Micro-Mechanical Modeling of Portland Cement ...

Micro-Mechanical Modeling of Portland Cement Concrete Mixture 1

Mohammad Jamal Khattak Associate Professor, Department of Civil Engineering, University of Louisiana at Lafayette, USA

Abstract- Micro-mechanical model for Portland cement concrete (PCC) mixture was developed using the advanced imaging and discrete element modeling (DEM) techniques. Shape-structural model of two-phased PCC (aggregate and matrix) was developed using cluster of small discrete disk-shaped particles for each phase. Shear and normal stiffness, static and sliding friction, and inter- particle contact bonds (cohesion/adhesion) were three contact models which were employed to model the constitutive behavior of the PCC mixture. An experimental study was conducted to validate the developed DEM model. The results of the compressive test simulation reasonably predicted the stressstrain behavior of PCC mixture. Index Terms— Micro-mechanical model, PFC2D, PCC, compressive strength, Discrete element model, constitutive behavior.

I. INTRODUCTION icro-mechanical modeling of construction materials Mhave gained substantial popularity over the last two decades due to advances in technology and computer power. Voigt Approximation (upper bound) [1] and Reuss Approximation (lower bound) [2] are the common micromechanical models with simplified assumptions and noninteraction of particles amongst various phases of composite materials. The Voigt model also called as a series model, in which the two phases of a composite material are subjected to uniform stress. Knowing the volume proportion of each phase, such as aggregate and matrix in Portland cement concrete (PCC) mixture, the effective elastic modulus of the composite material can be determined by the following equation:

Ec = EmVm + EaVa

(1)

Where Ec= Effective elastic modulus of PCC, Em=Elastic modulus of matrix (cement paste + fine aggregate), Ea= Elastic modulus of aggregate, Vm=Volume fraction of matrix, and Va= Volume fraction of aggregate. On the other hand the two phases of a composite material (PCC) in Reuss model are in parallel and subjected to same strain levels due to applied uniform stress. The effective elastic modulus of PCC can be determined using the following equation:

1 Vm Va = + Ec Em E a

(2)

The results of upper bound Voigt model are similar to that of hard particles in a soft matrix (natural aggregate PCC). However, the lower bound Reuss model is comparable to soft particles in hard matrix (lightweight aggregate PCC). Hirsch [3] combined both models (Eq 1 and 2) in series with their relative proportions. Later researchers modified the Hirsch’s model by using various combinations of lower and upper bound models and applied to PCC and hot mix asphalt (HMA) mixtures [4]. Other models that have been used for composite materials include but are not limited to Counto, Concrete, Paul, Hashin and Shtrikman, Composite spheres, and Mori-Tanaka models [5-10]. The discrete element modeling (DEM) was introduced by Cundall [11] for analysis of rock mechanics, and successfully applied to granular materials (soils) [12] and solid materials with bonded contact models [13]. DEM is a numerical technique, in which Newton’s second law of motion and a finite difference scheme are utilized to study the interaction among discrete particles in contact [13]. Amongst various discrete elements (DE) codes the particle flow code (PFC)2D has higher computation efficiency and the ability to model fracture behavior, interaction, as well as the interface conditions (adhesion) between the various phases of composite materials [13, 14]. Microfabric micromechanical DEM (MDEM) has been developed and successfully applied by Buttlar and You [14] to model the stiffness behavior of HMA mixtures. In this technique, aggregate and matrix phases of HMA are modeled by cluster of finite disk shaped DE. By modeling the aggregate and matrix with a mesh of small DEs called ‘‘microfabric,’’ it is possible to model complex aggregate shapes. MDEM was later adopted by Khattak et al [15] to predict the dynamic modulus of HMA mixture under indirect tension mode. MDEM technique was further extended by Khattab and Khattak et al [16] to predict the strength and elastic modulus of fiber-reinforced polymer composites. This paper presents the application of MDEM to model the stress-strain behavior of PCC mixtures under compressive loading. Micro-mechanical parameters required to develop DE model were determined using the mechanical characteristic obtained from laboratory experiments. Advanced imaging techniques were utilized to capture the actual shape and distribution of aggregate present in PCC mixture. The digital image was than transformed to build a synthetic shape structural and

constitutive model using the PFC2D code to conduct compressive test simulation. II. MICRO-MECHANICAL CONSTITUTIVE MODELS A. Contact Stiffness and Bonds Constitutive Models In the DEM approach, the complex behavior of a composite material is simulated by simple contact constitutive models. Shear and normal stiffness, static and sliding friction, and inter-particle contact bonds (cohesion/adhesion) are three contact models which are employed. The stiffness model provides an elastic relationship between the contact force and relative displacement between particles [13]. Fig. 1 illustrates two particles A and B in contact, where normal stiffnesses are shown to have magnitudes knA and knB, and shear stiffnesses are have magnitudes of ksA and ksB, respectively. For a linear contact model, the contact stiffness is computed by assuming that the stiffnesses of the two contacting particles act in series. The forcedisplacement law of the two particles in contact in a contact-stiffness model can be expressed using the following relationships [13].

