Check Valve with Valve Member Biased by Connectors Extending from a Valve Seat for Operation of a Subterranean Tool

A check valve has a valve member flexibly attached to a seat with resilient members that fold in response to flow above a predetermined value to shut the valve. The flexibly extending members store potential energy that is released to separate the valve member from the seat when the pressure applied against the seated valve member declines to the point that the potential energy stored in the flexible members is able to move the valve member off the seat. The valve assembly including the valve member, the seat and the flexible members can be a common material, or can be all one or more nonmetallic materials or involve composites or other easily drilled materials that are compatible with the operating conditions of pressure differential, temperature and composition of the well fluid. The check valve assembly can be pushed with pressure when closed to operate a subterranean tool.