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

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.

On Dynamics of Elastic Networks With Rigid Junctions Within Nonlinear Micro-Polar Elasticity

“…This case corresponds to symmetric material particles such as spheres. Dynamic properties introduced through two micro-inertia tensors could be taken into account considering material symmetry as in Eremeyev and Konopińska-Zmysłowska (2020); see also Vilchevskaya et al (2022), where other references on micro-inertia tensors could be found. Derived here formulae for the micro-inertia tensors complete the description by Eremeyev (2019) of a network undergoing large deformations within micro-polar elasticity.…”

Micromorphic theory as a model for blood in the microcirculation: correction and analysis

“…Definition (1.8) slightly deviates from the definition of the inertia tensor commonly used in classical mechanics [17]. However, the two formulations can be transformed into each other, as described in [36]. By the use of the inertia tensor i and gyration tensor G, the spin inertia per unit mass σ [6] is defined:…”

“…It characterizes the change in time of a particle's spin angular momentum (cf. [36]). The field quantities introduced in this section are used to formulate the governing balance and constitutive equations of a micromorphic fluid below.…”

Nonclassical continuum theories for fluent media incorporating rotation rates and their thermodynamic consistency

“…In Ref. [60], authors present an extended micropolar approach within framework of Eringen's micropolar, microstretch, and micromorphic theories and some variations. In Ref.…”