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Unified solid–fluid Lagrangian FEM model derived from hyperelastic considerations

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Abstract

When writing movement equations in stresses for continuous media, it makes no difference whether the media are solid or fluid. The fundamental difference in the solution of these two problems relies on the respective constitutive laws. For solids, shear stresses are related to shear strains, resulting the Navier–Cauchy equation. For fluids, shear stresses are related to the time rate of shear strains, resulting in the Navier–Stokes equation. For solid and fluid isothermal problems, the pressure is related to the volumetric change. Based on hyperelastic solid mechanics equations, we present an alternative total Lagrangian unified model to simulate free surface compressive viscous isothermal fluid flow and simple viscoelastic solids. The proposed model is based on the deformation gradient multiplicative decomposition, which enables to establish a consistent Lagrangian constitutive law for quasi-Newtonian and non-Newtonian fluids, as well as for Kelvin–Voigt-like solids. The proposed constitutive model and the resulting positional prismatic finite element formulation are explored in numerical examples.

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Funding

This research has been supported by the São Paulo Research Foundation, Brazil—Grant #2020/05393-4.

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  1. Structural Engineering Department, University of São Paulo at São Carlos School of Engineering, Av. Trabalhador Sãocarlense, 400, São Carlos, São Paulo, 13566-590, Brazil

    Humberto Breves Coda & Rodolfo André Kuche Sanches

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  1. Humberto Breves Coda
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Correspondence to Humberto Breves Coda.

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Appendix

Appendix

In this Appendix, we show the shear stress behavior as a function of the dimensionless parameter \(\gamma_{i}\) for constant strain time rate. Before doing so, it is important to mention that both Lagrangian and Eulerian time rates are considered, and that no units were used in the analysis.

The numerical test consists of a stretched unitary cube (linear approximation), see Fig. 

Fig. 39
figure 39

Numerical test—geometry, discretization, and stress calculation

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39, with nodes at \(x_{i} = 0\) constrained at \(x_{i}\) direction. Velocity is imposed by moving nodes at face \(x_{1} = 1\) to the right in two ways: (i) for Lagrangian analysis, the current positions are given by \(y_{1} = 1 + v \cdot t\), and (ii) for Eulerian analysis \(y_{1} = e^{(v \cdot t)}\) and, in this test, correspond to the longitudinal stretch \(\lambda\). The adopted face velocity is \(v = 1\), the time step is \(4.0 \times 10^{ - 4}\), the density is \(\rho = 0\), and viscosity \(\mu = 1.0 \times 10^{ - 3}\). The Cauchy shear stress is calculated as \(\sigma_{12} = (\sigma_{11} - \sigma_{22} )/2\).

In Fig. 

Fig. 40
figure 40

Cauchy shear stress calculated for Lagrangian strain velocity

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40, we present the behavior of the Cauchy shear stress \(\sigma_{12}\) as a function of \(\gamma_{1} = \gamma_{2}\) and \(\overline{G}_{1} = \overline{G}_{2}\) for Lagrangian velocity, and in Fig. 

Fig. 41
figure 41

Cauchy shear stress calculated for Eulerian strain velocity

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41 the shear stress is presented for the Eulerian velocity. The values of \(\overline{G}_{i}\) should also vary because constants \(\gamma_{i}\) divide the stress expression (47).

From Fig. 40, we conclude that a “Lagrangian” quasi-Newtonian fluid can be modeled by adopting \(\gamma_{i} = 1.0\) and \(\overline{G}_{i} = 1.5\mu\). From Fig. 41, we conclude that an “Eulerian” quasi-Newtonian fluid can be modeled by adopting \(\gamma_{i} = 0.5\) and \(\overline{G} = 3\mu\). In Examples 4.1 and 4.2, the Eulerian quasi-Newtonian relations are adopted.

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Coda, H.B., Sanches, R.A.K. Unified solid–fluid Lagrangian FEM model derived from hyperelastic considerations. Acta Mech 233, 2653–2685 (2022). https://doi.org/10.1007/s00707-022-03237-z

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