Evaluating the Effect of the Second Invariant of Deformation Tensor in The Axial and Azimuthal Shear Deformations

محورهای موضوعی : Mechanical Engineering

Amir Ghafouri Sayyad ¹ (Department of Mechanical Engineering, Science and Research Branch, Islamic Azad University, Tehran, Iran)
Ali Imam ² (Department of Mechanical Engineering, Science and Research Branch, Islamic Azad University, Tehran, Iran)
Shahram Etemadi Haghighi ³ (Department of Mechanical Engineering, Science and Research Branch, Islamic Azad University, Tehran, Iran)

تاریخ دریافت : 1400/01/30 تاریخ پذیرش : 1400/05/23 تاریخ انتشار : 1400/12/10

کلید واژه: Incompressible hyperelastic solids, Strain hardening, Azimuthal shear, Controllable deformation, Axial shear, Second invariant of the deformation tensor,

چکیده مقاله :

The purpose of the present paper is to investigate the effect of the second invariant of the deformation tensor on the axial and azimuthal shear deformation of an incompressible hyperelastic solid with various strain energy functions. To this end, the axial shear deformation of an incompressible cylinder with the modified Gent-Thomas, Gent-Thomas, Gent-Gent, and Carroll strain energies subjected to an axial shear traction is considered, where the displacement field is determined analytically for the first three models and numerically for the fourth model. The phenomenon of strain hardening at large elastic deformations, predicted either by the limiting chain extensibility condition for the modified Gent-Thomas and Gent-Gent models or phenomenologically by the Carroll model, is observed and it is shown that the second invariant of deformation increases the strain hardening experienced by such materials. Next, the azimuthal shear deformation of an incompressible annular wedge with the modified Gent-Thomas, Gent-Thomas, Gent-Gent, and Carroll models is considered, where the annular wedge is subjected to a controllable azimuthal shear deformation and the angular displacement is determined analytically for all the above models. Again, the second invariant of the deformation tensor is shown to have a significant effect on the azimuthal shear deformation as reflected in the increase of the strain hardening of the material in such deformation. In addition, the annular wedge with the modified Gent-Thomas and Carroll models is shown to have a higher resistance in azimuthal shear deformation than the other models mentioned above.

منابع و مأخذ:

• Hayes, M., Saccomandi, G., Topics in Finite Elasticity, Springer-Verlag, New York, USA, 2001, pp. 31-93, ISBN: 978-3-211-83336-0.


• Gent, A. N., Elastic Instabilities in Rubber, International Journal of Non-Linear Mechanics, Vol. 40, No. 2-3, 2005, pp. 165-175, DOI: 1016/j.ijnonlinmec.2004.05.006.


• Warne, D. A., Warne, P. G., Torsion in Nonlinearly Elastic Incompressible Circular Cylinders, International Journal of Non-Linear Mechanics, Vol. 86, 2016, pp. 158-166, DOI: 1016/j.ijnonlinmec.2016.08.008.


• Gent, A. N., Thomas, A. G., Forms for the Stored (Strain) Energy Function for Vulcanized Rubber, Journal of Polymer Science, Vol. 28, No. 118, 1958, pp. 625–628, DOI: 10.1002/pol.1958.1202811814.


• Pucci, E., Saccomandi, G., A Note on the Gent Model for Rubber-Like Materials, Rubber Chemistry and Technology, Vol. 75, No. 5, 2002, pp. 839-852, DOI: 5254/1.3547687.


• Carroll, M. M., A Strain Energy Function for Vulcanized Rubbers, Journal of Elasticity, Vol. 103, No. 2, 2011, pp. 173-187, DOI: 10.1007/s10659-010-9279-0.


• Gent, A. N., Extensibility of Rubber under Different Types of Deformation, Journal of Rheology, Vol. 49, No. 1, 2005, pp. 271-275, DOI: 1122/1.1835343.


• Gent A. N., 1996, A New Constitutive Relation for Rubber, Rubber Chemistry and Technology, Vol. 69, No. 1, 1996, pp. 59-61, DOI: 10.5254/1.3538357.


• Kilian, H. G., Equation of State of Real Networks, Polymer, Vol. 22, No. 2, 1981, pp. 209–217, DOI: 1016/0032-3861(81)90200-7.


• Edwards, S. F., Vilgis, T., The Effect of Entanglements in Rubber Elasticity, Polymer, Vol. 27, No. 4, 1986, pp. 483–492, DOI: 1016/0032-3861(86)90231-4.


• Zidi, M., Combined Torsion, Circular and Axial Shearing of a Compressible Hyperelastic and Prestressed, Journal of Applied Mechanics, Vol. 67, No. 1, 2000, pp. 33-40, DOI: 1115/1.321149.


• Ogden, R.W., Chadwick, P., and Haddon, E. W., Combined Axial and Torsional Shear of a Tube of Incompressible Isotropic Elastic Material, The Quarterly Journal of Mechanics and Applied Mathematics, Vol. 26, No. 1, 1973, pp. 23-41, DOI: 1093/qjmam/26.1.23.


• Jiang, X., Ogden, R. W., Some New Solutions for the Axial Shear of a Circular Cylindrical Tube of Compressible Elastic Material, International Journal of Non-linear Mechanics, Vol. 35, No. 2, 2000, pp. 361-369, DOI: 1016/S0020-7462(99)00041-4.


