# Finite Stretching and Shearing of an Internally Balanced Elastic Solid

- 182 Downloads
- 2 Citations

## Abstract

When seeking to separately describe elastic and inelastic effects in solids undergoing finite strain, it is typical to factor the deformation gradient \(\bf F\) into parts, say \(\mathbf{F}={\hat {\mathbf{F}}}{\mathbf{F}}^{*}\). Such a factorization can also be used to describe a notion of internal elastic balance. For equilibrium deformations this balance emerges naturally from an appropriately formulated energy minimization by considering the variation with respect to the multiplicative decomposition itself. The internal balance requirement is then a tensor relation between **F**, \({\hat {\mathbf{F}}}\) and **F** ^{∗}. Such theories holds promise for describing various substructural reconfigurations in solids. This includes the development of surfaces that localize slip once a threshold value of load is obtained. We consider an incompressible internally balanced elastic material with a constitutive law that is motivated by neo-Hookean behavior in the standard hyperelastic theory. The role of the internal balance relation is examined in uniaxial loading and also in simple shearing. In uniaxial loading the principle directions remain fixed. In simple shearing the principle directions change as the deformation proceeds. To address the latter, we establish a specific sense in which the internal balance relation can be solved analytically for general isochoric \(\bf F\).

## Keywords

Hyperelasticity Internal balance Neo-Hookean Substructural reconfiguration## Mathematics Subject Classification

74A20 74A60 74B20## Notes

### Acknowledgements

Helpful discussions with H. Tsai when early aspects of this treatment were first formulated are gratefully acknowledged. This publication was made possible by NPRP grant #4-1333-1-214 from the Qatar National Research Fund (a member of the Qatar Foundation). The statements made herein are solely the responsibility of the authors.

## References

- 1.Anand, L., Gurtin, M.E.: The decomposition \({\mathbf{F}=\mathbf{F}^{e} \mathbf{F}^{p}}\), material symmetry, and plastic irrotationality for solids that are isotropic-viscoplastic or amorphous. Int. J. Plast.
**21**, 1686–1719 (2005) zbMATHCrossRefGoogle Scholar - 2.Capriz, G.: Continua with Microstructure. Springer Tracts in Natural Philosophy, vol. 35. Springer, New York (1989) zbMATHGoogle Scholar
- 3.Choksi, R., Del Piero, G., Fonseca, I., Owen, D.: Structured deformations as energy minimizers in models of fracture and hysteresis. Math. Mech. Solids
**4**, 321–356 (1999) zbMATHMathSciNetCrossRefGoogle Scholar - 4.Del Piero, G., Owen, D.R.: Structured deformations of continua. Arch. Ration. Mech. Anal.
**124**, 99–155 (1993) zbMATHCrossRefGoogle Scholar - 5.Del Piero, G., Owen, D.R.: Integral-gradient formulae for structured deformations. Arch. Ration. Mech. Anal.
**131**, 121–138 (1995) zbMATHCrossRefGoogle Scholar - 6.Demirkoparan, H., Pence, T.J., Tsai, H.: Hyperelastic internal balance by multiplicative decomposition of the deformation gradient. Arch. Ration. Mech. Anal.
**214**, 923–970 (2014) zbMATHMathSciNetCrossRefGoogle Scholar - 7.Deseri, L., Owen, D.R.: Toward a field theory for elastic bodies undergoing disarrangements. J. Elast.
**70**, 197–236 (2003) zbMATHMathSciNetCrossRefGoogle Scholar - 8.Deseri, L., Owen, D.R.: Submacroscopically stable equilibria of elastic bodes undergoing disarrangements and dissipation. Math. Mech. Solids
**15**, 611–638 (2010) zbMATHMathSciNetCrossRefGoogle Scholar - 9.DiCarlo, A., Quiligotti, S.: Growth and balance. Mech. Res. Commun.
**29**, 449–456 (2002) zbMATHMathSciNetCrossRefGoogle Scholar - 10.Garikipati, K.: The kinematics of biological growth. Appl. Mech. Rev.
**62**(3), 030801 (2009) CrossRefADSGoogle Scholar - 11.Goriely, A., Amar, M.B.: On the definition and modeling of incremental, cumulative, and continuous growth laws in morphoelasticity. Biomech. Model. Mechanobiol.
**6**, 289–296 (2007) CrossRefGoogle Scholar - 12.Grillo, A., Federico, S., Wittum, G., Imatani, S., Giaquinta, G., Micunovic, M.: Evolution of a fibre-reinforced growing mixture. Nuovo Cimento C
**32**, 97–119 (2009) ADSGoogle Scholar - 13.Gurtin, M.E.: Configurational Forces as Basic Concepts of Continuum Physics. Springer, New York (2000) Google Scholar
- 14.Kröner, E.: Allgemeine kontinuumstheorie der versetzungen und eigenspannungen. Arch. Ration. Mech. Anal. 4, 273–334 (1959/60) Google Scholar
- 15.Lee, E.H.: Elastic plastic deformation at finite strain. J. Appl. Mech.
**36**, 1–6 (1969) zbMATHCrossRefADSGoogle Scholar - 16.Maugin, G.A.: The principle of virtual power: from eliminating metaphysical forces to providing an efficient modeling tool. Contin. Mech. Thermodyn.
**25**, 127–146 (2013). In memory of Paul Germain (1920–2009) MathSciNetCrossRefADSGoogle Scholar - 17.Owen, D.R., Paroni, R.: Second-order structured deformations. Arch. Ration. Mech. Anal.
**155**, 215–235 (2000) zbMATHMathSciNetCrossRefGoogle Scholar - 18.Rivlin, R.S.: Large elastic deformations of isotropic materials, iv. further development of the general theory. Philos. Trans. R. Soc. Lond. A
**241**(14-15), 379–397 (1948) zbMATHMathSciNetCrossRefADSGoogle Scholar - 19.Rodriguez, E.K., Hoger, A., McCulloch, A.D.: Stress-dependent finite growth in soft elastic tissues. J. Biomech.
**27**, 455–467 (1994) CrossRefGoogle Scholar - 20.Sawyers, K.S.: On the possible values of the strain invariants for isochoric deformations. J. Elast.
**7**, 99–102 (1977) CrossRefGoogle Scholar - 21.Stephenson, R.A.: On the uniqueness of the square-root of a symmetric, positive-definite tensor. J. Elast.
**10**, 213–214 (1980) zbMATHMathSciNetCrossRefGoogle Scholar - 22.Truesdell, C., Noll, W.: The Nonlinear Field Theories of Mechanics. Handbuch der Physik III/3. Springer, Berlin, Heidelberg, New York (1965) Google Scholar