12801 FACTA UNIVERSITATIS Series: Mechanical Engineering Vol. 22, No 2, 2024, pp. 329 - 342 https://doi.org/10.22190/FUME240605035H © 2024 by University of Niš, Serbia | Creative Commons License: CC BY-NC-ND Original scientific paper A FRACTAL-BASED APPROACH TO THE MECHANICAL PROPERTIES OF RECYCLED AGGREGATE CONCRETES Chun-Hui He1, Hua-Wei Liu1, Chao Liu1,2 1School of Civil Engineering, Xi'an University of Architecture and Technology, Xi’an, China 2School of Science, Xi'an University of Architecture and Technology, Xi’an, China ORCID iDs: Chun-Hui He https://orcid.org/0000-0003-0810-5248 Hua-Wei Liu https://orcid.org/0000-0003-4433-5075 Chao Liu https://orcid.org/0000-0002-9634-7791 Abstract. The mechanical properties of porous concrete, such as strength and durability, are significantly influenced by moisture transport, particularly in the case of recycled aggregate concretes. The pore distribution and pore size of the concrete, as well as the ambient temperature in the surrounding environment, exert a significant influence on the moisture transport. This paper establishes the fractal Fick’s law, the fractal Darcy law, and the fractal Richards equation. In conclusion, a fractal model for the diffusion and permeability in porous concrete is established. This study examines the mechanisms of moisture diffusion and water permeability in concrete. The comparison of the theoretical prediction with the experimental data indicates a high degree of congruence, thereby suggesting that the concrete’s relative humidity response can be predicted by the established model. This provides a foundation for the optimal design of concrete with required mechanical properties in special applications. Key words: Fractal dimension, Durability of concrete, Mathematical model, Moisture 1. INTRODUCTION Concrete is the most prevalent porous material utilized globally, and it is an enduring theme in mechanical engineering [1-3], and it might be also the very promising construction material in the Moon and the Mars by 3-D printing technology. The humidity within concrete plays a pivotal role in determining its mechanical, chemical, and thermal properties [4-6]. Consequently, the durability and reliability of concrete are significantly influenced by the transport of humidity within the concrete, especially for the recycled aggregate concretes. Given the vast quantity of demolished concrete waste present globally, the utilization of aggregates derived from this waste represents an environmentally friendly and cost-effective construction material. Received: June 05, 2024 / Accepted July 20, 2024 Corresponding author: Hua-Wei Liu, Chao Liu School of Civil Engineering, Xi'an University of Architecture and Technology, Xi’an, China E-mails: liuhuawei@xauat.edu.cn (H. W. Liu); chaoliu@xauat.edu.cn (C. Liu) https://orcid.org/0000-0003-0810-5248 https://orcid.org/0000-0003-4433-5075 https://orcid.org/0000-0002-9634-7791 330 C.-H. HE, H.-W. LIU, C. LIU In a surprising turn of events, the Telegraph reported in September 2023 that reinforced autoclaved aerated concrete (RAAC) had failed with only a lifespan of 30 to 40 years. This presents an urgent challenge for engineers and architects. This concrete material, which has been used in construction for decades, began to corrode and produce cracks. While this has caused some unfortunate accidents in the UK and other places, it has also led to some scientific discoveries. The primary factor contributing to its deterioration is the transformation of moisture, which results in corrosion. The volume of the corroded reinforcing steel bar can be increased by up to seven times [7], which is a remarkable phenomenon. The most concerning aspect is the absence of warning signs preceding damage, which further complicates the unpredictability of the failure. Although there has been considerable research into the transport of moisture in concrete using differential models [8], there has been little investigation into the underlying mechanisms in fractal spaces. The fractal diffusion theory [9,10] represents an effective tool for the analysis of moisture transfer in concrete. The intricate micro/nano-structure of the pores can be represented by a fractal network [11-13]. In this paper, we will present a mathematical model in a two-scale fractal space, as described in Refs. [14-17]. 