Carbon nanotube (CNT) is one of the most promising nanomaterials to increase the elastic modulus of cementitious composites. However, CNT characteristics, dispersion procedure, and matrix composition/ hydration are critical and correlated with a high degree of uncertainty in achieving a desirable perfor-mance. This paper proposes a mechanics-based model to predict the elastic modulus of CNT reinforcedcementitious materials based on the Halpin-Tsai equation. The current study employs a probabilistic approach to consider the inﬂuences of multiple experimental variables and various sources of uncertainty associated with them. To this end, ﬁrst, a Bayesian methodology is adopted to perform the model valida-tion, evaluation, and selection procedures using extensive literature test data. Then, the optimum ranges of variables are found to maximize the chance of achieving a certain target modulus (herein, 50% more than control) using a reliability analysis. The proposed model suggests that the highest elastic modulusis achieved when using CNT aspect ratio ranges from 400 to 800 and concentration between 0.08 and 0.18 c-wt%. Also, the experimentally observed elastic modulus and the proposed probabilistic model exhibit a similar trend. For example, the importance of hydration age increases as CNT concentrationincreases.Finally, the proposed model can be used to reliably predict the elastic modulus, guiding futureresearchers to design more efﬁciently. /C2112020 Elsevier Ltd. All rights reserved. 1. Introduction Elastic modulus correlates the stress and strain of a material in the elastic range. It is an important mechanical property because it governs the deformation of a material when it is subjected to exter- nal loads. Concerning concrete, the elastic modulus is important forpredicting the deﬂection of structures in service. It also governs the long-term structural behaviors such as creep and relaxation. There- fore, many researchers have tried to improve the elastic modulus using various types and sizes of materials (e.g., ﬁbers and particles) to produce cementitious composite [1–8] . Fibers, such as steel andcarbon, can normally improve the strength of the composite, thus minimizing the risk of cracking and helping to reduce the cross sec- tion dimensions [9]. To delay the initiation of cracking, current research tends to use small size ﬁbers, e.g. carbon nanotubes or car- bon nanoﬁbers. Recently, the use of carbon nanotubes (CNTs) as the reinforcement in cementitious materials (cement paste, cementmortar, and concrete) has attracted considerable attention of researchers [10–13] . Although CNT exhibits the elastic modulus in the order of 1 TPa[14,15] with tensile strength ranges from 11 to 63GPa [16], the elastic modulus of CNT reinfoirced composites is much lower than that of expected with two underlying factors: dis- persion quality and bond mechanisms of CTNs within cementitious composites [5,8] . Some researchers reported that the elastic modu- lus of CNT reinforced cementitious materials could be signiﬁcantly increased against the control [10,17] . While, other researchers https://doi.org/10.1016/j.conbuildmat.2020.122092 0950-0618/ /C2112020 Elsevier Ltd. All rights reserved.⇑Corresponding author. E-mail addresses: mahyar.ramezani@louisville.edu (M. Ramezani), young.kim@ louisville.edu (Y.H. Kim), z.sun@louisville.edu (Z. Sun).Construction and Building Materials 274 (2021) 122092 Contents lists available at ScienceDirect Construction and Building Materials journal homepage: www.elsevier.com/lo cate/conbuildmat reported negligible improvement or even degradation in the elastic modulus [18,19] . The elastic modulus can be affected by various fac- tors, including, but not limited to: different surface treatments [17], various types (MWCNTs and SWCNTs) [20], and fabrication proce- dures [18]. Due to these complexities and non-standardized fabrica- tion procedures, there is a challenge to suggest a model to reliably predict the elastic modulus of CNT reinforced cementitious materi- als. Another obstacle to CNT practical applications is the cost. In cementitious materials, multi-walled carbon nanotubes (MWCNTs) are mostly used due to their lower prices. From the past market records, the cost of MWCNTs was reduced signiﬁcantly from $7000/kg to $330/kg during the period 2004 to 2019. The adventof new technologies (e.g., catalyst and energy) may further decrease the MWCNT production costs. The MWCNT price is projected to be $135/kg in 2028 [21]. When CNTs are used to fabricate nanocomposites (e.g., metal, ceramic, and