During the landing roll on runway 31, the student pilot of the Cessna 172P, depressed the brakes and felt the left brake "sinking." He then applied both brakes again, and the airplane veered to the right, into the grass. As the airplane departed the right side of the runway, it struck a runway light and a taxiway sign. The student pilot reported 41 hours of total flight experience, 7 of which were as pilot-in-command. He reported no mechanical anomalies with the airplane. Examination of the airplane revealed substantial damage to the fuselage, and no preimpact anomalies with the brake system. The winds reported at the airport approximately 40 minutes prior to the accident, were from 300 degrees at 5 knots.