# A Cosserat Model of Elastic Solids Reinforced by a Family of Curved and Twisted Fibers

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Kirchhoff Rods

#### 2.1. Kinematics

#### 2.2. Strain-Energy Function

#### 2.3. Equilibrium Theory

_{1}, that the Lagrange multipliers ${f}_{\alpha}$ play the role of constitutively indeterminate transverse shear forces acting on a fiber cross section.

## 3. Cosserat Elasticity of Fiber-Reinforced Materials

_{1}in the form

_{2}. These in turn furnish ${\mathit{R}}^{t}\mathit{FD}=\lambda \mathit{D}$ and hence two constraints

#### 3.1. Kinematical and Constitutive Variables in Cosserat Elasticity

#### 3.2. Virtual Power and Equilibrium

#### 3.3. Fiber-Matrix Interaction

## 4. Material Symmetry

#### 4.1. Change of Reference Configuration

#### 4.2. Material Symmetry Transformations

## 5. Examples

#### 5.1. Matrix Energy

#### 5.2. Fiber Symmetry

#### 5.2.1. Transversely Hemitropic Fibers

#### 5.2.2. Transversely Orthotropic Fibers

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

## References

- Steigmann, D.J. Theory of elastic solids reinforced with fibers resistant to extension, flexure and twist. Int. J. Non-Linear Mech.
**2012**, 47, 734–742. [Google Scholar] [CrossRef] [Green Version] - Steigmann, D.J. Effects of fiber bending and twisting resistance on the mechanics of fiber-reinforced elastomers. In CISM Course: Nonlinear Mechanics of Soft Fibrous Tissues; Dorfmann, L., Ogden, R.W., Eds.; Springer: Wien, Austria; New York, NY, USA, 2015; Volume 559, pp. 269–305. [Google Scholar]
- Landau, L.D.; Lifshitz, E.M. Theory of Elasticity, 3rd ed.; Pergamon: Oxford, UK, 1986. [Google Scholar]
- Dill, E.H. Kirchhoff’s theory of rods. Arch. Hist. Exact Sci.
**1992**, 44, 1–23. [Google Scholar] [CrossRef] - Antman, S.S. Nonlinear Problems of Elasticity; Springer: Berlin, Germany, 2005. [Google Scholar]
- Cosserat, E.; Cosserat, F. Théorie des Corps Déformables; Hermann: Paris, France, 1909. [Google Scholar]
- Toupin, R.A. Theories of elasticity with couple stress. Arch. Ration. Mech. Anal.
**1964**, 17, 85–112. [Google Scholar] [CrossRef] - Truesdell, C.; Noll, W. The Non-Linear Field Theories of Mechanics. In Handbuch der Physik, Vol. III/3; Flügge, S., Ed.; Springer: Berlin, Germany, 1965. [Google Scholar]
- Reissner, E. Note on the equations of finite-strain force and moment stress elasticity. Stud. Appl. Math.
**1975**, 54, 1–8. [Google Scholar] - Reissner, E. A further note on finite-strain force and moment stress elasticity. Z. Angew. Math. Phys.
**1987**, 38, 665–673. [Google Scholar] [CrossRef] - Pietraszkiewicz, W.; Eremeyev, V.A. On natural strain measures of the nonlinear micropolar continuum. Int. J. Solids Struct.
**2009**, 46, 774–787. [Google Scholar] [CrossRef] [Green Version] - Neff, P. Existence of minimizers for a finite-strain micro-morphic elastic solid. Proc. Roy. Soc. Edinb. A
**2006**, 136, 997–1012. [Google Scholar] [CrossRef] [Green Version] - Akbarov, S.D.; Guz, A.N. Mechanics of Curved Composites; Kluwer: Dordrecht, The Netherlands, 2000. [Google Scholar]
- Dorfmann, L.; Ogden, R.W. CISM Course: Nonlinear Mechanics of Soft Fibrous Tissues; Dorfmann, L., Ogden, R.W., Eds.; Springer: Wien, Austria; New York, NY, USA, 2015; Volume 559. [Google Scholar]
- Basu, S.; Waas, A.M.; Ambur, D.R. Compressive failure of fiber composites under multi-axial loading. J. Mech. Phys. Solids
**2006**, 54, 611–634. [Google Scholar] [CrossRef] - Eremeyev, V.A.; Pietraszkiewicz, W. Material symmetry group of the non-linear polar-elastic continuum. Int. J. Solids Struct.
**2012**, 49, 1993–2005. [Google Scholar] [CrossRef] [Green Version] - Noll, W. A mathematical theory of the mechanical behavior of continuous media. Arch. Ration. Mech. Anal.
**1958**, 2, 197–226. [Google Scholar] [CrossRef] - Steigmann, D.J. The variational structure of a nonlinear theory for spatial lattices. Meccanica
**1996**, 31, 441–455. [Google Scholar] [CrossRef] [Green Version] - Spencer, A.J.M.; Soldatos, K.P. Finite deformations of fibre-reinforced elastic solids with fibre bending stiffness. Int. J. Non-Linear Mech.
**2007**, 42, 355–368. [Google Scholar] [CrossRef] [Green Version] - Healey, T.J. Material symmetry and chirality in nonlinearly elastic rods. Math. Mech. Solids
**2002**, 7, 405–420. [Google Scholar] [CrossRef] - Luo, C.C.; O’Reilly, O.M. On the material symmetry of elastic rods. J. Elast.
**2000**, 60, 35–56. [Google Scholar] [CrossRef] - Lauderdale, T.; O’Reilly, O.M. On the restrictions imposed by non-affine material symmetry groups for elastic rods: Application to helical substructures. Eur. J. Mech. A/Solids
**2007**, 26, 701–711. [Google Scholar] [CrossRef] - Chadwick, P. Continuum Mechanics: Concise Theory and Problems; Dover: New York, NY, USA, 1976. [Google Scholar]

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**MDPI and ACS Style**

Shirani, M.; Steigmann, D.J.
A Cosserat Model of Elastic Solids Reinforced by a Family of Curved and Twisted Fibers. *Symmetry* **2020**, *12*, 1133.
https://doi.org/10.3390/sym12071133

**AMA Style**

Shirani M, Steigmann DJ.
A Cosserat Model of Elastic Solids Reinforced by a Family of Curved and Twisted Fibers. *Symmetry*. 2020; 12(7):1133.
https://doi.org/10.3390/sym12071133

**Chicago/Turabian Style**

Shirani, Milad, and David J. Steigmann.
2020. "A Cosserat Model of Elastic Solids Reinforced by a Family of Curved and Twisted Fibers" *Symmetry* 12, no. 7: 1133.
https://doi.org/10.3390/sym12071133