaj I, Abd M, Jaffer H. Mechanical properties of micro and nano TiO2/epoxy composites. Int J Min Metall Mech Eng. 2013;1(2):93–7.
2. Büyüköztürk O Buehler M Lau D Tuakta C Structural solution using molecular dynamics: fundamentals and a case study of epoxy-silica interface Int J Solids Struct 2011 10.1016/j.ijsolstr.2011.03.018
Büyüköztürk O, Buehler M, Lau D, Tuakta C. Structural solution using molecular dynamics: fundamentals and a case study of epoxy-silica interface. Int J Solids Struct. 2011. 10.1016/j.ijsolstr.2011.03.018.10.1016/j.ijsolstr.2011.03.018
3. Forsat M Badnava S Mirjavadi S Barati M Hamouda A Small scale effects on transient vibrations of porous FG cylindrical nanoshells based on nonlocal strain gradient theory Eur Phys J Plus 2020 10.1140/epjp/s13360-019-00042-x
Forsat M, Badnava S, Mirjavadi S, Barati M, Hamouda A. Small scale effects on transient vibrations of porous FG cylindrical nanoshells based on nonlocal strain gradient theory. Eur Phys J Plus. 2020. 10.1140/epjp/s13360-019-00042-x.10.1140/epjp/s13360-019-00042-x
4. Forsat M Mirjavadi S Hamouda A Investigation of the effect of Larestan’s pipeline water on the mechanical properties of concretes containing granite aggregates Adv Civil Eng 2019 10.1155/2019/4197186
Forsat M, Mirjavadi S, Hamouda A. Investigation of the effect of Larestan’s pipeline water on the mechanical properties of concretes containing granite aggregates. Adv Civil Eng. 2019. 10.1155/2019/4197186.10.1155/2019/4197186
5. Gbadeyan O Adali S Bright G Sithole B The investigation of reinforcement properties of nano-CaCO3 synthesized from Achatina fulica snail shell through mechanochemical methods on epoxy nanocomposites Nanocomposites 2021 7 1 79 86 10.1080/20550324.2021.1936972
Gbadeyan O, Adali S, Bright G, Sithole B. The investigation of reinforcement properties of nano-CaCO3 synthesized from Achatina fulica snail shell through mechanochemical methods on epoxy nanocomposites. Nanocomposites. 2021;7(1):79–86. 10.1080/20550324.2021.1936972.10.1080/20550324.2021.1936972
6. Guo L Kitamura T Yan Y Sumigawa T Huang K Fracture mechanics investigation on crack propagation in the nano-multilayered materials Int J Solids Struct 2015 10.1016/j.ijsolstr.2015.03.025
Guo L, Kitamura T, Yan Y, Sumigawa T, Huang K. Fracture mechanics investigation on crack propagation in the nano-multilayered materials. Int J Solids Struct. 2015. 10.1016/j.ijsolstr.2015.03.025.10.1016/j.ijsolstr.2015.03.025
7. Haque MF Bi-directional effects on the tunnel-soil-pile interaction under seismic loadings J GeoEng 2022 17 3 165 173 10.6310/jog.202209_17(3).5
Haque MF. Bi-directional effects on the tunnel-soil-pile interaction under seismic loadings. J GeoEng. 2022;17(3):165–73. 10.6310/jog.202209_17(3).5.10.6310/jog.202209_17(3).5
8. Haque MF Analytical solution of Tunnel-Soil-Tunnel (TST) interaction under seismic excitation Geomech Geoeng 2024 10.1080/17486025.2023.2300474
Haque MF. Analytical solution of Tunnel-Soil-Tunnel (TST) interaction under seismic excitation. Geomech Geoeng. 2024. 10.1080/17486025.2023.2300474.10.1080/17486025.2023.2300474
9. Haque MF Predicting lateral displacement of sand for mat foundation under simplifed seismic loading Discover Civil Eng 2024 10.1007/s44290-024-00001-1
Haque MF. Predicting lateral displacement of sand for mat foundation under simplifed seismic loading. Discover Civil Eng. 2024. 10.1007/s44290-024-00001-1.10.1007/s44290-024-00001-1
