Extended micropolar approach within the framework of 3M theories and variations thereof

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.

Automated summary

In his work on generalized continuum mechanics, Eringen introduced the 3M theories of micromorphic, microstretch, and micropolar materials modeling, providing a comprehensive framework for studying continuum with different degrees of freedom. The extended micropolar theory further expands on this concept, allowing for a more flexible description of structural changes in material systems. By comparing these theories, the paper highlights their similarities and differences, showing the strengths and limitations of each approach.