polymer), several theoretical models for the elastic modulus exist in the literature. Many researchers attempted to use the conventional theoretical models such as Halpin-Tsai model [22,23] , Mori-Tanaka model [24], and Cox model [25] to recognize the mechanisms of CNTs that affect the elastic modulus of nanocomposites. The conventional models, however, are unable to accurately predict the elastic modulus of nanocompos- ites in most cases [26,27] , because these models disregard CNT dispersion and bonding characteristics within the hosting matrix. To overcome this, few researchers tried to modify the conven- tional models by including the CNT aggregation-related constants to ﬁt the model to the experimental test data [28–30] . However, these modiﬁed models are empirical (i.e., without a physical meaning) and underestimate the elastic modulus of CNT- reinforced nanocomposites below certain CNT concentrations [28]. Despite some modeling efforts, there are very few published studies that consider the dispersion and bonding mechanisms, simultaneously. Thus, it is essential to consider the inﬂuences of CNT dispersion and bonding states in both fabrication and model- ing procedures.In addition, the applicability of the conventional models for CNT reinforced cementitious materials is questionable and has not yet been examined. One of the main reasons is that many researchers reported inconsistent experimental test results con- cerning the inﬂuence of CNTs on the mechanical properties. In fact, the discrepancies are often explained by the degree of CNT dispersion within the matrix and their interfacial bond strength which are inﬂuenced by the interactions between multiple exper- imental variables [31]. Also, the inﬂuence of CNTs on the rate of cement hydration has been recently reported [31,32] , yet it was not considered as an important parameter in a mechanics-based model. The current study aims to propose a mechanics-based model for CNTs in cementitious materials. To this end, Section 2 pre- sents the base model (Halpin-Tsai model [22,23] ) and the required modiﬁcations (critical factors to consider the interac- tions between experimental variables [33–35] ) in order to improve its predictability for CNT cement composites. Besides, the proposed model is formulated in a probabilistic manner to improve its robustness against various uncertainty ( Section 3 ). Followed by compiling existing experimental test data obtained from the literature including the authors’ work ( Section 4 ), sev- eral probabilistic models are developed and the best model is selected ( Sections 5.1 and 5.2 ). Then, the performance of the pro- posed model is evaluated using two approaches: the probability of failure concept ( Section 5.3 ) and the sensitivity and important measures ( Section 5.4 ). The details are discussed in the following sections. 2. Theoretical background This section ﬁrst presents the available theoretical model used to predict the elastic modulus of composites containing CNTs. Then, the proposed critical factors are added to the model to pre- dict the elastic modulus of CNT reinforced cement composites.Notation Ec Elastic modulus of the composite ( GPa orksi) Em Elastic modulus of the matrix ( GPa orksi) Ef Elastic modulus of the ﬁber ( GPa orksiÞ Ef0 Equivalent elastic modulus of the ﬁber n Measure of the ﬁber geometry L Fiber length ( mm orin.) d Fiber diameter ( mm orin.) vf Volume fraction of the ﬁber vo Fiber orientation factor Un Fiber orientation angle an Fraction of ﬁbers with orientation angel Un j CNT concentration ( c-wt%) AR CNT aspect ratio UE Ultrasonication energy UEt Total ultrasonication energy ( J/ml) UEm Intensity of the ultrasonication energy ( J/ml/ min) UA Ultrasonication amplitude ts Ultrasonication duration (min) SP/CNTs Superplasticizer-to-CNTs ratio w/c Water-to-cement ratio t Hydration age of cement gD Dispersion factor gH Hydration age factor gM Matrix factor Cx;HðÞ Capacity model of interest (herein, Ec)x Set of measurable variables H Set of unknown model parameters CxðÞ Deterministic model cx;hðÞ Correction term (herein, gD,gHandgM) @ Standard deviation of the error term e Random variable with zero mean and unit variance @e Error term of the model LðHÞ Likelihood function pðHÞ Prior distribution k Normalizing factor fHðÞ Posterior distribution AIC Akaike Information Criterion BIC Bayesian Information Criterion Np Number of unknown model parameters Ns Number of samples gð/C1Þ Limit state function k Minimum design requirement Pf Probability of failure b Reliability index $Hfb Gradient