Thermal Conductivity of Sand-Tire Rubber Mixtures with Varying Tire Chip Fractions and Size Ratios as a Function of Void Ratio and Applied Vertical Stress

Thermal Conductivity of Sand-Tire Rubber Mixtures with Varying Tire Chip Fractions and Size Ratios as a Function of Void Ratio and Applied Vertical Stress | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Thermal Conductivity of Sand-Tire Rubber Mixtures with Varying Tire Chip Fractions and Size Ratios as a Function of Void Ratio and Applied Vertical Stress Jiseok Oh, Hyunwook Choo This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-4010884/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Sand–tire rubber mixtures are promising materials for thermal insulation. However, studies evaluating the impact of applied stress on the thermal conductivity ( k t ) of these mixtures are limited, despite the fact that energy storage tanks are typically located at deep depths where the sand–tire rubber mixtures may experience changes in the connectivity between sand particles under increasing stress. Therefore, in this study, thermal needle probe tests were conducted on sand–tire rubber mixtures with various size ratios ( SR = 0.3, 1.4, and 5.2) and tire chip fractions ( TF = 0.0, 0.1, 0.2, 0.4, and 1.0). To separate the impact of porosity and that of applied stress on k t of the tested sand–rubber mixtures, k t was measured as a function of porosity at very low stress levels. The k t values of the mixtures were then measured according to the applied vertical stress. The results of the tests performed at low stress levels demonstrated that k t of the tested mixtures decreased with increasing TF and decreasing SR because of the decrease in the number of sand-to-sand contacts. All mixtures showed a decrease in k t with increasing porosity; however, the dependence of k t on porosity was affected by TF and SR . With an increase in the applied stress, k t of all the mixtures increased. In particular, the tested mixtures with smaller SR showed even greater k t than pure sand at an applied vertical stress of 460 kPa, highlighting the significance of the applied stress on k t of the tested mixtures. Most notably, this study developed a novel thermal conductivity model that considers both the packing impact and pure stress impact, providing a more comprehensive framework for predicting the thermal conductivity of sand–tire rubber mixtures with varying SRs , TFs , porosities, and applied stresses. thermal conductivity sand–tire rubber mixtures size ratio tire chip fraction applied stress Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Figure 10 Figure 11 Figure 12 Figure 13 Figure 14 Figure 15 1 INTRODUCTION Over the past few decades, the number of discarded tires has increased significantly, owing to the growing demand for vehicles. approximately 1.5 billion tires are discarded globally each year, accounting for 2–3% of the total waste generated worldwide (Mashiri et al. 2015 ; Mohajerani et al. 2020 ). The incineration and disposal of these tires poses environmental problems, causing air, soil, and water pollution. Consequently, there is ongoing discussion on recycling these tires for sustainable environmental management (Sathiskumar and Karthikeyan 2019 ). In geotechnical engineering, efforts have been undertaken to reuse tire chips or crumb rubber as sustainable construction materials (Meles et al. 2014 ; Ryu et al. 2022 ). Tire chips have distinct properties (e.g., durability and low apparent density) compared with natural geomaterials because of the properties of their constituent minerals; thus, tire chips are commonly mixed with sand to improve their engineering properties and usability. For the successful application of sand–tire chip mixtures as construction materials, various engineering properties, including the frictional resistance, compressibility, damping, and stiffness of sand–tire rubber mixtures, have been studied (Lee et al. 2007 ; Ghaaowd et al. 2017 ; Noorzad and Raveshi 2017 ; Xiao et al. 2019 ; Ryu et al. 2022 ; Won et al. 2023 ). These previous studies have reported that sand–tire rubber mixtures exhibit a low unit weight, low stiffness, high frictional resistance, high compressibility, high hydraulic conductivity, and high damping capacity. Consequently, these mixtures are primarily used as lightweight backfill materials, road subbases, rail and highway construction materials, seismic isolation materials, and materials for slope stabilization and foundation reinforcement (Ahmed and Lovell 1993 ; Poh and Broms 1995 ; Bosscher et al. 1997 ; Tweedie et al. 1998 ; Rowe and McIsaac 2005 ; Aydilek et al. 2006 ). However, studies on the thermal conductivity of sand–tire rubber mixtures are limited, although tire rubber is a promising thermally insulating material (Yang et al. 2022 ). Given the increasing diversity of construction environments, understanding the thermal properties of geomaterials is crucial. This knowledge is essential for tasks such as thermal insulation, heat dissipation, and frost protection, making the thermal conductivity ( k t ) of soil a key variable to consider in the geotechnical design process. The k t of soil can be used in foundation design in extreme environments, predicting ground changes owing to global warming, calculating road pavement thickness, designing underground heating pipes, and geothermal power generation design, encompassing the safety management of underground structures (Benson et al. 1996 ; Loveridge and Powrie 2013 ; Bai and Niedzwecki 2014 ). Owing to the low k t of rubber minerals, sand–tire rubber mixtures have potential use as insulating materials, such as underground burial materials for liquefied gas pipelines and energy storage tanks and as materials for controlling freeze depth in road pavements (Christ et al. 2010 ; Zhao et al. 2014 ; Liu et al. 2020 ). The tire rubber fraction, size ratio ( SR ) between rubber and sand, porosity, and water content were selected as the main testing variables to evaluate the thermal conductivity of sand–tire rubber mixtures (Xiao et al. 2019 ; Liu et al. 2020 ; Cui et al. 2022 ; Yang et al. 2022 ). These previous studies reported that the thermal conductivity of a mixture is inversely proportional to the tire chip fraction ( TF ) and porosity, and directly proportional to the size ratio and water content. However, studies evaluating k t according to the applied vertical stress are limited (Cui et al. 2022 ). Tank thermal energy storage (TTES) or pit thermal energy storage (PTES) is typically located underground at depths of 5–15 m (Pourahmadiyan et al. 2023 ), resulting in vertical stresses acting on the soil beneath the TTES or PTES ranging from 150 to 300 kPa. Thus, the effect of applied vertical stress on sand–tire rubber mixtures should be investigated. Additionally, owing to the huge deformability of tire rubber, the sand–tire rubber mixtures experience a change in connectivity between sand particles with increasing applied stress (Lee et al. 2007 ; Kim and Santamarina 2008 ). Because k t via sand-to-sand contacts is higher than that via sand-to-rubber or rubber-to-rubber contacts, the hypothesis in this study is that sand–tire rubber mixtures can show significant dependency on the applied stress, leading to a change in k t of the sand–tire rubber mixtures that differs from that of natural geomaterials. This study aimed to evaluate the thermal conductivity of sand–tire chip mixtures with varying tire chip fractions and size ratios according to the porosity and applied vertical stress. Sand–tire chip mixtures with three different size ratios (0.3, 1.4, and 5.2) were prepared with varying tire chip fractions (0.0, 0.1, 0.2, 0.4, and 1.0) and relative densities (10–90%). Because the change in applied stress includes changes in both volume and contact area, k t measurements under a negligible stress level were performed first as a function of porosity. The k t of the mixtures were then measured according to the applied stress. In this manner, the k t estimation formulas for the packing impact and pure stress impact were separately suggested in this study. 2 THERMAL CONDUCTIVITY MODELS FOR DRY SOILS Soil is a three-phase material, including soil particles, water, and air; thus, the thermal conductivity ( k t ) of soil is determined by the volume fraction of each component, contact quality between soil particles, and the interaction of these three components with different thermal conductivities. Consequently, k t can generally be expressed as a function of the mineralogy, particle size, density (or porosity), water content, applied stress, and pore fluid characteristics (Yagi and Kunii 1957 ; Johansen 1977 ; Farouki and Farouki 1981 ; Brandon and Mitchell 1989 ; Becker et al. 1992 ; Midttomme and Roaldset 1998 ; Gangadhara Rao and Singh 1999 té and Konrad 2005 ; Yun and Santamarina 2008 ). Various models, including theoretical (e.g., effective medium theory model) (Mickley 1951 ; Woodside and Messmer 1961 ; Tarnawski et al. 2015 ), semi-empirical (e.g., normalized conductivity model) (De Vries 1963 ; Chen et al. 2012 ), and empirical models (Lu et al. 2007 ; Lu et al. 2014 ; Li et al. 2021 ) have been successfully developed to capture the measured k t of their own studies. However, in the case of the k t of dry granular materials, empirical models have shown better predictive performance compared to other models because of the significant difference in k t values between the constituent minerals of the soil particles (2–8 W/mK) and air (0.026 W/mK) (Farouki and Farouki 1981 ; Dong et al. 2015 ; Won et al. 2023 ). This implies that, in dry soil, the low thermal conductivity of air hinders the overall thermal conductivity, and the major path of heat transfer occurs through particle contact (Carslaw et al. 1962 ; Yun and Santamarina 2008 ; Choo et al. 2021 ). The thermal conductivity ( k t ) of soils in a dry state mainly depends on the thermal conductivity of the constituent minerals and the contact conditions between the particles (Li et al. 2021 ). The contact conditions can be assessed based on the coordination number (the average number of contacts per particle) captured by the porosity and the contact area captured by the applied stress (Mitchell and Soga 2005 ). Thus, k t of dry granular materials can be expressed as a function of porosity and applied stress (Yun and Santamarina 2008 ; Lee et al. 2016 ). However, previous studies have also demonstrated that changes in k t due to variations in applied stress are generally minor compared with those resulting from changes in porosity (Yun and Santamarina 2008 ; Choo et al. 2021 ; Bhatt et al. 2022 ). Therefore, most existing research has represented k t of dry soil as a sole function of porosity (Table 1 ). Table 1 shows three frequently employed k t estimation formulas for dry soils as a function of porosity, and the three models have different mathematical expressions: a power function (Johansen 1977 ), an exponential function (Côté and Konrad 2005 ), and a linear function (Lu et al. 2007 ). Table 1 Empirical equations for predicting the thermal conductivity of dry angular sand (or crushed rock). Reference Empirical equation Note Johansen ( 1977 ) \({k}_{t\left(dry\right)}=a\bullet {n}^{-b}\pm 25\%\) Angular sand ( a = 0.039, b = 2.2) Cote and Konrad (2005) \({k}_{t\left(dry\right)}=\chi \bullet {10}^{-\eta \bullet n}\) Angular sand ( χ = 1.7, η = 1.8) Lu et al. ( 2007 ) \({k}_{t\left(dry\right)}=-c\bullet n+d\) Soils with 0.2 < n < 0.6 ( c = 0.56, d = 0.51) Note: \({k}_{t\left(dry\right) }\) = dry thermal conductivity; n = porosity; and a , b , c , d , χ and η = fitting parameter. 3 MATERIALS AND METHODOLOGY 3.1 Materials Two sands (median particle size, D 50 = 0.76 mm for K-5 sand and D 50 = 0.21 mm for K-7 sand) and two tire chips ( D 50 = 1.09 mm for TC1 and D 50 = 0.21 mm for TC2) were employed in this study to prepare sand–tire rubber mixtures. The index properties of the unmixed materials are summarized in Table 2 , and their particle size distribution curves of unmixed materials are shown in Fig. 1 . Because the relative size between rubber and sand particles is a crucial factor in determining the overall mechanical properties of the mixtures (Lee et al. 2007 ; Kim and Santamarina 2008 ), mixtures with three different size ratios ( SR ), as defined in Eq. ( 1 ), were prepared by mixing K-5 and TC2 to obtain SR = 0.3, K-5 and TC1 to obtain SR = 1.4, and K-7 and TC2 to obtain SR = 5.2. Table 2 Physical properties of unmixed materials. Type of properties G s D 50 (mm) e max e min Sand K-5 2.66 0.76 1.05 0.69 K-7 2.65 0.21 1.07 0.68 Tire chip TC1 1.18 1.09 1.69 1.24 TC2 1.15 0.21 2.27 1.43 Testing method ASTM D854 ASTM D422 ASTM D4254 JGS 0161 Note: G s = specific gravity; D 50 = median grain size; e max and e min = maximum and minimum void ratios, respectively. $$Size ratio\left(SR\right)= \frac{{D}_{50\left(tire chip\right)}}{{D}_{50\left(sand\right)}}$$ 1 3.2 Thermal Conductivity Measurements For measuring the thermal conductivity of mixtures, the thermal needle probe method (ASTM D5334-14) was employed because of its simplicity, high accuracy, and short measurement duration (Naidu and Singh 2004 ). The probe sensor used for measuring the thermal conductivity was made of stainless steel, 30 mm in length and 1.3 mm in diameter, and was equipped with a heating wire and a thermistor that measured electrical resistance changes with temperature (East30Sensors Ltd.). As shown in Fig. 2 , the thermal needle probe was installed at the bottom of a thermally insulated oedometer cell with a diameter of 100 mm and height of 67 mm, allowing for thermal conductivity measurements according to the applied vertical stress. The thermal conductivity ( k t ) was determined based on the infinite line heat source theory (Carslaw et al. 1962 ): $${k}_{t}=\frac{C\bullet Q}{4\pi }\bullet \frac{\varDelta \text{l}\text{n}\left(t\right)}{\varDelta T}$$ 2 where C is the calibration coefficient, Q is the input power applied to the heating wire in W/m, and ∆ T is the temperature difference of the medium over the time interval ∆ t . ∆ t was set to 30 s in this study, and care was taken to ensure that the total temperature change did not exceed 2˚C to prevent errors due to convection. 3.3 Test Program As shown in Table 3 , the experiment was conducted along two main directions. Test case #1 focused on determining the changes in the thermal conductivity of the tested mixtures with varying tire chip fraction ( TF ), defined in Eq. ( 3 ), owing to the packing impact under a negligible stress level. Thus, samples with relative densities ranging from 10% (very loose state) to 90% (very dense state) were prepared to measure thermal conductivity. In the case of test case #2, focusing on determining changes in thermal conductivity due to the stress impact, the initial relative density of the sample was set to 30% (loose state) and 70% (dense state), and then the vertical stress was gradually increased from 0.5 to 460 kPa at a load increase ratio of 1. Thermal conductivity was measured at the end of each loading process. Samples with varying initial target relative densities were prepared using the dry funnel deposition method, followed by symmetric impacts with a rubber mallet. The thermal conductivity reported in this study is the average of three replicate tests. Table 3 Test matrix. Test case Size ratio Tire chip fraction Applied vertical stress (kPa) Initial relative density (%) #1 – Impact of packing state on thermal conductivity 0.3, 1.4, 5.2 0.0, 0.1, 0.2, 0.4, 1.0 0.5 (top cap weight) 10–90 #2 – Impact of applied stress on thermal conductivity 0.5–460 (LIR = 1) 30, 70 Note: LIR = loading increment ratio $$Tire chip fraction\left(TF\right)= \frac{{V}_{tire chip}}{Total Volume}$$ 3 where V tire chip is the volume of tire chip. 4. RESULTS AND DISCUSSIONS Section 4.1 describes the thermal conductivity of the tested sand–tire rubber mixtures with varying tire chip fractions ( TF ) and size ratio ( SR ) according to the porosity under very low stress levels (~ 0.5 kPa) resulting from the top cap weight. The effect of applied vertical stress, ranging from 0.5 to 460 kPa, on thermal conductivity of tested mixtures is presented in Section 4.2 . 