Geometric constitutive theory and frame invariance

The need for a proper geometric approach to constitutive theory in non-linear continuum mechanics (NLCM) is witnessed by lasting debates about basic questions concerning time-invariance, integrability, conservativeness and frame invariance. Our aim is to bring geometry to play a central role in theoretical and computational issues of NLCM. This demand is imposed by the present state of art, dominated by a mainly algebraic approach which, being a modified heritage of the linearized theory, is inadequate to manage concepts and methods in a non-linear framework. A proper definition of spatial and material fields and the statement of the ensuing covariance paradigm, provide a firm foundation to the theory of constitutive behavior in NLCM. The notion of constitutive frame invariance (CFI) is introduced as geometric correction to the formulation of material frame indifference (MFI). Standard models of constitutive behavior are critically discussed and compared with the ones consistent with the new approach. The outcome is a physically testable theory which eventually results in new effective computation tools for structural engineers.