# Incremental viscoelasticity at finite strains for the modelling of 3D concrete printing

## 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.

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1. 1.

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.

2. 2.

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 $$\varvec{S}_\infty = \lambda _\infty \log [J] \varvec{C}^{-1} + \mu _\infty (\varvec{1} - \varvec{C}^{-1})$$.

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Correspondence to B. Nedjar.