Air Traffic Control Fully Certified
Distraction; Situational Awareness; Communication Breakdown
Airspace Violation All Types; Deviation - Track / Heading All Types; Deviation / Discrepancy - Procedural Clearance; Inflight Event / Encounter CFTT / CFIT
Air Traffic Control Issued Advisory / Alert
Human Factors; Procedure
Aircraft X was issued takeoff clearance with a right turn to 090 degrees. The pilot read back those exact instructions. Upon seeing the aircraft still on runway heading; not commencing the right turn; I repeated the 090 heading and told the pilot to contact departure control. The pilot read back my clearance verbatim. After a few go-arounds; I received a call from departure control to transfer communications of Aircraft X. I knew I did; but tried again to no avail. I called back departure control and told them that I could not raise Aircraft X. Aircraft X was east bound; but far enough to the north that they may have infringed on airspace protecting the Willis and Hancock Towers downtown Chicago. I was informed later in my shift that Aircraft X may have had an airspace/obstruction violation. It appeared that Aircraft X had not initiated the right turn when instructed.The only recommendation that I have is that Pilots comply with ATC clearances when they are issued. If they are unable to do so; advise. If Aircraft X needed more altitude or time for the turn; the pilot needed to inform me. I issued the turn when I did because I knew that aircraft would be clear of that obstruction airspace. Even when I re-issued the turn I was confident the aircraft would have been clear of the airspace. Instead; the pilot turned when they desired and left the frequency; not switching to the assigned frequency causing confusion.
I was giving up the position to another controller. Aircraft X departed MDW and climbed via the SID to 030 on a 090 heading. I went over the traffic and gave the position up. Aircraft X had not checked on and was flying directly toward the Willis Tower a 3300 MVA area. The controller that relieved me called MDW local and asked them to try the aircraft again and climb them to 033 to get them above the MVA. MDW stated that they did not have the aircraft. The other MDW controllers tried the aircraft on their frequencies and the aircraft was tried on guard numerous times. The aircraft flew within .5 miles of the actual tower at 030. This was the second aircraft that session that went NORDO on a 090 heading off of MDW pointed directly into the tower MVA.
MDW Tower and C90 TRACON Controller reported that an aircraft departed MDW and was instructed to turn and contact Departure.  Aircraft did not turn and was not on either frequency.