Mesoscopic study of concrete I: generation of random aggregate structure and finite element mesh
Introduction
The composite behavior of concrete is exceedingly complex and up to now many details such as strain softening, microcrack propagation, failure mechanisms and size effects etc. are still far from being fully understood. Since it is difficult to look inside concrete to observe the actual crack propagation or to experimentally determine the microscopic stress field, it has become obvious that further progress based exclusively on experimental studies will be limited. For in-depth understanding, theoretical studies based on micromechanics analysis of the interaction between various components of concrete are required. One simple and effective micromechanics approach is the microplane model1, 2which deduces the macroscopic constitutive behavior of concrete from the stress–strain relations of a number of microplanes (contact layers between rigid aggregate particles) at arbitrary inclinations inside the concrete. A more advanced micromechanics approach is the rigid particle contact model[3]in which a random system of rigid particles and their contact planes are generated and the macroscopic behavior of the concrete evaluated in an incremental manner considering in each loading step the equilibrium conditions for each particle subjected to the inter-particle forces.
Both the above models do not take into account the actual shape, size, elasticity and possible failure of the aggregate particles, and the thickness and properties of the mortar layers between aggregate particles. For these details to be considered, the microstructure of concrete needs to be realistically simulated and the concrete analyzed as a multi-phase composite material. According to Zaitsev and Wittmann[4], there are several hierachical levels of such multi-phase microstructural analysis, namely: macrolevel, mesolevel, microlevel and nanolevel. At macrolevel, the concrete is regarded as a homogeneous material with smeared cracks. At mesolevel, the concrete is treated as consisting of a coarse aggregate, mortar matrix with fine aggregate dissolved in it, and interfacial zones between the aggregate and the mortar matrix. At microlevel, the mortar matrix of the previous level is subdivided into fine aggregate and hardened cement paste with pores embedded inside. At nanolevel, the hardened cement paste is further divided into big pores (air voids) and hardened cement paste with only small pores (capillary pores) in it. With the type of computer facilities available today and at the current state-of-the-art, mesoscopic level analysis appears to be the most practicable and useful approach for evaluating the composite behavior of concrete.
Bazant et al.[5], Schorn and Rode[6]and Schlangen and van Mier[7]have performed mesoscopic study of concrete using a truss model, a framework model and a lattice model, respectively. These models have the same characteristics that the continua constituting the coarse aggregate, mortar matrix and interfacial zones are all modeled by discrete one-dimensional members which are given nonlinear stress-strain relations under tension. A more elegant and rigorous approach has been developed by Wittmann and his co-workers8, 9, 10, 11. In their approach, the mesoscopic structure of concrete is much more realistically simulated and the coarse aggregate and mortar matrix are both modeled as continua using finite elements. More recently, Wang[12]developed a nonlinear finite element method for mesoscopic study of concrete taking into account not only cracking, but also stress relief near cracks and nonlinear behavior of the constituents under compression.
Mesoscopic analysis of concrete requires the generation of a random aggregate structure (RAS) in which the shape, size and distribution of the coarse aggregate closely resemble real concrete in the statistical sense. The shape depends on the aggregate type. Generally, gravel aggregates have a rounded shape while crushed rock aggregates an angular shape. Wittmann et al.[8]have generated rounded aggregates using the morphological law developed by Beddow and Meloy [[13]] and angular aggregates as polygons each having the number of edges and the corresponding angles randomly chosen. Others, however, just assumed that the aggregate particles are spherical5, 6, 7. Regarding simulation of the size and spatial distributions, most researchers including Bazant et al.[5], Schlangen and van Mier[7]and Wittmann et al.[8]used the take-and-place method while de Schutter and Taerwe[14]used the divide-and-fill method. However, despite exerted efforts, there are still many details of RAS that have so far not been considered, as will be addressed in this paper.
