Abstract
Within a 3D concrete printing process, the fresh concrete is aging due to hydration. One of the consequences from the purely mechanical point of view is that its constitutive relation must be defined in rate form. This restriction is taken into account in this contribution and, besides on the incremental elasticity, we moreover introduce the relaxation of the internal stresses in order to describe the creep at early age. On another hand, due to the soft nature of the material, the finite strain range is herein a priori assumed. Eventual structural instabilities during the printing process can therefore be predicted as well. On another hand, with regards to the incremental formulation of the boundary value problem, the kinematics must be adapted as well. We use for this the multiplicative decomposition of the actual deformation gradient into its known part at an earlier time and the relative deformation gradient with respect to the configuration at that time. Within a Lagrangian formulation, the incremental constitutive relations and evolution equations can then be ideally defined on the above mentioned intermediate configuration prior to be transported back to the reference configuration. In particular, the early age creep is here described through an internal variable approach the evolution of which is motivated by the generalized Maxwell model. In this work, this latter is adapted for incremental viscoelasticity. Model examples are proposed and the numerical efficiency of the proposed framework is illustrated through a set of representative simulations.
Similar content being viewed by others
Notes
We recall the definition of the Green-Lagrange strain tensor: \(\varvec{E} = \frac{1}{2} \left\{ \varvec{C} - \varvec{1} \right\} \), where \(\varvec{C} = \varvec{F}^T \varvec{F}\) is the right Cauchy-Green tensor and \(\varvec{1}\) is the second-order identity tensor.
For the hyperelastic version of the model of Eq. (13), the strain energy function would be \(W^\infty = \frac{1}{2} \lambda _\infty \log ^2 [J] - \mu _\infty \log [J] + \frac{1}{2} \mu _\infty (\varvec{C}:\varvec{1} - 3)\) in terms of the (total) right Cauchy-Green tensor \(\varvec{C}\). With the state law \(\varvec{S}_\infty = \frac{\partial W^\infty }{\partial \varvec{C}}\), this gives \(\varvec{S}_\infty = \lambda _\infty \log [J] \varvec{C}^{-1} + \mu _\infty (\varvec{1} - \varvec{C}^{-1})\).
References
Bažant ZP (ed) (1988) Mathematical modeling of creep and shrinkage of concrete. Wiley, New York
Benboudjema F, Torrenti JM (2008) Early-age behaviour of concrete nuclear containments. Nucl Eng Des 238(10):2495–2506
Craveiro F, Duarte JP, Bartolo H, Bartolo PJ (2019) Additive manufacturing as an enabling technology for digital construction: A perspective on construction 4.0. Autom Constr 103:251–267
De Schutter G, Taerwe L (1996) Degree of hydration based description of mechanical properties of early-age concrete. Mater Struct 29:335–344
Hauggaard AB, Damkilde L, Hansen PF (1999) Transitional thermal creep of early age concrete. J Eng Mech 125:458–465
Holzapfel GA (2000) Nonlinear solid mechanics. A continuum approach for engineering. Wiley, Chichester
Kaliske M, Rothert H (1997) Formulation and implementation of three-dimensional viscoelasticity at small and finite strains. Comput Mech 19:228–239
Kruger J, Zeranka S, van Zijl G (2019) 3D concrete printing: a lower bound analytical model for buildability performance quantification. Autom Constr 106:102904
Labonnote N, Ronnquist A, Manum B, Rüther P (2016) Additive construction: state-of-the-art, challenges and opportunities. Autom Constr 72(3):347–366
Lim S, Buswell RA, Le TT, Austin SA, Gibb AFG, Thorpe T (2012) Developments in construction-scale additive manufacturing processes. Autom Constr 21:262–268
Marsden JE, Hughes TJR (1983) Mathematical foundations of elasticity. Prentice-Hall, Englewood-Cliffs
Nair SAO, Alghamdi H, Arora A, Mehdipour I, Sant G, Neithalath N (2019) Linking fresh paste microstructure, rheology and extrusion characteristics of cementitious binders for 3D printing. J Am Ceram Soc 102:3951–3964
Nedjar B (2016) On constitutive models of finite elasticity with possible zero apparent Poisson’s ratio. Int J Solids Struct 91:72–77
Nedjar B (2021) On a geometrically nonlinear incremental formulation for the modeling of 3D concrete printing. Mech Res Commun 116:103748. https://doi.org/10.1016/j.mechrescom.2021.103748
Nedjar B, Baaser H, Martin RJ, Neff P (2018) A finite element implementation of the isotropic exponentiated Hencky-logarithmic model and simulation of the eversion of elastic tubes. Comput Mech 62(4):635–654
Ogden RW (1997) Non-linear elastic deformations. Dover, New York
Panda B, Lim JH, Tan MJ (2019) Mechanical properties and deformation behaviour of early age concrete in the context of digital construction. Compos B 165:563–571
Simo JC (1998) Numerical analysis and simulation of plasticity. In: Ciarlet P, Lions J (eds) Handbook of numerical analysis, vol 6. North-Holland, Amsterdam, pp 183–499
Simo JC, Hughes TJR (1998) Computational inelasticity. Springer, New York
Suiker ASJ (2018) Mechanical performance of wall structures in 3D printing processes: theory, design tools and experiments. Int J Mech Sci 137:145–170
Wolfs RJM, Bos FP, Salet TAM (2018) Early age mechanical behaviour of 3D printed concrete: numerical modelling and experimental testing. Cem Concr Res 106:103–116
Wolfs RJM, Bos FP, Salet TAM (2019) Triaxial compression testing on early age concrete for numerical analysis of 3D concrete printing. Cement Concr Compos 104:103344
Wolfs RJM, Suiker ASJ (2019) Structural failure during extrusion-based 3D printinf processes. Int J Adv Manuf Technol 104:565–584
Wriggers P (2008) Nonlinear finite element methods. Springer, Berlin
Zhang J, Wang J, Dong S, Han B (2019) A review of the current progress and application of 3D printed concrete. Compos A 125:105533
Zohdi TI (2017) Modeling and simulation of functionalized materials for additive manufacturing and 3d printing: continuous and discrete media: continuum and discrete element methods, vol 60. Springer, New York
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Nedjar, B. Incremental viscoelasticity at finite strains for the modelling of 3D concrete printing. Comput Mech 69, 233–243 (2022). https://doi.org/10.1007/s00466-021-02091-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00466-021-02091-5