2. FRACTAL MOISTURE TRANSFER The intricate porosity structure within the concrete will result in a negative pressure, which will induce the evaporation of the absorbed water. This will initiate the process of moisture diffusion in concrete, as the concrete surface has the highest relative humidity, while the center has the lowest. The diffusion process is a complex phenomenon that occurs in three dimensions [18 ]. In order to identify an analytical expression for this process, this paper examines the one-dimensional diffusion within concrete. The porous concrete is treated as a two-scale fractal space, the fractal Fick’s laws can be expressed, respectively, as DRH J D x    = −  (1) DRH J t x       = −   (2) where J is the diffusion flux, RHD represents the relative humidity, D is the diffusion coefficient, α is the two-scale fractality for the porosity, and ∂α/∂tα is the two-scale fractal derivative. So the fractal diffusion equation can be expressed as ( )D DRH RH D t x x        =    (3) When α=1, Eq. (1) turns out to be the traditional Richards equation, which was used to study diffusion process in soils [19]. https://baike.baidu.com/item/%E5%B8%8C%E8%85%8A%E5%AD%97%E6%AF%8D/4428067#3-1 A Fractal-based Approach to the Mechanical Propertis of Recycled Aggregate Concretes 331 2.1. Initial Surface Absorption When α=1, the initial surface absorption rates of water by the concrete can be expressed in the term of the Boltzmann variable [19] 1/2I t− (4) where I is the initial surface absorption rates, t is time. Nevertheless, Eq. (4) was found to be unsuitable for the concrete diffusion process. It should therefore be modified in accordance with the recommendation set out in [20]: nI t− (5) where n is an experimental constant. In order to investigate the physical significance of the variable n in Eq. (5), it is possible to introduce a generalized Boltzmann variable in a manner analogous to that for the traditional Richards equation [19]: /2x t  −= (6) where λ is the generalized Boltzmann variable. The initial surface absorption rate on the concrete surface can be expressed as /2I at −= (7) where a is a constant. The relationship between n and the fractality is therefore as follows 2n = (8) Kumar and Bhattacharjee [20] denoted n as the silting factor, now we see that it is relative to two-scale fractal dimensions. 2.2. Fractal Diffusion in the Concrete Following the initial surface absorption, the moisture diffuses from the concrete surface to the concrete interior. The diffusion process is primarily influenced by the geometry of the concrete pores. In order to study the properties of Eq. (3) in more depth, the two-scale transform is introduced [21,22] s t x    =  = (9) Eq. (3) becomes ( )D DRH RH D s     =    (10) The two-scale transform [21,22] is regarded as a physical interpretation of the fractional complex transform [23], which was originally proposed by Li & He [24]. It has been extensively employed in engineering, as evidenced by numerous references (see, for instance, Refs. [25-29]). The solution of Eq. (10) is https://baike.baidu.com/item/%E5%B8%8C%E8%85%8A%E5%AD%97%E6%AF%8D/4428067#3-1 332 C.-H. HE, H.-W. LIU, C. LIU 1/2 1/2 0 1/2 1/2 0 1 ( , ) ( ) 1 erf( ) 2 1 ( )erfc( ) 2 DRH s RH RH RH D s RH RH RH D s    − −   − −     = + − −    = + − (11) where RH0=RH(0,t), RH∞=RH(∞,t), erf is the error function and erfc is the complementary error function. The solution of Eq. (1) is obtained as follows. 1/2 /2 0 1/2 /2 0 1 ( , ) ( ) 1 erf( ) 2 1 ( )erfc( ) 2 DRH x t RH RH RH D t x RH RH RH D t x     − −   − −     = + − −    = + − (12) The diffusion process is influenced by the pore structure of concrete, and the diffusion process in turn alters the pore structure of concrete. In order to investigate the relationship between the diffusion coefficient and the diffusion process with the pore properties of concrete, a dimensionless analysis [16,17,30] is employed in this paper. The diffusion coefficient has a dimension of m2/s. By means of dimensionless analysis, it can be assumed that 0 0( , ) ( ) ( )m nD DdRH dRH D D V V dt dt =  (13) Here V0 is the pore volume, m and n are dimensionless exponents. When V0 = 0, there is no diffusion; and when V0 = 1, diffusion can be studied using the traditional theory of thermodynamics. Furthermore, changes in relative humidity also affect the diffusion process, which ends when the diffusion process tends to be stabilized, that is to say when dRHD/dt=0. The dimensionless equation is 2 1 3[ ] [ ]m nL T L T− −= (14) That means 2 / 3, 1m n= = (15) We, therefore, obtain 2/3 0( ) DdRH D V dt  (16) The rate of change of relative humidity is related to saturation. When saturated (Θ=1), dRHD/dt =0. The rate of change of relative humidity is greatest when Θ=0. Based on the above analysis, it can be assumed that (1 )aDdRH dt  − (17) where a is the experimental constant. Also the pore volume is proportional to porosity: 0V  (18) A Fractal-based Approach to the Mechanical Propertis of Recycled Aggregate Concretes 333 Based on the above analyses, the following equations were obtained for the diffusion coefficient versus porosity and saturation degree 2/3 (1 )aD k= − (19) where  is the concrete’s porosity, k and a are experimental constants. In literature, there was an empirical formula for soils’ diffusion [31]: 3/4 10/3(1 )D c= − (20) Here c is a dimension-related parameter. 