If the RAS generated is represented by a mesh of finite elements, then detailed analysis using the finite element method is possible. Creating finite element meshes is often a long and tedious process which is prone to human errors. To avoid such difficulty, some investigators6, 7, 12used the projection method which projects a regular mesh onto the RAS and assigns different material properties, respectively, to the elements according to their locations. Unfortunately, this method has the major drawbacks that the shape of the aggregate/mortar boundaries cannot be closely followed unless a very fine mesh is used and that the small thickness and anisotropic behavior of the interfacial zones cannot be correctly simulated. Sadouki and Wittmann[9]and de Schutter and Taerwe[14]have generated finite element meshes for RAS that follow the aggregate/mortar boundaries exactly. However, the meshes generated are not of very good quality because many badly shaped elements are contained in the meshes. It is also not clear how the procedures used can be computerized.
In this paper, a procedure for generating RAS for rounded and angular aggregates is proposed. Particular attention is paid to the shaping and sizing of the aggregate particles. Rounded aggregates are generated with shapes following Beddow and Meloy’s morphological law, while angular aggregates are generated as polygons with prescribed elongation ratios rather than just as randomly shaped polygons. Furthermore, instead of defining the size of a particle as its average diameter as previous researchers did, it is defined herein as the width of the particle to conform with the definition used in sieving analysis. An effort is also made to ensure that all aggregate particles are coated with a minimum thickness of mortar film. For the mesh generation, the advancing front approach is used15, 16. This is today considered one of the most generally applicable automatic meshing techniques for discretization of planar domains. However, since the mesoscopic structure of concrete consists of three phases, this approach cannot be directly applied because the phase boundaries must be exactly followed as element boundaries and the nodes for elements in different phases must coincide on the phase boundaries. These difficulties are overcome by treating the different phases as separate domains and using the same starting points along the phase boundaries.
Section snippets
Gravel aggregates
The shape of a gravel aggregate can be characterized by transforming the boundary contour of each particle into polar coordinates and expressing the polar radius r of the contour as a harmonic function of the polar angle θ, as given by:in which A0 is the average radius, Aj are the amplitudes of the Fourier frequencies and αj are the corresponding phase angles. For particles from the same source, Beddow and Meloy [13] proposed a morphological law that αj are random
The take-and-place method
The RAS to be generated consists of two parts: an assembly of randomly distributed aggregate particles and the mortar matrix filling the space between the particles. Since the layout of the mortar matrix is dependent entirely on the distribution of the aggregate particles, it needs not be separately considered. For a given type of aggregate whose shape has been prescribed, the major factors to be considered in the generation of an RAS are the size distribution and spatial distribution of the
Random aggregate structures generated
Some examples of RAS generated are presented in Fig. 3. The concrete sections shown are either 150×150 mm squares or 150 mm diameter circles. In all cases, the minimum and maximum sizes of the coarse aggregate are 5 and 20 mm, respectively. Fig. 3(a) shows an RAS of a rounded aggregate with shape following Beddow and Meloy’s morphological law and Ragg=0.45. The value of γ evaluated is 0.20. Fig. 3(b) shows an RAS of an angular aggregate with Ragg=0.45 generated by option A. The elongation ratio is
Basic principle
For the mesh generation, the advancing front method15, 16is used. This method generates triangular elements along a moving front which starts at the boundary of the domain and advances towards the interior until the whole domain is covered. Since the mesoscopic structure of concrete consists of three phases each of which is to be modeled by different elements, the three phases are treated as separate domains: the aggregate domain, the mortar matrix domain and the interface domain. The aggregate
Conclusions
A take-and-place method of generating random aggregate structures for rounded and angular aggregates is proposed. It incorporates a new definition of the size of a particle as the minimum width of a slot that the particle can pass straight through to conform with the definition used in sieving analysis. As other researchers did, rounded aggregates are generated with shapes following Beddow and Meloy’s morphological law. However, angular aggregates are generated as polygons whose shape is
Acknowledgements
The financial support of the Croucher Foundation of Hong Kong for the research work presented herein is gratefully acknowledged.
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