How to consider the Interfacial Transition Zones in the finite element modelling of concrete?
Introduction
This study aims to suggest a method to consider the presence of the Interfacial Transition Zone (ITZ) around aggregates in concrete. The ITZs are the thin thicknesses of cement paste around aggregates with different properties than the cement paste which form the matrix of concrete (called the bulk cement paste). The main theory suggests that it is due to the cement grains in contact with the aggregate boundary at the moment of mixing (called the wall effect) which limit the water motion in this area and this water, after hydration of the cement clinkers, results in the formation of the porosity [1], [2]. As a high quantity of water is maintained in this area, a high volume of porosity is created. The ITZ thickness (δITZ) is between 20 μm and 50 μm and some experimental studies have shown that it corresponds to the mean radius of cement grains [1]. The value of the thickness and the ITZ properties could also depend on the aggregate size [3] and its mineralogic composition [4]. The differences observed in numerical results in comparison with experimental measurements on the mechanical behaviour of concrete are often explained by the fact that the ITZ is not taken into account in the model. Should consideration of the ITZ in modelling be gained?
Standard codes and models used in civil engineering do not take it into account because these models are empirical and the ITZ effects are not of a magnitude relevant to engineering application. Recent numerical models do not allow going down to the fine scale of the ITZ and justify the calculation error by the ITZ effect which cannot be digitalized. Some attempts have been performed to consider the ITZ in numerical models [5], [6], [7], [8] and in analytical models [9], [10], [11], [12] to calculate the effective properties of concrete as a heterogeneous material. Garboczi and Berryman [7] and Zheng et al. [12] have suggested a homogenization of aggregates with ITZs which then form a new inclusion in the matrix. The homogenization of aggregate + ITZ is unsatisfactory because it leads to a decrease in the strength of the aggregate and it becomes impossible to check the hypothesis of the brittleness of the ITZ which could be the cause of the concrete failure. For modelling the ITZ properties can be calibrated on experimental observations [4], [13], [3]. Many authors have suggested some methods to calculate the ITZ properties by considering some empirical assumptions [14], [15], [16], [17], but without linking with the effective behaviour of concrete. Bentz and Garboczi [5] were the only ones who suggest a model which determines the ITZ properties according to the hydration of the cement paste, but the link with the mechanical properties was not established.
In this study, the calculation of the effective behaviour of concrete was performed according to the numerical characterization of the ITZ properties. Two parts of this study are the originality of this work: the calculation of the ITZ mechanical properties according to the hydration of cement and the implementation of the ITZs in a finite element (FE) calculation.
Nadeau [16] has calculated the ITZ properties by supposing an evolution of the water-to-cement (w/c) ratio according to the distance from the aggregate boundary [18]. By considering this hypothesis, the ITZ properties were calculated according to the age of the concrete, i.e. the cement hydration. The model of Bernard et al. [19], modified by Grondin et al. [20], was retained for the calculation of the cement hydration. The ITZ around an aggregate was supposedly formed by n layers having the same thickness. In each layer the w/c ratio is different, so the volume fraction of the residual clinkers and formed hydrates was calculated in each layer according to the local hydration kinetic. Then, the effective properties of the ITZs were calculated by homogenization [20] by averaging the local properties of each layers. The homogenization of the ITZ layers keeps the initial strength of the aggregate and allows defining a brittle area between the aggregate and the matrix. This homogenization method does not take into account the brittleness within the layers, but the main goal of this study is the consideration of a brittleness area around aggregate to show its effect on the global behaviour of the specimen.
The ITZ thickness is commonly between 20 μm and 50 μm. In a finite element model it is very numerically expensive to represent it explicitly, even less to decompose it in many layers with different properties. For instance, in a 2D representative elementary volume (REV) of concrete with dimensions 50 ∗ 50 mm2, if we would represent the ITZ by one finite element with a thickness of 30 μm, we have to mesh the REV by 2.7 · 106 elements. So, in order to take into account the ITZ properties in the calculation with a limitation of the finite elements, we have considered a three-phase material for concrete: aggregates, the bulk paste, and an effective mixed interphase (EMI). The EMI around an aggregate is formed with the ITZ and a part of the bulk cement paste. Its properties are homogenized with the effective properties of ITZs and the bulk cement paste properties according to the volume fractions of ITZs and of the bulk cement paste in EMI. Calculations were performed in the finite element code of calculation Cast3m to assess the damage of concrete with and without ITZs. The damage was obtained by considering a damage model based on the microplane theory [21].
First of all, the algorithm of the model is presented in detail. Second of all, a statistical study is shown in order to define the relevant method to take into account the ITZ properties in the modelling of concrete. Then, the calculation results are analysed for concrete under a tensile load.
Section snippets
Digital representation of the EMI in a FE modelling
The ITZs surrounding aggregates in concrete could be modelled by an interfacial law in a FE code [22], but we loss the information about the evolution of the properties into the ITZs. Without taking into account the effect of the aggregate size and its mineralogic properties, Nadeau [16] has suggested a model to calculate the volume fraction of cement fcitz through the ITZ at the first contact of cement and aggregates before added water:where fcb represents the
Determination of the ITZ properties according to hydration
Grondin et al. [20] have suggested a homogenization method, based on the self-consistent scheme, to calculate the elastic properties of cement pastes at early ages according to the hydration progress. The hydration model considers the chemical reactions between clinkers and water according to the initial w/c ratio. The calculations are made for each area around aggregates. So for ITZ, we have considered n layers with a specific w/c ratio in each layer. The same calculation is made for the bulk
Influence of the number of layers on the effective behaviour of the EMI
The Nadeau's model [16] shows that the more that the number of layers is high, the more that the determination of the ITZ properties is accurate. Fig. 4, Fig. 5 show the influence of the number of layers (1 to 10) on the values of the effective Young's modulus and the effective Poisson's coefficient of the ITZ with a thickness of 20 μm at different ages. The value of the bulk cement paste is given as a reference.
The age of concrete influences only the value of the effective Young's modulus of
The problem formulation
Tensile tests on concrete specimens have been modelled in order to check the interest to take into account the ITZs at the mesoscopic scale.
A volume V of concrete is formed by two media: a matrix defined by the medium Vm and inclusions defined by the medium Vi. A load F is applied on one of the surface boundary of V with the unit normal vector . This load implies local displacement fields , local strain fields and local stress fields in each point of V. The non-linear
Conclusion
This study has led to few observations and many perspectives:
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The influence of the ITZs is very localized and the choice of their properties has an influence on the effective mechanical behaviour: for modelling at the mesoscopic scale the results are closed to the experimental macroscopic mechanical tests when the ITZs have the same tensile strength of the bulk cement paste, and closed to experimental microstructure inspection tests when the ITZs have a lower tensile strength than the bulk
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