vector of bwith respect to Hf a Row vector of the negative normalized gradient of the limit state function Ju/C3z/C3 Jacobian of the probability transformation SD0Standard deviation diagonal matrixM. Ramezani, Young Hoon Kim and Z. Sun Construction and Building Materials 274 (2021) 122092 2 2.1. Halpin-Tsai model Amongst various theoretical models, the Halpin-Tsai equation is the most widely used model to predict the elastic modulus of com- posites because of its simplicity and relative high accuracy [36]. The Halpin-Tsai model has been extensively used to predict the elastic modulus of brittle and ductile materials containing CNTs such as polymers, metals, and ceramics [37–41] . It was also used to evaluate the effective elastic moduli of steel-ﬁber reinforced concrete [42]. The Halpin-Tsai model evaluates the elastic modulus of aligned short ﬁber reinforced composites in terms of the equiv- alent ﬁber and matrix by taking into account the volume fraction of ﬁbers, their geometry (i.e., shape and dimensions), and elastic properties. The general form of the Halpin-Tsai model can be expressed as [22]: Ec¼1þnEf0v f 1/C0Ef0vfEm ð1Þ where Ecis the elastic modulus of the composite ( GPa orksi),Emis the elastic modulus of the matrix ( GPa orksi),nis the measure of ﬁber geometry ( n¼2L=dfor cylindrical ﬁbers), Lis the ﬁber length (mmorin.),dis the ﬁber diameter ( mmorin.),Ef0is the equivalent elastic modulus of the ﬁber, and vfis the volume fraction of the ﬁber which can be calculated using the densities of ﬁber and matrix. The Ef0in the Halpin-Tsai model can be modiﬁed to take into account for randomly oriented short discontinuous ﬁbers as follows: Ef0¼voEf Em/C01 voEf Emþnð2Þ where vois the ﬁber orientation factor and Efis the elastic modulus of the ﬁber ( GPa orksiÞ. The vois used to take into account for the possibility of the distributions of discontinuous ﬁbers with different orientations as vo¼P nancos4Un[43], where anis the fraction of ﬁbers having orientation angle Unwith respect to the reference axis (P nan¼1). The value of vois one for aligned ﬁbers (the conven- tional Halpin-Tsai model) and is equal to 3/8 and 1/6 for randomly orientated ﬁbers in two and three dimensions, respectively [44]. Because CNTs are randomly oriented in three dimensions due to their nanoscale sizes, vo¼1=6 is considered in this study [28,29,45] . As mentioned earlier, the Halpin-Tsai model disregards the inﬂuence of CNTs on cement hydration process, their improper dis- persion, and inadequate adhesion between CNTs and matrix.Therefore, it might lead to inaccurate predictions of the elastic modulus of nanocomposites [26,27] . To correct this bias and also to understand the mechanisms of CNTs in cementitious materials, the authors’ previous study has focused on identifying the most important experimental variables and the interactions between them affecting the mechanical properties including the elastic modulus [31,32] . The identiﬁed important variables include CNT concentration and/or volume fraction ( jand/or vf), aspect ratio (AR), ultrasonication energy ( UE), ultrasonication amplitude ( UA), superplasticizer-to-CNTs ( SP/CNTs ) ratio, water-to-cement ( w/c) ratio, sand-to-cement ( s/c) ratio, and hydration age of cement ( t). 2.2. Critical factors Through extensive experimental and analytical work, three crit- ical factors (dispersion factor [ gD], hydration age factor [ gH], and matrix factor [ gM]) were proposed as follows to include the contri- butions of multiple variables associated with CNT characteristics, ultrasonication procedure, and matrix composition/hydration to the elastic modulus. 1)gDis to correlate the CNT dispersion as a function of AR, CNT volume fraction in aqueous solution ( vf/C0w),SP/CNTs ratio, and Ultrasonication Energy Indicator ( UEI). 2)gMis to include the contributions of w/cand s/cratios. 3)gHis to consider the contributions of jand t. A brief description of these critical factors is provided in this section. More details are provided in the authors’ previous study [31]. 