On General Measures of Deformation

Each par t ic le of a con t inuum is assigned a second order tensor which is t a k e n as a measure of the deformat ion of some neighborhood of the part icle, and which is de te rmined by a funct ional depending on the configurations of t h a t neighborhood. Two invar iance restr ict ions are imposed on the funct ional whose values are spat ial s t rain tensors, t h a t is, associated wi th the deformed configuration. The first r equ i rement is t h a t a t ime shif t and rigid t ransformat ion of t he deformed conf igurat ion leave the spat ia l deformat ion tensor unal te red re la t ive to it. The second requires t h a t i f part icles of dis t inct cont inua undergo the same deformat ion, t he corresponding deformat ion tensors should be the same. Fo r the special case in which the funct ional depends on the deformat ion in the smallest ne ighborhood of a part icle, the restr ict ions imply t h a t the deformat ion tensors associated wi th the deformed and reference configurations arc isotropic funct ions of the ]eft and r ight CAITClVZGREEN tensors, respect ively.