4.1 Thermal Conductivity of Tested Mixtures under Negligible Applied Stress 4.1.1 Effects of Porosity and Tire Chip Fraction on Thermal Conductivity Figure 3 shows the change in thermal conductivity ( k t ) of the tested mixtures with varying size ratios and tire chip fractions ( TF ) according to the porosity measured under a constant vertical stress (0.5 kPa) (Test case #1 in Table 3 ). It can be clearly observed in Fig. 3 that the k t of tested materials increased with decreasing porosity (or increasing packing density), regardless of SR s and TF s. A decrease in porosity resulted in an increase in the interparticle coordination number, leading to an increased number of paths for particle contact conduction. Additionally, a decrease in porosity implies a reduction in the proportion of air that hinders heat transfer within the soil, thereby enhancing heat conduction through particle contacts. Thus, k t increases with decreasing porosity (Johansen 1977 ; Chen 2008 ; Liu et al. 2020 ; Cui et al. 2022 ). Figure 3 also demonstrates that the measured thermal conductivity ( k t ) at a given porosity decreased with increasing tire chip fraction ( TF ) (Xiao et al. 2019 ; Liu et al. 2020 ; Cui et al. 2022 ; Yang et al. 2022 ). The mineral conductivity of rubber is around 0.25 W/mK; while the conductivity of minerals that make soil particles generally ranges from 1 to 5 W/mK (Côté and Konrad 2005 ). Therefore, in sand–tire rubber mixtures, the connectivity between sand particles, which have higher mineral conductivity, determines the overall k t (Xiao et al. 2019 ; Liu et al. 2020 ). Thus, the decrease in k t with increasing TF can be attributed to the transition in the dominant contact mode within the mixture from the highly conductive sand–to-sand contacts at low TF to the less conductive rubber-to-rubber and/or rubber-to-sand contacts at high TF (Lee et al. 2010 ; Liu et al. 2020 ). 4.1.2 Effect of Size Ratio on Thermal Conductivity Due to the difference in the thermal conductivity ( k t ) between pure sands (i.e., K-5 sand and K-7 sand), and between pure tire chips (i.e., TC1 and TC2), the normalized conductivity K N (Eq. ( 4 ), Yang et al. ( 2022 )) was plotted as a function of tire chip fraction ( TF ) at a relative density of 70% (Fig. 4 ). $${K}_{N}= \frac{{k}_{t}-{k}_{tire}}{{k}_{sand}-{k}_{tire}}$$ 4 where k tire is the thermal conductivity of the pure tire chip ( TF = 100%), and k sand is the thermal conductivity of pure sand ( TF = 0%). Figure 4 indicates that K N at a given TF increased with increasing size ratio, indicating that the use of sand–tire rubber mixtures with smaller SR results in superior thermal insulating properties. As mentioned above, owing to the significant difference in mineral conductivity between sand and rubber particles, the connectivity between sand particles determines the overall k t , implying that mixtures with larger SR have better connectivity between sand particles. Previous experimental and numerical studies (Perez et al. 2016 ; Xiao et al. 2019 ; Ryu et al. 2022 ; Won et al. 2023 ) also reported that the connectivity between sand particles increases with increasing SR . The SR -dependent connectivity change between sand particles is schematically shown in Fig. 5 . A smaller SR implies that tire chips are smaller than sand particles, positioning tire chips with lower thermal conductivity among sand particles with higher thermal conductivity, thereby easily disrupting sand-to-sand contacts under a low applied stress level. This results in an increased number of thermal conduction paths via sand-to-rubber or rubber-to-rubber contacts, leading to a decreased thermal conductivity. Conversely, a larger size ratio enhances the connectivity between sand particles, indicating that sand-to-sand contacts can be the main path for thermal conduction (Fig. 5 ) and consequently result in a relatively higher overall thermal conductivity with increasing SR (Lee et al. 2015 ; Lee et al. 2017 ; Xiao et al. 2019 ; Yang et al. 2022 ). 4.1.3 Thermal Conductivity Model The data of Fig. 3 , the measured thermal conductivity ( k t ) of sand-tire rubber mixtures under negligible stress level, was fitted with previous thermal conductivity models for dry soils (Table 1 ). The coefficient of determination (R 2 ) for the models of Johansen ( 1977 ), Cote and Konrad (2005), and Lu et al. ( 2007 ) were determined to be 0.995, 0.996, and 0.991, respectively, indicating that the relationship between k t and porosity of all tested mixtures could be accurately captured by either the power, exponential, or linear model. Because Cote and Konrad model showed slightly better performance than the other two models, consistent with the observations of Dong et al. ( 2015 ), the fitting parameters χ and η in Cote and Konrad model in Table 1 of tested mixtures with varying size ratios were plotted as a function of tire chip fraction ( TF ) in Fig. 6 . Note χ –factor indicates the k t of materials when porosity equals zero; thus, χ –factor reflects the inherent conductivity of the minerals and should decrease with increasing TF (Fig. 6 (a)). η –factor indicates the sensitivity of k t on porosity. As shown in Fig. 3 , the absolute value of the line slope decreased with increasing TF , indicating that the k t of sand–tire rubber mixtures with higher TF showed a diminished dependency on porosity (Fig. 6 (b)). The reason for this change in the porosity dependence with increasing TF is unclear. However, the results of Liu et al. ( 2020 ) support this observation. A likely explanation is that rubber is an insulating material, indicating that the replacement of the air phase with solid rubber particles owing to the decrease in porosity results in a limited increase in thermal conductivity, although the number of thermal conduction paths increases. Finally, it is notable that the determined χ and η of the tested pure sands and pure tire chips were comparable with those of Cote and Konrad (2005) for angular sand and peat, respectively. Because the fitting parameters χ and η in Cote and Konrad model of tested mixtures changed nonlinearly from those of pure sand to pure rubber (Fig. 6 ), the normalized χ and η ( χ N and η N ) were defined as: $${\chi }_{N}= \frac{\chi -{\chi }_{tire}}{{\chi }_{sand}-{\chi }_{tire}}$$ 4 $${\eta }_{N}= \frac{\eta -{\eta }_{tire}}{{\eta }_{sand}-{\eta }_{tire}}$$ 5 where χ tire and χ sand are the χ factors of pure tire chip and pure sand, respectively; and η tire and η sand are the respective η exponents of pure tire chip and pure sand. Figure 7 shows the variations in χ N and η N of the tested mixtures with varying size ratios according to tire chip fraction. For a tire chip fraction based on volume ( TF ≤ 40%), good linear relations occur between χ N and TF , and between η N and TF . $${\chi }_{N}= 1-{b}_{\chi }\bullet TF$$ 6 $${\eta }_{N}= 1-{b}_{\eta }\bullet TF$$ 7 where b χ and b η are the fitting parameters. It can be clearly observed in Fig. 7 that b χ and b η are SR -dependent; thus, the determined b χ and b η were plotted as a function of SR in Fig. 8 . Figure 8 also includes the experimental results of Cui et al. ( 2022 ). Figure 8 demonstrates that b χ and b η decreased nonlinearly with increasing SR , which is attributed to the increased connectivity between sand particles with increasing SR , and the following power models can capture the relationships between b χ and SR , and between b η and SR : $${b}_{\chi }= 1.58\bullet {SR}^{-0.22}$$ 8 $${b}_{\eta }= 0.65\bullet {SR}^{-0.34}$$ 9 Substituting Equations ( 4 )–( 7 ) into the Cote and Konrad model in Table 1 yields $${k}_{t}=\left[\left(1-{b}_{\chi }\bullet TF\right)\left({\chi }_{sand}-{\chi }_{tire}\right)+{\chi }_{tire}\right]\bullet {10}^{-\left[\left(1-{b}_{\eta }\right)\left({\eta }_{sand}-{\eta }_{tire}\right)+{\eta }_{tire}\right]\bullet n}$$ 10 where b χ and b η are given by Equations ( 8 ) and ( 9 ), respectively. The novelty of the newly developed Eq. ( 10 ) lies in the fact that the k t of sand–tire rubber mixtures with varying SR s, TF s, and porosities can be estimated based on k t of pure sand and pure tire rubber. Figure 9 shows a comparison between the measured and estimated k t values based on Eq. ( 10 ), and a good agreement can be observed between the two. The prediction performance of Eq. ( 10 ) is governed by the reliability of the input parameters b χ and b η . Although several studies have evaluated the k t of sand–tire rubber mixtures according to size ratios and tire chip fractions (Lee et al. 2015 ; Xiao et al. 2019 ); measuring k t under varying porosities has been limited. Thus, Equations ( 8 ) and ( 9 ) were derived based on limited experimental results. Further studies are recommended to verify the reliability of Equations ( 8 )–( 10 ). 4.2 Thermal Conductivity of Tested Mixtures according to Applied Stress 4.2.1 Effect of Applied Stress, Tire Chip Fraction, and Size Ratio on Thermal Conductivity Figure 10 shows the change in thermal conductivity of the tested mixtures with varying size ratios and tire chip fractions according to the applied vertical stress (Test case #2 with initial relative density = 70% in Table 3 ). Figure 10 demonstrates that irrespective of SR s and TF s, the measured k t in all the mixtures increased with the applied vertical stress because of the formation of a more interconnected network for heat to flow through (Yun and Santamarina 2008 ). Similar to the results shown in Fig. 3 , at low stress levels (< 60 kPa), a clear decrease in k t with increasing TF at a given stress level was observed. However, as the vertical stress further increased, the difference in k t based on TF at the same stress level gradually diminished, and k t of all the mixtures approached that of pure sand, implying that the tested sand–tire rubber mixtures can develop sand-to-sand thermal conduction paths at higher stress levels (> 200 kPa) (Fig. 5 ). Similar to the results shown in Fig. 4 , the measured k t for the mixtures with SR = 0.3 was smaller than that for mixtures with SR = 5.2 at low stress levels; however, Fig. 10 clearly demonstrates the opposite effect of SR on k t at high stress levels. In particular, the tested mixtures with SR = 0.3 at TF = 20% and 40% showed even greater k t than pure sand at a vertical stress of 460 kPa (Fig. 10 (a)). This observation suggests that once the sand–tire rubber mixtures with small SR experience high stress levels, the insulating characteristics can disappear, suggesting that the applied stress should be an important parameter when utilizing sand–tire rubber mixtures as insulating materials, and the use of larger tire chips can guarantee thermal insulation properties regardless of stress levels. The increase in applied stress involves both changes in contact area and coordination number (Johnson and Johnson 1987 ; Choo and Burns 2015 ; Choo and Lee 2021 ; Ryu et al. 2022 ). Thus, the change in the void ratio, which reflects the change in the coordination number, is plotted as a function of the applied vertical stress in Fig. 11 . Figure 11 (a) shows that the compressibility (i.e., the decrease in void ratio with increasing stress) increased with increasing TF because of the change in the interaction between sand and rubber components, and the elastic nature of rubber particles (Ryu et al. 2022 ). Most notably, the void ratio of mixtures with SR = 0.3 at TF = 20% and 40% becomes smaller than the minimum void ratio of pure sand (K-5 sand) at high stress levels (> 200 kPa), reflecting the significant particle rearrangement and the consequent contact mode change due to the squeezing/distortion of soft rubber particles. Thus, these mixtures can exhibit greater k t than pure sand at high stress levels owing to the development of a sand-to-sand conduction path and a significant decrease in the air volume fraction (Fig. 10 (a)). The compressibility of the mixtures increased with decreasing SR at a given TF because SR determines the connectivity between sand particles (Fig. 11 (b)). As the SR decreases, the smaller tire rubber particles can easily be located in the pore space between the larger sand particles, resulting in a significant disruption of the direct contact between sand particles at low stress levels. This causes soft rubber particles to actively participate in the load-carrying skeleton, resulting in a large volume contraction with increasing applied stress (Ryu et al. 2022 ). In contrast, in the case of the tested mixtures with SR = 5.2, the applied stress could be transmitted through sand-to-sand contacts, resulting in a limited change in the contact mode and a consequent limited increase in k t (Fig. 10 (c)). 4.2.2 Pure Stress Impact on Thermal Conductivity The thermal conductivity ( k t ) increase in Fig. 10 includes the k t increase due to the decrease in porosity (i.e., k t based on packing impact) and k t increase due to the increase in contact area (i.e., k t based on pure stress impact). Thus, to examine the effect of pure stress on k t , the measured k t of mixtures with SR = 1.4 at TF = 0% and 40% (Fig. 10 (b)) is plotted as a function of porosity in Fig. 12 (a). The open symbols in Fig. 12 indicate the measured k t under negligible stress (Fig. 3 (b), Test case #1 in Table 3 ), and the filled symbols indicate k t with increasing applied stress (Fig. 10 (b), Test case #2 in Table 3 ). The dotted lines in Fig. 12 (a) represent the estimated k t values based on Eq. ( 10 ). Thus, the difference in k t corresponding to the filled symbols and the trendline at a given porosity (i.e., △ in Fig. 12 (a)) can be considered as the increase in k t due to the pure stress impact. This △ is plotted as a function of the applied vertical stress in Fig. 12 (b) for mixtures with SR = 1.4 at an initial relative density of 70%. Figure 12 (b) shows that k t based on pure stress impact can be approximated as a power function of applied vertical stress ( σ’ v ): $${k}_{t}=\alpha \bullet {\left(\frac{{\sigma {\prime }}_{v}}{1kPa}\right)}^{\beta }$$ 11 where α and β are the fitting parameters. Note the α –factor indicates the k t based on pure stress impact when the applied stress is very small ( σ’ v = 1 kPa), and β- exponent indicates the stress-sensitivity. Figure 13 shows the variations of α and β of tested materials according to tire chip fraction ( TF ). It can be observed in Figs. 13 (a) and 13(b) that the α –factor decreased but β- exponent increased with increasing TF until TF ~ 20%, and then α –factor and β- exponent approached those of pure tire rubber with a further increase in TF . Note the sand-rubber mixtures at low TF show sand-like behavior (Kim and Santamarina 2008 ; Lee et al. 2010 ; Perez et al. 2016 ; Ryu et al. 2022 ); thus, α decreased with increasing TF at low TF (< 10–20%) because the mineral conductivity of rubber is much smaller than that of sand particles. In addition, at low TF , the direct contacts between sand particles, which were initially disrupted due to the presence of tire rubber particles, can develop with increasing stress due to the squeezing/distortion of rubber particles (Ryu et al. 2022 ; Won et al. 2023 ) (Fig. 5 ); thus, β- exponent increased with increasing TF at low TF . At a high TF ( TF > 20–40%), the sand–rubber mixtures can exhibit rubber-like behavior, implying that the engineering properties will be determined by the interaction between rubber particles. Thus, α and β approached those of pure tire rubber with increasing TF at high TF . Because the connectivity between sand particles at low stress level increases with increasing size ratio, the greater α but smaller β at a given TF can be observed for the mixtures with larger SR (Fig. 13 ). In addition, in the case of mixtures with larger SR (i.e., mixtures with SR = 5.2), the smaller sand particles can penetrate rubber surface with increasing applied stress (Fig. 5 ) (Xiao et al. 2019 ), resulting in a limited increase in connectivity between sand particles, and a consequent decrease in β with increasing SR . Figure 13 also demonstrates that loose specimens showed smaller α , but greater β because loose specimen experiences greater volume change during loading and the consequent greater contact mode change. Finally, it is noteworthy that the determined α for all tested materials were very small, ranging from 0.001 to 0.095, indicating that the k t contributed by pure stress impact is negligible at low stress levels. It is also notable that the determined β values in Fig. 13 are comparable with those of theoretical or numerical thermal conduction model, ranging from 0.3 to 0.67 (Weidenfeld et al. 2004 ; Garrett and Ban 2011 ; Choo et al. 2012 ). 