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A multi-scale numerical approach for a consistent understanding and modelling of structural concrete

Final Report Summary - MSAXSC (A multi-scale numerical approach for a consistent understanding and modelling of structural concrete)

The main purpose of this project was the development of numerical models, based on the finite-element method, to carry out multi-scale analysis of concrete and reinforced concrete structures subjected to damage and cracking, and their application to problems of practical relevance, such as the shear failure of reinforced concrete beams and the alkali-silica reaction induced damage in massive structures.
The work was subdivided in three main tasks:
(1) 2D and 3D mesoscale model of damage and cracking in multi-phase solid materials;
(2) Interaction between steel reinforcement and concrete;
(3) Multi-scale approach to incorporate meso-structural details into the macro-scale.

Two different approaches were adopted for the mesoscale modelling. A discrete approach based on the cohesive method, and a continuum approach, using a damage model. The former one was particularly effective to describe the interaction between the crack lips under different loading conditions, whereas the latter one was preferred to determine the loss of stiffness due to progressive damage at the mesoscale, within a multi-scale analysis framework. The strategy implemented for the cohesive method is based on the insertion of cohesive elements on-the-fly, along the edges of solid elements, with a procedure allowing mesh topological changes. Therefore, it requires a continuous update of the unknowns and of the tangent stiffness matrix during the step-by-step solution. A specific cohesive law was implemented, which allows defining different behaviors in opening and sliding mode, as it is typical for concrete, with a coupling with a penalty parameter approach to deal with contact. As regards the continuum approach, a local damage model was used, characterized by a saw-tooth softening law. In this framework, the explicit representation of the material microstructure and the use of distinct constitutive laws for different phases and interface regions, allows the modelling of multi-phase solid materials. Such tools have been implemented and optimized for quasi-static loading conditions, for both two-dimensional and three-dimensional geometries. They are also available for parallel simulations.
As regards the interaction between reinforcement and concrete, the embedded model was chosen, because of its versatility and ease-of-use in modelling complex reinforcement arrangements, with many bars, not necessarily straight. Both perfect and partial bond between reinforcement and concrete are possible. Such a method allows the analysis of reinforced concrete members at the structural scale, where it can be coupled with a damage law or the cohesive method to model material deterioration.
The multi-scale analysis was tackled with the FE2 homogenization technique. An algorithm was implemented to deal with the interaction between the macro-scale, where the material is considered as homogeneous, and the mesoscale, where a representative volume element is used to explicitly model the microstructure. Such an approach is based on a continuous transfer of information across scales: deformations are passed from the macro- to the mesoscale, where the evolution of damage is computed, and the resulting stiffness is transferred back to update the macroscale model. A local damage model was used at both scales.

As far as the scientific objectives of the project are concerned, the implemented methodologies have been used to analyze the progressive damage in a representative volume of concrete at the mesoscale and its effects on the structural response. Focus has been addressed to the study of the crack formation and the interaction between the crack surfaces under different loading conditions (pure mode I and mixed mode), permitting to derive a relation between the geometrical and mechanical parameters of the microstructure and the tractions transferred across cracks subjected to shear. This is a key point for a complete understanding of the mechanism of shear strength of reinforced concrete beams. A specific study was also carried out at the mesoscale level to analyze the effects of residual stresses in concrete. The introduction of eigenstresses in the concrete matrix, mimicking the effect of the long-term drying shrinkage, permitted to explain partially the appearance of residual deformations during tensile cyclic loading, for which no exhaustive explanations were provided up to now.
The multi-scale approach was used to analyze the long term effect of the alkali-silica reaction in the dams, in terms of material deterioration and loss of stiffness. In this context, the damage induced by the pressure exerted by the products of the alkali-silica reaction is fully resolved at the level of the representative volume of concrete, by taking into account the correct degree of confinement, derived from the macro-scale. Then, the response is homogenized and transferred to the structural level, where the equilibrium is solved with respect to applied loads and boundary conditions. It is worth noting that such an upscale approach from the mesoscale to the structural scale is a novelty with respect to previous reported work on the multi-scale analysis applied to the problem of alkali-silica reaction, which focused on the coupling between the micro- and the mesoscale. The proposed study permits to assess to which extent the alkali-silica reaction can undermine the stability of a dam, helping to identify the correct time of intervention on existing structures.
The effectiveness and the versatility of the implemented methodologies is proved by their successful application to other kind of problems. For instance, the extrinsic cohesive method has been applied to study dynamic fragmentation in glass. This represents an innovative result from the modelling and computational point of view, since it is the first attempt to numerically model the fragmentation of tempered glass with an explicit representation of cracks, needed to identify the fragments. Besides, based on the obtained results, the effect of the plate thickness on the number of fragments, an aspect that had not yet been fully understood, has been quantified.

The newly implemented methods and the obtained results represent a significant advancement in the meso- and multi-scale modelling of concrete structures. They are relevant for the scientific community engaged in the development of Standards, and for private and public societies in charge of the maintenance and repair of existing structures.

All the aforementioned modelling tools have been implemented in the open-source finite-element library Akantu, developed by the team of the host institution (Computational Solid Mechanics Laboratory, Ecole Polytechnique Fédérale de Lausanne). Besides the scientific aspects, the contribution to the development of Akantu is a significant asset of the present project. Such a code constitutes a high quality contribution to European excellence and competitiveness. In fact, it is a useful tool for researches in a wide range of fields, from material science, to mechanics of materials and structural mechanics. The code Akantu may have also applications in the industry, where it can be used to perform numerical simulations supporting the development of new materials, the interpretation of experimental tests, and the design of structures and components. In this regard, Mauro Corrado was strongly committed in activities to promote the use of Akantu outside the academic field. Together with other colleagues, he supervised the development of the first stage of a graphical user interface, that will make Akantu more attractive for the industry.