Fn = n.K n .U n

(3)

exceeds the strength of the bond (Sn and/or Ss), the bond breaks, and separation or frictional sliding can occur. The friction force Fr is given by Fr= µFn, where µ is coefficient of friction between the contacting particles [13]. B. Micro-mechanical Model Parameters The micro-mechanical parameters used for modeling are derived from macro-mechanical characteristics of the composite materials. In general the macro properties such as elastic modulus and strengths for each component of the composite materials are either determined in the laboratory or obtained from the manufacturer. Once the macro properties are known the micro parameters are calculated as follows [13]:

k n = 2.E.t ,

k s = 2.G.t

(7)

E 2(1 + ν )

(8)

G=

S n = 2.σ t .R.t , S s = 2.τ .R.t

(9)

Where E and G are the elastic and shear moduli, respectively; the σt and τ are the tensile and shear strengths, respectively; ν is the Poisson ratio, R is the radius of particle (disk) and t is the thickness of particle (unit thickness is used). III. TEST MATERIALS AND METHODS

∆Fs = −k s.∆U s (4) k A .k B K n = An n B (5) kn + kn

ks =

k sA .k sB k sA + k sB

(6)

A. Materials Ordinary Portland cement was utilized to construct PCC mixture for compression test. The cement and lime stone aggregates were obtained from the local contractor located in Lafayette, LA. The specific gravities of cement, fine and course aggregate were 3.15, 2.65, and 2.66, respectively. Conventional mix design was conducted based on ACI Stanadard practice 211.1 to determine proportions of materials. The mix design is shown in Table 1. Cylindrical samples for PCC and matrix and dog bone samples for matrix were prepared and tested at the end of 28 days curing time. Table 1. PCC mix proportion

Fig. 1 Schematic of contact and bond models Equation 3 relates the total normal force (Fn) to the total normal displacement (Un) through contact normal stiffness (Ks), where n is the total number of contacts. Equation 4 relates the incremental shear force (∆Fs) to the incremental shear displacement (∆Us) through contact shear stiffness (kn). The normal and shear strength between two contacting balls can be simulated using simple contactbond models, which are applied at the contact point (Fig. 1). When the tensile and/or shear stress at a contact

Water (kg) 194

Cement (kg) 432

Fine Aggr (kg) 606

Coarse Aggr (kg) 1013

Air (m3) 0.151 (6%)

B. Compression Test This test was conducted similar to ASTM C39 test method for compressive strength of molded PCC cylinders. A 150 mm in diameter by 300 mm high PCC specimen was loaded at an axial strain rate of 0.5 mm/min using the material testing system (MTS). Three surfacemounted LVDT were attached at 120o apart from each

other at a gauge length of 150 mm. Real time data acquisition was performed and time, load and deformations were recorded. This facilitated in capturing the complete stress-strain response of the samples. Three PCC specimens were tested under compression. Similarly three matrix specimens of 76 mm diameter and 152 mm high were tested under compressive loads using the above procedure. C. Dogbone Tensile Test The tensile strength of matrix (cement + fine aggregate) was determined using the test method, described in AASHTO T132 with slight modification. It normally involves the direct tension testing of a small briquette matrix samples. The dogbone-shaped briquette is 76 mm long and has a 645 mm2 cross section at mid length. Special self aligning grips allowed passive gripping of the specimen in the test machine and ensured uniform loading. Traditionally, the test is conducted in load controlled mode. However, in this study a strain control mode of loading was applied at a rate of 0.5 mm/min. Triplicate samples of matrix cured for 28 days were tested.

component, the DEM simulations of uniaxial compressive tests were conducted using the digital synthetic specimens in PFC2D as shown in Fig. 2. The simulated specimens contained up to 20,000 disk-shaped particles with a radius of 0.127 mm. Compressive loading was applied to the top loading plate of the synthetic specimen in PFC2D, while the bottom loading plate of the specimen was kept fixed. After a certain continuous incremental compressive loading, the response of each aggregate and each matrix particle can be monitored. Fig. 3a illustrates the compressive (black) and tensile (red) contact force chains as the result of the compressive test simulation in PFC2D. Fig. 3b demonstrates the contact force chains for the enlarged portion of Fig. 3a.

