PROBABILISTIC MICROMECHANICAL MODEL OF ENGINEERED CEMENTITIOUS COMPOSITES (ECC)

Transcription

1 PROBABILISTIC MICROMECHANICAL MODEL OF ENGINEERED CEMENTITIOUS COMPOSITES (ECC) J. Li (1) and E.-H. Yang (2) (1) Interdisciplinary Graduate School, Nanyang Technological University, Singapore (2) School of Civil & Environmental Engineering, Nanyang Technological University, Singapore Abstract Two failure criteria of Engineered Cementitious Composites (ECC) with high strain hardening capacity were introduced based on micromechanics, which provides guidelines for tailoring the material. Since the nature of heterogeneity of the cementitious composite affects the possibility of failure of the ultimate strain hardening behavior, it is a necessity to perform the reliability assessment for the material. In this paper, a simplified procedure is proposed for evaluating the probability of strain hardening behavior of ECC for preliminary design applications. To overcome the complexity of the failure criteria, a nonparametric regression algorithm known as multivariate adaptive regression splines (MARS) has been used to relate the criteria index to various micromechanical parameters in terms of fiber, matrix and interface. Probabilistic assessments on the strain hardening behavior of ECC were carried out using the First-order reliability method (FORM) based on the built MARS model. 1. INTRODUCTION The unique feature of ECCs is that they can be designed for specific requirements using the micromechanics-based strain hardening model. The model can be used to optimize mix composition of fiber, matrix and interface to achieve maximum tensile ductility while minimizing the fiber content. Specific tailoring of fiber types, fiber oiling coating, and flaw size of matrix was successfully realized with the design strategy [1-3]. The two fundamental criteria for each crack to attain strain hardening behavior are written as follows [4]. (1) (2) where J b is the complementary energy, J tip is the matrix toughness, σ fc is the first cracking strength, σ 0 is the bridging strength. As practical composite design guideline, the current theory implicitly assumes: 1) uniform fiber distribution, 2) uniform identical initial flaw size involved in the composite, and in addi- 111

2 tion, 3) no variation in micromechanical parameters, such as the bond properties of the fiber and matrix. However, material variations exist as a result of the non-ideal processing conditions and limitations. It is unavoidable that the number of fibers across any particular crack fluctuates due to the non-uniformity of the fiber dispersion. With such fluctuation of the fiber distribution, the fiber/matrix bond is affected as well, since fiber clumping in some areas could cause a reduction of the bond. The variability results in that the complementary energy J b and the bridging strength σ 0 have a range instead of a fixed value. On the other hand, there exists distributed flaw size in the cement-based matrix, which is the result of air entrapped and entrained during the composite processing. Therefore, the matrix toughness J tip and the first cracking strength σ fc will become random variables. As a result, to consider the random nature of pre-existing flaw size and fiber distribution, large margins between the complementary energy J b and the matrix toughness J tip, as well as the first cracking strength σ fc and the bridging strength σ 0 are preferred. Through a number of experiments, it has been demonstrated that for PE-ECC with J b /J tip,>3 and σ 0 /σ fc >1.2 to produce saturated strain hardening behavior, while σ 0 /σ fc is to be larger than 1.45 for PVA-ECC [5, 6]. The case is quite similar to the conventional structural issue involves a factor of safety in which failure is assumed to occur when the load of the system exceeds the resistance. To address this kind of problems, the probabilistic design approach is generally employed to take the stochastic nature of the design parameters into account. In addition, the saturation of multiple cracking is not a necessary condition of strain hardening behavior according to previous researchers work [7-9]. So the probabilistic-based micromechanical model should be extended to cover both saturated and unsaturated ECCs. In this paper, instead of the empirical method expressed in the form of fixed value as mentioned, the two criterion index are taken as random variables as well as all the design micromechanical parameters. The failure is interpreted as no strain hardening behavior of ECCs. 2. DETERMINISTIC MICROMECHANICAL MODEL ECC is designed using a well-defined tool called micromechanical model. The two fundamental requirements for strain-hardening behavior are as Eqns. (1) and (2). Therein,, and can be calculated using Eqns. (3) to (5) [4], respectively. (3) (4),c (5) where σ 0 is the maximum bridging stress corresponding to the opening δ 0, K m is the matrix fracture toughness, and E m is the matrix Youngs modulus, c is the pre-existing internal flaw size. σ B (δ) which describes the relationship between the bridging stress σ B across the crack and the crack opening δ [4], is the key to establish the deterministic micromechanical model. Analytical tools of fracture mechanics, micromechanics and probabilistics are used to derive σ B (δ). As a result, the σ B (δ) curve is expressible as a function of micromechanical parameters listed in Fig. 1. And the procedure for computing the σ B (δ) relationship can be realized numerically. 112

