The mathematical foundation and corresponding physical sense in large numbers of constitutive models of geotechnical material are important to understand and use them. The mathematical foundation and condition of all kinds of geotechnical constitutive models are pointed out。The keystone is whether or not there is unique potential field in which the stresses（strains）or their increments are its gradient vectors. The mathematical conditions include coaxiality of principal stresses and strains and their increments; symmetry of modulus matrix; rank of the matrix equal to 1; the curl of gradient vector equal to 0. The generalized potential theory is proposed in which unique potential function or 2-3 potential functions with linearly independent gradient vectors are used to determine the stress-strain relationship. The theory is provided large capacity and with wider application field for geotechnical material with complex mechanical properties.