General aviation Pilot reported a near midair collision in the traffic pattern. After landing the reporter attempted to speak with the other pilot about the incident but was unsuccessful.

Narrative

On DATE around XA:30 I was on the approach to land at ZZZ airport approaching from the south. Due to overcast ceiling at about 2000 ft. and terrain as high as 2140 ft. in the vicinity of the airport I decided to enter the pattern at base leg. As I switched to CTAF I've heard 2 more airplanes announcing their positions in the traffic pattern: one used a callsign 'Aircraft Z' and announced entering left downwind for a fullstop landing Runway XX; the other one used a callsign 'Aircraft Y' and was announcing 'left close traffic; Runway XX'. I made initial announcement of my position and intentions as 'ZZZ traffic; Aircraft X; 7 miles south of ZZZ airport; enter left base; XX; fullstop; ZZZ traffic'. I also turned on strobe lights; landing and taxi lights to make my airplane more visible. As I was getting closer to the airport - I made same announcement for 5 miles south position; scanning for traffic; and as I got to about 3 miles south I made announcement of my position and intentions again. By this time airplane with callsign 'Aircraft Z' landed and announced 'clear the active runway' and airplane with callsign 'Aircraft Y' announced his position as 'turning left downwind; XX'. With my airplane in landing configuration and approach speed of 90 MPH I entered the left base leg for Runway XX and announced my intention as 'ZZZ traffic; Aircraft X; left base; XX; fullstop; ZZZ traffic'. All this time I was scanning for a Cessna which used callsign 'Aircraft Y' on the downwind and got that airplane in sight on the downwind leg approximately 9 o`clock about 1000 ft. away converging from the left. I kept watching that Cessna and expected pilot of that airplane to extend his downwind leg however pilot of 'Aircraft Y' started left descending turn heading straight at me. I immediately communicated on the radio 'Aircraft Y; stop your turn; stop your turn' however pilot of that Cessna ignored my transmission and kept turning doing nothing to avoid collision. I immediately initiated climbing turn to avoid midair collision and was able to evade that Cessna passing left of my airplane less than 100 ft. I announced 'go around' joined upwind leg as aircraft using callsign 'Aircraft Y' turned final approach and landed. I flew left traffic pattern announcing my position and intentions for every leg and landed on Runway XX. On the ground I saw lady stepping out from the left door of [a] Cessna. I approached that lady to talk about near midair collision situation we just had. Lady who I think was a pilot of that airplane stayed silent however her passenger started acting aggressively before I was able to say a single word so I walked away to deescalate the conflict. As a corrective action for myself I put: avoid using non-standard pattern entries if other traffic is present in the traffic pattern. Try to establish radio communication with airplane converging and clarify intentions while staying at safe distance.