4.2.3 Thermal Conductivity Model The measured thermal conductivity ( k t ) shown in Fig. 13 was compared with the estimated k t based on Eq. ( 10 ) to assess the error incurred when predicting k t without considering the applied stress, and Fig. 14 shows the mean percent error (MPE) and maximum percent error. Within the permissible error range of 20% (Johansen 1977 té and Konrad 2005 ; Yang et al. 2022 ), the thermal conductivity of pure materials and mixtures with a low TF can be reasonably predicted using Eq. ( 10 ). In other words, k t of dry granular material is primarily determined by porosity, with the effect of applied stress being secondary (Yun and Santamarina 2008 ; Choo et al. 2021 ; Bhatt et al. 2022 ). However, Fig. 14 shows that the percentage difference between the measured and estimated k t increased with increasing TF and decreasing SR , and the mixtures with SR = 0.3 and 1.4 at TF = 20% and 40% exhibited the maximum difference, ranging from 20–45%. This suggests that sand–tire rubber mixtures undergo significant changes in contact mode with increasing stress, underscoring the need for a new model. The difference between the measured and estimated k t values in Fig. 14 originated from the intentional neglect of k t change according to the applied stress. Thus, the total thermal conductivity ( k t (total)) can be expressed as the sum of the thermal conductivity based on the packing impact ( k t (packing), Cote and Konrad model in Table 1 ) and that based on the pure stress impact ( k t (stress), Eq. ( 11 )), as follows: k t (total) = k t (packing) + k t (stress) = \(\chi \bullet {10}^{-\eta \bullet n}+\alpha \bullet {\left(\frac{{\sigma {\prime }}_{v}}{1kPa}\right)}^{\beta }\) (12) where χ and η for sand–tire rubber mixtures are given in Eq. ( 10 ). α and β in Eq. ( 11 ) or (12) exhibited a nonlinear trend of decreasing (or increasing) followed by increasing (or decreasing) with respect to tire chip fraction, and their relationships were influenced by the packing density and SR (Fig. 13 ). Thus, quantitatively representing α and β is very challenging. Instead, this study examined the relationship between α and β of the tested mixtures (Fig. 15 ) to provide insights for predicting the values of α and β in sand–rubber mixtures. Figure 15 also includes the data of previous studies (Choo et al. 2012 ; Lee et al. 2016 ; Cui et al. 2022 ). It can be observed in Fig. 15 that an inverse relationship can be found between α and β . The tested pure sand showed an α of approximately 0.02 and β of approximately 0.4. Increasing TF until TF ~ 20% results in a decrease in α but increase in β (Fig. 15 ). A further increase in TF results in an increase in α but decrease in β towards the values of pure rubber. Thus, based on these observations, it is deemed feasible to roughly predict the α and β of mixtures according to TF . 5 CONCLUSIONS This study comprehensively investigated the thermal conductivity ( K t ) of sand-tire rubber mixtures with varying size ratios ( SR ) and tire chip fractions ( TF ) according to porosity and applied vertical stress. To separate the impact of porosity and that of applied stress on K t of the tested sand–tire rubber mixtures, K t was measured as a function of porosity at very low stress levels. The K t values of the mixtures were then measured according to the applied vertical stress. The key findings of this study are summarized as follows. (1) At very low stress levels, K t primarily depended on the porosity, with mixtures exhibiting lower conductivity as the tire chip fraction increased and the size ratio decreased. (2) Because the dependence of K t on porosity was affected by TF and SR , a new formula for estimating K t based on the packing impact was proposed to incorporate the effects of TF and SR . (3) As the applied stress increased, the K t of the tested mixtures increased owing to the formation of a sand-to-sand thermal conduction path resulting from the squeezing/distortion of soft rubber particles. (4) The tested mixtures with smaller SR had K t higher than that of pure sand at a vertical stress of 460 kPa, indicating that the applied stress is an important parameter when using sand–tire rubber mixtures as insulating materials, and the use of larger tire chips can guarantee thermal insulation properties regardless of stress levels. (5) The maximum difference between the measured K t of the tested mixtures with increasing applied stress and the estimated K t based on the packing impact ranged from 20–45%; thus, a new formula for estimating K t based on the pure stress impact was proposed. Thus, this study introduced a novel thermal conductivity model that considers both the packing and pure stress impacts. Declarations Author Contribution The authors confirm contribution to the paper as follows: study conception and design: H. Choo data collection: J. Oh analysis and interpretation of results: H. Choo; J. Ohdraft manuscript preparation: J. Ohrevised manuscript preparation: H. ChooAll authors reviewed the results and approved the final version of the manuscript. ACKNOWLEDGMENT This research was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (RS-2023-00208844 References Ahmed, I., and Lovell, C. (1993). Rubber soils as lightweight geomaterials. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-4010884","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":277375346,"identity":"d86a336d-b202-408b-acbd-1f491d2c34b5","order_by":0,"name":"Jiseok Oh","email":"","orcid":"","institution":"Hanyang University","correspondingAuthor":false,"prefix":"","firstName":"Jiseok","middleName":"","lastName":"Oh","suffix":""},{"id":277375347,"identity":"f46c801e-0225-4cd2-99aa-2e26847a181d","order_by":1,"name":"Hyunwook Choo","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA3klEQVRIiWNgGAWjYDACCQYGZgYGmwQkITaitKSRruUwCVr4ZzcffFzYdj6Pf3aP4eeCXwzy/A1saR/wWnLnWLLxzLbbxRJ3zhhLz+xjMJxxgO3wDHxaDCRyzKR5t91ObLiRYyDN28PAuIGBvRmvw6BaziXOv5Fj/BuoxZ5YLQcSN9wAMnh+MCRuYGA7jFeLxI20ZGPef8mJG2+klVnzNkgkzzjMloxXC/+M5IOPec7YJc67kbz5Ns8fG9v+9jZjvFqQAIcBA2MbJJqIBewPGBj+EK98FIyCUTAKRg4AABewRmUdpPdmAAAAAElFTkSuQmCC","orcid":"","institution":"Hanyang University","correspondingAuthor":true,"prefix":"","firstName":"Hyunwook","middleName":"","lastName":"Choo","suffix":""}],"badges":[],"createdAt":"2024-03-04 06:59:49","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-4010884/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-4010884/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":52545428,"identity":"1bed32c0-7322-4c68-ab7a-72b6f7729748","added_by":"auto","created_at":"2024-03-12 18:21:18","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":24631,"visible":true,"origin":"","legend":"\u003cp\u003eParticle distribution curves of the tested unmixed materials.\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-4010884/v1/491e97fd0804fad71aceec2c.png"},{"id":52545419,"identity":"6f69f4ca-fd39-4278-8ad6-bb3d12f968fd","added_by":"auto","created_at":"2024-03-12 18:21:16","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":19599,"visible":true,"origin":"","legend":"\u003cp\u003eTest setup for thermal conductivity measurements.\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-4010884/v1/99a817f45bdd7270df562a06.png"},{"id":52545398,"identity":"b85b7f42-753d-466d-9b0f-27e04853af5b","added_by":"auto","created_at":"2024-03-12 18:21:07","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":241552,"visible":true,"origin":"","legend":"\u003cp\u003eMeasured thermal conductivity according to porosity: (a) mixtures with size ratio (\u003cem\u003eSR\u003c/em\u003e = median particle size of tire chip / median particle size of sand) = 0.3; (b) mixtures with \u003cem\u003eSR\u003c/em\u003e = 1.4; and (c) mixtures with \u003cem\u003eSR\u003c/em\u003e = 5.2.\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-4010884/v1/731b0a9f1c9d62b15328db05.png"},{"id":52545423,"identity":"016c0352-a570-4eca-8a84-f62860dcbf57","added_by":"auto","created_at":"2024-03-12 18:21:17","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":27633,"visible":true,"origin":"","legend":"\u003cp\u003eVariation of normalized thermal conductivity as a function of tire chip fraction. Note, normalized \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e or \u003cem\u003eK\u003c/em\u003e\u003csub\u003e\u003cem\u003eN\u003c/em\u003e\u003c/sub\u003e = Equation (3) and \u003cem\u003eSR\u003c/em\u003e = size ratio.\u003c/p\u003e","description":"","filename":"4.png","url":"https://assets-eu.researchsquare.com/files/rs-4010884/v1/b1882ddefc0e4965b1af8873.png"},{"id":52545418,"identity":"6a1c48c2-8cf4-4e54-8f22-251665b058cd","added_by":"auto","created_at":"2024-03-12 18:21:16","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":1491078,"visible":true,"origin":"","legend":"\u003cp\u003eSchematic drawing of the changes in connectivity between sand particles according to size ratios and stress levels.\u003c/p\u003e","description":"","filename":"5.png","url":"https://assets-eu.researchsquare.com/files/rs-4010884/v1/bd93e547533cd30d49517624.png"},{"id":52545426,"identity":"191db672-9f92-4d77-bd4e-d77bd8d47ffc","added_by":"auto","created_at":"2024-03-12 18:21:17","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":293838,"visible":true,"origin":"","legend":"\u003cp\u003eVariations of fitting parameters in Cote and Konrad model in Table 1 of tested mixtures with varying size ratios according to of tire chip fraction: (a) variation of \u003cem\u003eχ\u003c/em\u003e–factor and (b) variation of \u003cem\u003eη\u003c/em\u003e–exponent.\u003c/p\u003e","description":"","filename":"6.png","url":"https://assets-eu.researchsquare.com/files/rs-4010884/v1/3f1d1d27eb7f5626fd4e7321.png"},{"id":52545407,"identity":"61f03a2f-642b-4041-beaf-051acc0271c8","added_by":"auto","created_at":"2024-03-12 18:21:11","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":249478,"visible":true,"origin":"","legend":"\u003cp\u003eVariations of normalized \u003cem\u003eχ\u003c/em\u003e(Equation 4) and \u003cem\u003eη\u003c/em\u003e (Equation 5) of tested mixtures with varying size ratios (\u003cem\u003eSR\u003c/em\u003e) according to tire chip fraction.\u003c/p\u003e","description":"","filename":"7.png","url":"https://assets-eu.researchsquare.com/files/rs-4010884/v1/0e15074bfd37a2c6f1e6787a.png"},{"id":52545422,"identity":"912e8dda-d1a8-49d4-b36e-2bf3ca8d754d","added_by":"auto","created_at":"2024-03-12 18:21:16","extension":"png","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":16234,"visible":true,"origin":"","legend":"\u003cp\u003eVariations of fitting parameters (\u003cem\u003eb\u003c/em\u003e\u003csub\u003e\u003cem\u003eχ\u003c/em\u003e\u003c/sub\u003e and \u003cem\u003eb\u003c/em\u003e\u003csub\u003e\u003cem\u003eη\u003c/em\u003e\u003c/sub\u003e) in Equations (6) and (7) as a function of size ratio.\u003c/p\u003e","description":"","filename":"8.png","url":"https://assets-eu.researchsquare.com/files/rs-4010884/v1/1fa80626548ed194aef8a395.png"},{"id":52545424,"identity":"d8f6cfef-39aa-45d7-b545-6d45dc59e75e","added_by":"auto","created_at":"2024-03-12 18:21:17","extension":"png","order_by":9,"title":"Figure 9","display":"","copyAsset":false,"role":"figure","size":17712,"visible":true,"origin":"","legend":"\u003cp\u003eComparison between the measured and estimated thermal conductivities (\u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e). Note estimated \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e = Equation (10).\u003c/p\u003e","description":"","filename":"9.png","url":"https://assets-eu.researchsquare.com/files/rs-4010884/v1/422c7f70711c00dc279dc99b.png"},{"id":52545405,"identity":"a83c1394-40c1-4918-ae0e-32dcb34e8ac2","added_by":"auto","created_at":"2024-03-12 18:21:11","extension":"png","order_by":10,"title":"Figure 10","display":"","copyAsset":false,"role":"figure","size":339289,"visible":true,"origin":"","legend":"\u003cp\u003eVariations in thermal conductivity (\u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e) of tested mixtures with varying size ratios (\u003cem\u003eSR\u003c/em\u003e) and tire chip fractions (\u003cem\u003eTF\u003c/em\u003e) according to applied vertical stress: (a) mixtures with \u003cem\u003eSR\u003c/em\u003e = 0.3; (b) mixtures with \u003cem\u003eSR\u003c/em\u003e = 1.4; and (c) mixtures with \u003cem\u003eSR\u003c/em\u003e = 5.2.\u003c/p\u003e","description":"","filename":"10.png","url":"https://assets-eu.researchsquare.com/files/rs-4010884/v1/0e554b891466edef0a450132.png"},{"id":52545415,"identity":"905297c8-eeb0-4677-90ca-cdff36a836fb","added_by":"auto","created_at":"2024-03-12 18:21:16","extension":"png","order_by":11,"title":"Figure 11","display":"","copyAsset":false,"role":"figure","size":259766,"visible":true,"origin":"","legend":"\u003cp\u003eVariations in void ratio of tested mixtures according to applied vertical stress: (a) mixtures with varying \u003cem\u003eTFs\u003c/em\u003e at \u003cem\u003eSR\u003c/em\u003e = 0.3 and (b) mixtures with varying \u003cem\u003eSRs\u003c/em\u003e at \u003cem\u003eTF\u003c/em\u003e = 40%. \u003cem\u003ee\u003c/em\u003e\u003csub\u003e\u003cem\u003emax\u003c/em\u003e\u003c/sub\u003e = maximum void ratio; and \u003cem\u003ee\u003c/em\u003e\u003csub\u003e\u003cem\u003emin\u003c/em\u003e\u003c/sub\u003e = minimum void ratio.\u003c/p\u003e","description":"","filename":"11.png","url":"https://assets-eu.researchsquare.com/files/rs-4010884/v1/065a893df27c6df1527e5e27.png"},{"id":52545464,"identity":"ee6d169d-a2e6-4bd4-a924-30d9b4da2cee","added_by":"auto","created_at":"2024-03-12 18:21:23","extension":"png","order_by":12,"title":"Figure 12","display":"","copyAsset":false,"role":"figure","size":246685,"visible":true,"origin":"","legend":"\u003cp\u003ePure stress impact on the measured thermal conductivity: (a) determination of \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e based on pure stress impact and (b) variation of \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e based on pure stress impact according to applied vertical stress.\u003c/p\u003e","description":"","filename":"12.png","url":"https://assets-eu.researchsquare.com/files/rs-4010884/v1/679a51d7efb3f9f714704826.png"},{"id":52545412,"identity":"0d3901cb-0f4f-4a18-803f-a940088f3d55","added_by":"auto","created_at":"2024-03-12 18:21:15","extension":"png","order_by":13,"title":"Figure 13","display":"","copyAsset":false,"role":"figure","size":353528,"visible":true,"origin":"","legend":"\u003cp\u003eVariations of \u003cem\u003eα\u003c/em\u003e(figures (a) and (b)) and \u003cem\u003eβ\u003c/em\u003e (figures (c) and (d)) of tested mixtures according to tire chip fraction.\u003c/p\u003e","description":"","filename":"13.png","url":"https://assets-eu.researchsquare.com/files/rs-4010884/v1/6a92e74a910b4ee08bb9eb66.png"},{"id":52545395,"identity":"df235f88-201e-4cf7-b121-1cb9e539d9e1","added_by":"auto","created_at":"2024-03-12 18:21:04","extension":"png","order_by":14,"title":"Figure 14","display":"","copyAsset":false,"role":"figure","size":278635,"visible":true,"origin":"","legend":"\u003cp\u003eMaximum percent error and mean percent error of thermal conductivity according to tire chip fraction.\u003c/p\u003e","description":"","filename":"14.png","url":"https://assets-eu.researchsquare.com/files/rs-4010884/v1/81a04e0b6c36dc97231fe51a.png"},{"id":52545432,"identity":"7135a2cd-4358-47e8-bedc-d77c0cdc781f","added_by":"auto","created_at":"2024-03-12 18:21:20","extension":"png","order_by":15,"title":"Figure 15","display":"","copyAsset":false,"role":"figure","size":216554,"visible":true,"origin":"","legend":"\u003cp\u003eRelationship between \u003cem\u003eα\u003c/em\u003eand \u003cem\u003eβ\u003c/em\u003e of tested mixtures and previous studies.