2.2.1. Dispersion factor ( gD) CNTs are generally pre-dispersed in water using a surfactant- assisted ultrasonication process [10,17] . In cementitious materials, poly-carboxylate based superplasticizers ( SP) are considered to be the most effective surfactant to disperse CNTs [10,11,17] . In an ultrasonication process, there is a direct relationship between the ultrasonication amplitude and the rate of the ultra- sonication energy delivered (i.e., the intensity of ultrasonication energy). The higher intensity of ultrasonication energy could lead to a faster CNT dispersion, but it also shortens the length of CNTs after optimum dispersion achieves, reducing their load carrying capacity. Therefore, UEI was proposed as UEI¼UEt UEm/C16/C17UA where UEtis the total ultrasonication energy ( J/ml),UEmis the intensity of the ultrasonication energy ( J/ml/ min), and UAis the ultrasonica- Table 1 Critical factors affecting the elastic modulus. CriticalfactorEquation Variable (unit) Mechanism gD 1/C0AR/C2vf/C0w UEI/C2SP=CNTsðÞhigD¼dispersion factor AR¼CNT aspect ratio (unitless) vf/C0w¼CNT volume fraction in aqueous solution (unitless) UEI¼ultrasonication energy indicator (index) SP/CNTs = superplasticizer-to-CNTs ratio (unitless)Ifvf/C0worAR¼0 (i.e., no CNTs added), gD¼1 (perfect dispersion). Asvf/C0wand/or ARincreases, gDgets smaller. AsUEIand/or SP/CNTs ratio increases, gDgradually reaches one. Beyond certain UEIand/or SP/CNTs ratio, gDgets constant. gM1þs=c ðÞ w=cðÞgM¼matrix factor s=c¼sand-to-cemet ratio (unitless) w=c¼water-to-cement ratio (unitless)Asw/cratio decreases, the value of gMincreases. Ass/cratio increases, the value of gMincreases. gH t 4þ0:85t/C16/C17 j gH¼hydration age factor t¼age of specimen (days) j¼CNT concentration ( c- wt%)If CNT is not added ( j=0c-wt%),gH¼1. At low CNT concentrations ( j/C200.1c-wt%), the inﬂuence of tongHis minimal. Asjincreases, gHreaches unity ( gH¼1) at later ages (i.e., a higher value of t). Note: c-wt% = based on weight percent of cement powder.M. Ramezani, Young Hoon Kim and Z. Sun Construction and Building Materials 274 (2021) 122092 3 tion amplitude which is normalized between 0 and 1. The UEIis to balance between the ultrasonication energy and amplitude to yield a similar degree of CNT dispersion and comparable elastic modulus [31]. In the case of a high UA, a lower amount of UEtmight be uti- lized to attain the same UEIas incorporating lower UAwith higher UEtdoes. Besides, as ARand/or vf/C0wincreases, providing a uniform dispersion of CNTs gets challenging (see Table 1 ). Also, up to cer- tain limits, incorporating a higher UEIand/or SP/CNTs ratio leads to a better dispersion, beyond which CNT dispersion quality is not improved [10,46] . 2.2.2. Matrix factor ( gM) If improperly designed, uniform dispersion of CNTs in aqueous solution (water+surfactant) does not necessarily lead to a homoge- neous distribution of CNTs within the cementitious materials [47]. Increasing the density of the cement matrix (higher s/cratio and/or lower w/cratio) may maintain the quality of CNT dispersion (see Table 1 ) by blocking the free movement of CNTs. 2.2.3. Hydration age factor ( gH) CNTs were found to absorb water and slowly release the retained water over time [20] due to their hydrophilic nature [48]. This might hinder the rate of progress in the hydration pro- cess of cement when using high CNT concentrations (see Table 1 ). 2.2.4. Applicability of the proposed critical factors To validate the applicability of the proposed critical factors to predict the elastic modulus, a simple regression model was developed: Ec¼gDXgHYgMZEm ð3Þ where X,Y, and Zare the unknown model parameters to ﬁt the data. The unknown model parameters were calibrated using the maxi- mum likelihood parameter estimation function against 84 experi- mental test data from the literature including the authors’ work (see the solid circles in Fig. 1 )[31]. The mean values of the cali- brated X,Y,and Zwere reported to be 6.32, 0.43, and 0.22, respec- tively [31]. A comparison between the predicted (using Eq. (3)) and the experimental elastic modulus is demonstrated in Fig. 1 . The 1:1 line (dashed line in Fig. 1 ) indicates that the predicted elastic mod- ulus is identical to its measured experimental value. Also, the solid lines represent ± 20% error. Although Eq. (3)shows an acceptableperformance to predict the elastic modulus of the investigated data (see the solid circles in Fig. 1 ), it is unable to reliably predict the elastic modulus of the new dataset (see the open triangles in Fig. 1 ). Therefore, the robustness of the regression model is ques- tionable. Besides, the regression model cannot capture the mecha- nisms of CNTs affecting the elastic modulus. In real applications, the elastic modulus increases as ARand/or vfincreases up to certain thresholds beyond which the elastic modulus degrades due to the dispersion issues. However, when using Eq. (3), the predicted elastic modulus tends to be higher at lower ARand/or vf. This might be explained by the fact that as ARand/or vf/C0wdecreases, the value ofgDbecomes larger (see Table 