arXiv subjects

Lev Steinberg

Publications and source records attributed to Lev Steinberg.

7 recordsLinked to original sources

arXiv2026

A variationally consistent mesoscopic Cosserat theory with distributed defects and configurational forces

We develop a variationally consistent mesoscopic extension of Cosserat elasticity motivated by the breakdown of compatibility in classical formulations. By admitting compatibility-breaking perturbations, the classical theory ceases to remain closed under admissible variations, necessitating an enlargement of the constitutive framework. This leads naturally to a formulation in which torsion and curvature are treated as independent distributed measures of defects. The theory is constructed using a Palatini-type variational approach, with the coframe and connection as independent fields. The resulting Euler--Lagrange equations yield both the standard balance laws and defect-related excitation fields. Material invariance gives rise to configurational forces and moments, which emerge as Noether currents and are directly linked to defect transport governed by the Bianchi identities. The framework provides a unified description of defect kinematics, configurational mechanics, and microstructural evolution. Illustrative examples and numerical evaluations demonstrate how defect transport generates configurational forces and highlight the underlying Maxwell-type structure of the theory. The proposed formulation offers a consistent geometric foundation for the analysis of structured solids with evolving internal geometry and provides a basis for future developments in defect dynamics and dissipative processes.

math-ph↗

arXiv2026

A Variational Formulation of Classical Cosserat Elasticity with Independent Coframe and Rotational Connection

We present a geometric formulation of classical Cosserat elasticity in which the coframe and rotational connection are treated as independent variational fields. In contrast to conventional metric-based approaches, this formulation makes the underlying geometric structure explicit and separates translational and rotational degrees of freedom at the level of the action. The governing equations are obtained directly as Euler--Lagrange equations and yield the Cosserat force and moment balance laws without imposing compatibility constraints a priori.It is further shown that configurational balances arise from invarianceof the action under material translations and rotations via Noether's theorems, providing an explicit variational interpretation of micropolar mechanics. A metric-free linearization recovers the classical strain and wryness measures and establishes equivalence with standard tensorial formulations under appropriate constitutive assumptions. The proposed framework clarifies the role of the connection field, which remains implicit in classical theories, and provides a geometrically explicit variational framework.for Cosserat continua.The formulation also provides a natural foundation for generalized incompatible Cosserat continua and mesoscopic defect theories

math-ph↗

arXiv2022

Simulation of Dislocation in Cosserat Elastic Plates

In this article we present the numerical simulation of a dislocation incorporated into a Cosserat plate. The simulation is based on the mathematical model for bending of Cosserat elastic plates recently developed by the authors. The dislocation is modeled by a sequence of domains that converge to the point of the dislocation and by a residual force distributed around that point. The resulted plate deformation is calculated using the Finite Element method. We also discuss a possible effect of the dislocation on a hole incorporated into the plate.

cond-mat.soft↗

arXiv2018

Model Validation and Vibration Analysis of Cosserat Plates

In this paper we present the validation of our recently published mathematical model for the dynamics of Cosserat elastic plates. The validation is based on the comparison with the exact solution of the 3-dimensional Cosserat elastodynamics. The preliminary computations of eigenfrequencies show the high agreement with the exact values. The computations allow us to detect the splitting of the frequencies of vibrations (micro vibration) depending on the orientation of micro elements. This provided us with a powerful tool for distinguishing between the frequencies of the micro and macro vibrations of the plate.

math.NA↗

arXiv2016

Finite Element Method for Cosserat Plates

This paper presents the Finite Element Method for Cosserat plates. The mathematical model for Cosserat elastic plates is based on the calculation of the optimal value of the splitting parameter. We discuss the existence and uniqueness of the weak solution and the convergence of the proposed FEM. The Finite Element analysis of the clamped Cosserat plates of different shapes under different loads is provided. We present the numerical validation of the proposed FEM by estimating the order of convergence, when comparing the main kinematic variables with the analytical solution. We also consider the numerical analysis of plates with circular holes. We show that as expected the stress concentration factor around the hole is smaller than the classical value and smaller holes exhibit less stress concentration compared to larger ones.

math.NA↗

arXiv2009

Elastic Plate Deformation with Transverse Variation of Microrotation

The purpose of this paper is to present a new mathematical model for the deformation of thin Cosserat elastic plates. Our approach, which is based on a generalization of the classical Reissner plate theory, takes into account the transverse variation of microrotation of the plates. The model assumes polynomial approximations over the plate thickness of asymmetric stress, couple stress, displacement, and microrotation, which are consistent with the elastic equilibrium, boundary conditions and the constitutive relationships. Based on the generalized Hellinger-Prange -Reissner variational principle and strain-displacement relation we obtain the complete theory of Cosserat plate. We also proved the solution uniqueness for the plate boundary value problem.

math-ph↗

arXiv2009

Elastic Plates Motions with Transverse Variation of Microrotation

The purpose of this paper is to present a new mathematical model for the dynamics of thin Cosserat elastic plates. Our approach, which is based on a generalization of the classical Reissner-Mindlin plate theory, takes into account the transverse variation of microrotation and corresponding microintertia of the the elastic plates. The model assumes polynomial approximations over the plate thickness of asymmetric stress, couple stress, displacement, and microrotation, which are consistent with the elastic equilibrium, boundary conditions and the constitutive relationships. Based on the generalized Hellinger-Prange-Reissner variational principle for the dynamics and strain-displacement relation we obtain the complete dynamic theory of Cosserat plate. Key words: Cosserat materials, elastic plates, transverse microrotation, variational principle, elastodynamics

math-ph↗