Canonical frame-indifferent transport operators with the four-dimensional formalism of differential geometry

Abstract

To say that a constitutive model has to verify “the principle of material objectivity” to ensure its frame-indifference has become a common wisdom. Objective transports are thus defined to serve as tensor rates. These operators are in particular applied to the Cauchy stress tensor. They are used as time derivatives to describe non-linear or dissipative phenomena observed during the finite transformations of a material continuum. Because an infinite number of such transports may be constructed and shown to be objective, the selection of the appropriate transport and its validity still constitutes an open and debatable question.Differential geometry, within its four-dimensional formalism, has proven its ability to describe physical fields and their variations in space and time while ensuring the covariance of any physical law. This description is here applied to the motion of a material continuum within the classical hypotheses of Newtonian physics. In this context, we show that the rate of a tensor as seen by a point of space–time is uniquely defined by the covariant rate; this quantity is not invariant with respect to superposed rigid body motions. The rate of a tensor as seen by a moving particle of matter is uniquely defined by the Lie derivative of the tensor. This operator is invariant with respect to superposed rigid body motions. Both, the covariant rate and the Lie derivative are independent of the observer and could thus be used in a constitutive model within a four-dimensional formalism. We show next that the projection of the Lie derivative of the Cauchy stress tensor within an inertial 3D Cartesian frame corresponds to Truesdell’s transport and that the other 3D objective stress transports, if they have the dimension of a rate, do not correspond to a time derivative of this tensor. The Truesdell transport is thus the only objective transport that represents a frame-indifferent time derivative of the Cauchy stress tensor.