Quasi-brittle porous material: Simulated effect of stochastic air void structure on compressive strength
Graphical abstract
Introduction
The strength and behavior of a material depends on its internal structure. In particular, in the case of a porous material, it is known that porosity affects strongly e.g. elastic modulus, yield behavior and tensile and compressive strength. Concrete and other cement-based materials are widely used porous materials where porosity has a crucial role in the suitability of the materials for their numerous uses. For this reason, and in addition to the need to make these materials more environmentally friendly by utilizing additives, the effect of the porosity on the behavior of the material has been extensively studied for several decades. The importance of porosity, pore distribution and potential additives for the compressive strength of cement-based materials has been clarified in several studies, see for example, [[1], [2], [3], [4], [5], [6], [7], [8], [9], [10]]. These studies have shown that porosity has a strong effect on compressive strength and, in addition, in the case of additives, changes in compressive strength come at least in part due to a change in porosity. The phenomenological, experiment based, equations introduced by Balshin [11], Ryshkewitch [12], Schiller [13] and Hasselman [14] have been utilized to describe the dependence of the compressive strength of cement-based materials on porosity for decades. Analytical approaches have also been developed to describe this dependence [15,16].
The pores of cement-based materials are often classified into different categories based on their size and type: e.g. gel (<10 nm), capillary (0.005–10 μm), and air (macropores 0.005–5 mm) pores. Also, the cracks due to shrinkage at aggregate-cement interface are common. The ranges of pore categories are not strictly defined and different sources cite slightly different ranges, see, e.g., [[17], [18], [19], [20], [21], [22], [23]]. Capillary pores and other larger pores are found to be responsible for reduction in strength of cement paste, while the effect of gel pores are noticed to be negligible [1,4]. Several measurement methods have been applied to the porous structure characterization of cement-based materials, such as mercury intrusion porosimetry (MIP), vapor sorption, scanning electron microscopy, and X-ray tomography. Frequently used mercury intrusion as well as vapor sorption techniques are indirect methods, which mean that the interpretation of the results usually requires some assumptions and theoretical simplifications [[24], [25], [26], [27]]. Mercury intrusion technique allows pore sizes over a broad range to be measured, whereas the vapor sorption techniques are more sensitive to gel scale pores. In MIP, the closed pores remain nonintruded, while for those large pores, which are only accessible by very narrow throats, the size is misinterpreted; this mechanism is commonly referred to as the “ink bottle” effect [22]. To observe a direct physical structure of the microstructure different imaging techniques have been utilized. Scanning electron microscopy produce images of microstructure features from 2D cross sections with resolution of submicron ranging from capillar porosity scale [23,28,29]. For the acquisition of 3D information of the pores, some studies using X-ray computed tomography have been presented [[30], [31], [32]]. Micro-tomography provides resolution in the order of few microns.
In recent decades, the finite element method has been extensively utilized in the study of the behavior of cement-based heterogeneous materials. These studies are focused on, e.g., the dependence of the elastic properties on the aggregate content and on the shape and distribution of the aggregate, see for example, [33,34]. In addition to the elastic properties, the strength and fracture behavior of the material and its dependence on aggregate properties have been considered, e.g., in references [[35], [36], [37], [38], [39]]. The most common approach in these studies is to divide the cement-based material into the cement matrix, the aggregate, and the interfacial transaction zone, each of which has its own material properties. In addition to the effect of aggregate, the effect of porosity on the mechanical behavior of cement-based material has also been studied by the finite element method [40,41] and by the discrete element method [42].
This paper aims to investigate the effect of variations in porous structure on the compressive strength of quasi-brittle material through a numerical approach. X-ray microtomography was used to determine 3D pore structure of two mortar samples, without and with 10% green liquor dregs (GLD). GLDs, which are side streams of pulping mills containing environmentally hazardous metals, are a recycling challenge. Its treatment and recycling are under investigation by several parties. However, in this study, the role of GLD as an additive of mortar is limited to the generation of different structures for porosity studies. To simulate the compressive strength, 128 cubes with the sides of 0.5 mm and different pore volume ratios, distributions and positionings were constituted based on the measured pore size distributions. Due to the cube size of the simulations and the measurement resolution the pore sizes were limited to between 20 μm and 500 μm. The statistically relevant number of pore structures was used in the simulations to reveal the dependence of compressive strength solely on porosity, without interference with other factors due to composition and manufacturing process.
