3D concrete printing: A lower bound analytical model for buildability performance quantification
Introduction
3D printing of concrete (3DPC) is set to revolutionise the construction industry by yielding unparalleled aesthetics, quality control, cost-effectiveness and reduced construction times. Not only does it enjoy significant attention in the academia [[1], [2], [3], [4]], but also in industry where several companies are embracing this Industry 4.0 technology [[5], [6], [7], [8]]. However, as this technology is based on additive manufacturing i.e. the successive addition of material layers to form an object, more emphasis is required on material rheology which is defined as the branch of physics that describes the deformation and flow of matter. Typically, for conventional concrete casting, the rheology of concrete is assessed via a slump test. However, this simple characterisation provides insufficient information regarding the rheology of the concrete and its appropriateness for 3D printing. A small feasible rheological domain exists for cementitious materials to be 3D printable. Thus, a more detailed rheological characterisation via a rheometer is required for optimum 3D printability. Kruger et al. [9] proposed a novel approach for thixotropic characterisation of cementitious materials with the use of a rheometer, that specifically appertains to 3DPC.
In addition, material behaviour under loading is of critical importance for 3DPC. It directly influences buildability, defined as a material's ability to retain its shape after several layers have been deposited onto each other. It is important to note that geometrical nonlinearity i.e. elastic buckling can also be a failure mechanism in defining the buildability with a material, and not just physical nonlinearity itself. Several mechanical parameters are required to describe this behaviour. However, these parameters are difficult to obtain for concrete in its plastic state, as it rapidly gains strength and stiffness, especially over the first few hours after mixing. Hence, several tests are required over time to obtain the time dependent mechanical properties in the fresh state. In addition, some tests are not possible to conduct when the concrete's viscosity is too low.
Most models predicting buildability performance of materials for 3DPC are based on the fresh state mechanical properties of the material. Wolfs et al. [10] developed a numerical model to analyse the mechanical behaviour of concrete in the fresh state for 3DPC purposes. A finite element analysis (FEA) is conducted using the Mohr-Coulomb failure criterion and the results compared to an experimental 3DPC procedure. The model yielded reasonable results, with a 27.5% over-prediction of the total number of filament layers compared to the experimental results. Importantly, the correct mode of failure is predicted. Suiker [11] proposed a mechanistic model that considers both elastic buckling and plastic collapse of a 3DPC straight wall structure in the plastic state. An experimental procedure is conducted and the results compared to that of the model, which under predicted the total number of filament layers by 10%. Roussel [12] proposed an analytical model which also accounts for both physical and geometrical nonlinearity. However, as mentioned, all of the aforementioned models are based on the fresh state mechanical properties of materials (except Roussel's model that is partly based on material rheology), which are onerous to quantify experimentally.
In this research, an analytical constitutive model for buildability performance quantification is proposed based on rheological material parameters rather than the mechanical properties of materials. These parameters are obtainable for a material with any viscosity via a rheometer. This approach has the further benefit of providing valuable information regarding the appropriateness of the material for 3D printing, such as the degree of thixotropic behaviour, and can aid in material quality control. In contrast to the models based on mechanical properties, this model only accounts for physical nonlinearity, in particular plastic flow, and does not account for geometrical nonlinearity such as elastic buckling. A simplified approach is adopted to replicate Mohr-Coulomb failure, however, without determining the parameters required for said failure criterion. The ultimate aim for this analytical model is to be a simple and practical design tool to conservatively, yet accurately, determine the number of layers a 3D print can achieve depending on the rheology of the material. An experimental verification process is conducted by performing a 3D print until failure occurs which is then compared to the results obtained via the proposed model.
Section snippets
Model development
A synopsis of the model that is developed in this section is presented in Fig. 1. The material resistance to failure is characterised based on a thixotropy model developed by Kruger et al. [9]. The building rate is related to the 3DPC process and determines the normal stress on the critical bottom layer by upper layer depositions. A Mohr-Coulomb plastic failure criterion is presented in Section 2.3 and simplified via strength correction factors based on the aspect ratios of filament layers.
Experimental verification
An experimental verification procedure is conducted and presented in this section. A circular hollow column is 3D printed until failure occurs, of which the number of attained layers is then compared to the number of layers predicted with the model developed in Section 2. This also serves as an example for the practical application of the model.
Conclusion
In this research, a lower bound analytical model is developed for buildability performance quantification of 3D printable materials. The model only accounts for plastic yielding of the bottom critical layer and not for buckling in the fresh concrete state. The material strength evolution after extrusion is depicted via a bi-linear thixotropy model, which acts as the basis of the model. Thereafter, a linear building rate is defined that depicts the rate of stress increase due to the accruement
Notations
The following parameters are applicable to this model in its complete state:
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Athix is the structuration rate of the material (Pa/s).
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FAR the strength correction factor that accounts for confinement (unitless)
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g the gravitational constant taken as 9.81 m/s2
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hl the height of a filament layer (mm)
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lp the constant path length per filament layer (mm)
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NL the total number of filament layers (always rounded up)
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ρ the density of the material (kg/m3)
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Rthix the re-flocculation rate of the material (Pa/s)
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t the
Declaration of Competing Interest
None.
Acknowledgements
The research is funded by The Concrete Institute (TCI) and the Department of Trade and Industry of South Africa under THRIP Research Grant TP14062772324.
Contribution of authors
This work is part of Jacques Kruger's PhD research, executed under the supervision of Professor Gideon van Zijl and Doctor Stephan Zeranka. Mr. Kruger conducted all experimental work, the analysis, interpretation and presentation of data, as well as the writing of the manuscript. Prof. van Zijl and Dr. Zeranka assisted in
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