3 Figure 1: Micromechanical parameters of ECC 3. FRAMEWORK OF PROBABILISTIC MICROMECHANICAL MODEL When the probabilistic method is introduced into the structural engineering, it is usual to write the performance function as Eqn. (6) [10].,,,, 0 (6) where R is the function (model) that describes the capacity or strength of the structure, S is the function (model) that describes the load of the structure, and X i are random variables. In general this function is called the limit state function, and most of the basic variables are not deterministic but stochastic. Therefore it is inevitable to express the relationship (6) in terms of probability. The design formula becomes: 0 (7) where P{ } is probability that the failure event will occur, P acc is accepted value of the probability of failure, is distribution function of the standard normal distribution, and β is reliability index. The incoherency of raw materials or the variety of mixing process of ECCs will lead to large heterogeneity. By virtue of the micromechanical model, the heterogeneity can be expressed by micromechanical parameters as variables X i. Correspondingly, J b, σ 0 and J tip, σ fc in those two criteria can be regarded as R and S, respectively. Therefore, the same concept could be exploited in the development of the probabilistic micromechanical model of ECCs. Nevertheless, the function of,, and,, should be explicit and closed form of the input variables while J b and σ 0 are realized numerically. Such difficulties can be dealt with using numerical approaches to find out substitution model, such as the response surface method (RSM), the artificial neural networks (ANN) or the nonparametric regression methods. After obtaining the limit states functions, given the statistics of the input variables, probabilistic assessment approaches such as First-order reliability method (FORM) and the Monte Carlo simulation (MCS) can be used to derive the probability. In summary, the design framework of probabilistic micromechanical model is depicted as Fig

4 Step I: To create database for J b and σ 0 with micromechanical model (Section 2) Step II: To establish the substitution model for J b and σ 0 with MARS (Section 4) Step III: To conduct probabilistic assessment for some cases with FORM Figure 2: Framework of probabilistic-based micromechanical model 4. DETAILS OF MARS Multivariate adaptive regression splines (MARS) is a flexible regression modeling introduced by Friedman in 1991 [11]. Extensive applications of MARS include estimating the deformation of asphalt mixtures, analyzing shaking table tests of reinforced soil wall and analysis of geotechnical engineering systems [12-14]. It is a nonlinear and nonparametric method to model the nonlinear responses between the inputs and outputs of a system taking the form of linear combination of basic functions and their interactions, and is expressed as (8) where the coefficient c are constants estimated by the least-squares method, is a basis function (BF). BFs are piecewise linear and piecewise cubic functions. For simplicity, only piecewise linear is expressed here, which is of the form max 0, with a knot occurring at value t. Formally, max 0, (9) 0 The knot marks the end point of each segment. The MARS modeling is a data-driven process, generating basis functions by searching in a stepwise manner. And knot locations are selected through an adaptive regression algorithm. 5. PROBABILISTIC ASSESSMENT The same procedure should be made use of to conduct the probabilistic assessment for those two failure criteria of ECCs. Thereby only the first criteria will be demonstrated for brevity following the framework of probabilistic micromechanical model as Fig Substitution model of J b The dataset acquiring from the deterministic micromechanical model consists of 120 groups of data. Table 1 lists the MARS model to predict J b. Low R 2 is obtained as a result of small database but high dimensions. A vast of data pool is better to be generated in order to get optimal MARS model in which R 2 is approaching to 1. Table 2 lists the BFs of the MARS model for J b and their corresponding equations. x1 ~ x8 are corresponding micromechanical 114

5 parameters which is less than thirteen because MARS model can automatically prunes the model by removing the extraneous variables with the lowest contribution to find an optimal model. Table 1: MARS model to predict J b Outputs MARS model Type of BFs Piecewise linear No. of BFs 19 Max interaction 2 R GCV Table 2: Basis functions and corresponding equations of MARS model (cont. on next page) Basis function Equation BF1 max(0, 2.58e+010-x1) BF2 max(0, x4-0.03) BF3 max(0, x8-1.5e+006) * max(0, x5-9e+008) BF4 max(0, x4) * max(0, x5-9e+008) BF5 max(0, x4) * max(0, 9e+008 -x5) BF6 max(0,x1-2.58e+010)*max(0,x8-1.11e+006) BF7 max(0, x1-2.58e+010) * max(0, 1.11e+006 -x8) BF8 max(0, 1.5e+006 -x8) * max(0, x10-0.2) BF9 max(0, 5 -x7) * max(0, 3.22e+006 -x8) BF10 max(0, x1-2.58e+010) * max(0, 2 -x7) BF11 max(0, 5 -x7) * max(0, x8-3.11e+006) BF12 max(0, 5 -x7) * max(0, 3.11e+006 -x8) BF13 max(0, 5 -x7) * max(0, x ) BF14 max(0, x8-1.05e+006) BF15 max(0, x7-0.43) BF16 max(0, 1.05e+006 -x8) * max(0, x7) BF17 max(0, x4) * max(0, 4e+006 -x8) BF18 max(0, x4) * max(0, x7-2.29) BF19 max(0, x4) * max(0, x7) 115

