Harmonic shape for a fluid inclusion in a soft elastic matrix under plane deformation
摘要
Harmonicity principle is a strategy of reducing stress concentration around holes or inclusions in an elastic solid for given external far-field loading, and the use of this principle would lead to optimal shapes of the holes or inclusions that minimize the stress concentration in the solid in certain common situations (such shapes are commonly referred to as ‘harmonic shapes’). In this paper, we follow this principle and focus on identifying the harmonic shape of a macroscale fluid inclusion embedded in a soft elastic solid under plane deformation for a remote biaxial loading. Considering that the soft solid usually undergoes relatively large deformations and therefore the fluid pressure-induced traction acting on the surrounding solid would experience a directional change during deformation, we particularly incorporate such a directional change in the identification of the harmonic shape of the inclusion. We show that as opposed to the classical harmonic shape for a fluid inclusion (corresponding to the case in which the surrounding solid is relatively stiff and the directional change in the fluid pressure-induced traction during deformation is neglected), the modified harmonic shape identified here remains still elliptical but occupies a different aspect ratio. Basically, the modified harmonic shape features its dependence on the magnitude of the remote loading for a fixed ratio between the two components of the loading. We input the classical and modified harmonic shapes, respectively, into the large-deformation finite element model of a fluid inclusion in a soft hyperelastic Neo-Hookean solid, and confirm from the numerical simulations that the modified harmonic shape is indeed more accurate than the classical counterpart in meeting the initial harmonicity principle. We present also a few numerical examples to illustrate the differences between the current modified solution and the classical counterpart in determining the aspect ratio of the harmonic shape as well as corresponding internal pressure.