Bifurcation instability of substrate-supported metal films under biaxial in-plane tension

Abstract

The stretchability of metal materials is often limited by the onset and development of necking instability. For instance, necking of lithium metal often occurs at low strains and thus hinders its practical applications in stretchable lithium batteries. Substrate/metal bilayers are emerging as a promising solution to the stringent stretchability requirement of metal electrodes and current collectors in flexible and stretchable batteries. So far, a comprehensive understanding of the bifurcation instability of substrate-supported metal layers under arbitrary biaxial in-plane tensile loading still remains elusive. Most existing theoretical and numerical studies of the bifurcation instability of substrate-supported metal layers assume either plane strain condition or single-necking mode (i.e., a single diffusive neck occurs). However, in conducted experiments, substrate/metal bilayers are subjected to uniaxial tensile loading and formation of multiple necks is observed during the tests. This paper presents an all-wavelength bifurcation analysis to understand the deformation instability of substrate/metal bilayers under arbitrary biaxial tensile loadings, from equi-biaxial tension, to plane-strain tension, and to uniaxial tension. Two representative bilayer structures are investigated, namely, a metal layer supported by a plastic substrate and a metal layer supported by an elastomer substrate. The analysis predicts three bifurcation modes of substrate/metal bilayers, including single-necking mode, multiple-necking mode, and surface mode. The results quantitatively demonstrate the bifurcation retardation effect of the supporting substrate: the stiffer/thicker is the substrate, the higher is the bifurcation limit. More importantly, it is further shown that there exists a theoretical upper bound of the bifurcation limit of a substrate/metal bilayer structure, which has not been reported before. Understandings from the present study may shed light on the optimal design of substrate/metal bilayer structures with enhanced deformability under complex biaxial loading conditions.