2.3. Fractal Permeation in the Concrete In this paper, permeation is considered to occur when there is a pressure gradient, which is induced by the liquid-gas interface inside the concrete, the interface-induced force can be explained by the geometrical potential theory [32,33]. Darcy's law is the most basic tool for studying permeation phenomena, but it cannot analyze the effect of concrete pore properties (pore size and distribution) on permeation properties. The fractal Darcy's law can be expressed as A d P Q K dx   = (21) According to the definition of the fractal derivative, Eq. (21) can be approximated expressed as 1lim lim (1 ) ( ) ( )x r x r A d P A P A P A dP Q K K K K r dxdx x x           − +  →  →   = = =   +   (22) where Q is permeation flux, ΔP permeation pressure, r mean radius of pores, Γ gamma function, K permeability coefficient, A pore area, μ viscosity coefficient. In the fractal space, the pore area scales as 2A r  (23) The fractal Darcy's law can be expressed as 1Q cr P+= −  (24) where c is a constant. The liquid-gas interface induced pressure can be described by the Kelvin equation [34]: ln K RT P RH M   = − (25) where RHK is the relative humidity induced by the permeation process, R ideal gas constant (J/mol). M molar mass of water (kg/mol); T temperature (K); ρ density of water (kg/m). Applying the approximate formula, ln(1+x)=x, the following approximate relationship can be obtained ln ln(1 (1 )) (1 )K K KRH RH RH= − −  − − (26) https://baike.baidu.com/item/%E5%B8%8C%E8%85%8A%E5%AD%97%E6%AF%8D/4428067#3-4 https://baike.baidu.com/item/%E5%B8%8C%E8%85%8A%E5%AD%97%E6%AF%8D/4428067#3-18 334 C.-H. HE, H.-W. LIU, C. LIU Eq. (21) becomes 1 1ln (1 )K K RT RT Q cr RH cr RH M M   + +=  − − (27) According to the law of conservation of mass in the two-scale fractal space, we obtain that ( ) 0 Q t x       + =   (28) where ω is the ratio of water in liquid form to the total water in the concrete, it can be expressed as Water K Water Moisture V RH V V  = = +  (29) where Vwater and VMoisture are, respectively the volume for water and moisture. The mass conservation equation can then be written as 1( ) ( (1 )) 0K K RH RT cr RH Mt x        +  − − =   (30) or ( ) ( ) 0K c KRH D RH t x       + =   (31) Here Dc is given as follows 1 c RT D cr M +  = (32) The general solution of Eq. (31) is ( )K cRH f x D t = − (33) where f is a continuous function. According to the initial and final conditions, we can get  0 1 1 0 1 ( , ) ( ) exp ( ) ( ) exp ( ) K cRH x t RH RH RH D x D t RT RH RH RH D x cr t M        +   = + − −   = + − −    (34) where D1 and c are experimental parameters. 2.4. Coupled Transport of Moisture Diffusion and Water Permeation The processes of moisture diffusion and water permeation are studied separately in the aforementioned text, although they actually occur simultaneously. The relative humidity within the concrete can be approximately expressed as a result of the coupling of diffusion and permeation. ( , ) ( , ) (1 ) ( , )D KRH x t aRH x t a RH x t= + − (35) https://baike.baidu.com/item/%E5%B8%8C%E8%85%8A%E5%AD%97%E6%AF%8D/4428067#3-25 A Fractal-based Approach to the Mechanical Propertis of Recycled Aggregate Concretes 335 where a is the weight function, the early stage is dominated by diffusion, the final stage is dominated by permeation, assuming that at t=t0, the diffusion and permeation contribute equally to RH, so we can choose the following weight function 0 0 t a t t = + (36) where t0 is determined according to the actual experiment. Thus we obtain the coupled transport equation 0 0 0 1/2 /20 0 0 1 0 1 0 ( , ) ( , ) ( , ) 1 ( ) 1 erf( ) 2 ( )exp ( ) D K t t RH x t RH x t RH x t t t t t t RH RH RH D t x t t t RT RH RH RH D x cr t t t M      − −   +   = + + +    = + − −  +        + + − −  +    (37) 3. THEORETICAL ANALYSIS The experimental water box was designed with dimensions of 1000mm × 680mm × 600mm. The box was maintained at a temperature of 293 K (20 ℃), concrete samples were placed within the box for the purpose of measuring relative humidity at two locations, 2 cm and 4 cm beyond the concrete surface. The relative humidity was monitored at one- minute intervals until 2880 minutes had elapsed, the samples were tested for both the recycled aggregate concrete and natural aggregate concrete. Detailed experiment process was given in Ref. [35]. 