Behavior Of Concrete At Meso Scale Biology
In this paper, a methodology is proposed for simulation of behvaiour of concrete at meso scale by incorporating heterogeneity in 2-Dimensional geometrical model. The models are used for studying the mechanical properties of concrete in meso-scale. Nonlinear crushing and cracking material property of concrete is incorporated. The geometrical models of concrete is generated using randomly sized aggregates based on a given particle size distribution. During random placement of randomly sized aggregate, proper care has been taken so that none of the aggregates intersects each other, satisfies the particle size distribution as assumed and are within the matrix space. Three geometrical models are generated with different volume fractions of aggregates and then, these models are analyzed under tensile loads using nonlinear finite element software. Principle stress distribution, crack formation and load-displacement characteristics of the heterogeneous concrete are studied by using the special elements to capture cracking/crushing concrete. The study will be helpful for understanding and describing the engineering properties of cement based systems and will facilitate to analytically evaluate the influence of aggregates, their size and their distribution on the characteristics of concrete.
Keywords. Meso-scale; Finite Element Method; Tensile loading; Particle size distribution; Random aggregate structure; Crack model.1. Introduction
Concrete is the most extensively used material in constructions, because of its good strength and durability properties when compared to its cost. It is used to construct a large variety of structures and its components such as foundations, columns, beams, slabs, domes, walls, shear walls, bricks etc. Concrete is a composite material consisting of primarily aggregate, cement and water. Aggregate primarily consist of coarse gravel or crushed rocks such as limestone, or granite in addition to fine aggregate such as sand. Cement commonly consist of Portland cement along with admixtures such as fly ash, Ground granulated blast furnace slag, or volcanic ash/silica fume etc. Because of various materials and proportions of mixture in concrete are used by the engineers, the composite behavior of concrete is extremely complex to predict its properties from experiments and most of the time it is case specific. Hence, reliable predictions of the behavior of the composite material (in the present case, concrete) based on numerical simulation (models) will be helpful and reduce experimental burden.
Models on the other hand are helpful tools for understanding and describing the engineering properties of cement based systems. Models are also useful to decrease the number of trial-and-error cycles when a new material or material modification is attempted. For example, if the aggregate property (such as strength of coarse-aggregate is increased) in the concrete is changed, the mechanical properties of the new composite can be evaluated using models which can reduce the experimental effort. With the development of new computers and parallel computing systems, the analytical/numerical models have become increasingly robust and important, and they are able to provide promising results as well .
Models are categorized into overall kinetics, particle kinetics, hybrid kinetics and integrated kinetics in the case of hydration of cement-based systems. Models are also categorized based on the characteristic length scale. Multi-scale models are classified into macro-level, meso-level, micro-level, sub micro-level and nano-level . At meso-level, aggregate and the mortar matrix are explicitly differentiated and at this level finite element programs and lattice methods are proved to be useful for describing and simulating the material behavior, including the formation of cracks under different types of loading.
There are many meso-scopic models of concrete that were developed in the recent past by many researchers to understand the behavior of this composite material. Meso-scale models consisting of spherical aggregates are generated by Wriggers and Moftah  using Monte Carlo simulations. Re-generation algorithm was proposed to generate the models in case of high volume fractions of aggregates and the numerical simulations of these models with experimental data were compared. Ghouse et al.  used a unit-cell approach to simulate the numerical properties of concrete, by varying shape, size and volume fraction of aggregates in the unit-cell. Influence of Interfacial Transition Zone (ITZ) in the unit-cell model with circular aggregates was brought out. Bazant et al.  studied the influence of material composition on the behavior of concrete by developing meso-scale models using truss model to simulate the spread of cracking and its localization. A frame work model has been used in  to study the damage behavior in concrete. Schlangen and van Mier  presented a lattice model which seems to give promising results in the failure mechanism and crack face bridging in concrete. Wang et al.  and Kwan et al.  simulated the non-linear finite element analysis on mesoscoic models after generating the random aggregate structure and finite element mesh. The stress to strain transfer ratio for composite system is experimentally determined by Bhattacharya et al.  from the flexural load-deflection characteristics of beams with different compositions of composite. It is shown that the static flexural response of a layered beam can be predicted employing a third-order zigzag theory. The study has shown the procedure to handle heterogeneous materials at meso scale.
