Inverse Lagrangian Formulation for the Deformation of Hyperelastic Solids

Although the field equations of solid mechanics are commonly written in the undeformed configuration, neither fundamental theories nor mathematics prohibits the analysis from being carried out in a deformed configuration. In this letter, following recent developments in inverse deformation problems, the governing equations for static hyperelasticity problems are formulated in the current configuration after deformation. An inverse mapping from the deformed geometry to the original one is solved as the unknown field. Such an approach, herein referred to as the inverse Lagrangian formulation, is exemplified with several applications in addition to the inverse problem of known deformed geometry by design and determination of the original geometry. Applications are found in steady-state fluid-structure interaction problems, such as the design of microfluidic devices. More importantly, the method is also found to be useful in analyzing mechanical instability and bifurcation problems, as the inverse mapping from a buckled state to the original remains unique.