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Computational electro-chemo-mechanics of lithium-ion battery electrodes at finite strains

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Abstract

A finite strain theory for electro-chemo-mechanics of lithium ion battery electrodes along with a monolithic and unconditionally stable finite element algorithm for the solution of the resulting equation systems is proposed. The chemical concentration and the displacement fields are introduced as independent variables for the formulation diffusion-mechanics coupling. The electrochemistry of the surface reaction kinetics is imposed at the boundary in terms of the Butler–Volmer kinetics. The intrinsic coupling arises from both stress-assisted diffusion in electrodes and ion mass flux induced volumetric deformation. We demonstrate the theoretical modeling aspects and algorithmic performance through representative initial boundary value problems. The proposed finite strain theory is especially well suited for electrode materials like silicon which exhibit large volume changes during lithium insertion/ extraction. We demonstrate the inadequacy of small-strain theories for diffusion-mechanics coupling in silicon based anode materials. The proposed numerical algorithm shows excellent performance, demonstrated for 2D and 3D representative numerical examples.

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Notes

  1. Explicit definition of pressure gradient: \({\mathcal C}^0\) continuous shape functions are poor in the computation of the pressure gradient term, which is needed for the computation of Li\(^+\) ion flux (44)\(_2\). In order to overcome this difficulty, we propose a semi-implicit finite element scheme based on an explicit definition of the pressure gradient, in a sense of a selective staggered scheme. Here, we employ a projection algorithm that allows a straightforward computation of the gradient in terms of \({\mathcal C}^0\)-shapes. To this end, an additional negative pressure field \(\mathfrak {p}\) is introduced and locally defined by the residual

    $$\begin{aligned} \mathfrak {r}{:}= \mathfrak {p}+\frac{1}{3}\hbox {tr}\tilde{\varvec{\sigma }}_n=0 \end{aligned}$$
    (65)

    in terms of the stress \(\tilde{\varvec{\sigma }}_n\) at time \(t_n\), yielding the weak and incremental form

    $$\begin{aligned} G^{\mathfrak {p}} = \int _{{\mathcal B}} \delta \mathfrak {p}\cdot \mathfrak {r}\,dV = 0 \quad \text{ and }\quad \varDelta G_\mathrm{ext }^{\mathfrak {p}} = \int _{{\mathcal B}} \delta \mathfrak {p}\varDelta \mathfrak {p}\,dA \, . \end{aligned}$$
    (66)

    Introducing the \({\mathcal C}^0\) element interpolation \(\mathfrak {p}^h({\varvec{ X }}) = {\varvec{ N }}^e_\mathfrak {p}({\varvec{ X }}) {\varvec{ d }}_\mathfrak {p}\) consistent with (62), one obtains the finite element residual and tangent matrix

    $$\begin{aligned} {\varvec{ R }}_{\mathfrak {p}} = \mathop {{\mathbf {\mathsf{{A}}}}}\limits _{e=1}^{N_e} \int _{\partial {\mathcal B}^{el}} \!\! {\varvec{ N }}_\mathfrak {p}^{e} \mathfrak {r}\,dV \quad \text{ and }\quad {\varvec{ K }}_{\mathfrak {p}\mathfrak {p}} = \mathop {{\mathbf {\mathsf{{A}}}}}\limits _{e=1}^{N_e} \int _{\partial {\mathcal B}^{h}_{el}} \!\! {\varvec{ N }}_\mathfrak {p}^{e\,T} {\varvec{ N }}_\mathfrak {p}^{e} \,dV \, . \end{aligned}$$
    (67)

    in addition to (64). Then, the pressure gradient is locally approximated by

    $$\begin{aligned} \nabla p({\varvec{ X }}) {:}= {\varvec{ B }}^e_\mathfrak {p}({\varvec{ X }}) {\varvec{ d }}_\mathfrak {p}\end{aligned}$$
    (68)

    and constant within the times step \([t_n,t_{n+1}]\) under consideration. The semi-implicit schme that uses this approximation is very robust.

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Acknowledgments

Support for this research was provided by the German Research Foundation (DFG) for the Cluster of Excellence Exc 310 Simulation Technology at the University of Stuttgart.

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Authors and Affiliations

  1. Institute of Applied Mechanics (CE), Chair I, University of Stuttgart, Pfaffenwaldring 7, 70569, Stuttgart, Germany

    Christian Miehe

  2. Department of Mechanical Engineering, Middle East Technical University, Dumlupınar Bulvarı 1, 06800, Ankara, Turkey

    Hüsnü Dal

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Correspondence to Christian Miehe.

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Dal, H., Miehe, C. Computational electro-chemo-mechanics of lithium-ion battery electrodes at finite strains. Comput Mech 55, 303–325 (2015). https://doi.org/10.1007/s00466-014-1102-5

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  • DOI: https://doi.org/10.1007/s00466-014-1102-5

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