# Analytical solution of the uniaxial extension problem for the relaxed micromorphic continuum and other generalized continua (including full derivations)

@article{Rizzi2021AnalyticalSO, title={Analytical solution of the uniaxial extension problem for the relaxed micromorphic continuum and other generalized continua (including full derivations)}, author={Gianluca Rizzi and Hassam Khan and Ionel‐Dumitrel Ghiba and Angela Madeo and Patrizio Neff}, journal={Archive of Applied Mechanics}, year={2021}, volume={93}, pages={5 - 21} }

We derive analytical solutions for the uniaxial extension problem for the relaxed micromorphic continuum and other generalized continua. These solutions may help in the identification of material parameters of generalized continua which are able to disclose size effects.

## 11 Citations

### Primal and mixed finite element formulations for the relaxed micromorphic model

- BiologyComputer Methods in Applied Mechanics and Engineering
- 2022

### Size-effects of metamaterial beams subjected to pure bending: on boundary conditions and parameter identification in the relaxed micromorphic model

- EngineeringArXiv
- 2022

We devote this paper to model the size-eﬀects of metamaterial beams under bending with the aid of the relaxed micromorphic continuum. We analyze ﬁrst the size-dependent bending stiﬀness of…

### Lagrange and $H(\operatorname{curl},{\cal B})$ based Finite Element formulations for the relaxed micromorphic model

- Computer Science
- 2021

The characteristic length effect on the different components of the model is analyzed and it is revealed how the size-effect property is captured via this characteristic length.

### A non-classical Bernoulli-Euler beam model based on a simplified micromorphic elasticity theory

- Engineering
- 2021

### A benchmark strain gradient elasticity solution in two-dimensions for verifying computational approaches by means of the finite element method

- EngineeringMathematics and Mechanics of Solids
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In elasticity, microstructure-related deviations may be modeled by strain gradient elasticity. For so-called metamaterials, different implementations are possible for solving strain gradient…

### Higher order Bernstein-Bézier and Nédélec finite elements for the relaxed micromorphic model

- Computer ScienceArXiv
- 2023

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### Mechanics of size-dependent materials

- EngineeringArchive of Applied Mechanics
- 2022

The main driving force in modern technology is to make systems of smaller dimensions and novel materials. This poses enormous technological challenges to manufacturing procedures, both at structural…

### The consistent coupling boundary condition for the classical micromorphic model: existence, uniqueness and interpretation of parameters

- Materials ScienceContinuum Mechanics and Thermodynamics
- 2022

We consider the classical Mindlin–Eringen linear micromorphic model with a new strictly weaker set of displacement boundary conditions. The new consistent coupling condition aims at minimizing…

### Experimental evaluation of micromorphic elastic constants in foams and lattices

- EngineeringZeitschrift für angewandte Mathematik und Physik
- 2022

### Lagrange and $$H({\text {curl}},{{\mathcal {B}}})$$ based finite element formulations for the relaxed micromorphic model

- Computer ScienceComputational Mechanics
- 2022

The characteristic length effect on the different components of the model is analyzed and how the size-effect property is captured via this characteristic length parameter is revealed.

## References

SHOWING 1-10 OF 38 REFERENCES

### Analytical solution of the cylindrical torsion problem for the relaxed micromorphic continuum and other generalized continua (including full derivations)

- GeologyMathematics and Mechanics of Solids
- 2021

We solve the St. Venant torsion problem for an infinite cylindrical rod whose behaviour is described by a family of isotropic generalized continua, including the relaxed micromorphic and classical…

### Analytical solutions of the cylindrical bending problem for the relaxed micromorphic continuum and other generalized continua

- EngineeringContinuum Mechanics and Thermodynamics
- 2021

We consider the cylindrical bending problem for an infinite plate as modeled with a family of generalized continuum models, including the micromorphic approach. The models allow to describe length…

### Mechanics of Micromorphic Continua

- Mathematics
- 1968

The theory of micromorphic continua, developed by Eringen and his co-workers, is recapitulated and extended. Master equations are obtained in the form of integral operators from which all order…

### Extension, Bending and Torsion of Anisotropic Microstretch Elastic Cylinders

- Engineering
- 2000

In this paper, we consider the linear theory of anisotropic microstretch elastic solids and study the problem of extension, bending, and torsion of right cylinders with general cross-sections. The…

### Mechanics of extended continua: modeling and simulation of elastic microstretch materials

- Engineering
- 2006

The investigation of microstretch and micromorphic continua (which are prominent examples of so-called extended continua) dates back to Eringens pioneering works in the mid 1960, cf. (Eringen in…

### A unifying perspective: the relaxed linear micromorphic continuum

- Mathematics
- 2014

We formulate a relaxed linear elastic micromorphic continuum model with symmetric Cauchy force stresses and curvature contribution depending only on the micro-dislocation tensor. Our relaxed model is…

### Transparent anisotropy for the relaxed micromorphic model: macroscopic consistency conditions and long wave length asymptotics

- Computer Science
- 2016

### Null‐Lagrangians and the indeterminate couple stress model

- Chemistry, Mathematics
- 2016

The aims of this note is to present a new model based on a new representation of the curvature energy in the indeterminate couple stress model and to discuss some related choices from the literature.…

### A numerical study for linear isotropic Cosserat elasticity with conformally invariant curvature

- Mathematics
- 2009

We investigate the numerical response of the linear Cosserat model with conformal curvature. In our simulations we compare the standard Cosserat model with a novel conformal Cosserat model in torsion…