On April 6, 1995, about 1600 Alaska daylight time, a wheel equipped Piper PA-12 airplane, N3680M, sustained substantial damage while landing at the Soldotna Airport, Soldotna, Alaska. The certificated flight instructor and private pilot aboard were not injured. The 14 CFR Part 91 instructional flight operated in visual meteorological conditions without a flight plan. The flight instructor reported he was giving instruction to the private pilot in tail wheel familiarization. During the first of a planned series of touch-and-go landings on runway 07 at the Soldotna Airport, the airplane bounced, and then veered to the left after touching down again. The flight instructor said right rudder was applied to correct the condition, and the airplane then headed off the left side of the runway toward a 6-8 foot high snow bank. The instructor stated that power was applied to get the airplane to turn right. The airplane turned right, tipping up on the left wheel and wing tip. Power was reduced to idle and the airplane stopped in the middle of the runway. During the hard right turn, the left main wheel hub failed. The airplane was equipped with 29" racing slick "Tundra Tires." The flight instructor noted that the airplane was not equipped with brakes where he was seated (rear tandem seat), and that he felt he was unable to maintain directional control of the airplane because he was unable to apply differential braking.