10. Haque MF Ansary MF Analytical formulations of tunnel–soil–pile interaction under seismic excitations Earthquake Eng Resilience 2024 3 1 72 104 10.1002/eer2.67
Haque MF, Ansary MF. Analytical formulations of tunnel–soil–pile interaction under seismic excitations. Earthquake Eng Resilience. 2024;3(1):72–104. 10.1002/eer2.67.10.1002/eer2.67
11. Horstemeyer M Farkas D Kim S Tang T Potirniche G Nanostructurally small cracks (NSC): A review on atomistic modeling of fatigue Int J Fatigue 2010 32 1473 1502 10.1016/j.ijfatigue.2010.01.006
Horstemeyer M, Farkas D, Kim S, Tang T, Potirniche G. Nanostructurally small cracks (NSC): A review on atomistic modeling of fatigue. Int J Fatigue. 2010;32:1473–502. 10.1016/j.ijfatigue.2010.01.006.10.1016/j.ijfatigue.2010.01.006
12. Lau D Büyüköztürk O Fracture characterization of concrete/epoxy interface affected by moisture Mech Mater 2010 10.1016/j.mechmat.2010.09.001
Lau D, Büyüköztürk O. Fracture characterization of concrete/epoxy interface affected by moisture. Mech Mater. 2010. 10.1016/j.mechmat.2010.09.001.10.1016/j.mechmat.2010.09.001
13. Lau D Broderick K Buehler M Büyüköztürk O A robust nanoscale experimental quantification of fracture energy in a bilayer material system PNAS 2014 111 33 11990 11995 10.1073/pnas.1402893111 25097263
Lau D, Broderick K, Buehler M, Büyüköztürk O. A robust nanoscale experimental quantification of fracture energy in a bilayer material system. PNAS. 2014;111(33):11990–5. 10.1073/pnas.1402893111.25097263 10.1073/pnas.1402893111
14. Lau D Büyüköztürk O Buehler M Characterization of the intrinsic strength between epoxy and silica using a multiscale approach J Mater Res 2012 10.1557/jmr.2012.96
Lau D, Büyüköztürk O, Buehler M. Characterization of the intrinsic strength between epoxy and silica using a multiscale approach. J Mater Res. 2012. 10.1557/jmr.2012.96.10.1557/jmr.2012.96
15. Lurie S Belov P Gradient effects in fracture mechanics for nano-structured materials Eng Fract Mech 2014 10.1016/j.engfracmech.2014.07.032
Lurie S, Belov P. Gradient effects in fracture mechanics for nano-structured materials. Eng Fract Mech. 2014. 10.1016/j.engfracmech.2014.07.032.10.1016/j.engfracmech.2014.07.032
16. Maio U Gaetano D Greco F Lonetti P Blasi P Pranno A The reinforcing effect of nano-modified epoxy resin on the failure behavior of FRP-Plated RC structures Buildings 2023 10.3390/buildings13051139
Maio U, Gaetano D, Greco F, Lonetti P, Blasi P, Pranno A. The reinforcing effect of nano-modified epoxy resin on the failure behavior of FRP-Plated RC structures. Buildings. 2023. 10.3390/buildings13051139.10.3390/buildings13051139
17. Mirahmadi H Azimi M Mirjavadi S Calculation of stress intensity factor for functionally graded cylinders with two radial cracks using the weight function method Theor Appl Fract Mech 2016 10.1016/j.tafmec.2016.06.004
Mirahmadi H, Azimi M, Mirjavadi S. Calculation of stress intensity factor for functionally graded cylinders with two radial cracks using the weight function method. Theor Appl Fract Mech. 2016. 10.1016/j.tafmec.2016.06.004.10.1016/j.tafmec.2016.06.004
18. Mirjavadi S Afshari B Barati M Hamouda A Transient response of porous inhomogeneous nanobeams due to various impulsive loads based on nonlocal strain gradient elasticity Int J Mech Mater Des 2020 16 57 68 10.1007/s10999-019-09452-2
Mirjavadi S, Afshari B, Barati M, Hamouda A. Transient response of porous inhomogeneous nanobeams due to various impulsive loads based on nonlocal strain gradient elasticity. Int J Mech Mater Des. 2020;16:57–68. 10.1007/s10999-019-09452-2.10.1007/s10999-019-09452-2