\u003c/p\u003e","description":"","filename":"15.png","url":"https://assets-eu.researchsquare.com/files/rs-4010884/v1/af8eb892893e554ac1f9b488.png"},{"id":61191760,"identity":"9ec2c437-ee1f-41e2-834c-1e2c9f68c9de","added_by":"auto","created_at":"2024-07-26 20:12:27","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":5883930,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-4010884/v1/7e505b66-6784-41ab-9d06-219c7b2ff399.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Thermal Conductivity of Sand-Tire Rubber Mixtures with Varying Tire Chip Fractions and Size Ratios as a Function of Void Ratio and Applied Vertical Stress","fulltext":[{"header":"1 INTRODUCTION","content":"\u003cp\u003eOver the past few decades, the number of discarded tires has increased significantly, owing to the growing demand for vehicles. approximately 1.5\u0026nbsp;billion tires are discarded globally each year, accounting for 2\u0026ndash;3% of the total waste generated worldwide (Mashiri et al. \u003cspan citationid=\"CR43\" class=\"CitationRef\"\u003e2015\u003c/span\u003e; Mohajerani et al. \u003cspan citationid=\"CR48\" class=\"CitationRef\"\u003e2020\u003c/span\u003e). The incineration and disposal of these tires poses environmental problems, causing air, soil, and water pollution. Consequently, there is ongoing discussion on recycling these tires for sustainable environmental management (Sathiskumar and Karthikeyan \u003cspan citationid=\"CR56\" class=\"CitationRef\"\u003e2019\u003c/span\u003e). In geotechnical engineering, efforts have been undertaken to reuse tire chips or crumb rubber as sustainable construction materials (Meles et al. \u003cspan citationid=\"CR44\" class=\"CitationRef\"\u003e2014\u003c/span\u003e; Ryu et al. \u003cspan citationid=\"CR55\" class=\"CitationRef\"\u003e2022\u003c/span\u003e). Tire chips have distinct properties (e.g., durability and low apparent density) compared with natural geomaterials because of the properties of their constituent minerals; thus, tire chips are commonly mixed with sand to improve their engineering properties and usability. For the successful application of sand\u0026ndash;tire chip mixtures as construction materials, various engineering properties, including the frictional resistance, compressibility, damping, and stiffness of sand\u0026ndash;tire rubber mixtures, have been studied (Lee et al. \u003cspan citationid=\"CR35\" class=\"CitationRef\"\u003e2007\u003c/span\u003e; Ghaaowd et al. \u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e2017\u003c/span\u003e; Noorzad and Raveshi \u003cspan citationid=\"CR50\" class=\"CitationRef\"\u003e2017\u003c/span\u003e; Xiao et al. \u003cspan citationid=\"CR62\" class=\"CitationRef\"\u003e2019\u003c/span\u003e; Ryu et al. \u003cspan citationid=\"CR55\" class=\"CitationRef\"\u003e2022\u003c/span\u003e; Won et al. \u003cspan citationid=\"CR60\" class=\"CitationRef\"\u003e2023\u003c/span\u003e). These previous studies have reported that sand\u0026ndash;tire rubber mixtures exhibit a low unit weight, low stiffness, high frictional resistance, high compressibility, high hydraulic conductivity, and high damping capacity. Consequently, these mixtures are primarily used as lightweight backfill materials, road subbases, rail and highway construction materials, seismic isolation materials, and materials for slope stabilization and foundation reinforcement (Ahmed and Lovell \u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1993\u003c/span\u003e; Poh and Broms \u003cspan citationid=\"CR52\" class=\"CitationRef\"\u003e1995\u003c/span\u003e; Bosscher et al. \u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e1997\u003c/span\u003e; Tweedie et al. \u003cspan citationid=\"CR58\" class=\"CitationRef\"\u003e1998\u003c/span\u003e; Rowe and McIsaac \u003cspan citationid=\"CR54\" class=\"CitationRef\"\u003e2005\u003c/span\u003e; Aydilek et al. \u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2006\u003c/span\u003e). However, studies on the thermal conductivity of sand\u0026ndash;tire rubber mixtures are limited, although tire rubber is a promising thermally insulating material (Yang et al. \u003cspan citationid=\"CR64\" class=\"CitationRef\"\u003e2022\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eGiven the increasing diversity of construction environments, understanding the thermal properties of geomaterials is crucial. This knowledge is essential for tasks such as thermal insulation, heat dissipation, and frost protection, making the thermal conductivity (\u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e) of soil a key variable to consider in the geotechnical design process. The \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e of soil can be used in foundation design in extreme environments, predicting ground changes owing to global warming, calculating road pavement thickness, designing underground heating pipes, and geothermal power generation design, encompassing the safety management of underground structures (Benson et al. \u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e1996\u003c/span\u003e; Loveridge and Powrie \u003cspan citationid=\"CR40\" class=\"CitationRef\"\u003e2013\u003c/span\u003e; Bai and Niedzwecki \u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e2014\u003c/span\u003e). Owing to the low \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e of rubber minerals, sand\u0026ndash;tire rubber mixtures have potential use as insulating materials, such as underground burial materials for liquefied gas pipelines and energy storage tanks and as materials for controlling freeze depth in road pavements (Christ et al. \u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e2010\u003c/span\u003e; Zhao et al. \u003cspan citationid=\"CR66\" class=\"CitationRef\"\u003e2014\u003c/span\u003e; Liu et al. \u003cspan citationid=\"CR39\" class=\"CitationRef\"\u003e2020\u003c/span\u003e). The tire rubber fraction, size ratio (\u003cem\u003eSR\u003c/em\u003e) between rubber and sand, porosity, and water content were selected as the main testing variables to evaluate the thermal conductivity of sand\u0026ndash;tire rubber mixtures (Xiao et al. \u003cspan citationid=\"CR62\" class=\"CitationRef\"\u003e2019\u003c/span\u003e; Liu et al. \u003cspan citationid=\"CR39\" class=\"CitationRef\"\u003e2020\u003c/span\u003e; Cui et al. \u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e2022\u003c/span\u003e; Yang et al. \u003cspan citationid=\"CR64\" class=\"CitationRef\"\u003e2022\u003c/span\u003e). These previous studies reported that the thermal conductivity of a mixture is inversely proportional to the tire chip fraction (\u003cem\u003eTF\u003c/em\u003e) and porosity, and directly proportional to the size ratio and water content. However, studies evaluating \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e according to the applied vertical stress are limited (Cui et al. \u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e2022\u003c/span\u003e). Tank thermal energy storage (TTES) or pit thermal energy storage (PTES) is typically located underground at depths of 5\u0026ndash;15 m (Pourahmadiyan et al. \u003cspan citationid=\"CR53\" class=\"CitationRef\"\u003e2023\u003c/span\u003e), resulting in vertical stresses acting on the soil beneath the TTES or PTES ranging from 150 to 300 kPa. Thus, the effect of applied vertical stress on sand\u0026ndash;tire rubber mixtures should be investigated. Additionally, owing to the huge deformability of tire rubber, the sand\u0026ndash;tire rubber mixtures experience a change in connectivity between sand particles with increasing applied stress (Lee et al. \u003cspan citationid=\"CR35\" class=\"CitationRef\"\u003e2007\u003c/span\u003e; Kim and Santamarina \u003cspan citationid=\"CR32\" class=\"CitationRef\"\u003e2008\u003c/span\u003e). Because \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e via sand-to-sand contacts is higher than that via sand-to-rubber or rubber-to-rubber contacts, the hypothesis in this study is that sand\u0026ndash;tire rubber mixtures can show significant dependency on the applied stress, leading to a change in \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e of the sand\u0026ndash;tire rubber mixtures that differs from that of natural geomaterials.\u003c/p\u003e \u003cp\u003eThis study aimed to evaluate the thermal conductivity of sand\u0026ndash;tire chip mixtures with varying tire chip fractions and size ratios according to the porosity and applied vertical stress. Sand\u0026ndash;tire chip mixtures with three different size ratios (0.3, 1.4, and 5.2) were prepared with varying tire chip fractions (0.0, 0.1, 0.2, 0.4, and 1.0) and relative densities (10\u0026ndash;90%). Because the change in applied stress includes changes in both volume and contact area, \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e measurements under a negligible stress level were performed first as a function of porosity. The \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e of the mixtures were then measured according to the applied stress. In this manner, the \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e estimation formulas for the packing impact and pure stress impact were separately suggested in this study.\u003c/p\u003e"},{"header":"2 THERMAL CONDUCTIVITY MODELS FOR DRY SOILS","content":"\u003cp\u003eSoil is a three-phase material, including soil particles, water, and air; thus, the thermal conductivity (\u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e) of soil is determined by the volume fraction of each component, contact quality between soil particles, and the interaction of these three components with different thermal conductivities. Consequently, \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e can generally be expressed as a function of the mineralogy, particle size, density (or porosity), water content, applied stress, and pore fluid characteristics (Yagi and Kunii \u003cspan citationid=\"CR63\" class=\"CitationRef\"\u003e1957\u003c/span\u003e; Johansen \u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e1977\u003c/span\u003e; Farouki and Farouki \u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e1981\u003c/span\u003e; Brandon and Mitchell \u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e1989\u003c/span\u003e; Becker et al. \u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e1992\u003c/span\u003e; Midttomme and Roaldset \u003cspan citationid=\"CR46\" class=\"CitationRef\"\u003e1998\u003c/span\u003e; Gangadhara Rao and Singh \u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e1999\u003c/span\u003et\u0026eacute; and Konrad \u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e2005\u003c/span\u003e; Yun and Santamarina \u003cspan citationid=\"CR65\" class=\"CitationRef\"\u003e2008\u003c/span\u003e). Various models, including theoretical (e.g., effective medium theory model) (Mickley \u003cspan citationid=\"CR45\" class=\"CitationRef\"\u003e1951\u003c/span\u003e; Woodside and Messmer \u003cspan citationid=\"CR61\" class=\"CitationRef\"\u003e1961\u003c/span\u003e; Tarnawski et al. \u003cspan citationid=\"CR57\" class=\"CitationRef\"\u003e2015\u003c/span\u003e), semi-empirical (e.g., normalized conductivity model) (De Vries \u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e1963\u003c/span\u003e; Chen et al. \u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e2012\u003c/span\u003e), and empirical models (Lu et al. \u003cspan citationid=\"CR41\" class=\"CitationRef\"\u003e2007\u003c/span\u003e; Lu et al. \u003cspan citationid=\"CR42\" class=\"CitationRef\"\u003e2014\u003c/span\u003e; Li et al. \u003cspan citationid=\"CR38\" class=\"CitationRef\"\u003e2021\u003c/span\u003e) have been successfully developed to capture the measured \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e of their own studies. However, in the case of the \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e of dry granular materials, empirical models have shown better predictive performance compared to other models because of the significant difference in \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e values between the constituent minerals of the soil particles (2\u0026ndash;8 W/mK) and air (0.026 W/mK) (Farouki and Farouki \u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e1981\u003c/span\u003e; Dong et al. \u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e2015\u003c/span\u003e; Won et al. \u003cspan citationid=\"CR60\" class=\"CitationRef\"\u003e2023\u003c/span\u003e). This implies that, in dry soil, the low thermal conductivity of air hinders the overall thermal conductivity, and the major path of heat transfer occurs through particle contact (Carslaw et al. \u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e1962\u003c/span\u003e; Yun and Santamarina \u003cspan citationid=\"CR65\" class=\"CitationRef\"\u003e2008\u003c/span\u003e; Choo et al. \u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e2021\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eThe thermal conductivity (\u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e) of soils in a dry state mainly depends on the thermal conductivity of the constituent minerals and the contact conditions between the particles (Li et al. \u003cspan citationid=\"CR38\" class=\"CitationRef\"\u003e2021\u003c/span\u003e). The contact conditions can be assessed based on the coordination number (the average number of contacts per particle) captured by the porosity and the contact area captured by the applied stress (Mitchell and Soga \u003cspan citationid=\"CR47\" class=\"CitationRef\"\u003e2005\u003c/span\u003e). Thus, \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e of dry granular materials can be expressed as a function of porosity and applied stress (Yun and Santamarina \u003cspan citationid=\"CR65\" class=\"CitationRef\"\u003e2008\u003c/span\u003e; Lee et al. \u003cspan citationid=\"CR37\" class=\"CitationRef\"\u003e2016\u003c/span\u003e). However, previous studies have also demonstrated that changes in \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e due to variations in applied stress are generally minor compared with those resulting from changes in porosity (Yun and Santamarina \u003cspan citationid=\"CR65\" class=\"CitationRef\"\u003e2008\u003c/span\u003e; Choo et al. \u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e2021\u003c/span\u003e; Bhatt et al. \u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e2022\u003c/span\u003e). Therefore, most existing research has represented \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e of dry soil as a sole function of porosity (Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e). Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e shows three frequently employed \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e estimation formulas for dry soils as a function of porosity, and the three models have different mathematical expressions: a power function (Johansen \u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e1977\u003c/span\u003e), an exponential function (C\u0026ocirc;t\u0026eacute; and Konrad \u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e2005\u003c/span\u003e), and a linear function (Lu et al. \u003cspan citationid=\"CR41\" class=\"CitationRef\"\u003e2007\u003c/span\u003e).\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eEmpirical equations for predicting the thermal conductivity of dry angular sand (or crushed rock).