Acta Mechanica Sinica

, Volume 34, Issue 2, pp 359–370 | Cite as

Two-dimensional analysis of progressive delamination in thin film electrodes

  • Mei Liu
  • Bo Lu
  • Dong-Li Shi
  • Jun-Qian Zhang
Research Paper


By employing the two-dimensional analysis, i.e., plane strain and plane stress, a semi-analytical method is developed to investigate the interfacial delamination in electrodes. The key parameters are obtained from the governing equations, and their effects on the evolution of the delamination are evaluated. The impact of constraint perpendicular to the plane is also investigated by comparing the plane strain and plane stress. It is found that the delamination in the plane strain condition occurs easier, indicating that the constraint is harmful to maintain the structure stability. According to the obtained governing equations, a formula of the dimensionless critical size for delamination is provided, which is a function of the maximum volumetric strain and the Poisson’s ratio of the active layer.


Lithium-ion battery Thin film electrode Delamination Critical size 

List of symbols

x (and \(\eta )\), z (and \(\zeta )\)

Cartesian coordinates (m)

L and h

Length and thickness of the active layer (m)

\(L_{c} \)

Critical size for delamination (m)


Molar concentration of lithium-ions \((\hbox {mol}\cdot \hbox {m}^{-3})\)


Stoichiometric saturation concentration \((\hbox {mol}\cdot \hbox {m}^{-3})\)


Average concentration \((\hbox {mol}\cdot \hbox {m}^{-3})\)

\(\varphi \)

Inhomogeneity of concentration \((\hbox {mol}\cdot \hbox {m}^{-1})\)

\(N_x\) and \(Q_{xz}\)

Normal and shear force resultant \((\hbox {N}\cdot \hbox {m}^{-1})\)

\(M_x \)

Bending moment of active layer (N)

\(M_{c} \)

Diffusion-induced bending moment (N)


Surface current density \((\hbox {A}\cdot \hbox {m}^{-2})\)


Electrochemical load factor


Diffusivity of lithium-ions \((\hbox {m}^{2}\cdot \hbox {s}^{-1})\)

\(\varOmega \)

Partial molar volume of active material \((\hbox {m}^{3}\cdot \hbox {mol}^{-1})\)

p (and \(\sigma _{n})\)

Interfacial normal stress \((\hbox {N}\cdot \hbox {m}^{-2})\)

q (and \(\sigma _{t} )\)

Interfacial shear stress \((\hbox {N}\cdot \hbox {m}^{-2})\)

\(\sigma _{nc} \) and \(\sigma _{tc} \)

Interfacial normal and shear strength \((\hbox {N}\cdot \hbox {m}^{-2})\)

\(\sigma _x , \sigma _z , \tau _{xz} \)

Stresses \((\hbox {N}\cdot \hbox {m}^{-2})\)

\(\varepsilon _x , \varepsilon _z , \varepsilon _{xz}\)


\(\delta _{n} \) and \(\delta _{t} \)

Interfacial opening and sliding displacement (m)

\(\delta _{nc} \) and \(\delta _{tc} \)

Critical opening and sliding displacement (m)

\(\varGamma \)

Interfacial fracture toughness \((\hbox {J}\cdot \hbox {m}^{-2})\)

\(\upsilon _\mathrm{p} \) and \(\upsilon _\mathrm{s} \)

Poisson’s ratio of active material and substrate

\(E_\mathrm{p} \) and \(E_\mathrm{s} \)

Young’s modulus of active material and substrate \((\hbox {N}\cdot \hbox {m}^{-2})\)


Dimensionless Young’s modulus

\(u_\mathrm{p}\) and \(w_\mathrm{p} \)

Displacements of the active layer (m)

\(u_0 \)

Mid-plane displacement (m)

\(u_\mathrm{s} \) and \(w_\mathrm{s} \)

Displacements of substrate surface (m)

\(L_1 \) and \(L_2 \)

Rigid displacements of the active layer and substrate (m)


Rigid displacement difference between active layer and substrate


Bending stiffness of active layer \((\hbox {N}\cdot \hbox {m}\))


Shear modulus of the active layer \((\hbox {N}\cdot \hbox {m}^{-2})\)

\(\mu \)

Friction coefficient


State of charge (SOC)


Time (s)


Time to delamination onset


Faraday constant \((\hbox {s}\cdot \hbox {A}\cdot \hbox {mol}^{-1})\)

\(H_1 \)

Sliding displacement related to the interfacial stresses

\(\bar{{F}}_{p} , \bar{{F}}_{q} , \bar{{G}}_{p} , \bar{{G}}_{q} \)

Kernel functions for integral equations

The others which have “\(^{-}\)” on top stand for the corresponding dimensionless symbols. For example, \(\overline{c}\) stands for the dimensionless version of c



The project was supported by the National Natural Science Foundation of China (Grants 11332005 and 11172159) and the Shanghai Municipal Education Commission of China (Grant 13ZZ070).


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Copyright information

© The Chinese Society of Theoretical and Applied Mechanics; Institute of Mechanics, Chinese Academy of Sciences and Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Shanghai Institute of Applied Mathematics and MechanicsShanghai UniversityShanghaiChina
  2. 2.Department of MechanicsShanghai UniversityShanghaiChina
  3. 3.Shanghai Key Laboratory of Mechanics in Energy EngineeringShanghai UniversityShanghaiChina
  4. 4.Materials Genome InstituteShanghai UniversityShanghaiChina

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