Influence of aggregate size and volume fraction on shrinkage induced micro-cracking of concrete and mortar
Introduction
Drying of cement based composites, such as concrete and mortar, induces cracking if shrinkage of the constituents is either internally or externally restrained. For example, non-uniform drying leads to a moisture gradient, which results in non-uniform shrinkage of the specimen. Surface regions shrink faster than the inner bulk material, which results in surface cracking [2], [17]. Additionally, shrinkage might be restrained by aggregates within the composite [14]. Aggregate restrained shrinkage can lead to micro-cracking, which strongly influences the transport properties of the material [4], [18], [28]. However, the evolution of micro-cracks and their dependence on the size and volume fraction of aggregates is not fully understood yet. In [28] it was observed that permeability increases with increasing aggregate size at a constant aggregate fraction. This result is surprising, since an increase of the aggregate size at a constant aggregate fraction is usually accompanied by a decrease in the volume of interfacial transition zones (ITZs), which are known to be more porous than the cement paste. One hypothesis is that an increase of aggregate diameter at constant aggregate fraction results in an increase of micro-crack width, which is closely related to permeability [27]. The objective of this work was to establish whether this size effect really occurs. This will undoubtedly enhance the understanding of the link between micro-structure and macro-property, in particular the effect of micro-cracking on mass transport, which is a critical aspect for predicting durability and service-life. Shrinkage induced micro-cracking has been investigated recently with a lattice model [25]. However, to the authors' knowledge, the influence of the size of the aggregates on micro-cracking has not been analysed before.
In the present work, shrinkage induced micro-cracking of concrete was analysed by means of the nonlinear finite element method. In concrete, stress-free cracks form by a complex nonlinear fracture process, during which energy is dissipated in zones of finite size. There are three main approaches to describe fracture process zones within the finite element framework. Continuum approaches describe the evolution of cracks as zones of inelastic strains by means of higher-order constitutive models such as integral-type nonlocal models [3], [23]. In hybrid approaches, cracks are modelled as displacement discontinuities, which are embedded into the continuum description [7], [19], [20]. Finally, discrete approaches represent the nonlinear fracture process by means of the failure of discrete elements, such as trusses and beams [13], [24]. In recent years, one type of discrete lattice approach based on the Voronoi tessellation has been shown to be very suitable for fracture simulations [5], [6]. This lattice approach is robust and computationally efficient. With specially designed constitutive laws, the results obtained with this modelling approach are mesh-independent. In the present study, this lattice approach was used in combination with a damage-plasticity constitutive model, which was designed to result in mesh-independent responses [10].
The present study is based on several simplifications. Shrinkage is represented by an Eigenstrain, which was uniformly applied to the cement matrix only. This is representative of autogenous shrinkage, but does not fully represent transient non-uniform shrinkage due to moisture gradients which may lead to more cracking near the surface than in the center of a concrete element. Furthermore, the only inclusions considered were aggregates, which were embedded in a uniform cement paste. Micro-cracking due to other inclusions, e.g. unhydrated cement and calcium hydroxide crystals, was not considered. Aggregates were assumed to be separated from the cement paste by interfacial transition zones (ITZs), which were modelled to be weaker and more brittle than the cement matrix [16]. The mechanical response of aggregates was assumed to be elastic. Furthermore, the study was limited to two-dimensional plane stress analyses, in which aggregates are idealised as cylindrical inclusions of constant diameter.
Section snippets
Experimental results
This section summarises the findings from a recent experimental study by Wong et al. [28] as background to and motivation for the modelling work presented in this paper. Concretes and mortars with a range of aggregate contents, w/c ratio, binder type, curing period and preconditioning temperature were tested for oxygen diffusivity, oxygen permeability and water sorptivity. Thames Valley gravel (5–12.7 mm, 2% porosity) and sand (< 5 mm, 1% porosity) were used as coarse and fine aggregates
Modelling approach
In the present work, shrinkage induced cracking was described by means of a lattice approach [5] combined with a damage-plasticity constitutive law [10]. Here, the modelling approach is briefly reviewed. Nodes are placed randomly in the specimen constrained by a minimum distance dm, i.e. the smaller dm is, the smaller the average element length (Fig. 2a). Based on these randomly placed nodes, the spatial arrangement of lattice elements is determined by a Voronoi tessellation. The cross-sections
Nonlinear finite element analysis of shrinkage induced micro-cracking
Shrinkage induced micro-cracking was analysed by means of the nonlinear finite element approach described above. The elements representing the cement paste were subjected to an incrementally applied uniform shrinkage strain up to εs = 0.5 % (Eq. (5)). This value was chosen for the simulation to represent a relatively severe shrinkage of the neat cement paste on first-drying to low humidities. For example, Helmuth and Turk [12] reported first-drying shrinkage values of 0.35–0.70% for neat cement
Discussion
The edge length L of the specimen (Eq. (10)) depends on the aggregate diameter ϕ and the volume fraction ρ. Thus, the specimen area for which crack length, crack opening and hydraulic conductivity were evaluated, differed depending on the micro-structure considered. Comparison of the results for the different micro-structures is only valid if the average properties are independent of the specimen area. Here it is demonstrated that this is the case. It is assumed that the square crack pattern,
Conclusions
In the present work the influence of aggregate size and volume fraction on shrinkage induced micro-cracking was studied numerically by means of the nonlinear finite element method. The work resulted in the following conclusions:
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The length of micro-cracks decreases with increasing aggregate diameter and increasing volume fraction.
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The average crack width increases with increasing aggregate diameter and decreasing volume fraction.
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The permeability, which is related to the cube of the crack width,
Acknowledgements
The simulations were performed with the object-oriented finite element package OOFEM [21], [22] extended by the present authors. HSW and NRB acknowledge the financial support from the Engineering and Physical Sciences Research Council, UK.
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