This paper presents the Finite Element Method for Cosserat plates. The mathematical model for Cos... more 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.
The purpose of this paper is to present a new mathematical model for the dynamics of thin Cossera... more 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
Abstract: We study material deformations of polycrystals, rocks and other elastic materials that ... more Abstract: We study material deformations of polycrystals, rocks and other elastic materials that display dif-ferent types of imperfections with a typical size of 1 m. Stress–strain relationships of these materials depend on the processing history and exhibit common behavior, including nonlinearity, hysteresis, etc. We focus our study on the continuous distribution of singularities in the strain field, which we describe in terms of surface densities and fluxes. We define the mass mesodensity tensor and deduce the constitutive rela-tionships between the strain singularity current and the linear mesomomentum. Based on the modification of Peach–Koehler formula we consider the constitutive relation between the line mesostress tensor and the strain singularities density. These constitutive relationships allow us to model stresses in mesoelastic materials.
The purpose of this paper is to present a new mathematical model for the dynamics of thin Cossera... more 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.
The purpose of this paper is to present a new mathematical model for the deformation of thin Coss... more 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.
In this paper we present the validation of our recently published mathematical model for the dyna... more 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.
The purpose of this paper is to present a new mathematical model for the deformation of thin Coss... more 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.
In this article we present the numerical simulation of a dislocation incorporated into a Cosserat... more 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.
The purpose of this paper is to present a new mathematical model for the dynamics of thin Cossera... more 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.
Page 1. The linearization of inverse spectral problems for nonhomogeneous elastic bodies This art... more Page 1. The linearization of inverse spectral problems for nonhomogeneous elastic bodies This article has been downloaded from IOPscience. Please scroll down to see the full text article. 1996 Inverse Problems 12 483 (http://iopscience.iop.org/0266-5611/12/4/010) ...
This paper presents the Finite Element Method for Cosserat plates. The mathematical model for Cos... more 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.
The purpose of this paper is to present a new mathematical model for the dynamics of thin Cossera... more 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
Abstract: We study material deformations of polycrystals, rocks and other elastic materials that ... more Abstract: We study material deformations of polycrystals, rocks and other elastic materials that display dif-ferent types of imperfections with a typical size of 1 m. Stress–strain relationships of these materials depend on the processing history and exhibit common behavior, including nonlinearity, hysteresis, etc. We focus our study on the continuous distribution of singularities in the strain field, which we describe in terms of surface densities and fluxes. We define the mass mesodensity tensor and deduce the constitutive rela-tionships between the strain singularity current and the linear mesomomentum. Based on the modification of Peach–Koehler formula we consider the constitutive relation between the line mesostress tensor and the strain singularities density. These constitutive relationships allow us to model stresses in mesoelastic materials.
The purpose of this paper is to present a new mathematical model for the dynamics of thin Cossera... more 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.
The purpose of this paper is to present a new mathematical model for the deformation of thin Coss... more 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.
In this paper we present the validation of our recently published mathematical model for the dyna... more 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.
The purpose of this paper is to present a new mathematical model for the deformation of thin Coss... more 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.
In this article we present the numerical simulation of a dislocation incorporated into a Cosserat... more 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.
The purpose of this paper is to present a new mathematical model for the dynamics of thin Cossera... more 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.
Page 1. The linearization of inverse spectral problems for nonhomogeneous elastic bodies This art... more Page 1. The linearization of inverse spectral problems for nonhomogeneous elastic bodies This article has been downloaded from IOPscience. Please scroll down to see the full text article. 1996 Inverse Problems 12 483 (http://iopscience.iop.org/0266-5611/12/4/010) ...
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