A well drilling system can include a drilling tool with at least one component which is displaced by a material that changes shape. The material can be a shape memory material. The material may change shape in response to a temperature change. The component can be a drill bit cutter, a depth of cut control surface, a gauge surface or a stabilizer surface. A method of controlling a drilling operation can include configuring a drilling tool with a material which changes shape, and the material displacing at least one component of the drilling tool during the drilling operation. Displacement of the component can be controlled downhole to maintain drilling parameters (such as torque, vibration, steering performance, etc.) in desired ranges.