Section snippets
Preparation of mortar samples
Mortar samples with dimensions 40 mm × 40 mm × 160 mm were prepared according to the standard EN 196-1. Two different mortars were produced: Sample 1 consisted of CEN standard sand, water and cement and in Sample 2 10% of cement weight was replaced by dried and milled green liquor dregs. The six parallel samples with size approximately 5 mm × 5 mm × 5 mm were cut from the produced samples.
X-ray tomography imaging and image analysis
Internal 3D-structure of mortar samples was measured using SkyScan 1172 X-ray microtomographic scanner;
Constitutive model
The constitutive model used is the damaged plasticity model available in Abaqus [46]. The original model is presented in the reference [47] and its modifications in the reference [48]. The stress-strain relation is governed bywhere σ, d, D0el, ε and εpl are the Cauchy stress, scalar stiffness degradation variable, undamaged elastic stiffness of the material, total strain and plastic strain, respectively. The stress-strain behavior corresponding to the undamaged (d = 0) material
Simulation
In the simulations, the side of the subcube was 0.5 mm. An example of structure is shown in Fig. 4. The x3-directional displacement was set to zero at the bottom of the sample and 2 × 10−3 mm displacement was applied into negative x3-direction at the top of the sample, see Fig. 4. Displacements in the x1- and x2-directions were not restricted at any point in the subcube to ensure equal freedom in the x1- and x2-directions for each node. Examples of the simulated stress-strain behavior of eight
Results
The simulation procedure was performed on 128 subcube specimens. The measured pore distributions of Sample 1 and Sample 2 were the basis for two sets of structures, S1 and S2; 64 subcube specimens were taken from each. The simulated maximum stresses as a function of porosity and number of pores in the subcube specimens are presented in Fig. 12 (see also Table A.5, Table A.6 (Appendix)). The number of pores divides the subcubes into two separate groups; the number of pores in S1 varies between 0
Discussions
The simulations were performed on CSC's (the Finnish IT center for science) new supercomputer Puhti. For one standard sample simulation, a wall clock time of 36 h was set aside, during which time all samples exceeded the maximum stress point. The computation time required to simulate the entire compressible distance was considerably longer, varying from several days to two weeks. The wall clock time needed for the simulations was decreased by paralleling the computation, see Fig. 16. In the
Conclusions
A large number of simulations were performed to reveal the compressive strength-porosity relation for quasi-brittle material at low porosity region from zero to 3.5%. 128 stochastic structures that included only variability due to porosity, pore distribution and pore placement were generated for simulations based on measured distributions. Practical factors related to material, measurement method and simulation approach limited the porosity, the side length and the number of pores of a single
CRediT authorship contribution statement
Anna-Leena Erkkilä: Conceptualization, Methodology, Software, Formal analysis, Writing - Original Draft, Writing - review & editing, Visualization
Teemu Leppänen: Conceptualization, Methodology, Software, Validation, Formal analysis, Writing - Original Draft, Writing - Review & Editing, Visualization
Jussi Virkajärvi: Formal analysis, Data Curation, Writing - Original Draft and Review
Joni Parkkonen: Investigation, Data Curation, Writing - Original Draft
Leena Turunen: Resources, Project
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
This research was supported by Regional Council of Central Finland and European Regional Development Fund (ERDF) - Sustainable Bioresidual Concrete. Simulations were performed by CSC's (the Finnish IT center for science) Puhti supercomputer and commercial software Abaqus which was licensed to CSC. Samples were prepared in the concrete testing laboratory of JAMK University of Applied Science.
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