Mesoscale models for concrete: Homogenisation and damage behaviour

Abstract

In this paper three-dimensional geometrical models for concrete are generated taking the random structure of aggregates at the mesoscopic level into consideration. The generation process is based upon Monte Carlo's simulation method wherein the aggregate particles are generated from a certain aggregate size distribution and then placed into the concrete specimen in such a way that there is no intersection between the particles. For high volume fractions of aggregates, new algorithms for generating realistic concrete models are proposed.

The generated geometrical models are then meshed using the aligned approach in which the finite element boundaries are coincident with materials interfaces and therefore there are no material discontinuities within the elements.

The finite element method (FEM) is used in the direct computation of the effective properties of concrete. The results obtained from the numerical simulations and the subsequent homogenisation are then compared with experimental data. Furthermore numerical simulations of the damage and fracture process of concrete are performed using an isotropic damage model to model the progressive degradation of concrete. Finally, a concrete block is investigated where numerical and experimental results are discussed.

Introduction

Concrete is the most widely used construction material in the world due to its good strength and durability relative to its cost. It has a great variety of applications in the field of civil engineering. In building construction, concrete is used for footings, foundations, columns, beams, girders, wall slabs, and all important building elements.

Concrete has a highly heterogeneous microstructure and its composite behaviour is exceedingly complex. Therefore, reliable predictions of the behaviour of the material based exclusively on experimental studies have become limited. For obtaining a deeper understanding, theoretical studies based on micromechanics analysis of the interaction between various components of concrete have been developed for deducing the macroscopic constitutive behaviour of concrete. However, the microstructure and properties of the individual components of concrete and their effects on the macroscopic material behaviour have not been taken into account. For such details to be included into the computational analysis, concrete needs to be analysed as a multi-scale composite material where the microstructure is realistically simulated. Numerical simulations, coupled with theory and experiment, are considered to be an extremely important tool for successfully examining material properties by means of computational materials science.

In the numerical simulation of concrete at a mesoscopic level it is evident that several parameters such as the shape, size and distribution of coarse aggregates within the mortar matrix significantly influence the mechanical behaviour of concrete. The aim of this work is to generate random mesostructure models of the concrete material at the mesoscopic level based upon given parameters and probability distribution of aggregate particles coordinates and sizes. Then, the generated mesostructure models are to be meshed using an automatic meshing preprocessor for computational testing of the material with the finite element method (FEM) in order to analyse the effects of the mesostructure constituents on the macroscopic response of the concrete material.

Several mesoscopic models of concrete structure have been developed for studying the influence of material composition on the overall behaviour. Bazant et al. [1] developed a truss model to simulate realistically the spread of cracking and its localisation. Schorn and Rode [2] studied damage processes of concrete using a framework model. A lattice model presented by Schlangen and van Mier [3] seemed to be a promising tool for the simulation of typical failure mechanism and crack face bridging in concrete. Another approach for simulating the structure of concrete by a finite element mesh has been developed by Wittmann and his co-workers [4], [5], [6]. In this approach, mechanical and nonmechanical properties can be more realistically simulated for concrete with different compositions. Wang et al. [7] proposed a procedure for generating random aggregate structures based on the Monte Carlo random sampling principle and then developed a method of mesh generation for studying the nonlinear behaviour of concrete.

In the generation of random concrete structures the shape of aggregate particles has to be taken into account in order to study the effect of aggregate shapes upon the mechanical behaviour of concrete. Aggregate shapes have a significant influence on the stress distribution within the concrete material and, therefore, on the cracks initiation and damage accumulation up to the macroscopic failure. Two-dimensional models were generated by Zaitsev and Wittmann [8] to study crack patterns in concrete using polygonal and circular shapes for aggregates. Wang et al. [7] and Wittman et al. [5] have generated rounded aggregate particles following the morphological law developed by Beddow and Meloy [9]. For angular shapes of aggregate particles, Wang et al. [7] adjusted shapes of randomly generated polygons to have prescribed elongation ratios. However, for three-dimensional models, Bazant et al. [1], Guidoum and Navi [10], Schlangen and van Mier [3] and Schorn and Rode [2] just assumed that aggregate particles have spherical shapes. More recently, Häfner et al. [11], Leite et al. [12] and Zohdi [13] have used ellipsoid functions with varying parameters for obtaining various aggregate shapes. Garboczi [14] described a mathematical procedure using spherical harmonic functions to completely characterise concrete aggregate particles based on three-dimensional images acquired via X-ray tomography.

Aggregate size distribution plays an essential role in concrete mix design and optimisation. A proper selection of aggregate size distribution affects the main properties of concrete such as workability of concrete mix, mechanical strength, permeability and durability. The size distribution of aggregate particles can be either described by means of grading curves or obtained from sieve analysis. Fuller curve is one of the most acceptable grading curves which provide optimal aggregate packing and the best properties of concrete. Consequently, most researchers including Schlangen [15], Schlangen and van Mier [3], van Mier et al. [16] and Wittmann et al. [6] applied the Fuller curve to geometrical modelling of concrete.

In regard to the simulation of aggregate spatial distribution, different techniques have been developed to generate a mesoscopic structure that resembles the real concrete. Bazant et al. [1], Schlangen and van Mier [3], Wittmann et al. [5] and Wang et al. [7] adopted the take-and-place method for generating random particle models of low particle volume fractions. De Schutter and Taerwe [17] used another approach for the generation of random concrete structures based on the divide-and-fill method. A stochastic-heuristic algorithm was developed by Leite et al. [12] for generating a more realistic three-dimensional structure of concrete. For achieving very high aggregate volume fraction, van Mier and van Vliet [18] used an alternative algorithm called the random particle drop method for arrangement of aggregate particles. A different approach called the distinct element method was developed and extensively applied by Cundall and Strack [19] for simulating the behaviour of granular solids such as sand.

Numerical analysis of heterogeneous materials using the FEM requires the discretisation of the created mesoscopic models. Different meshing techniques have been applied for the discretisation of complex microstructures. Aligned meshing approaches have the advantage of explicitly representing the boundaries between particles and matrix. However, the mesh generation of random heterogeneous materials is rather a tedious process in three dimensions. The aligned approach has been used by several authors, e.g. [17], [20], [21], [22], [7], [6] for generating finite element meshes in two-dimensional cases. In recent years, it has been recognised that the interfacial transition zone (ITZ) between aggregates and mortar has a great influence on the initiation of microcracks in concrete. Wang et al. [7] developed a mesh generation method based on the advancing front approach in which the ITZ domains are modelled using Goodman type elements. However, Eckardt et al. [20] disregarded the ITZ and assumed that the bond between aggregates and mortar is rigid.

Schlangen and van Mier [3] and Schorn and Rode [2] used a projection method of a regular mesh onto the random aggregate structure. In the models of van Mier and van Vliet [18], van Mier et al. [16], Schlangen [15] and Schlangen and van Mier [3] the aggregates, matrix and ITZ properties are assigned to a lattice of beam elements. A similar technique has been applied by Leite et al. [12] in which the models are idealised as planar trusses and frames for two-dimensional analyses while for three-dimensional analyses the models are idealised as space structures. An alternative approach has been introduced by Zohdi and Wriggers [23] who used cubic meshes for testing mechanical responses of random heterogeneous materials. In this an unaligned approach the number of integration points is increased in order to better capture the geometry in elements with material discontinuities. Also, a quite good approximation of the geometry of heterogeneous materials has been recently obtained by Löhnert [24] using the hanging node concept in which the mesh is refined close to the geometrical boundaries of the inclusions and the cohesive zone.