In trying to interpret this question, I think partly it's asking whether there exists detailed music theory of rhythm, beyond just the basics of rhythmic notation and meter that one would encounter in a class on the fundamentals of music. And yes, there is a huge amount of scholarship out there about various theories of rhythm and meter (some of which dates back centuries).
Yes, there are patterns of rhythm, and there are various functions ascribed to it. For example, one can begin with discussion of the origin of accent, the differentiation of accent into various categories (e.g., metric -- produced by regularity in rhythmic patterns -- vs. agogic -- produced by length of a note -- vs. articulation -- produced by attack and/or volume). Accent influences creation of meter, as well as divergences from it. Just as melodic and harmonic composition consists of creating patterns and fulfilling expectations at times, while diverging from and denying expectations at other times, rhythm and meter require a similar interplay.
Take, for example, the opening phrase of the show tune "Anything Goes": it begins by a pattern of regular rhythms that generate a steady beat and set up the expectation for a metric regularity, then the rhythm begins to use syncopation and subvert that metric regularity, until eventually the rhythm of the melody stops suddenly, pauses, and then provides a regular succession of short notes followed by a long note (agogic accent) that is not syncopated, confirms the metric regularity, and thus concludes the phrase.
For another obvious example of rhythm taking on an active functional role in music, think of the closing bars of major classical symphonies, where there are often repeated long chords on strong beats, often followed by a very long note on the tonic, effectively an extreme agogic accent. The confirmation of the standard pulse of the piece, followed by slowing and longer notes on strong beats helps to give a rhythmic feeling of "closure."
Anyhow, there are lots of more detailed theories of rhythm out there. There are plenty of historical ones, but I assume you may be most interested in theories that are still actively discussed today. The first of these is probably Cooper and Meyer's The Rhythmic Structure of Music (1960), which built on Meyer's theories of musical expectation and related it to rhythmic structure, giving a sort of architectonic approach to meter building up from basic patterns. Lerdahl and Jackendoff in A Generative Theory of Tonal Music (1983) develop more hierarchical relationships of meter and rhythm happening on multiple levels and large spans of music, while Lawrence Kramer (The Time of Music, 1988) finds interest in the structure of the rhythm building up to tiny musical moments. Theorists such as Thomas Clifton (Music as Heard, 1983) and Chris Hasty (Meter as Rhythm, 1997) come from various philosophical perspectives on how notions of rhythm and experience of music give rise to meter and other emergent phenomena of musical structure in the first place, while other theorists like Justin London (Hearing in Time, 2001) begin from a psychological and cognitive perspective on musical perception of rhythm and meter.
There's a lot more that's happened in the past couple decades, but that lays out some of the major perspectives on theories of rhythm developed in the late 20th century and can serve as a jumping off point. Recently, interest has turned back to questions of how larger sensations of rhythmic units impact things like form. Anyhow, there's a lot out there. There are definitely plenty of theories about how rhythm can be "functional" and have lots of nuance, though I think there's less consistency about what approach is standard, unlike with harmony, where Roman numeral analysis and related notions of function have become dominant (no pun intended) in Anglo-American music theory.