Autodesk's Jeff Bartels explains how to assign render materials in a Code Set Style. This allows you to visualize a Civil 3D corridor model within InfraWorks.
From the Jeff Bartels YouTube Channel.
and the Civil Immersion blog.
The basic mechanics of how to publish from Civil 3D to Infraworks are explained well.
You should recognize you probably want a specific publish drawing that: probably DREFs the design corridor; includes the imported materials; and contains a Code Set Style built for this expressed purpose.
Jeff mentions the mission critical issue of the Code Set Style Link Style assignment. This Style name is what Infraworks reads and assigns to the rule. In Civil 3D the material assignment is separate from the Link Style assignment in a Code Set Style.
But Infraworks reads the entire current state of the Code Set Style on import. Yes. Jeff says this, but it is easy to get confused by the differences in Style meanings.