  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 global-scale physics are often desired to predict how external
bodies interact with soft media, as occurs in diverse situations from impact
and penetration problems to locomotion over natural terrains. This work
proposes a systematic program to develop three-dimensional reduced-order models
for soft materials from a fundamental basis using continuum symmetries and
rheological principles. In particular, we derive a reduced-order technique for
modeling intrusion in granular media which we term three-dimensional Resistive
Force Theory (3D-RFT), which is capable of accurately and quickly predicting
the resistive stress distribution on arbitrary-shaped intruding bodies. 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 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.
