When we measure the undrained strength of clay via CPT or triaxial test for instance, is there a way to correlate this to drained strength? How? I ask because measuring drained strength takes very long time but undrained strength can be measured quickly
They are certainly correlated but there is no simple relationship. A commonly used rule of thumb, where I live, is $c'=0.1c_u$ for overconsolidated clays and zero otherwise, but reality is much more complex. See for example this article. To the best of my knowledge, no accurate and generic relationship has been found.