Principle of Material Indifference

Material bodies consist of matter whose behavior is modeled by constitutive equations. Because it is impossible to capture all of the nuances, special ideal materials are devised that approximate the phenomena. Their behavior is governed by constitutive equations.

When I searched the literature for basic modeling principles of material bodies, I found a very useful account by Noll2 on the invariancy of constitutive equa­tions. It was enshrined later in the new edition of the Handbuch der Physik, jointly authored by Tmesdell and Noll.3 These constitutive equations satisfy three principles:

1) Coordinate invariance: Constitutive equations are independent of coordinate systems.

2) Dimensional invariance: Constitutive equations are independent of the unit system employed.

3) Material indifference: Constitutive equations are independent of the observer. Or expressed in other words, the constitutive equations of materials are invariant under spatial rigid rotations and translations.

Material interactions do not depend on the coordinate system used for their numerical evaluations. As an example, the airflow over an aircraft wing and the resulting pressure distribution exist a priori, without specification of a coordinate system. You could record it in aircraft coordinates or, via telemetry, in ground coordinates. In both cases you would calculate the same lift. Or consider the thrust vector of a turbojet engine. It could be measured in aircraft or engine coordinates. The resultant force is still the same.

Does it matter whether you use metric or English strain gauges to record the thrust? You will get different numbers, but certainly the aircraft responds to the thrust unfettered by human schemes of measuring units. Physical phenomena tran­scend the artificiality of units.

The principle of material indifference, or, more precisely, the principle of mate­rial /rame-indifference, as Truesdell and Noll3 call it, is tantamount to the general theory of material behavior. It asserts “that the response of a material is the same for all observers.”3 Let the captain delight in the bulge of the sails or a dockside bystander conclude that a stiff easterly blows. Their emotions may be different, but, nevertheless, the bulge has not budged.

You may be part of an international calibration team. You take that norm-sphere and measure its drag in the wind tunnel at the University of Florida and then travel to Stuttgart, Germany, and repeat your test. If the measurements differ, you would not explain the discrepancy by the fact that the facilities are separated by 4000 miles and tilted by 67 deg with respect to each other (different longitude and latitude); rather, you would look for physical differences in the tunnels.

The Principle of Material Indifference (PMI) is the cornerstone of mathemati­cal modeling of dynamic systems. It will enable us to formulate the equations of motions of aerospace vehicles in an invariant form and serve us to prove several theorems.