6 Table 2 continued J b = e-008*BF *BF e-013*BF e-006*BF e-005*BF e-015*BF e-014*BF *BF e- 005*BF e-010*BF e-006*BF e-005*BF *BF e-005*BF *BF e-005*BF *BF *BF *BF Probabilistic assessment To perform the probabilistic asssessment, first order reliability method (FORM) is selected to assess the failure probability of this mix. In the FORM procedure, the performance functions and MARS models are incorporated into an EXCEL spreadsheet using the approach developed by Low and Tang [15]. Using this method, the reliability index β and the probability of failure P f can be determined. One group of mix is designed as Table 3. The micromechanical parameters of G d, τ 0, β were derived from single fiber pullout test, meanwhile matrix toughness test was conducted to get K m in Table 4. For simple demonstration, other parameters are assumed as constant values for the first condition and normal distributions for the second condition, but in fact, these parameters might have other distributions, e.g. exponential or Weibull distribution. Fig. 3 and Fig. 4 illustrates two conditions with different distribution of input variables, while the second condition can reflect higher heterogeneity with normal distribution of variables. Table 3: Mix proportions and micromechanical parameters Cement Slag Water/B Sand/B Fiber % B means binder (= cement + slag) Table 4: Micromechanical parameters G d (J/m 2 ) τ 0 (Mpa) β K m 0.49± ± ± ±0.30 For the first condition calculated with Figure 3, the probability of failure is 9.4% which means this mix is not quite reliable to get strain hardening behavior. For the second condition as Figure 4, the probability of failure is 29.7%, much larger than the first condition. That is, when the variations of all the parameters are considered, it is more likely to fail to meet the energy criterion. Due to the random nature of the composite material, relatively lower probability of failure should be considered to ensure the formation of multiple cracking. Generally, 1%, 2% and 5% will be recommended. A large number of case studies are needed to confirm this value for ECCs. 116

7 Figure 3: FORM for the first condition Figure 4: FORM for the second condition 6. CONCLUTIONS AND OUTLOOK This paper presents a procedure to perform probabilistic assessment of strain hardening behavior of ECC step by step, with emphasizing how and why the idea was introduced. The main conclusion is that the framework of probabilistic micromechanical model was set up based on MARS and reliability analysis method, like FORM. MARS was used to find out a substitution model for the function of R and S since their analytical expressions are quite complex or implicitly expressed numerically. Furthermore, MARS takes the advantages of no prior knowledge of the form of the function required to know. For probabilistic assessment, FORM which is the simplest way to do analysis was proposed. 117

8 From the theory of probabilistics, there are other alternatives with the same aim of MARS and FORM. For example, the artificial neural networks (ANN) is also successfully applied to a number of civil engineering problems. Comparative work could be carried out to come up with more approximate substitution model with the true model. While Monte Carlo simulation (MCS) is more accurate than FORM to assess the failure of probability due to less assumptions of the distributions of input variables. More details should be noted and improved in the procedure, such as the agreement of MARS model with true model and true distribution of each input variable. Further works to polish these issues will be accomplished in the future. REFERENCES [1] En-Hua Yang and V.C. Li, Strain-hardening fiber cement optimization and component tailoring by means of a micromechanical model. Construction and Building Materials, 2010(24): p [2] Victor C. Li, et al., Interface Tailoring for Strain-Hardening Polyvinyl Alcohol Engineered Cementitious Composite (PVA-ECC). ACI Materials Journal, (5): p [3] Shuxin Wang and V.C. Li, Tailoring of pre-existing flaws in ECC matrix for saturated strain hardening, in Fracture Mechanics of Concrete Structures, Li et al (eds) [4] En-Hua Yang, et al., Fiber-Bridging Constitutive Law of Engineered Cementitious Composites. Journal of Advanced Concrete Technology, (1): p [5] Wu, C., Micromechanical taikoring of PVA-ECC for structural applications, in civil and environmental engineering. 2001, Unviversity of Michigan. [6] Kanda, T., Design of engineered cementitious composites for ductile seismic resistant elements, in civil and environmental engineering. 1998, University of Michigan. [7] Mustafa Sahmaran, et al., Influence of Aggregate Type and Size on Ductility and Mechanical Properties of Engineered Cementitious Composites. ACI Materials Journal, (3): p [8] Mo Li and V.C. Li, High-Early-Strength Engineered Cementitious Composites for Fast, Durable Concrete Repair-Material Properties. ACI Materials Journal, (1): p [9] En-Hua Yang, Yingzi Yang, and V.C. Li, Use of High Volumes of Fly Ash to Improve ECC Mechanical Properties and Material Greenness. ACI Materials Journal, (6): p [10] Probablistic Methods for Durability Design. 1999, The European Union-Brite EuRam III. [11] Feiedman, J.H., Multivariate Adaptive Regression Splines. The Annals of Statistics, (1): p [12] Mohammad Reza Mirzahosseini, et al., Permanent deformation analysis of asphalt mixtures using soft computing techniques. Expert Systems with Applications, (5): p [13] Saman Zarnani, Magdi M. El-Emam, and R.J. Bathurst, Comparison of numerical and analytical solutions for reinforced soil wall shaking table tests. Geomechanics and Engineerin, (4): p [14] W.G. Zhang and A.T.C. Goh, Multivariate adaptive regression splines for analysis of geotechnical engineering systems. Computers and Geotechnics, : p [15] B.K. Low and W.H. Tang, Efficient Spreadsheet Algorithm for First-Order Reliability Method. Journal of Engineering Mechanics, : p