The critical resolved shear stress, for a dislocation percolating through penetrable obstacles with various pinning strengths, was derived analytically. The stability of dislocation-obstacle configurations was determined by evaluating the configurational entropy. The analytical results implied a superposition law for the concentration of pinning obstacles on the dislocation line, and a Pythagorean-type hardening rule for the critical resolved shear stress when the obstacle strengths were weak and almost equal.

Solid-Solution Hardening by Point-Like Obstacles of Different Kinds. M.Hiratani, V.V.Bulatov: Philosophical Magazine Letters, 2004, 84[7], 461-70