\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"3\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eReference\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eEmpirical equation\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eNote\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eJohansen (\u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e1977\u003c/span\u003e)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({k}_{t\\left(dry\\right)}=a\\bullet {n}^{-b}\\pm 25\\%\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eAngular sand\u003c/p\u003e \u003cp\u003e(\u003cem\u003ea\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0.039, \u003cem\u003eb\u003c/em\u003e\u0026thinsp;=\u0026thinsp;2.2)\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eCote and Konrad (2005)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({k}_{t\\left(dry\\right)}=\\chi \\bullet {10}^{-\\eta \\bullet n}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eAngular sand\u003c/p\u003e \u003cp\u003e(\u003cem\u003eχ\u003c/em\u003e\u0026thinsp;=\u0026thinsp;1.7, \u003cem\u003eη\u003c/em\u003e\u0026thinsp;=\u0026thinsp;1.8)\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eLu et al. (\u003cspan citationid=\"CR41\" class=\"CitationRef\"\u003e2007\u003c/span\u003e)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({k}_{t\\left(dry\\right)}=-c\\bullet n+d\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eSoils with 0.2\u0026thinsp;\u0026lt;\u0026thinsp;\u003cem\u003en\u003c/em\u003e\u0026thinsp;\u0026lt;\u0026thinsp;0.6\u003c/p\u003e \u003cp\u003e(\u003cem\u003ec\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0.56, \u003cem\u003ed\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0.51)\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colspan=\"3\" nameend=\"c3\" namest=\"c1\"\u003e \u003cp\u003eNote: \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({k}_{t\\left(dry\\right) }\\)\u003c/span\u003e\u003c/span\u003e= dry thermal conductivity; \u003cem\u003en\u003c/em\u003e\u0026thinsp;=\u0026thinsp;porosity; and \u003cem\u003ea\u003c/em\u003e, \u003cem\u003eb\u003c/em\u003e, \u003cem\u003ec\u003c/em\u003e, \u003cem\u003ed\u003c/em\u003e, \u003cem\u003eχ\u003c/em\u003e and \u003cem\u003eη\u003c/em\u003e\u0026thinsp;=\u0026thinsp;fitting parameter.\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e"},{"header":"3 MATERIALS AND METHODOLOGY","content":"\u003cdiv id=\"Sec4\" class=\"Section2\"\u003e \u003ch2\u003e3.1 Materials\u003c/h2\u003e \u003cp\u003eTwo sands (median particle size, \u003cem\u003eD\u003c/em\u003e\u003csub\u003e\u003cem\u003e50\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;0.76 mm for K-5 sand and \u003cem\u003eD\u003c/em\u003e\u003csub\u003e\u003cem\u003e50\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;0.21 mm for K-7 sand) and two tire chips (\u003cem\u003eD\u003c/em\u003e\u003csub\u003e\u003cem\u003e50\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;1.09 mm for TC1 and \u003cem\u003eD\u003c/em\u003e\u003csub\u003e\u003cem\u003e50\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;0.21 mm for TC2) were employed in this study to prepare sand\u0026ndash;tire rubber mixtures. The index properties of the unmixed materials are summarized in Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e, and their particle size distribution curves of unmixed materials are shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e. Because the relative size between rubber and sand particles is a crucial factor in determining the overall mechanical properties of the mixtures (Lee et al. \u003cspan citationid=\"CR35\" class=\"CitationRef\"\u003e2007\u003c/span\u003e; Kim and Santamarina \u003cspan citationid=\"CR32\" class=\"CitationRef\"\u003e2008\u003c/span\u003e), mixtures with three different size ratios (\u003cem\u003eSR\u003c/em\u003e), as defined in Eq.\u0026nbsp;(\u003cspan refid=\"Equ1\" class=\"InternalRef\"\u003e1\u003c/span\u003e), were prepared by mixing K-5 and TC2 to obtain \u003cem\u003eSR\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0.3, K-5 and TC1 to obtain \u003cem\u003eSR\u003c/em\u003e\u0026thinsp;=\u0026thinsp;1.4, and K-7 and TC2 to obtain \u003cem\u003eSR\u003c/em\u003e\u0026thinsp;=\u0026thinsp;5.2.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab2\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003ePhysical properties of unmixed materials.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"6\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colspan=\"2\" nameend=\"c2\" namest=\"c1\"\u003e \u003cp\u003eType of properties\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003e\u003cem\u003eG\u003c/em\u003e\u003csub\u003e\u003cem\u003es\u003c/em\u003e\u003c/sub\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u003cem\u003eD\u003c/em\u003e\u003csub\u003e\u003cem\u003e50\u003c/em\u003e\u003c/sub\u003e (mm)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003e\u003cem\u003ee\u003c/em\u003e\u003csub\u003e\u003cem\u003emax\u003c/em\u003e\u003c/sub\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003e\u003cem\u003ee\u003c/em\u003e\u003csub\u003e\u003cem\u003emin\u003c/em\u003e\u003c/sub\u003e\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003eSand\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eK-5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e2.66\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e0.76\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e1.05\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e0.69\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eK-7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e2.65\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e0.21\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e1.07\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e0.68\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003eTire chip\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eTC1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e1.18\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e1.09\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e1.69\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e1.24\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eTC2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e1.15\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e0.21\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e2.27\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e1.43\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c2\" namest=\"c1\"\u003e \u003cp\u003eTesting method\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eASTM D854\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eASTM D422\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eASTM D4254\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003eJGS 0161\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003ctfoot\u003e \u003ctr\u003e\u003ctd colspan=\"6\"\u003eNote: \u003cem\u003eG\u003c/em\u003e\u003csub\u003e\u003cem\u003es\u003c/em\u003e\u003c/sub\u003e = specific gravity; \u003cem\u003eD\u003c/em\u003e\u003csub\u003e\u003cem\u003e50\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;median grain size; \u003cem\u003ee\u003c/em\u003e\u003csub\u003e\u003cem\u003emax\u003c/em\u003e\u003c/sub\u003e and \u003cem\u003ee\u003c/em\u003e\u003csub\u003e\u003cem\u003emin\u003c/em\u003e\u003c/sub\u003e = maximum and minimum void ratios, respectively.\u003c/td\u003e\u003c/tr\u003e \u003c/tfoot\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003e \u003cdiv id=\"Equ1\" class=\"Equation\"\u003e \u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e\n$$Size ratio\\left(SR\\right)= \\frac{{D}_{50\\left(tire chip\\right)}}{{D}_{50\\left(sand\\right)}}$$\u003c/div\u003e \u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\u003c/div\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec5\" class=\"Section2\"\u003e \u003ch2\u003e3.2 Thermal Conductivity Measurements\u003c/h2\u003e \u003cp\u003eFor measuring the thermal conductivity of mixtures, the thermal needle probe method (ASTM D5334-14) was employed because of its simplicity, high accuracy, and short measurement duration (Naidu and Singh \u003cspan citationid=\"CR49\" class=\"CitationRef\"\u003e2004\u003c/span\u003e). The probe sensor used for measuring the thermal conductivity was made of stainless steel, 30 mm in length and 1.3 mm in diameter, and was equipped with a heating wire and a thermistor that measured electrical resistance changes with temperature (East30Sensors Ltd.). As shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e, the thermal needle probe was installed at the bottom of a thermally insulated oedometer cell with a diameter of 100 mm and height of 67 mm, allowing for thermal conductivity measurements according to the applied vertical stress. The thermal conductivity (\u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e) was determined based on the infinite line heat source theory (Carslaw et al. \u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e1962\u003c/span\u003e):\u003c/p\u003e \u003cp\u003e \u003cdiv id=\"Equ2\" class=\"Equation\"\u003e \u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ2\" name=\"EquationSource\"\u003e\n$${k}_{t}=\\frac{C\\bullet Q}{4\\pi }\\bullet \\frac{\\varDelta \\text{l}\\text{n}\\left(t\\right)}{\\varDelta T}$$\u003c/div\u003e \u003cdiv class=\"EquationNumber\"\u003e2\u003c/div\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003ewhere \u003cem\u003eC\u003c/em\u003e is the calibration coefficient, Q is the input power applied to the heating wire in W/m, and ∆\u003cem\u003eT\u003c/em\u003e is the temperature difference of the medium over the time interval ∆\u003cem\u003et\u003c/em\u003e. ∆\u003cem\u003et\u003c/em\u003e was set to 30 s in this study, and care was taken to ensure that the total temperature change did not exceed 2˚C to prevent errors due to convection.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec6\" class=\"Section2\"\u003e \u003ch2\u003e3.3 Test Program\u003c/h2\u003e \u003cp\u003eAs shown in Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e, the experiment was conducted along two main directions. Test case #1 focused on determining the changes in the thermal conductivity of the tested mixtures with varying tire chip fraction (\u003cem\u003eTF\u003c/em\u003e), defined in Eq.\u0026nbsp;(\u003cspan refid=\"Equ3\" class=\"InternalRef\"\u003e3\u003c/span\u003e), owing to the packing impact under a negligible stress level. Thus, samples with relative densities ranging from 10% (very loose state) to 90% (very dense state) were prepared to measure thermal conductivity. In the case of test case #2, focusing on determining changes in thermal conductivity due to the stress impact, the initial relative density of the sample was set to 30% (loose state) and 70% (dense state), and then the vertical stress was gradually increased from 0.5 to 460 kPa at a load increase ratio of 1. Thermal conductivity was measured at the end of each loading process. Samples with varying initial target relative densities were prepared using the dry funnel deposition method, followed by symmetric impacts with a rubber mallet. The thermal conductivity reported in this study is the average of three replicate tests.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab3\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 3\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eTest matrix.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"5\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eTest case\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eSize ratio\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eTire chip fraction\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eApplied vertical stress (kPa)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eInitial relative density (%)\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e#1 \u0026ndash; Impact of packing state on thermal conductivity\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003e0.3, 1.4, 5.2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003e0.0, 0.1, 0.2, 0.4, 1.0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e0.5 (top cap weight)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e10\u0026ndash;90\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e#2 \u0026ndash; Impact of applied stress on thermal conductivity\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e0.5\u0026ndash;460 (LIR\u0026thinsp;=\u0026thinsp;1)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e30, 70\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003ctfoot\u003e \u003ctr\u003e\u003ctd colspan=\"5\"\u003eNote: LIR\u0026thinsp;=\u0026thinsp;loading increment ratio\u003c/td\u003e\u003c/tr\u003e \u003c/tfoot\u003e \u003c/table\u003e\u003c/div\u003e \u003cdiv id=\"Equ3\" class=\"Equation\"\u003e \u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ3\" name=\"EquationSource\"\u003e\n$$Tire chip fraction\\left(TF\\right)= \\frac{{V}_{tire chip}}{Total Volume}$$\u003c/div\u003e \u003cdiv class=\"EquationNumber\"\u003e3\u003c/div\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003ewhere \u003cem\u003eV\u003c/em\u003e\u003csub\u003etire chip\u003c/sub\u003e is the volume of tire chip.\u003c/p\u003e \u003c/div\u003e"},{"header":"4. RESULTS AND DISCUSSIONS","content":"\u003cp\u003eSection \u003cspan refid=\"Sec8\" class=\"InternalRef\"\u003e4.1\u003c/span\u003e describes the thermal conductivity of the tested sand\u0026ndash;tire rubber mixtures with varying tire chip fractions (\u003cem\u003eTF\u003c/em\u003e) and size ratio (\u003cem\u003eSR\u003c/em\u003e) according to the porosity under very low stress levels (~\u0026thinsp;0.5 kPa) resulting from the top cap weight. The effect of applied vertical stress, ranging from 0.5 to 460 kPa, on thermal conductivity of tested mixtures is presented in Section \u003cspan refid=\"Sec12\" class=\"InternalRef\"\u003e4.2\u003c/span\u003e.\u003c/p\u003e \u003cdiv id=\"Sec8\" class=\"Section2\"\u003e \u003ch2\u003e4.1 Thermal Conductivity of Tested Mixtures under Negligible Applied Stress\u003c/h2\u003e \u003cdiv id=\"Sec9\" class=\"Section3\"\u003e \u003ch2\u003e4.1.1 Effects of Porosity and Tire Chip Fraction on Thermal Conductivity\u003c/h2\u003e \u003cp\u003eFigure \u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e shows the change in thermal conductivity (\u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e) of the tested mixtures with varying size ratios and tire chip fractions (\u003cem\u003eTF\u003c/em\u003e) according to the porosity measured under a constant vertical stress (0.5 kPa) (Test case #1 in Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e). It can be clearly observed in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e that the \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e of tested materials increased with decreasing porosity (or increasing packing density), regardless of \u003cem\u003eSR\u003c/em\u003es and \u003cem\u003eTF\u003c/em\u003es. A decrease in porosity resulted in an increase in the interparticle coordination number, leading to an increased number of paths for particle contact conduction. Additionally, a decrease in porosity implies a reduction in the proportion of air that hinders heat transfer within the soil, thereby enhancing heat conduction through particle contacts. Thus, \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e increases with decreasing porosity (Johansen \u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e1977\u003c/span\u003e; Chen \u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e2008\u003c/span\u003e; Liu et al. \u003cspan citationid=\"CR39\" class=\"CitationRef\"\u003e2020\u003c/span\u003e; Cui et al. \u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e2022\u003c/span\u003e).