Instructor Pilot reported a NMAC with another aircraft in the traffic pattern at 2R2 non-towered airport.

Narrative

On Date; my student and I were conducting a landing lesson at Hendricks County Airport (2R2). There was another plane in the pattern; which we kept visual separation and good communication with. My student and I saw another plane approaching the area from the northeast on ADS-B but was not making any calls. As he continued inbound we assumed he was just exiting out from underneath the Class Charlie airspace. The plane then began to enter a left crosswind at around 500-600 feet AGL for Runway 18. My student and I were on the left downwind and the other plane in the pattern had just departed Runway 18. The other plane called his left crosswind and the other plane said they had just taken off [Runway] 18; the other plane did not say anything and just kept flying on crosswind joining the left downwind for 18. A few minutes later; my student and I departed 18 and made our traffic calls as needed. Once we were turning crosswind the other airplane made a call saying 'departing Runway 18' but did not say if he was staying in the pattern or exiting the area. As my student and I joined the left downwind leg for 18 the other aircraft immediately turned crosswind after takeoff. As doing so he was headed our direction and joined the downwind right behind us; no more than .5 mile separation. As he did so; I asked on comms of he had us insight and he said nothing. At this point my student and I came to the realization that he was gaining on us and I made the decision to take controls and execute a short approach into 18. My student was looking for the traffic as we were turning and he said he was no more than 200 feet away from us. As far as suggestions go; I believe there is not much more my student or I could have done. We did not know the intentions of the other pilot being he was not communicating with us. We were unable to climb to avoid traffic due to the Class Charlie airspace above.