3.1. Humidity Response According to the experiment data, we obtain the following humidity response: 0.4116 293 2 0. 6 4116 293 4 293 0.35685 0.35 5 2 9 4 8 2 3 0.24160 1 erf( ) 0.24261 1 erf( ) 0.23455 1 erf( ) 0.24711 1 erf( 0.75840 98.61 0.75739 113.36 0.76545 5 6 ) 2.53 0.75289 56.5 N N R R RH RH t t t t RH RH − − − − − − − − − − − − = −  = −    +   + +  = −  = + −           (38) In the subscripts, N and R represent, respectively, natural aggregate concrete and recycled aggregate concrete. The value 293 represents temperature in Kelvin, while the values 2 and 4 represent, respectively, the depth from the concrete surface in centimeters. 336 C.-H. HE, H.-W. LIU, C. LIU Fig. 1 Humidity change of concrete samples with temperature of 20 ℃ (293.15 K): Comparison between experimental results and calculated curves. (a)N-293-2; (b)N- 293-4; (c)R-293-2; (d)R-293-4. From Fig. 1, it can be seen that the experimental data basically match the theoretical results. The least squares fitting method was used in the curve fitting process, and because there is a sudden change in the initial response, the calculated curve does not have a good fit with the experimental curve, but the general trend is the same. Sample R-293-2 appeared obviously two times response, the beginning of the response is slow, when t=10000s, the second response occurs. This is due to the diffusion-permeability coupling. Sample N-293-2 shows a similar situation, but it is not very obvious, with a faster response at the beginning, but the response becomes slower after t=100000 s. In contrast, samples N-293-4 and R-293-4 do not show the phenomenon of two responses, suggesting that diffusion dominates at x=4 cm, but it can be reasonably assumed that diffusion-permeability coupling will occur subsequently, the place near surface has a faster coupling than the center of the concrete. 3.2. Space Diffusion Process According to the coefficients in Eq. (38), the average values of x=2 cm and x=4 cm are taken, and then re-fitting based on the experimental data gives the following results. A Fractal-based Approach to the Mechanical Propertis of Recycled Aggregate Concretes 337 0.4116 0.21 3 30 293 0.2472 29 0. 5685 3 0.757 0 0.24289 226.0301 0.75 11 1 erf( ) 0.24 3083 19 e .rf( )17 132 17 N R R t x t x H RH − − − −   +  = −  = −+       (39) According to the fitting results Eq. (39), Fig. 2 and Fig. 3 give the diffusion process of the specimens with time and space, and the trend is basically the same, but there are errors. The fitting results Eq. (39) indicate that water transport is not a simple one-dimensional transport problem, but is closer to three-dimensional transport. Fig. 2 Theoretical prediction for the diffusion process for the natural aggregate concrete at temperature of 20℃ (293.15K): (a) distance-time-relative humidity relationship; (b) time-relative humidity relationship; (c) distance-relative humidity relationship Fig. 2 shows the diffusion process with time and depth for natural aggregate concrete at 20 °C (293.15 K). Since there are only two data sets for depth (x=2 cm and x=4 cm), there is a large deviation in the depth prediction. Fig. 2(b) shows that the fitted curves at x=2 cm and x=4 cm are in general agreement with the experimental data, and the trend for x=1 cm versus 3 cm is plotted according to Eq. (39). Fig. 2(c) shows that the relative humidity fits better at x=4 cm and shows a large error at x=2 cm for further improvement. 338 C.-H. HE, H.-W. LIU, C. LIU Fig. 3 shows the diffusion process with time and depth for the recycled aggregate concrete at 20 °C (293.15 K). Fig. 3(c) shows that the relative humidity is in good agreement at x=4 cm, with a large error at x=2 cm for further improvement. Fig. 3 Theoretical prediction for the diffusion process for the recycled aggregate concrete at temperature of 20 ℃ (293.15 K): (a) distance-time-relative humidity relationship; (b) time-relative humidity relationship; (c) distance-relative humidity relationship 3.3 Couple of the Fractal Diffusion Process and the Fractal Permeation Process In view of the above analyses, the shallower the depth, the more likely to appear the coupling phenomenon of diffusion and permeation, in the depth of the farther, mainly diffusion is dominant. Considering both diffusion and permeation, the coupled transport equation for humidity is given as:     5 0.4116 0.1914 293 5 0.1914 5 0.8232 5 10 ( , ) 0.75789 0.24211 1 erf(203.2086 ) 10 0.75789 0.24211exp 4.9741( 1.1797 10 ) 10 NRH x t t x t t x t t − − −  = + − +  + + − −  + (40) A