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Soft materials often display complex behaviors that transition through apparent solid- and fluid-like regimes. While a growing number of microscale simulation methods exist for these materials, reduced-order models that encapsulate the macroscale physics are often desired to predict how external bodies interact with soft media. Such an approach could provide direct insights in diverse situations from impact and penetration problems to locomotion over natural terrains. This work proposes a systematic program to develop three-dimensional (3D) reduced-order models for soft materials from a fundamental basis using continuum symmetries and rheological principles. In particular, we derive a reduced-order, 3D resistive force theory (3D-RFT), which is capable of accurately and quickly predicting the resistive stress distribution on arbitrary-shaped bodies intruding through granular media. Aided by a continuum description of the granular medium, a comprehensive set of spatial symmetry constraints, and a limited amount of reference data, we develop a self-consistent and accurate 3D-RFT. We verify the model capabilities in a wide range of cases and show that it can be quickly recalibrated to different media and intruder surface types. The premises leading to 3D-RFT anticipate application to other soft materials with strongly hyperlocalized intrusion behavior. 

While terrestrial locomotors often contend with permanently deformable substrates like sand, soil, and mud, principles of motion on such materials are lacking. We study the desert-specialist shovel-nosed snake traversing a model sand and find body inertia is negligible despite rapid transit and speed dependent granular reaction forces. New surface resistive force theory (RFT) calculation reveals how wave shape in these snakes minimizes material memory effects and optimizes escape performance given physiological power limitations. RFT explains the morphology and waveform-dependent performance of a diversity of non-sand-specialist snakes but overestimates the capability of those snakes which suffer high lateral slipping of the body. Robophysical experiments recapitulate aspects of these failure-prone snakes and elucidate how re-encountering previously deformed material hinders performance. This study reveals how memory effects stymied the locomotion of a diversity of snakes in our previous studies (Marvi et al., 2014) and indicates avenues to improve all-terrain robots.