\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eFigure \u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e also demonstrates that the measured thermal conductivity (\u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e) at a given porosity decreased with increasing tire chip fraction (\u003cem\u003eTF\u003c/em\u003e) (Xiao et al. \u003cspan citationid=\"CR62\" class=\"CitationRef\"\u003e2019\u003c/span\u003e; Liu et al. \u003cspan citationid=\"CR39\" class=\"CitationRef\"\u003e2020\u003c/span\u003e; Cui et al. \u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e2022\u003c/span\u003e; Yang et al. \u003cspan citationid=\"CR64\" class=\"CitationRef\"\u003e2022\u003c/span\u003e). The mineral conductivity of rubber is around 0.25 W/mK; while the conductivity of minerals that make soil particles generally ranges from 1 to 5 W/mK (C\u0026ocirc;t\u0026eacute; and Konrad \u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e2005\u003c/span\u003e). Therefore, in sand\u0026ndash;tire rubber mixtures, the connectivity between sand particles, which have higher mineral conductivity, determines the overall \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e (Xiao et al. \u003cspan citationid=\"CR62\" class=\"CitationRef\"\u003e2019\u003c/span\u003e; Liu et al. \u003cspan citationid=\"CR39\" class=\"CitationRef\"\u003e2020\u003c/span\u003e). Thus, the decrease in \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e with increasing \u003cem\u003eTF\u003c/em\u003e can be attributed to the transition in the dominant contact mode within the mixture from the highly conductive sand\u0026ndash;to-sand contacts at low \u003cem\u003eTF\u003c/em\u003e to the less conductive rubber-to-rubber and/or rubber-to-sand contacts at high \u003cem\u003eTF\u003c/em\u003e (Lee et al. \u003cspan citationid=\"CR34\" class=\"CitationRef\"\u003e2010\u003c/span\u003e; Liu et al. \u003cspan citationid=\"CR39\" class=\"CitationRef\"\u003e2020\u003c/span\u003e).\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec10\" class=\"Section3\"\u003e \u003ch2\u003e4.1.2 Effect of Size Ratio on Thermal Conductivity\u003c/h2\u003e \u003cp\u003eDue to the difference in the thermal conductivity (\u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e) between pure sands (i.e., K-5 sand and K-7 sand), and between pure tire chips (i.e., TC1 and TC2), the normalized conductivity \u003cem\u003eK\u003c/em\u003e\u003csub\u003e\u003cem\u003eN\u003c/em\u003e\u003c/sub\u003e (Eq.\u0026nbsp;(\u003cspan refid=\"Equ5\" class=\"InternalRef\"\u003e4\u003c/span\u003e), Yang et al. (\u003cspan citationid=\"CR64\" class=\"CitationRef\"\u003e2022\u003c/span\u003e)) was plotted as a function of tire chip fraction (\u003cem\u003eTF\u003c/em\u003e) at a relative density of 70% (Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e).\u003c/p\u003e \u003cp\u003e \u003cdiv id=\"Equ4\" class=\"Equation\"\u003e \u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ4\" name=\"EquationSource\"\u003e\n$${K}_{N}= \\frac{{k}_{t}-{k}_{tire}}{{k}_{sand}-{k}_{tire}}$$\u003c/div\u003e \u003cdiv class=\"EquationNumber\"\u003e4\u003c/div\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003ewhere \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003etire\u003c/em\u003e\u003c/sub\u003e is the thermal conductivity of the pure tire chip (\u003cem\u003eTF\u003c/em\u003e\u0026thinsp;=\u0026thinsp;100%), and \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003esand\u003c/em\u003e\u003c/sub\u003e is the thermal conductivity of pure sand (\u003cem\u003eTF\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0%). Figure\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e indicates that \u003cem\u003eK\u003c/em\u003e\u003csub\u003e\u003cem\u003eN\u003c/em\u003e\u003c/sub\u003e at a given \u003cem\u003eTF\u003c/em\u003e increased with increasing size ratio, indicating that the use of sand\u0026ndash;tire rubber mixtures with smaller \u003cem\u003eSR\u003c/em\u003e results in superior thermal insulating properties. As mentioned above, owing to the significant difference in mineral conductivity between sand and rubber particles, the connectivity between sand particles determines the overall \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e, implying that mixtures with larger \u003cem\u003eSR\u003c/em\u003e have better connectivity between sand particles. Previous experimental and numerical studies (Perez et al. \u003cspan citationid=\"CR51\" class=\"CitationRef\"\u003e2016\u003c/span\u003e; Xiao et al. \u003cspan citationid=\"CR62\" class=\"CitationRef\"\u003e2019\u003c/span\u003e; Ryu et al. \u003cspan citationid=\"CR55\" class=\"CitationRef\"\u003e2022\u003c/span\u003e; Won et al. \u003cspan citationid=\"CR60\" class=\"CitationRef\"\u003e2023\u003c/span\u003e) also reported that the connectivity between sand particles increases with increasing \u003cem\u003eSR\u003c/em\u003e.\u003c/p\u003e \u003cp\u003eThe \u003cem\u003eSR\u003c/em\u003e-dependent connectivity change between sand particles is schematically shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e. A smaller \u003cem\u003eSR\u003c/em\u003e implies that tire chips are smaller than sand particles, positioning tire chips with lower thermal conductivity among sand particles with higher thermal conductivity, thereby easily disrupting sand-to-sand contacts under a low applied stress level. This results in an increased number of thermal conduction paths via sand-to-rubber or rubber-to-rubber contacts, leading to a decreased thermal conductivity. Conversely, a larger size ratio enhances the connectivity between sand particles, indicating that sand-to-sand contacts can be the main path for thermal conduction (Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e) and consequently result in a relatively higher overall thermal conductivity with increasing \u003cem\u003eSR\u003c/em\u003e (Lee et al. \u003cspan citationid=\"CR36\" class=\"CitationRef\"\u003e2015\u003c/span\u003e; Lee et al. \u003cspan citationid=\"CR33\" class=\"CitationRef\"\u003e2017\u003c/span\u003e; Xiao et al. \u003cspan citationid=\"CR62\" class=\"CitationRef\"\u003e2019\u003c/span\u003e; Yang et al. \u003cspan citationid=\"CR64\" class=\"CitationRef\"\u003e2022\u003c/span\u003e).\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec11\" class=\"Section3\"\u003e \u003ch2\u003e4.1.3 Thermal Conductivity Model\u003c/h2\u003e \u003cp\u003eThe data of Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e, the measured thermal conductivity (\u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e) of sand-tire rubber mixtures under negligible stress level, was fitted with previous thermal conductivity models for dry soils (Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e). The coefficient of determination (R\u003csup\u003e2\u003c/sup\u003e) for the models of Johansen (\u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e1977\u003c/span\u003e), Cote and Konrad (2005), and Lu et al. (\u003cspan citationid=\"CR41\" class=\"CitationRef\"\u003e2007\u003c/span\u003e) were determined to be 0.995, 0.996, and 0.991, respectively, indicating that the relationship between \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e and porosity of all tested mixtures could be accurately captured by either the power, exponential, or linear model. Because Cote and Konrad model showed slightly better performance than the other two models, consistent with the observations of Dong et al. (\u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e2015\u003c/span\u003e), the fitting parameters \u003cem\u003eχ\u003c/em\u003e and \u003cem\u003eη\u003c/em\u003e in Cote and Konrad model in Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e of tested mixtures with varying size ratios were plotted as a function of tire chip fraction (\u003cem\u003eTF\u003c/em\u003e) in Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e. Note \u003cem\u003eχ\u003c/em\u003e\u0026ndash;factor indicates the \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e of materials when porosity equals zero; thus, \u003cem\u003eχ\u003c/em\u003e\u0026ndash;factor reflects the inherent conductivity of the minerals and should decrease with increasing \u003cem\u003eTF\u003c/em\u003e (Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e(a)). \u003cem\u003eη\u003c/em\u003e\u0026ndash;factor indicates the sensitivity of \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e on porosity. As shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e, the absolute value of the line slope decreased with increasing \u003cem\u003eTF\u003c/em\u003e, indicating that the \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e of sand\u0026ndash;tire rubber mixtures with higher \u003cem\u003eTF\u003c/em\u003e showed a diminished dependency on porosity (Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e(b)). The reason for this change in the porosity dependence with increasing \u003cem\u003eTF\u003c/em\u003e is unclear. However, the results of Liu et al. (\u003cspan citationid=\"CR39\" class=\"CitationRef\"\u003e2020\u003c/span\u003e) support this observation. A likely explanation is that rubber is an insulating material, indicating that the replacement of the air phase with solid rubber particles owing to the decrease in porosity results in a limited increase in thermal conductivity, although the number of thermal conduction paths increases. Finally, it is notable that the determined \u003cem\u003eχ\u003c/em\u003e and \u003cem\u003eη\u003c/em\u003e of the tested pure sands and pure tire chips were comparable with those of Cote and Konrad (2005) for angular sand and peat, respectively.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eBecause the fitting parameters \u003cem\u003eχ\u003c/em\u003e and \u003cem\u003eη\u003c/em\u003e in Cote and Konrad model of tested mixtures changed nonlinearly from those of pure sand to pure rubber (Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e), the normalized \u003cem\u003eχ\u003c/em\u003e and \u003cem\u003eη\u003c/em\u003e (\u003cem\u003eχ\u003c/em\u003e\u003csub\u003e\u003cem\u003eN\u003c/em\u003e\u003c/sub\u003e and \u003cem\u003eη\u003c/em\u003e\u003csub\u003e\u003cem\u003eN\u003c/em\u003e\u003c/sub\u003e) were defined as:\u003cdiv id=\"Equ5\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ5\" name=\"EquationSource\"\u003e\n$${\\chi }_{N}= \\frac{\\chi -{\\chi }_{tire}}{{\\chi }_{sand}-{\\chi }_{tire}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e4\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ6\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ6\" name=\"EquationSource\"\u003e\n$${\\eta }_{N}= \\frac{\\eta -{\\eta }_{tire}}{{\\eta }_{sand}-{\\eta }_{tire}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e5\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere \u003cem\u003eχ\u003c/em\u003e\u003csub\u003e\u003cem\u003etire\u003c/em\u003e\u003c/sub\u003e and \u003cem\u003eχ\u003c/em\u003e\u003csub\u003e\u003cem\u003esand\u003c/em\u003e\u003c/sub\u003e are the \u003cem\u003eχ\u003c/em\u003e factors of pure tire chip and pure sand, respectively; and \u003cem\u003eη\u003c/em\u003e\u003csub\u003e\u003cem\u003etire\u003c/em\u003e\u003c/sub\u003e and \u003cem\u003eη\u003c/em\u003e\u003csub\u003e\u003cem\u003esand\u003c/em\u003e\u003c/sub\u003e are the respective \u003cem\u003eη\u003c/em\u003e exponents of pure tire chip and pure sand. Figure\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e shows the variations in \u003cem\u003eχ\u003c/em\u003e\u003csub\u003e\u003cem\u003eN\u003c/em\u003e\u003c/sub\u003e and \u003cem\u003eη\u003c/em\u003e\u003csub\u003e\u003cem\u003eN\u003c/em\u003e\u003c/sub\u003e of the tested mixtures with varying size ratios according to tire chip fraction. For a tire chip fraction based on volume (\u003cem\u003eTF\u003c/em\u003e\u0026thinsp;\u0026le;\u0026thinsp;40%), good linear relations occur between \u003cem\u003eχ\u003c/em\u003e\u003csub\u003e\u003cem\u003eN\u003c/em\u003e\u003c/sub\u003e and \u003cem\u003eTF\u003c/em\u003e, and between \u003cem\u003eη\u003c/em\u003e\u003csub\u003e\u003cem\u003eN\u003c/em\u003e\u003c/sub\u003e and \u003cem\u003eTF\u003c/em\u003e.\u003c/p\u003e \u003cp\u003e \u003cdiv id=\"Equ7\" class=\"Equation\"\u003e \u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ7\" name=\"EquationSource\"\u003e\n$${\\chi }_{N}= 1-{b}_{\\chi }\\bullet TF$$\u003c/div\u003e \u003cdiv class=\"EquationNumber\"\u003e6\u003c/div\u003e\u003c/div\u003e \u003cdiv id=\"Equ8\" class=\"Equation\"\u003e \u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ8\" name=\"EquationSource\"\u003e\n$${\\eta }_{N}= 1-{b}_{\\eta }\\bullet TF$$\u003c/div\u003e \u003cdiv class=\"EquationNumber\"\u003e7\u003c/div\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003ewhere \u003cem\u003eb\u003c/em\u003e\u003csub\u003e\u003cem\u003eχ\u003c/em\u003e\u003c/sub\u003e and \u003cem\u003eb\u003c/em\u003e\u003csub\u003e\u003cem\u003eη\u003c/em\u003e\u003c/sub\u003e are the fitting parameters. It can be clearly observed in Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e that \u003cem\u003eb\u003c/em\u003e\u003csub\u003e\u003cem\u003eχ\u003c/em\u003e\u003c/sub\u003e and \u003cem\u003eb\u003c/em\u003e\u003csub\u003e\u003cem\u003eη\u003c/em\u003e\u003c/sub\u003e are \u003cem\u003eSR\u003c/em\u003e-dependent; thus, the determined \u003cem\u003eb\u003c/em\u003e\u003csub\u003e\u003cem\u003eχ\u003c/em\u003e\u003c/sub\u003e and \u003cem\u003eb\u003c/em\u003e\u003csub\u003e\u003cem\u003eη\u003c/em\u003e\u003c/sub\u003e were plotted as a function of \u003cem\u003eSR\u003c/em\u003e in Fig.\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e8\u003c/span\u003e. Figure\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e8\u003c/span\u003e also includes the experimental results of Cui et al. (\u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e2022\u003c/span\u003e). Figure\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e8\u003c/span\u003e demonstrates that \u003cem\u003eb\u003c/em\u003e\u003csub\u003e\u003cem\u003eχ\u003c/em\u003e\u003c/sub\u003e and \u003cem\u003eb\u003c/em\u003e\u003csub\u003e\u003cem\u003eη\u003c/em\u003e\u003c/sub\u003e decreased nonlinearly with increasing \u003cem\u003eSR\u003c/em\u003e, which is attributed to the increased connectivity between sand particles with increasing \u003cem\u003eSR\u003c/em\u003e, and the following power models can capture the relationships between \u003cem\u003eb\u003c/em\u003e\u003csub\u003e\u003cem\u003eχ\u003c/em\u003e\u003c/sub\u003e and \u003cem\u003eSR\u003c/em\u003e, and between \u003cem\u003eb\u003c/em\u003e\u003csub\u003e\u003cem\u003eη\u003c/em\u003e\u003c/sub\u003e and \u003cem\u003eSR\u003c/em\u003e:\u003c/p\u003e \u003cp\u003e \u003cdiv id=\"Equ9\" class=\"Equation\"\u003e \u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ9\" name=\"EquationSource\"\u003e\n$${b}_{\\chi }= 1.58\\bullet {SR}^{-0.22}$$\u003c/div\u003e \u003cdiv class=\"EquationNumber\"\u003e8\u003c/div\u003e\u003c/div\u003e \u003cdiv id=\"Equ10\" class=\"Equation\"\u003e \u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ10\" name=\"EquationSource\"\u003e\n$${b}_{\\eta }= 0.65\\bullet {SR}^{-0.34}$$\u003c/div\u003e \u003cdiv class=\"EquationNumber\"\u003e9\u003c/div\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eSubstituting Equations (\u003cspan refid=\"Equ5\" class=\"InternalRef\"\u003e4\u003c/span\u003e)\u0026ndash;(\u003cspan refid=\"Equ8\" class=\"InternalRef\"\u003e7\u003c/span\u003e) into the Cote and Konrad model in Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e yields\u003cdiv id=\"Equ11\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ11\" name=\"EquationSource\"\u003e\n$${k}_{t}=\\left[\\left(1-{b}_{\\chi }\\bullet TF\\right)\\left({\\chi }_{sand}-{\\chi }_{tire}\\right)+{\\chi }_{tire}\\right]\\bullet {10}^{-\\left[\\left(1-{b}_{\\eta }\\right)\\left({\\eta }_{sand}-{\\eta }_{tire}\\right)+{\\eta }_{tire}\\right]\\bullet n}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e10\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere \u003cem\u003eb\u003c/em\u003e\u003csub\u003e\u003cem\u003eχ\u003c/em\u003e\u003c/sub\u003e and \u003cem\u003eb\u003c/em\u003e\u003csub\u003e\u003cem\u003eη\u003c/em\u003e\u003c/sub\u003e are given by Equations (\u003cspan refid=\"Equ9\" class=\"InternalRef\"\u003e8\u003c/span\u003e) and (\u003cspan refid=\"Equ10\" class=\"InternalRef\"\u003e9\u003c/span\u003e), respectively. The novelty of the newly developed Eq.\u0026nbsp;(\u003cspan refid=\"Equ11\" class=\"InternalRef\"\u003e10\u003c/span\u003e) lies in the fact that the \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e of sand\u0026ndash;tire rubber mixtures with varying \u003cem\u003eSR\u003c/em\u003es, \u003cem\u003eTF\u003c/em\u003es, and porosities can be estimated based on \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e of pure sand and pure tire rubber. Figure\u0026nbsp;\u003cspan refid=\"Fig9\" class=\"InternalRef\"\u003e9\u003c/span\u003e shows a comparison between the measured and estimated \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e values based on Eq.