Green studied mathematics at Jesus College, Cambridge, where he attended lectures by Sydney Goldstein, Arthur Eddington and G. I. Taylor.
He was from 1968 Sedleian Professor of Natural Philosophy at the University of Oxford (where the elasticity theorist Augustus Edward Hough Love was one of his predecessors).
Green was in his lifetime one the U.K.'s leading scientists, who in his career increasingly concerned himself with linear and after World War II nonlinear elasticity theory.
In elasticity theory he worked with Wolfgang Zerna, who was in Newcastle in 1948/1949 as part of an academic exchange programme between the U.K. and Germany.
The Green-Naghdi theory of water waves, as named by R. Cengiz Ertekin, is one of the outstanding contributions of the close collaboration of Green and Paul M. Naghdi.