The Rate Constitutive Equations and Their Validity for Progressively Increasing Deformation

Abstract

The rate constitutive equations based on upper convected, lower convected, Jaumann, Truesdell and Green–Naghdi stress rates, etc. [1, 2, 3, 4, 5, 6, 7, 8, 9, 10] have been used for the solid matter and polymeric liquids when the mathematical models are derived employing conservation laws in the Eulerian description. The rate constitutive equations provide relationship between convected time derivatives of the stress tensor and the convected time derivatives of strain tensor through the constitution of the matter. It is well known that if one assumes constant properties of the matter, then the use of various rate constitutive equations yield different responses [9] when the strains and strain rates are not infinitesimal (finite deformation) even though the rate constitutive equations are objective or frame invariant. This has been rationalized by the argument that in order for the different rate constitutive equations to produce the same response, the material tensor must be different for each case [9]. Our point of view is rather pragmatic in the sense that if all rate constitutive models describe the behavior of the same matter in which material constants do not change during deformation, then surely there must be some inherent assumptions in their derivations that are responsible for this anomaly. The covariant and contravariant convected bases in the current configuration of the deforming matter provide two possible alternate means of defining the convected time derivatives of the contravariant and covariant Cauchy stress tensors as well as the strain tensors. The relationship between the convected time derivatives of the stress tensor, material tensor and convected time derivative of the strain tensor result in the rate constitutive equations. Thus there are at least two obvious approaches for deriving rate constitutive equations: one based on covariant description referred to as lower convected rate constitutive equations and the other based on a contravariant description referred to as upper convected rate constitutive equations. It can be shown that the other rate constitutive equations available (in the literature) can also be derived using these two basic descriptions by modifications that are either justified based on the physics of the deforming matter or mathematical manipulations. When the strains and the strain rates are small (closer to infinitesimal assumption), there is isomorphism (or equivalence) between the covariant and contravariant descriptions, hence in this case the two descriptions will yield identical results even though the explicit forms of the rate equation expressions in the two descriptions are different. It is shown that when the strains and the strain rates are finite (finite deformation), the isomorphism or equivalence between the two descriptions is lost. The paper demonstrates that with progressively increasing deformation leading to finite deformation only the contravariant description has physical basis, hence the rate constitutive equations derived using contravariant description remain valid whereas all others become spurious. Detailed mathematical development of the rate constitutive equations based on covariant, contravariant descriptions and others based on these two bases are presented to illustrate their validity and limitations. Numerical examples are also presented using Giesekus constitutive model for dense polymeric liquids (polymer melts) to demonstrate the validity of contravariant basis and the failures of covariant and others commonly used in the literature.