CHAPTER 2 • Governing Equations 2.1 MATHEMATICAL DESCRIPTION OF SHAPE CHANGES IN SOLIDS 2.1.1 Displacement and Velocity Fields 2.1.2 Displacement Gradient and Deformation Gradient Tensors 2.1.3 Deformation Gradient Resulting from Two Successive Deformations 2.1.4 The Jacobian ofthe Deformation Gradient 2.1.5 Lagrange Strain Tensor 2.1.6 Eulerian Strain Tensor 2.1.7 Infinitesimal Strain Tensor 2.1.8 Engineering Shear Strains 2.1.9 Decomposition ofInfinitesimal Strain into Volumetrie and Deviatorie Parts 2.1.1 0 Infinitesimal Rotation Tensor 2.1.11 Principal Values and Directions of the Infinitesimal Strain Tensor 2.1.12 Cauchy-Green Deformation Tensors 7 v vi • Contents 2.2 2.3 2.4 2.1.13 Rotation Tensor and Left and Right Stretch Tensors 2.1.14 Principal Stretches 2.1.15 Generalized Strain Measures 2.1.16 The Velocity Gradient 2.1.17 Stretch Rate and Spin Tensors 2.1.18 Infinitesimal Strain Rate and Rotation Rate 2.1.19 Other Deformation Rate Measures 2.1.20 Strain Equations of Compatibility for Infinitesimal Strains MATHEMATICAL DESCRIPTION OF INTERNAL FORCES IN SOLIDS

An analysis of the dependence of electric potential on the state of stress of a lithiated-silicon electrode is presented. Based on the Larch\'e and Cahn chemical potential for a solid solution, a thermodynamic argument is made for the existence of the stress-potential coupling in lithiated-silicon; based on the known properties of the material, the magnitude of the coupling is estimated to be ca. 60 mV/GPa in thin-film geometry. An experimental investigation is carried out on silicon thin-film electrodes in which the stress is measured in situ during electrochemical lithiation and delithiation. By progressively varying the stress through incremental delithiation, the relation between stress change and electric-potential change is measured to be 100 - 120 mV/GPa, which is of the same order of magnitude as the prediction of the analysis. The importance of the coupling is discussed in interpreting the hysteresis observed in potential vs. state-of-charge plots, and the role of stress in modifying the maximum charge capacity of a silicon electrode under stress.Comment: 21 pages, 5 figure

a b s t r a c tWe formulate the continuum field equations and constitutive equations that govern deformation, stress, and electric current flow in a Li-ion half-cell. The model considers mass transport through the system, deformation and stress in the anode and cathode, electrostatic fields, as well as the electrochemical reactions at the electrode/electrolyte interfaces. It extends existing analyses by accounting for the effects of finite strains and plastic flow in the electrodes, and by exploring in detail the role of stress in the electrochemical reactions at the electrode-electrolyte interfaces. In particular, we find that that stress directly influences the rest potential at the interface, so that a term involving stress must be added to the Nernst equation if the stress in the solid is significant. The model is used to predict the variation of stress and electric potential in a model 1-D half-cell, consisting of a thin film of Si on a rigid substrate, a fluid electrolyte layer, and a solid Li cathode. The predicted cycles of stress and potential are shown to be in good agreement with experimental observations.

A two-dimensional model of a surface initiated rolling contact fatigue crack has been developed. The model takes into account the effects of frictional locking between the faces of the crack, and the influence of fluid pressure acting on the crack faces. The model has been used to investigate three possible mechanisms for propagating the cracks: mode II crack growth due to the cyclic shear stresses caused by repeated rolling contact; crack growth due to fluid forced into the crack by the load; and crack growth due to fluid trapped inside the crack. The predictions of the theory are compared with the behaviour of contact fatigue cracks.