Fractal-based Approach to the Mechanical Propertis of Recycled Aggregate Concretes 339     5 0.35685 0.2237 293 5 0.2237 5 0.7137 5 10 ( , ) 0.75917 0.2408 1 erf(153.2464 ) 10 0.75917 0.24083exp 5.0816( 5.2324 10 ) 10 RRH x t t x t t x t t − − −  = + − +  + + − −  + (41) In the case of water transport, diffusion is the dominant when t < 105. When t >105, water transport is dominated by permeation. The calculation curves of the diffusion-permeability coupled transport model for each specimen are shown in Figs. 4 and 5. Fig. 4 Diffusion-permeability coupling for the natural aggregate concrete at temperature of 20 ℃ (293.15 K): (a) distance-time-relative humidity relationship; (b) time-relative humidity relationship; (c) distance-relative humidity relationship Fig. 4 shows the calculated curves of the diffusion-permeability coupling model for natural aggregate concrete at 20 °C (293.15 K), and the overall fitting accuracies are relatively high to meet the engineering requirements. Fig. 4(b) shows that the fitted curves at x=2 cm and x=4 cm are in good agreement with the test data, so the predicted curves at x=1 cm and x=3 cm have some credibility. Fig. 4(c) also shows a significant improvement in accuracy at x=4 cm, indicating that the improved model has been used for engineering predictions, it can be a better illustration of the humidity response at t=3000 s, 6000 s, 9000 s, 12000 s and 15000 s, respectively. 340 C.-H. HE, H.-W. LIU, C. LIU Fig. 5 Diffusion-permeability coupling for the recycled aggregate concrete at temperature of 20 ℃ (293.15 K): (a) distance-time-relative humidity relationship; (b) time-relative humidity relationship; (c) distance-relative humidity relationship Fig. 5 shows the calculation curves of the diffusion-permeability coupling model of recycled aggregate concrete at 20 °C (293.15 K), and the overall fitting accuracy is also very good, and the accuracy basically meets the engineering requirements. Fig. 5(b) shows that the fitting curves at x=2 cm and x=4 cm have some deviation from the experimental data, which is mainly due to the special pore structure of recycled aggregate concrete and the internal interfaces, how to improve its accuracy needs further theoretical and experimental research. Fig. 5(c) also shows that the accuracy at x=2 cm needs to be further improved, which indicates that the closer to the surface of the concrete, the more obvious the penetration effect through the internal interface of recycled aggregate concrete. In summary, the diffusion-permeability coupling model can fully show the relative humidity changing process with time and depth, and the fitted curve is basically consistent with the experimental data, which indicates that the moisture transport model established in this paper has a certain degree of credibility. The diffusion-permeability coupling is a complex phenomenon. This paper employs weighting functions to decouple the two processes, thereby creating a framework for developing a comprehensive model of diffusion-permeability coupling in the future. Furthermore, the weighting functions may be enhanced by incorporating additional factors. A Fractal-based Approach to the Mechanical Propertis of Recycled Aggregate Concretes 341 4. CONCLUSION In this paper, the coupling mechanism of water vapor diffusion and permeation inside concrete is investigated based on the diffusion and permeation equations in the two-scale fractal space. The relative humidity response model of concrete was established, and its theoretical results were compared with experimental data, showing a high accuracy of the established model. The conclusions are as follows: (1) This paper presents a new approach to fractal modifications of Fick's law and Darcy’s law in two-scale fractal space. (2) A coupled model for the moisture diffusion and water permeation is proposed. Though the one-dimensional model is already doing a great job of predicting the internal humidity response characteristics of concrete, there is still plenty of room for improvement. (3) The weight function is easy to determine based on the experimental data. In order to leverage the potential of this approach, it is imperative that the mechanical field addresses the challenges of translating this theoretical analysis into practical applications. The present theory offers a promising avenue for the design of porous concretes with the specified mechanical properties. Additionally, it offers a novel approach to addressing porous problems that arise in non-continuum mechanics, material science, and soil mechanics in the context of fractal space. 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