Book contents
- Network Materials
- Network Materials
- Copyright page
- Contents
- Preface
- 1 Introduction
- 2 Fibers and Fiber Bundles
- 3 Fiber Interactions
- 4 Network Structure and Geometric Parameters
- 5 Affine Deformation
- 6 Mechanical Behavior and Relation to Structural Parameters
- 7 Constitutive Formulations for Network Materials
- 8 Strength and Toughness of Network Materials
- 9 Time Dependent Behavior
- 10 Networks with Fiber Surface Interactions
- 11 Composite Networks
- Index
- References
5 - Affine Deformation
Published online by Cambridge University Press: 15 September 2022
Book contents
- Network Materials
- Network Materials
- Copyright page
- Contents
- Preface
- 1 Introduction
- 2 Fibers and Fiber Bundles
- 3 Fiber Interactions
- 4 Network Structure and Geometric Parameters
- 5 Affine Deformation
- 6 Mechanical Behavior and Relation to Structural Parameters
- 7 Constitutive Formulations for Network Materials
- 8 Strength and Toughness of Network Materials
- 9 Time Dependent Behavior
- 10 Networks with Fiber Surface Interactions
- 11 Composite Networks
- Index
- References
Summary
Affine models have been used traditionally to describe the deformation of networks. Due to their prevalence, this chapter is dedicated to the review of such formulations. The chapter begins with a brief review of finite kinematics of continua and the definition of stress measures. Further, the affine deformation is defined and several parameters used to quantify the degree of nonaffinity are introduced. An expression is derived to quantify the evolution of preferential fiber orientation during affine deformation. Several constitutive models based on the affine deformation assumption are discussed: The affine models for molecular networks of flexible and semi-flexible filaments, and the affine model for athermal networks. The stress–optical law is reviewed, and its relation to the affine deformation models is discussed.
Keywords
- Type
- Chapter
- Information
- Network MaterialsStructure and Properties, pp. 128 - 145Publisher: Cambridge University PressPrint publication year: 2022
References
Bancelin, S., Lynch, B., Bonod-Bidaud, C., et al. (2015). Ex-vivo multiscale quantitation of skin biomechanics in wild-type and genetically-modified mice using multiphoton microscopy. Sci. Rep. 5, 17635.Google Scholar
Bazant, Z. P. & Oh, B. H. (1985). Microplane model for progressive fracture of concrete and rock. ASCE J. Eng. Mech. 111, 559–582.Google Scholar
Carol, I., Jirasek, M. & Bazant, Z. P. (2004). A framework for microplane models at large strain, with application to hyperelasticity. Int. J. Sol. Struct. 41, 511–557.Google Scholar
Cox, H. L. (1952). The elasticity and strength of paper and other fibrous materials. Br. J. Appl. Phys. 3, 72–81.Google Scholar
Gasser, T. C., Ogden, R. W. & Holzapfel, G. A. (2006). Hyperelastic modelling of atrial layers with distributed collagen fiber orientations. J. Roy. Soc. Interfaces 3, 15–35.Google Scholar
Graessley, W. W. (1975). Elasticity and chain dimensions in Gaussian networks. Macromolecules 8, 865–868.CrossRefGoogle Scholar
Gusev, A. A. (2019). Numerical estimates of the topological effect in the elasticity of Gaussian polymer networks and their exact theoretical description. Macromolecules 52, 3244–3251.Google Scholar
Guth, E. & Mark, H. (1934). Zur innermolekularen, Statistik, insbesondere bei Kettenmolekiilen I. Monats. F. Chem. 65, 93–121.Google Scholar
Hatami-Marbini, H. & Picu, R. C. (2008). Scaling of non-affine deformation in random semiflexible fiber networks. Phys. Rev. E 77, 062103.Google Scholar
Higgs, P. G. & Ball, R. C. (1988). Polydisperse polymer networks: elasticity, orientational properties, and small angle neutron scattering. J. de Physique 49, 1785–1811.Google Scholar
James, H. M. & Guth, E. (1943). Theory of the elastic properties of rubber. J. Chem. Phys. 11, 455–481.Google Scholar
Komori, T. & Itoh, M. (1991a). A new approach to the theory of the compression of fiber assemblies. Textile Res. J. 61, 420–428.Google Scholar
Komori, T. & Itoh, M. (1991b). Theory of the general deformation of fiber assemblies. Textile Res. J. 61, 588–594.CrossRefGoogle Scholar
Kuhn, W. (1934). Uber die Gestalt fadenformiger Molekule in Losungen. Kolloid-Z 68, 2–15.Google Scholar
Lee, D. H. & Carnaby, G. A. (1992). Compressional energy of random fiber assembly, part I: Theory. Textile Res. J. 62, 185–191.Google Scholar
Lin, T. S., Wang, R., Johnson, J. A. & Olsen, B. D. (2019). Revisiting the elasticity theory of Gaussian phantom networks. Macromolecules 52, 1685–1694.Google Scholar
Lin, Y. C., Yao, N. Y., Broedersz, C. P., et al. (2010). Origins of elasticity in intermediate filament networks. Phys. Rev. Lett. 104, 058101.Google Scholar
Nesbitt, S., Scott, W., Macione, J. & Kotha, S. (2015). Collagen fibrils in skin orient in the direction of applied uniaxial load in proportion to stress while exhibiting differential strains around hair follicles. Materials 8, 1841–1857.CrossRefGoogle ScholarPubMed
Rubinstein, M. & Colby, R. H. (2003). Polymer physics. Oxford University Press, Oxford.Google Scholar
Rubinstein, M. & Panyukov, S. (2002). Elasticity of polymer networks. Macromolecules 35, 6670–6686.Google Scholar
Ruland, A., Chen, X., Khansari, A., et al. (2018). A contactless approach for monitoring the mechanical properties of swollen hydrogels. Soft Matt. 14, 7228–7236.Google Scholar
Sakai, T., Akagi, Y., Matsunaga, T., et al. (2010). Highly elastic and deformable hydrogel formed from tetraarm polymers. Macromol. Rapid Commun. 31, 1954–1959.Google Scholar
Tehrani, M. & Sarvestani, A. (2017). Effect of chain length distribution on mechanical behavior of polymeric networks. Eur. Poly. J. 87, 136–146.Google Scholar