\u0026nbsp;(\u003cspan refid=\"Equ11\" class=\"InternalRef\"\u003e10\u003c/span\u003e), and a good agreement can be observed between the two.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe prediction performance of Eq.\u0026nbsp;(\u003cspan refid=\"Equ11\" class=\"InternalRef\"\u003e10\u003c/span\u003e) is governed by the reliability of the input parameters \u003cem\u003eb\u003c/em\u003e\u003csub\u003e\u003cem\u003eχ\u003c/em\u003e\u003c/sub\u003e and \u003cem\u003eb\u003c/em\u003e\u003csub\u003e\u003cem\u003eη\u003c/em\u003e\u003c/sub\u003e. Although several studies have evaluated the \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e of sand\u0026ndash;tire rubber mixtures according to size ratios and tire chip fractions (Lee et al. \u003cspan citationid=\"CR36\" class=\"CitationRef\"\u003e2015\u003c/span\u003e; Xiao et al. \u003cspan citationid=\"CR62\" class=\"CitationRef\"\u003e2019\u003c/span\u003e); measuring \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e under varying porosities has been limited. Thus, Equations (\u003cspan refid=\"Equ9\" class=\"InternalRef\"\u003e8\u003c/span\u003e) and (\u003cspan refid=\"Equ10\" class=\"InternalRef\"\u003e9\u003c/span\u003e) were derived based on limited experimental results. Further studies are recommended to verify the reliability of Equations (\u003cspan refid=\"Equ9\" class=\"InternalRef\"\u003e8\u003c/span\u003e)\u0026ndash;(\u003cspan refid=\"Equ11\" class=\"InternalRef\"\u003e10\u003c/span\u003e).\u003c/p\u003e \u003c/div\u003e \u003c/div\u003e \u003cdiv id=\"Sec12\" class=\"Section2\"\u003e \u003ch2\u003e4.2 Thermal Conductivity of Tested Mixtures according to Applied Stress\u003c/h2\u003e \u003cdiv id=\"Sec13\" class=\"Section3\"\u003e \u003ch2\u003e4.2.1 Effect of Applied Stress, Tire Chip Fraction, and Size Ratio on Thermal Conductivity\u003c/h2\u003e \u003cp\u003eFigure \u003cspan refid=\"Fig10\" class=\"InternalRef\"\u003e10\u003c/span\u003e shows the change in thermal conductivity of the tested mixtures with varying size ratios and tire chip fractions according to the applied vertical stress (Test case #2 with initial relative density\u0026thinsp;=\u0026thinsp;70% in Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e). Figure\u0026nbsp;\u003cspan refid=\"Fig10\" class=\"InternalRef\"\u003e10\u003c/span\u003e demonstrates that irrespective of \u003cem\u003eSR\u003c/em\u003es and \u003cem\u003eTF\u003c/em\u003es, the measured \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e in all the mixtures increased with the applied vertical stress because of the formation of a more interconnected network for heat to flow through (Yun and Santamarina \u003cspan citationid=\"CR65\" class=\"CitationRef\"\u003e2008\u003c/span\u003e). Similar to the results shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e, at low stress levels (\u0026lt;\u0026thinsp;60 kPa), a clear decrease in \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e with increasing \u003cem\u003eTF\u003c/em\u003e at a given stress level was observed. However, as the vertical stress further increased, the difference in \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e based on \u003cem\u003eTF\u003c/em\u003e at the same stress level gradually diminished, and \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e of all the mixtures approached that of pure sand, implying that the tested sand\u0026ndash;tire rubber mixtures can develop sand-to-sand thermal conduction paths at higher stress levels (\u0026gt;\u0026thinsp;200 kPa) (Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e).\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eSimilar to the results shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e, the measured \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e for the mixtures with \u003cem\u003eSR\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0.3 was smaller than that for mixtures with \u003cem\u003eSR\u003c/em\u003e\u0026thinsp;=\u0026thinsp;5.2 at low stress levels; however, Fig.\u0026nbsp;\u003cspan refid=\"Fig10\" class=\"InternalRef\"\u003e10\u003c/span\u003e clearly demonstrates the opposite effect of \u003cem\u003eSR\u003c/em\u003e on \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e at high stress levels. In particular, the tested mixtures with \u003cem\u003eSR\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0.3 at \u003cem\u003eTF\u003c/em\u003e\u0026thinsp;=\u0026thinsp;20% and 40% showed even greater \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e than pure sand at a vertical stress of 460 kPa (Fig.\u0026nbsp;\u003cspan refid=\"Fig10\" class=\"InternalRef\"\u003e10\u003c/span\u003e(a)). This observation suggests that once the sand\u0026ndash;tire rubber mixtures with small \u003cem\u003eSR\u003c/em\u003e experience high stress levels, the insulating characteristics can disappear, suggesting that the applied stress should be an important parameter when utilizing sand\u0026ndash;tire rubber mixtures as insulating materials, and the use of larger tire chips can guarantee thermal insulation properties regardless of stress levels.\u003c/p\u003e \u003cp\u003eThe increase in applied stress involves both changes in contact area and coordination number (Johnson and Johnson \u003cspan citationid=\"CR31\" class=\"CitationRef\"\u003e1987\u003c/span\u003e; Choo and Burns \u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e2015\u003c/span\u003e; Choo and Lee \u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e2021\u003c/span\u003e; Ryu et al. \u003cspan citationid=\"CR55\" class=\"CitationRef\"\u003e2022\u003c/span\u003e). Thus, the change in the void ratio, which reflects the change in the coordination number, is plotted as a function of the applied vertical stress in Fig.\u0026nbsp;\u003cspan refid=\"Fig11\" class=\"InternalRef\"\u003e11\u003c/span\u003e. Figure\u0026nbsp;\u003cspan refid=\"Fig11\" class=\"InternalRef\"\u003e11\u003c/span\u003e(a) shows that the compressibility (i.e., the decrease in void ratio with increasing stress) increased with increasing \u003cem\u003eTF\u003c/em\u003e because of the change in the interaction between sand and rubber components, and the elastic nature of rubber particles (Ryu et al. \u003cspan citationid=\"CR55\" class=\"CitationRef\"\u003e2022\u003c/span\u003e). Most notably, the void ratio of mixtures with \u003cem\u003eSR\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0.3 at \u003cem\u003eTF\u003c/em\u003e\u0026thinsp;=\u0026thinsp;20% and 40% becomes smaller than the minimum void ratio of pure sand (K-5 sand) at high stress levels (\u0026gt;\u0026thinsp;200 kPa), reflecting the significant particle rearrangement and the consequent contact mode change due to the squeezing/distortion of soft rubber particles. Thus, these mixtures can exhibit greater \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e than pure sand at high stress levels owing to the development of a sand-to-sand conduction path and a significant decrease in the air volume fraction (Fig.\u0026nbsp;\u003cspan refid=\"Fig10\" class=\"InternalRef\"\u003e10\u003c/span\u003e(a)).\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe compressibility of the mixtures increased with decreasing \u003cem\u003eSR\u003c/em\u003e at a given \u003cem\u003eTF\u003c/em\u003e because \u003cem\u003eSR\u003c/em\u003e determines the connectivity between sand particles (Fig.\u0026nbsp;\u003cspan refid=\"Fig11\" class=\"InternalRef\"\u003e11\u003c/span\u003e(b)). As the \u003cem\u003eSR\u003c/em\u003e decreases, the smaller tire rubber particles can easily be located in the pore space between the larger sand particles, resulting in a significant disruption of the direct contact between sand particles at low stress levels. This causes soft rubber particles to actively participate in the load-carrying skeleton, resulting in a large volume contraction with increasing applied stress (Ryu et al. \u003cspan citationid=\"CR55\" class=\"CitationRef\"\u003e2022\u003c/span\u003e). In contrast, in the case of the tested mixtures with \u003cem\u003eSR\u003c/em\u003e\u0026thinsp;=\u0026thinsp;5.2, the applied stress could be transmitted through sand-to-sand contacts, resulting in a limited change in the contact mode and a consequent limited increase in \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e (Fig.\u0026nbsp;\u003cspan refid=\"Fig10\" class=\"InternalRef\"\u003e10\u003c/span\u003e(c)).\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec14\" class=\"Section3\"\u003e \u003ch2\u003e4.2.2 Pure Stress Impact on Thermal Conductivity\u003c/h2\u003e \u003cp\u003eThe thermal conductivity (\u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e) increase in Fig.\u0026nbsp;\u003cspan refid=\"Fig10\" class=\"InternalRef\"\u003e10\u003c/span\u003e includes the \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e increase due to the decrease in porosity (i.e., \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e based on packing impact) and \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e increase due to the increase in contact area (i.e., \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e based on pure stress impact). Thus, to examine the effect of pure stress on \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e, the measured \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e of mixtures with \u003cem\u003eSR\u003c/em\u003e\u0026thinsp;=\u0026thinsp;1.4 at \u003cem\u003eTF\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0% and 40% (Fig.\u0026nbsp;\u003cspan refid=\"Fig10\" class=\"InternalRef\"\u003e10\u003c/span\u003e(b)) is plotted as a function of porosity in Fig.\u0026nbsp;\u003cspan refid=\"Fig12\" class=\"InternalRef\"\u003e12\u003c/span\u003e (a). The open symbols in Fig.\u0026nbsp;\u003cspan refid=\"Fig12\" class=\"InternalRef\"\u003e12\u003c/span\u003e indicate the measured \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e under negligible stress (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e(b), Test case #1 in Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e), and the filled symbols indicate \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e with increasing applied stress (Fig.\u0026nbsp;\u003cspan refid=\"Fig10\" class=\"InternalRef\"\u003e10\u003c/span\u003e(b), Test case #2 in Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e). The dotted lines in Fig.\u0026nbsp;\u003cspan refid=\"Fig12\" class=\"InternalRef\"\u003e12\u003c/span\u003e(a) represent the estimated \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e values based on Eq.\u0026nbsp;(\u003cspan refid=\"Equ11\" class=\"InternalRef\"\u003e10\u003c/span\u003e). Thus, the difference in \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e corresponding to the filled symbols and the trendline at a given porosity (i.e., △ in Fig.\u0026nbsp;\u003cspan refid=\"Fig12\" class=\"InternalRef\"\u003e12\u003c/span\u003e(a)) can be considered as the increase in \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e due to the pure stress impact. This △ is plotted as a function of the applied vertical stress in Fig.\u0026nbsp;\u003cspan refid=\"Fig12\" class=\"InternalRef\"\u003e12\u003c/span\u003e(b) for mixtures with \u003cem\u003eSR\u003c/em\u003e\u0026thinsp;=\u0026thinsp;1.4 at an initial relative density of 70%. Figure\u0026nbsp;\u003cspan refid=\"Fig12\" class=\"InternalRef\"\u003e12\u003c/span\u003e(b) shows that \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e based on pure stress impact can be approximated as a power function of applied vertical stress (\u003cem\u003eσ\u0026rsquo;\u003c/em\u003e\u003csub\u003e\u003cem\u003ev\u003c/em\u003e\u003c/sub\u003e):\u003c/p\u003e \u003cp\u003e \u003cdiv id=\"Equ12\" class=\"Equation\"\u003e \u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ12\" name=\"EquationSource\"\u003e\n$${k}_{t}=\\alpha \\bullet {\\left(\\frac{{\\sigma {\\prime }}_{v}}{1kPa}\\right)}^{\\beta }$$\u003c/div\u003e \u003cdiv class=\"EquationNumber\"\u003e11\u003c/div\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003ewhere \u003cem\u003eα\u003c/em\u003e and \u003cem\u003eβ\u003c/em\u003e are the fitting parameters. Note the \u003cem\u003eα\u003c/em\u003e\u0026ndash;factor indicates the \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e based on pure stress impact when the applied stress is very small (\u003cem\u003eσ\u0026rsquo;\u003c/em\u003e\u003csub\u003e\u003cem\u003ev\u003c/em\u003e\u003c/sub\u003e = 1 kPa), and \u003cem\u003eβ-\u003c/em\u003eexponent indicates the stress-sensitivity.\u003c/p\u003e \u003cp\u003eFigure \u003cspan refid=\"Fig13\" class=\"InternalRef\"\u003e13\u003c/span\u003e shows the variations of \u003cem\u003eα\u003c/em\u003e and \u003cem\u003eβ\u003c/em\u003e of tested materials according to tire chip fraction (\u003cem\u003eTF\u003c/em\u003e). It can be observed in Figs.\u0026nbsp;\u003cspan refid=\"Fig13\" class=\"InternalRef\"\u003e13\u003c/span\u003e(a) and 13(b) that the \u003cem\u003eα\u003c/em\u003e\u0026ndash;factor decreased but \u003cem\u003eβ-\u003c/em\u003eexponent increased with increasing \u003cem\u003eTF\u003c/em\u003e until \u003cem\u003eTF\u003c/em\u003e\u0026thinsp;~\u0026thinsp;20%, and then \u003cem\u003eα\u003c/em\u003e\u0026ndash;factor and \u003cem\u003eβ-\u003c/em\u003eexponent approached those of pure tire rubber with a further increase in \u003cem\u003eTF\u003c/em\u003e. Note the sand-rubber mixtures at low \u003cem\u003eTF\u003c/em\u003e show sand-like behavior (Kim and Santamarina \u003cspan citationid=\"CR32\" class=\"CitationRef\"\u003e2008\u003c/span\u003e; Lee et al. \u003cspan citationid=\"CR34\" class=\"CitationRef\"\u003e2010\u003c/span\u003e; Perez et al. \u003cspan citationid=\"CR51\" class=\"CitationRef\"\u003e2016\u003c/span\u003e; Ryu et al. \u003cspan citationid=\"CR55\" class=\"CitationRef\"\u003e2022\u003c/span\u003e); thus, \u003cem\u003eα\u003c/em\u003e decreased with increasing \u003cem\u003eTF\u003c/em\u003e at low \u003cem\u003eTF\u003c/em\u003e (\u0026lt;\u0026thinsp;10\u0026ndash;20%) because the mineral conductivity of rubber is much smaller than that of sand particles. In addition, at low \u003cem\u003eTF\u003c/em\u003e, the direct contacts between sand particles, which were initially disrupted due to the presence of tire rubber particles, can develop with increasing stress due to the squeezing/distortion of rubber particles (Ryu et al. \u003cspan citationid=\"CR55\" class=\"CitationRef\"\u003e2022\u003c/span\u003e; Won et al. \u003cspan citationid=\"CR60\" class=\"CitationRef\"\u003e2023\u003c/span\u003e) (Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e); thus, \u003cem\u003eβ-\u003c/em\u003eexponent increased with increasing \u003cem\u003eTF\u003c/em\u003e at low \u003cem\u003eTF\u003c/em\u003e. At a high \u003cem\u003eTF\u003c/em\u003e (\u003cem\u003eTF\u003c/em\u003e\u0026thinsp;\u0026gt;\u0026thinsp;20\u0026ndash;40%), the sand\u0026ndash;rubber mixtures can exhibit rubber-like behavior, implying that the engineering properties will be determined by the interaction between rubber particles. Thus, \u003cem\u003eα\u003c/em\u003e and \u003cem\u003eβ\u003c/em\u003e approached those of pure tire rubber with increasing \u003cem\u003eTF\u003c/em\u003e at high \u003cem\u003eTF\u003c/em\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eBecause the connectivity between sand particles at low stress level increases with increasing size ratio, the greater \u003cem\u003eα\u003c/em\u003e but smaller \u003cem\u003eβ\u003c/em\u003e at a given \u003cem\u003eTF\u003c/em\u003e can be observed for the mixtures with larger \u003cem\u003eSR\u003c/em\u003e (Fig.