Abstract
As part of his groundbreaking work on generalized continuum mechanics, Eringen proposed what he called 3M theories, namely the concept of micromorphic, microstretch, and micropolar materials modeling. The micromorphic approach provides the most general framework for a continuum with translational and (internal) rotational degrees of freedom (DOF), whilst the rotational DOFs of micromorphic and micropolar continua are subjected to more and more constraints. More recently, an “extended” micropolar theory has been presented by one of the authors: Eringen’s 3M theories were children of solid mechanics based on the concept of the indestructible material particle. Extended micropolar theory was formulated both ways for material systems as well as in spatial description, which is useful when describing fluid matter. The latter opens the possibility to model situations and materials with a continuum point that on the microscale consists no longer of the same elementary units during a physical process. The difference culminates in an equation for the microinertia tensor, which is no longer a kinematic identity. Rather it contains a new continuum field, namely an independent production term and, consequently, establishes a new constitutive quantity. This makes it possible to describe processes of structural change, which are difficult if not impossible to be captured within the material particle model. This paper compares the various theories and points out their communalities as well as their differences.
Highlights
It is interesting to note that the concepts of a director in the liquid crystal theory proposed by Ericksen and Leslie are closely linked to microinertia as well, even though this is not explicitly said so
The linear velocity and the angular velocity vector at the continuum level of extended micropolar theory are introduced through the amount of linear and angular momentum stored in the Representative Volume Element (RVE), which are both obtained for a discrete ensemble of elementary particles
The main difference between the fields of 3M theories and Extended” Micropolar Theory (EMT) was explained: The fields of 3M theories are based on the concept of a subcontinuum within a material point
Summary
The term Generalized Continuum Theories (GCTs) was coined by Eringen and Maugin, see discussions in [22, 23], in order to emphasize the need for a mathematical framework capable of describing the physical behavior of matter with an inner structure, i.e., internal length scales. The kinematics of extended micropolar theory will essentially be identical to the corresponding 3M version In this context, we will present the concept of directors required to described the internal rotational degree of freedom of the continuum point. We will present the concept of directors required to described the internal rotational degree of freedom of the continuum point This will be followed by the introduction of all relevant continuum fields of 3M and extended micropolar theory. The following sections are dedicated to the continuum balances of 3M and extended theories In context with the latter, we will again emphasize the concept of spatial description. EMT uses the concept of an open representative volume element containing matter It is a theory emphasizing the needs of fluid mechanics. It introduces a new constitutive quantity, which can be used to model processes of structural change not covered by 3M theories
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