IV. IMAGE-BASED MICRO-STRUCTURAL MODEL The PCC specimen was thoroughly surface cleaned using water after diamond saw cut and allowed to air dry at room temperature. The PCC specimen was scanned to obtain a digital image of the cross-sectional area as shown in Fig. 2. The specimen image was trimmed to a 25 mm x 50 mm sized image for processing and PFC2D modeling. The brightness and contrast of the digital image was adjusted to create a black and white digital image that would interpret matrix and aggregate phases, respectively. The digital image was processed through a routine developed in C++ programming to establish a numerical logical matrix of the digital image. The logical matrix was used to establish the lists of aggregate and matrix pixels. This matrix was processed to sort the aggregates and matrix particles and establish x- and y-coordinates for particles. Such routines were also used to create the consistent coding for PFC2D with appropriate ball radii, heights, widths as well as identification of interface particles between matrix and aggregate. All such files were run though PFC2D to produce a heterogeneous synthetic PCC image-based shape-structural model as shown in Fig. 2. The lower-right corner of the synthetic model was also zoomed in for details of particles.

Matrix

Aggregate

V. RESULTS AND DISCUSSION A. DEM Simulations The micro-mechanical model parameters were determined from the macro-mechanical characteristics obtained from the laboratory test results (Table 2). After the micro-mechanical properties were assigned to each

Figure 2 Image processing and synthetic PFC2D model

Table 2. Macro- and micro-mechanical properties Limestone

Matrix

CV2 (%)

PC C

Elastic Modulus, (GPa) 501 13.8 1.0 26.8 Modulus of Rupture, (MPa) 9.21 Tensile strength, (MPa) 2.91 4.9 Comp. Strength, (MPa) 40.8 2.1 36.6 kn (GPa-m) 100 27.7 ks (GPa-m) 42 11.5 Sn (N) 2340 751 Ss (N) 1170 376 1. Obtained from manufacturer, 2. Coefficient of variation

CV2 (%)

50

22 5.1 -

Compressive Stress,kPa..

Parameters

of the PCC mixtures. The strength and strain values at failure are similar however, the elastic modulus (secant: initial slope of curve) as predicted by simulation exhibits lower value than the test data. Test Results

Simulation Results

40 30 20 10 0 0

0.1

0.2 Strain, %

0.3

0.4

Fig. 4 Stress-strain curve as a result of simulation

(a)

(b)

Fig. 3 Force chains of contact normal and shear forces Due to the heterogeneous nature of PCC mixtures local tensile stress occured due to the applied compressive stress. Once the tensile stress exceeded the bond strength between particles the bond broke and resulted in the initiation of micro crack. These localized mico cracks interconnected together and grew into macro cracks which further caused the complete failure of the specimen. The stress-strain response of the PCC synthetic model was monitor and recorded. B. Comparison of Test Data and DEM Simulation Compressive test was conducted on triplicate samples of PCC mixtures. The modulus and tensile strength were determined as shown in Table 2. It was found that the average elastic modulus of PCC was 26.8 GPa with coefficient variation (CV) of 22%. The compressive strength was 36.6 MPa with CV of 5.1%. The typical stress-strain results from laboratory test data and simulations are shown in Fig. 4. It can be seen that the PFC2D simulation reasonably predicts mechanical behavior

Various images taken from sliced PCC sample were used for the simulations. The average strength as obtained by the simulation was 32.5 MPa with CV of 7%. The elastic modulus value of 22.7 GPa with CV of 10 % was 15% lower than the actual test data. Since, small images (25 mm x 50 mm) were used for simulations, the volume fraction and distribution of aggregate largely affected the simulation results. Furthermore, the difference in stressstrain response could be attributed to 2D nature of the modeling with fewer particle to particle contacts as appose to actual 3D laboratory testing. A calibration scheme needs to be developed which can facilitate improved predictions. It is believed simulation of large sized image will represent accurate shape and size distribution of PC mixture which can improve the predictions. However, such simulation will increase the number of particles thus increasing the simulation time. Additionally, calibration factor based on aggregate fraction needs to be established. VI. CONCLUSION AND RECOMMENDATIONS Micro-mechanical model of PCC mixture was developed based on advanced imaging and DEM techniques. Actual shape, distribution and structure of aggregate were captured by high resolution scanner. A synthetic heterogeneous model of PCC in PFC2D was developed based on elastic constitution models. Uniaxial virtual compressive test simulations were conducted. Based on the laboratory tests and simulation results the following conclusions and recommendation can be drawn. 1. The elastic constitutive behavior of PCC mixtures was well captured by the developed DEM using PFC2D. The contact force chains between particles exhibited a reasonable distribution of shear and normal contact forces.

2. The average elastic modulus and compressive strength of PCC mixture using the simulation were lower than the actual laboratory test results but within the CV values. 3. It is recommended to develop calibration scheme to better predict and refine the developed DEM model for PCC. VII. ACKNOWLEDGEMENT The authors wish to express their sincere thanks to the University of Louisiana at Lafayette for using their facility and financial support. Special thanks are also extended to Mr. Mark Leblanc, laboratory assistant for assisting in experimentations. REFERENCES [1]

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