1 ). To overcome these limitations, in this study, the proposed analytical factors ( gD,gH, and gM; see Table 1 ) are added to the Halpin-Tsai model to predict the elastic modulus: Ec¼gDgHgM1þnEf0v f 1/C0Ef0vfEm ð4Þ In addition, Eq. (4)is formulated in a probabilistic manner to account for the high degree of uncertainty while incorporating CNTs, statistical uncertainty in estimation of the unknown model parameters, model error associated with inexact form of the model, and missing of the possible inﬂuential variables [49,50] . The proposed probabilistic model is then used to identify the most important variables and their optimum ranges using a reliability analysis. This beneﬁts future researchers and designers to identify which variables should be changed to increase the elastic modulus. 3. Methodology This section presents the background used to construct the probabilistic model. The details are as follows. 3.1. Formulation of the proposed probabilistic model This section presents the probabilistic approach based on the available deterministic Halpin-Tsai model (Eq. (1)). Additional cor- rection terms (herein, gD,gH, and gM) and model error are intro- duced to the deterministic expression to account for inherent bias and to capture the remaining variability in residuals, respec- tively [51]. The probabilistic model relates the quantity of interest (herein, Ec) to a set of measurable variables (deﬁned as x¼ðx1;x2;x3;/C1/C1/C1 Þ) such as AR,j,UEI,SP/CNTs ratio, w/cratio, s/c ratio, t, etc. The probabilistic capacity model can be written as follows: Cx;HðÞ ¼ CxðÞ þ cx;hðÞ þ @e ð5Þ where Cx;HðÞ is the capacity model of interest (herein, Ec),xis a set of measurable variables, H¼h;@ðÞ ;h¼ðh1;h2;h3;/C1/C1/C1 Þdenotes a set of unknown model parameters used to ﬁt the data, CxðÞis the selected deterministic model, cx;hðÞ is the correction term (herein, gD,gH, and gM),@is a standard deviation of the error term taken from Var Cx;HðÞ½/C138 ¼ @2,eis a random variable with zero mean and unit variance, and @eis the error term of the model. There are two assumptions in the formulation of the probabilistic model: 1) @is assumed to be independent of x¼ðx1;x2;x3;/C1/C1/C1 Þ (i.e., homoscedasticity assumption) and 2) ehas the normal distribution (i.e., normality assumption). The ﬁnal form of the probabilistic model used to predict the elastic modulus is shown below: Ec¼gDðxD;hDÞ/C2gHðxH;hHÞ/C2gMðxM;hMÞ/C21þnEf0vf/C0mh1 1/C0Ef0vf/C0mh1Emþ@e ð6Þ Fig. 1. Experimental versus predicted elastic modulus using Eq. (3).M. Ramezani, Young Hoon Kim and Z. Sun Construction and Building Materials 274 (2021) 122092 4 where vf/C0mis the volume fraction of CNTs within the cement matrix. Note that because vf/C0mused in cementitious materials is very low ( vf/C0m¼0.000038–0.001277; based on the database), the predicted elastic modulus using the deterministic Halpin-Tsai equation is almost the same as the elastic modulus of the matrix. Therefore, vf/C0mh1is used to ﬁt the data. Also gDxD;hD ðÞ ¼ 1/C0ARh2/C2vf/C0wh4 UEIh3/C2SP=CNTsðÞh5/C16/C17hi ,gHxH;hH ðÞ ¼t 4þ0:85t/C16/C17 jh6, and gMxM;hM ðÞ ¼1þs=c ðÞh7 w=cðÞh8are critical factors after adding the unknown model parameters. 3.2. Bayesian approach A Bayesian methodology is used to correct the impact of each variable in a probabilistic manner. In a Bayesian methodology, the concept of probability is interpreted as the degree of belief of a likelihood of an event by bringing prior knowledge into consider- ation. The unknown model parameters in Bayesian approach depend on the values of the experimental test data and are esti- mated using the following updating rule [52]: fðHÞ¼kLðHÞpðHÞð 7Þ where His the vector of unknown parameters, LðHÞis the likeli- hood function that captures the information from data, pðHÞis prior distribution that represents the information available before collecting the data, k¼RLðHÞpðHÞfðHÞ/C2/C3/C01is the normalizing fac- tor and fHðÞis the posterior distribution that shows the updated state of information about H. Once fHðÞ is known, the posterior mean vector ( MH) and the covariance matrix ( RHH) could be obtained. Because there is no prior information about the unknown model parameters, a non-informative prior distribution pðHÞ/1 @is used [51]. If new data becomes available in the future, Eq. (7)can still be used to update the estimates of the unknown model param- eters (i.e., means, variances, and correlation coefﬁcients) by using the current fHðÞas a new previous distribution [50]. 4. Database for constructing probabilistic model A total of 105 data from 11 different