19. Mirjavadi S Forsat M Barati M Khan I Analysis of post-buckling of higher-order graphene oxide reinforced concrete plates with geometrical imperfection Adv Concrete Construct 2020 9 4 397 406 10.12989/acc.2020.9.4.397
Mirjavadi S, Forsat M, Barati M, Khan I. Analysis of post-buckling of higher-order graphene oxide reinforced concrete plates with geometrical imperfection. Adv Concrete Construct. 2020;9(4):397–406. 10.12989/acc.2020.9.4.397.10.12989/acc.2020.9.4.397
20. Mirjavadi S Forsat M Barati M Khan I Finite element based post-buckling analysis of refined graphene oxide reinforced concrete beams with geometrical imperfection Comput Concrete 2020 25 4 283 291 10.12989/cac.2020.25.4.283
Mirjavadi S, Forsat M, Barati M, Khan I. Finite element based post-buckling analysis of refined graphene oxide reinforced concrete beams with geometrical imperfection. Comput Concrete. 2020;25(4):283–91. 10.12989/cac.2020.25.4.283.10.12989/cac.2020.25.4.283
21. Sahar N Hong S-I Kohn D Micro- and nano-structural analyses of damage in bone Micron 2005 36 617 629 10.1016/j.micron.2005.07.006 16169739
Sahar N, Hong S-I, Kohn D. Micro- and nano-structural analyses of damage in bone. Micron. 2005;36:617–29. 10.1016/j.micron.2005.07.006.16169739 10.1016/j.micron.2005.07.006
22. Tuakta C Büyüköztürk O Conceptual model for prediction of FRP-concrete bond strength under moisture cycles J Compos Constr 2011 15 5 743 756 10.1061/(ASCE)CC.1943-5614.0000210
Tuakta C, Büyüköztürk O. Conceptual model for prediction of FRP-concrete bond strength under moisture cycles. J Compos Constr. 2011;15(5):743–56. 10.1061/(ASCE)CC.1943-5614.0000210.10.1061/(ASCE)CC.1943-5614.0000210
23. Yazman Ş Uyaner M Karabörk F Akdemir A Effects of nano reinforcing/matrix interaction on chemical, thermal and mechanical properties of epoxy nanocomposites J Compos Mater 2021 55 28 4257 4272 10.1177/00219983211037059
Yazman Ş, Uyaner M, Karabörk F, Akdemir A. Effects of nano reinforcing/matrix interaction on chemical, thermal and mechanical properties of epoxy nanocomposites. J Compos Mater. 2021;55(28):4257–72. 10.1177/00219983211037059.10.1177/00219983211037059
24. Yousri O Abdellatif M Bassioni G Effect of Al2O3 Nanoparticles on the mechanical and physical properties of epoxy composite Arab J Sci Eng 2017 10.1007/s13369-017-2955-7
Yousri O, Abdellatif M, Bassioni G. Effect of Al2O3 Nanoparticles on the mechanical and physical properties of epoxy composite. Arab J Sci Eng. 2017. 10.1007/s13369-017-2955-7.10.1007/s13369-017-2955-7
25. Zhou H Liu H-Y Zhou H Zhang Y Gao X Mai Y-W On adhesive properties of nano-silica/epoxy bonded single-lap joints Mater Design 2016 10.1016/j.matdes.2016.01.055
Zhou H, Liu H-Y, Zhou H, Zhang Y, Gao X, Mai Y-W. On adhesive properties of nano-silica/epoxy bonded single-lap joints. Mater Design. 2016. 10.1016/j.matdes.2016.01.055.10.1016/j.matdes.2016.01.055
26. Zhou Y Zhang J Zhang R Liu E Xue X Xing X Zhang Q Effect of nano-Al2O3/epoxy resin composite on the shear strength recovery of fractured rock masses with various crack widths and SCA interfacial treatments Case Stud Const Mater 2023 10.1016/j.cscm.2022.e01715
Zhou Y, Zhang J, Zhang R, Liu E, Xue X, Xing X, Zhang Q. Effect of nano-Al2O3/epoxy resin composite on the shear strength recovery of fractured rock masses with various crack widths and SCA interfacial treatments. Case Stud Const Mater. 2023. 10.1016/j.cscm.2022.e01715.10.1016/j.cscm.2022.e01715