• Kanner, L. M., Horgan, C. O., Inhomogeneous Shearing of Strain Stiffening Rubber-Like Hollow Circular Cylinders, International Journal of Solids and Structures, Vol. 45, No. 20, 2008, pp. 5464-5482, DOI: 1016/j.ijsolstr.2008.05.030.


• Horgan, C. O., Saccomandi, G., Pure Axial Shear of Isotropic, Incompressible Nonlinearly Elastic Materials With Limiting Chain Extensibility, Journal of Elasticity Vol. 57, No. 3, 1999, pp. 307-319, DOI:1023/A:1007639129264.


• Benjamin, C. C., Myneni, M., Muliana, A., and Rajagopal, K. R., Motion of a Finite Composite Cylindrical Annulus Comprised of Nonlinear Elastic Solids Subject to Periodic Shear, International Journal of Non-Linear Mechanics, Vol. 113, 2019, pp. 31-43, DOI: 1016/j.ijnonlinmec.2019.03.010.


• Horgan, C. O., Saccomandi, G., and Sgura, I., A Two-Point Boundary-Value Problem for the Axial Shear of Hardening Isotropic Incompressible Nonlinearly Elastic Materials, SIAM Journal on Applied Mathematics, Vol. 62, No. 5, 2002, pp. 1712-1727, DOI: 1137/S0036139901391963.


• Steinmann, P., Hossain, M., and Possart, G., Hyperelastic Models for Rubber-Like Materials: Consistent Tangent Operators and Suitability for Treloar’s Data, Archive of Applied Mechanics, Vol. 82, No. 9, 2012, pp. 1183-1217, DOI: 1007/s00419-012-0610-z.


• Destrade, M., Saccomandi, G., and Sgura, I., Methodical Fitting for Mathematical Models of Rubber-Like Materials, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 473, No. 2198, 2017, pp. 20160811, DOI: 1098/rspa.2016.0811.


• Zidi, M., Azimuthal Shearing and Torsion of a Compressible Hyperelastic and Prestressed Tube, International Journal of Non-Linear Mechanics, Vol. 3, No. 2, 2000, pp. 201-209, DOI: 1016/S0020-7462(99)00008-6.


• Zidi, M., Large Shearing of a Prestressed Tube, Journal of Applied Mechanics, Vol. 67, No. 1, 2000, pp. 1-4, DOI:1115/1.321175.


• Horgan, C. O., Saccomandi, G., Pure Azimuthal Shear of Isotropic, Incompressible Hyperelastic Materials with Limiting Chain Extensibility, International Journal of Non-Linear Mechanics, Vol. 3, No. 3, 2001, pp. 465-475, DOI: 1016/S0020-7462(00)00048-2.


• Dagher, M. A., Soldatos, K. P., Pure Azimuthal Shear Deformation of an Incompressible Tube Reinforced by Radial Fibres Resistant in Bending, The IMA Journal of Applied Mathematics, Vol. 79, No. 5, 2014, pp. 848–868, DOI: 1093/imamat/hxu013.


• Bustamante, R., Some Universal Solutions for a Class of Incompressible Elastic Body that is not Green Elastic: The Case of Large Elastic Deformations, The Quarterly Journal of Mechanics and Applied Mathematics, Vol. 73, No. 2, 2020, pp. 177-199, DOI: 1093/qjmam/hbaa006.


• El Hamdaoui, M., Merodio, J., and Ogden, R. W., Deformation Induced Loss of Ellipticity in an Anisotropic Circular Cylindrical Tube, Journal of Engineering Mathematics, Vol. 109, No. 1, 2018, pp. 31-45, DOI: 1007/s10665-017-9904-z.


• Horgan, C. O., Saccomandi, G., Helical Shear for Hardening Generalized Neo-Hookean Elastic Materials, Mathematics and Mechanics of Solids, Vol. 8, No. 5, 2003, pp. 539-559, DOI: 10.1177/10812865030085007.


• Anssari-Benam, A., Bucchi, A., A Generalised Neo-Hookean Strain Energy Function for Application to The Finite Deformation of Elastomers, International Journal of Non-Linear Mechanics, Vol. 128, 2021, pp. 103626, DOI: 1016/j.ijnonlinmec.2020.103626.


• Ghafouri Sayyad, A., Imam, A., and Etemadi Haghighi, S., Evaluating the Response of a Modified Gent-Thomas Strain Energy Function Having Limiting Chain Extensibility Condition in Torsion and Azimuthal Shear, Polymers and Polymer Composites, 2021, DOI: 1177/09673911211003394.


• Marckmann, G., Verron, E., Comparison of Hyperelastic Models for Rubber-Like Materials, Rubber Chemistry and Technology, 79, No. 5, 2006, pp. 835-858, DOI: DOI:10.5254/1.3547969.


• Ogden, R. W., Non-Linear Elastic Deformations, Ellis Horwood, Chichester, England, 1984, pp. 224-326, ISBN: 0-8-5312-273-3.


• Ogden, R.W., Saccomandi, G., and Sgura, I., Fitting Hyperelastic Models to Experimental Data, Computational Mechanic, Vol. 34, No. 6, 2004, pp. 484-502, DOI: 1007/s00466-004-0593-y.


• Abramowitz, M., Stegun I. A., Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables, Dover, New York, USA, 1965, pp. 885-887, ISBN: 978-0486612720.