A 3D Lattice Model For Fracture Of Concrete : A Multiscale Approach

Abstract

It is quite well known that fracture behavior of concrete is complex and is influenced by several factors. Apart from material properties, geometric parameters influence fracture behavior and one notable phenomenon is size effect. The existence of the size effect in concrete is well known and various attempts to model the behavior is well documented in literature. However the approach by Bazant to describe the size effect behavior in concrete has received considerable attention. The major advantage of developing the size effect law for concrete is the ability to describe the fracture behavior (namely failure strength) of large size structures inaccessible to laboratory testing. The prediction of size effect is done on the basis of laboratory testing of small size geometrically similar structures. In all the models developed earlier heterogeneity of concrete has not been quantitatively simulated. Hence, the complete description considering heterogeneity in concrete is attempted using the lattice model to understand size effect behavior in concrete. In the present study, a detailed description of the heterogeneity in concrete is at- tempted by 3D lattice structure. Analytical treatment to gain insights to fracture behavior is difficult and hence a numerical approach capable of handling the het- erogeneous nature of the material is adopted. A parametric study is performed to understand the influence of various model parameters like mesh size, failure criterion, softening model. The conventional size effect studies for 2D geometrically similar structures are performed and a comparison is done with experimentally observed behavior. The variation of fracture process zone with respect to structure size is observed as the reason for size effect. The influence of variation in properties of ag- gregate, matrix and interface are studied to explain the deviation in pre-peak and post-peak response. A statistical study is performed to establish the size dependence of linear regression parameters (Bf ‘t and D0) which are used in Bazant size effect law. An analytical framework is also proposed to substantiate the above results. Size effect in concrete is generally attributed to the effect of depth viz. the dimension in the plane of loads. However although the effect of thickness viz. a dimension in a plane perpendicular to that of the loads is not considered in concrete. The same is quite well known in fracture of metals. Therefore the variation in grading of aggregates along with the influence of thickness on fracture behavior is analysed. To understand the thickness effect a comparison of 2D and 3D geometrically similar structures is performed to understand the effect of thickness on fracture parameters. Heterogeneity is a matter of scale. A material may be homogeneous at a coarser scale while at a finer scale it is heterogeneous. Hence only way to capture the effect of the behavior at micro level on the behavior at meso level particularly in a heterogeneous material like concrete is by a multi-scale modelling. The best numerical tool for multiscale model of a heterogeneous material is lattice model. The heterogeneous nature of concrete is not just due to the presence of aggregates but is evident right from the granular characteristics of cement. The hydration of cement grain leads to the development of products with varying mechanical and chemical properties. As the micro-crack initiation and development of thermal cracking is observed at the micron level, understanding of hydration behavior in concrete can be thought of as a pre-requisite for complete understanding of fracture behavior. The properties of matrix and interface observed during hydration modelling can also be used as an input for fracture predictions at upper scale models (eg. mesoscale). This can also be used to study the coupling of scales to understand the multi-scale fracture behavior in concrete. A numerical model is hence developed to study the hydration of concrete. Due to the existence of complex mechanisms governing the hydration behavior in con- crete and the large number of parameters affecting its rate, the hydration of a grain is assumed to proceed in isolation. A single particle hydration model is developed to study the hydration of isolated grain. A shrinking core model usually used to describe the burning of coal is adopted as a base model for analytically describing the hydra- tion behavior. The shrinkage core model in literature is modified to be applicable to hydration of cement matrix. The effect of particle diameter as well as changing water concentration is incorporated into the model whereas the influence of reduction in pore sizes as well as the effect due to embedding of particles and the constraint due to hydration of neighbouring particles is accounted using correction factor. The effect of temperature on rate of hydration is considered to be independent of the physical and chemical aspects of grain. Hence a temperature function developed using Arrhe- nius equation and activation energy is incorporated separately. The porous nature of reaction products affects the diffusivity leading to the development of tortuous path for flow of water through the hydrated portion. Knowing the tortuosity it is possible to obtain the diffusivity which in turn can be used as an input to the lattice model. An algorithm is developed to determine the tortuosity in diffusion of water through the reaction products. The tortuosity depends on the distribution of pores in the hydrated system. This requires the use of simulation technique to generate the initial position of voids. A simulation technique is also required to generate the initial con- figuration of hydrating cement system. In order to generate the initial configurations of such systems a numerical technique to generate a large scale assembly of particles is proposed. In the present work, parameters of Bazant's size effect law Bf’t and D0 are shown to depend on structure size and heterogeneity. The span to thickness ratio of the structure increases fracture energy and also substantially influences the response of structure. The variation in failure load occurring due to the heterogeneous nature of the material is shown to follow a normal distribution. The fracture behavior of a material is seen to be influenced strongly by the variation in the strength of matrix and interface. The model proposed to describe the hydration process of cement can be used to determine the properties of matrix and interface. The degree of hydration as well as the embedded centre plane area can be adopted as a measure of strength of matrix and interface. The understanding of the hydration process and the wall effect around the aggregate surface can possibly improve our ability to predict the strength of interface. The material strength of the interface is certainly a necessary input to the lattice model. Infact experimental determination of interface strength is a lot more complicated than the present numerical approach. The only weakness of the present numerical approach is the assumption regarding certain empirical constants which of course may be improved further. Understanding of material behavior can be further improved if a molecular dynamics approach is adopted to describe the hydration behavior of cement. The approach via molecular dynamics is suggested as a problem for future research.