\u0026nbsp;\u003cspan refid=\"Fig13\" class=\"InternalRef\"\u003e13\u003c/span\u003e). In addition, in the case of mixtures with larger \u003cem\u003eSR\u003c/em\u003e (i.e., mixtures with \u003cem\u003eSR\u003c/em\u003e\u0026thinsp;=\u0026thinsp;5.2), the smaller sand particles can penetrate rubber surface with increasing applied stress (Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e) (Xiao et al. \u003cspan citationid=\"CR62\" class=\"CitationRef\"\u003e2019\u003c/span\u003e), resulting in a limited increase in connectivity between sand particles, and a consequent decrease in \u003cem\u003eβ\u003c/em\u003e with increasing \u003cem\u003eSR\u003c/em\u003e. Figure\u0026nbsp;\u003cspan refid=\"Fig13\" class=\"InternalRef\"\u003e13\u003c/span\u003e also demonstrates that loose specimens showed smaller \u003cem\u003eα\u003c/em\u003e, but greater \u003cem\u003eβ\u003c/em\u003e because loose specimen experiences greater volume change during loading and the consequent greater contact mode change. Finally, it is noteworthy that the determined \u003cem\u003eα\u003c/em\u003e for all tested materials were very small, ranging from 0.001 to 0.095, indicating that the \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e contributed by pure stress impact is negligible at low stress levels. It is also notable that the determined \u003cem\u003eβ\u003c/em\u003e values in Fig.\u0026nbsp;\u003cspan refid=\"Fig13\" class=\"InternalRef\"\u003e13\u003c/span\u003e are comparable with those of theoretical or numerical thermal conduction model, ranging from 0.3 to 0.67 (Weidenfeld et al. \u003cspan citationid=\"CR59\" class=\"CitationRef\"\u003e2004\u003c/span\u003e; Garrett and Ban \u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e2011\u003c/span\u003e; Choo et al. \u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e2012\u003c/span\u003e).\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec15\" class=\"Section3\"\u003e \u003ch2\u003e4.2.3 Thermal Conductivity Model\u003c/h2\u003e \u003cp\u003eThe measured thermal conductivity (\u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e) shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig13\" class=\"InternalRef\"\u003e13\u003c/span\u003e was compared with the estimated \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e based on Eq.\u0026nbsp;(\u003cspan refid=\"Equ11\" class=\"InternalRef\"\u003e10\u003c/span\u003e) to assess the error incurred when predicting \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e without considering the applied stress, and Fig.\u0026nbsp;\u003cspan refid=\"Fig14\" class=\"InternalRef\"\u003e14\u003c/span\u003e shows the mean percent error (MPE) and maximum percent error. Within the permissible error range of 20% (Johansen \u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e1977\u003c/span\u003et\u0026eacute; and Konrad \u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e2005\u003c/span\u003e; Yang et al. \u003cspan citationid=\"CR64\" class=\"CitationRef\"\u003e2022\u003c/span\u003e), the thermal conductivity of pure materials and mixtures with a low \u003cem\u003eTF\u003c/em\u003e can be reasonably predicted using Eq.\u0026nbsp;(\u003cspan refid=\"Equ11\" class=\"InternalRef\"\u003e10\u003c/span\u003e). In other words, \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e of dry granular material is primarily determined by porosity, with the effect of applied stress being secondary (Yun and Santamarina \u003cspan citationid=\"CR65\" class=\"CitationRef\"\u003e2008\u003c/span\u003e; Choo et al. \u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e2021\u003c/span\u003e; Bhatt et al. \u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e2022\u003c/span\u003e). However, Fig.\u0026nbsp;\u003cspan refid=\"Fig14\" class=\"InternalRef\"\u003e14\u003c/span\u003e shows that the percentage difference between the measured and estimated \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e increased with increasing \u003cem\u003eTF\u003c/em\u003e and decreasing \u003cem\u003eSR\u003c/em\u003e, and the mixtures with \u003cem\u003eSR\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0.3 and 1.4 at \u003cem\u003eTF\u003c/em\u003e\u0026thinsp;=\u0026thinsp;20% and 40% exhibited the maximum difference, ranging from 20\u0026ndash;45%. This suggests that sand\u0026ndash;tire rubber mixtures undergo significant changes in contact mode with increasing stress, underscoring the need for a new model.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe difference between the measured and estimated \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e values in Fig.\u0026nbsp;\u003cspan refid=\"Fig14\" class=\"InternalRef\"\u003e14\u003c/span\u003e originated from the intentional neglect of \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e change according to the applied stress. Thus, the total thermal conductivity (\u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e (total)) can be expressed as the sum of the thermal conductivity based on the packing impact (\u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e (packing), Cote and Konrad model in Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e) and that based on the pure stress impact (\u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e (stress), Eq.\u0026nbsp;(\u003cspan refid=\"Equ12\" class=\"InternalRef\"\u003e11\u003c/span\u003e)), as follows:\u003c/p\u003e \u003cp\u003e \u003cem\u003ek\u003c/em\u003e \u003csub\u003e \u003cem\u003et\u003c/em\u003e \u003c/sub\u003e (total)\u0026thinsp;=\u0026thinsp;\u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e (packing)\u0026thinsp;+\u0026thinsp;\u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e (stress) = \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\chi \\bullet {10}^{-\\eta \\bullet n}+\\alpha \\bullet {\\left(\\frac{{\\sigma {\\prime }}_{v}}{1kPa}\\right)}^{\\beta }\\)\u003c/span\u003e\u003c/span\u003e (12)\u003c/p\u003e \u003cp\u003ewhere \u003cem\u003eχ\u003c/em\u003e and \u003cem\u003eη\u003c/em\u003e for sand\u0026ndash;tire rubber mixtures are given in Eq.\u0026nbsp;(\u003cspan refid=\"Equ11\" class=\"InternalRef\"\u003e10\u003c/span\u003e). \u003cem\u003eα\u003c/em\u003e and \u003cem\u003eβ\u003c/em\u003e in Eq.\u0026nbsp;(\u003cspan refid=\"Equ12\" class=\"InternalRef\"\u003e11\u003c/span\u003e) or (12) exhibited a nonlinear trend of decreasing (or increasing) followed by increasing (or decreasing) with respect to tire chip fraction, and their relationships were influenced by the packing density and \u003cem\u003eSR\u003c/em\u003e (Fig.\u0026nbsp;\u003cspan refid=\"Fig13\" class=\"InternalRef\"\u003e13\u003c/span\u003e). Thus, quantitatively representing \u003cem\u003eα\u003c/em\u003e and \u003cem\u003eβ\u003c/em\u003e is very challenging. Instead, this study examined the relationship between \u003cem\u003eα\u003c/em\u003e and \u003cem\u003eβ\u003c/em\u003e of the tested mixtures (Fig.\u0026nbsp;\u003cspan refid=\"Fig15\" class=\"InternalRef\"\u003e15\u003c/span\u003e) to provide insights for predicting the values of \u003cem\u003eα\u003c/em\u003e and \u003cem\u003eβ\u003c/em\u003e in sand\u0026ndash;rubber mixtures. Figure\u0026nbsp;\u003cspan refid=\"Fig15\" class=\"InternalRef\"\u003e15\u003c/span\u003e also includes the data of previous studies (Choo et al. \u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e2012\u003c/span\u003e; Lee et al. \u003cspan citationid=\"CR37\" class=\"CitationRef\"\u003e2016\u003c/span\u003e; Cui et al. \u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e2022\u003c/span\u003e). It can be observed in Fig.\u0026nbsp;\u003cspan refid=\"Fig15\" class=\"InternalRef\"\u003e15\u003c/span\u003e that an inverse relationship can be found between \u003cem\u003eα\u003c/em\u003e and \u003cem\u003eβ\u003c/em\u003e. The tested pure sand showed an \u003cem\u003eα\u003c/em\u003e of approximately 0.02 and \u003cem\u003eβ\u003c/em\u003e of approximately 0.4. Increasing \u003cem\u003eTF\u003c/em\u003e until \u003cem\u003eTF\u003c/em\u003e\u0026thinsp;~\u0026thinsp;20% results in a decrease in \u003cem\u003eα\u003c/em\u003e but increase in \u003cem\u003eβ\u003c/em\u003e (Fig.\u0026nbsp;\u003cspan refid=\"Fig15\" class=\"InternalRef\"\u003e15\u003c/span\u003e). A further increase in \u003cem\u003eTF\u003c/em\u003e results in an increase in \u003cem\u003eα\u003c/em\u003e but decrease in \u003cem\u003eβ\u003c/em\u003e towards the values of pure rubber. Thus, based on these observations, it is deemed feasible to roughly predict the \u003cem\u003eα\u003c/em\u003e and \u003cem\u003eβ\u003c/em\u003e of mixtures according to \u003cem\u003eTF\u003c/em\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003c/div\u003e"},{"header":"5 CONCLUSIONS","content":"\u003cp\u003eThis study comprehensively investigated the thermal conductivity (\u003cem\u003eK\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e) of sand-tire rubber mixtures with varying size ratios (\u003cem\u003eSR\u003c/em\u003e) and tire chip fractions (\u003cem\u003eTF\u003c/em\u003e) according to porosity and applied vertical stress. To separate the impact of porosity and that of applied stress on \u003cem\u003eK\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e of the tested sand\u0026ndash;tire rubber mixtures, \u003cem\u003eK\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e was measured as a function of porosity at very low stress levels. The \u003cem\u003eK\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e values of the mixtures were then measured according to the applied vertical stress. The key findings of this study are summarized as follows.\u003c/p\u003e \u003cp\u003e(1) At very low stress levels, \u003cem\u003eK\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e primarily depended on the porosity, with mixtures exhibiting lower conductivity as the tire chip fraction increased and the size ratio decreased.\u003c/p\u003e \u003cp\u003e(2) Because the dependence of \u003cem\u003eK\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e on porosity was affected by \u003cem\u003eTF\u003c/em\u003e and \u003cem\u003eSR\u003c/em\u003e, a new formula for estimating \u003cem\u003eK\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e based on the packing impact was proposed to incorporate the effects of \u003cem\u003eTF\u003c/em\u003e and \u003cem\u003eSR\u003c/em\u003e.\u003c/p\u003e \u003cp\u003e(3) As the applied stress increased, the \u003cem\u003eK\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e of the tested mixtures increased owing to the formation of a sand-to-sand thermal conduction path resulting from the squeezing/distortion of soft rubber particles.\u003c/p\u003e \u003cp\u003e(4) The tested mixtures with smaller \u003cem\u003eSR\u003c/em\u003e had \u003cem\u003eK\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e higher than that of pure sand at a vertical stress of 460 kPa, indicating that the applied stress is an important parameter when using sand\u0026ndash;tire rubber mixtures as insulating materials, and the use of larger tire chips can guarantee thermal insulation properties regardless of stress levels.\u003c/p\u003e \u003cp\u003e(5) The maximum difference between the measured \u003cem\u003eK\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e of the tested mixtures with increasing applied stress and the estimated \u003cem\u003eK\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e based on the packing impact ranged from 20\u0026ndash;45%; thus, a new formula for estimating \u003cem\u003eK\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e based on the pure stress impact was proposed. Thus, this study introduced a novel thermal conductivity model that considers both the packing and pure stress impacts.\u003c/p\u003e"},{"header":"Declarations","content":"\u003ch2\u003eAuthor Contribution\u003c/h2\u003e\u003cp\u003eThe authors confirm contribution to the paper as follows: study conception and design: H. Choo data collection: J. Oh analysis and interpretation of results: H. Choo; J. Ohdraft manuscript preparation: J. Ohrevised manuscript preparation: H. ChooAll authors reviewed the results and approved the final version of the manuscript.\u003c/p\u003e\u003cp\u003eACKNOWLEDGMENT\u003c/p\u003e\n\u003cp\u003eThis research was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (RS-2023-00208844\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eAhmed, I., and Lovell, C. (1993). 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Study on the water-heat coupled phenomena in thawing frozen soil around a buried oil pipeline. \u003cem\u003eApplied thermal engineering\u003c/em\u003e,\u003cem\u003e 73\u003c/em\u003e(2), 1477-1488. https://doi.org/https://doi.org/10.1016/j.applthermaleng.2014.06.017\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"thermal conductivity, sand–tire rubber mixtures, size ratio, tire chip fraction, applied stress","lastPublishedDoi":"10.21203/rs.3.rs-4010884/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-4010884/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eSand\u0026ndash;tire rubber mixtures are promising materials for thermal insulation. However, studies evaluating the impact of applied stress on the thermal conductivity (\u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e) of these mixtures are limited, despite the fact that energy storage tanks are typically located at deep depths where the sand\u0026ndash;tire rubber mixtures may experience changes in the connectivity between sand particles under increasing stress. Therefore, in this study, thermal needle probe tests were conducted on sand\u0026ndash;tire rubber mixtures with various size ratios (\u003cem\u003eSR\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0.3, 1.4, and 5.2) and tire chip fractions (\u003cem\u003eTF\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0.0, 0.1, 0.2, 0.4, and 1.0). To separate the impact of porosity and that of applied stress on \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e of the tested sand\u0026ndash;rubber mixtures, \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e was measured as a function of porosity at very low stress levels. The \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e values of the mixtures were then measured according to the applied vertical stress. The results of the tests performed at low stress levels demonstrated that \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e of the tested mixtures decreased with increasing \u003cem\u003eTF\u003c/em\u003e and decreasing \u003cem\u003eSR\u003c/em\u003e because of the decrease in the number of sand-to-sand contacts. All mixtures showed a decrease in \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e with increasing porosity; however, the dependence of \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e on porosity was affected by \u003cem\u003eTF\u003c/em\u003e and \u003cem\u003eSR\u003c/em\u003e. With an increase in the applied stress, \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e of all the mixtures increased. In particular, the tested mixtures with smaller \u003cem\u003eSR\u003c/em\u003e showed even greater \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e than pure sand at an applied vertical stress of 460 kPa, highlighting the significance of the applied stress on \u003cem\u003ek\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e of the tested mixtures. Most notably, this study developed a novel thermal conductivity model that considers both the packing impact and pure stress impact, providing a more comprehensive framework for predicting the thermal conductivity of sand\u0026ndash;tire rubber mixtures with varying \u003cem\u003eSRs\u003c/em\u003e, \u003cem\u003eTFs\u003c/em\u003e, porosities, and applied stresses.\u003c/p\u003e","manuscriptTitle":"Thermal Conductivity of Sand-Tire Rubber Mixtures with Varying Tire Chip Fractions and Size Ratios as a Function of Void Ratio and Applied Vertical Stress","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-03-12 18:19:27","doi":"10.21203/rs.3.rs-4010884/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"5b694868-3b37-4e86-b4b7-8339b0b58043","owner":[],"postedDate":"March 12th, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2024-07-26T20:04:13+00:00","versionOfRecord":[],"versionCreatedAt":"2024-03-12 18:19:27","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-4010884","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-4010884","identity":"rs-4010884","version":["v1"]},"buildId":"qtupq5eGEP_6zYnWcrvyt","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}