Shape of the aggregates also plays a major role on the behavior of concrete properties and they also have a great influence on stress distribution, crack initiation etc. Hafner et al.  had a detailed study on geometries of inclusion-matrix (aggregates). Sine-functions to ellipsoids, to create random and complex shapes of aggregates were applied. Zaitsev and Wittmann  generated the two-dimensional models to study crack patterns in concrete using the aggregates with polygonal- and circular- shapes. Liete et al. ,  simulated the meso-level models using lattice structures to understand the fracture processes of concrete. In the research works (, , , ), ellipsoid shapes of aggregates were assumed for the simulation of three dimensional concrete meso-level models while the other researchers like Bazant et al. , Schorn and Rode  and Schlangen and van Mier  assumed the aggregates as spherical.
At meso-level, continuum models and lattice models are used to study the behavior of concrete. Using the lattice model, damage processes of specimens with tensile loads are studied by using framework model where small struts were used  or by assuming beam elements  or truss elements [13-14]. 3-D two-cell representation of laminates with periodic fibre arrays was proposed by Zhang et al  where the matrix was assumed to be a nonlinear visco-elastic material and the fibre as an elastic one. The modelling of material nonlinearity, crack initiation and propagation were incorporated into a finite element model. Numerical simulations were under tensile loading and found to be in good agreement with the reported results obtained from experiments.
From the study it is found that the proposed models are proved to predict well for homogeneous/laminated materials. Highly heterogeneous and nonlinear brittle material like concrete has been paid negligible attention. Further, the available models need huge computational effort and hence, are not well suited to study the non-linear behavior of concrete. On the other hand, continuum models are generated in the form of two dimensional or three dimensional geometrical models as a representative volume element (RVE) and then numerical simulation is performed on these models to predict the properties of concrete.
In this study, two-dimensional continuum models of concrete are generated and numerical simulation is performed on these models to predict the non-linear behavior of concrete. The geometric models are 100X100mm cells consisting of circular/spherical aggregates with diameters ranging from 16 to 32 mm.2. Generation of geometric models
To generate two-dimensional geometric models, an algorithm is developed in MATLAB. Algorithm is based on Monte-Carlo simulation . To develop this algorithm, it is required to assign the size of the aggregates, and shape of the aggregates. But, the size of the aggregates is not constant for all the aggregate and they only depend on aggregate size distribution (from sieve analysis) whereas the shape of each aggregate is assumed to be circular for simplicity. The following steps are involved to generate the geometric models:
Step 1: Generation of aggregates:
In this step, the aggregates for the required volume following an aggregate size distribution graph are generated. A typical aggregate size distribution graph is used in the present study as shown in Fig. 1, to demonstrate the proposed methodology. From the distribution, two co-ordinates and their respective volume fractions in the whole aggregates are determined. From these two values the range and the volume for which the aggregates are o be generated can be evaluated. The volume of aggregates in a particular range is given by:
Where & are the volume of the aggregates that is to be generated in range 1 and total volume of aggregates respectively.& are the lower and upper limits of diameters of aggregate in that particular range respectively. & are the lower and upper limits of percentages of aggregates in that particular range respectively.
Now, the volume of aggregate to be generated in a particular range is known, the random sizes within the range of the aggregates will be generated. As, we have assumed that all the aggregates are of circular in shape, here size of the aggregate represents the diameter of the aggregate. The size of each aggregate is given by:
Where, is the uniformly distributed random number between 0 and 1.
Hence, volume of aggregate generated is given by:
Now, this volume is subtracted from the volume that is to be generated in this particular range and then one more aggregate is generated using the similar process. This looping must be continued until the remaining volume () will be less than the area of the second aggregate. Then this remaining volume is added to the next range and again aggregates are generated and this process must continue until the last range. The last remaining volume in the last range is converted in to one aggregate itself, so that the total volume of aggregates generated will be equal to total volume of aggregates required. The algorithm for generation of aggregates in first range, is as shown:
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