Abstract
A nonlinear model of an isotropic elastic material is proposed, which is a generalization of the Murnagan model, in which the expansion of the specific potential energy of the strains in a series in powers of the Genki logarithmic strain tensor is used. The physical meaning of the constants included in the obtained relations is determined. Along with bulk modulus K and shear modulus G, constants c1, c2, c3 associated with third-order moduli of elasticity were used: the constant c1 reflects the nonlinear dependence of the hydrostatic stress on volumetric deformation, the constant c2 reflects the dilatation effect, and the constant c3 reflects the deviation of the stress state angle from the deformed state angle. The article is dedicated to the blessed memory of outstanding scientists A. A. Ilyushin and L. A. Tolokonnikov, whose ideas were developed in this work.
Similar content being viewed by others
References
V. V. Novozhilov, Foundations of the Nonlinear Theory of Elasticity (Gostekhizdat, Moscow, 1948; Courier Corporation, Massachusetts, 1999).
L. A. Tolokonnikov, “The Connection Between Stresses and Strains in the Nonlinear Theory of Elasticity,” Prikl. Mat. Mekh. 20 (3), 439–444 (1956).
A. I. Lurie, Nonlinear Theory of Elasticity (Nauka, Moscow, 1980) [in Russian].
G. Montella, S. Govindjee, and P. Neff, “The Exponentiated Hencky Strain Energy in Modelling Tire Dderived Material for Moderately Large Deformations,” J. Engng Mat. Tech. Trans. ASME. (2015). Available at arXiv:1509.06541
L. A. Mihai and A. Goriely, “How to Characterize a Nonlinear Eelastic Material? A Reviewon nonlinear Constitutive Parameters in Isotropic Finite Elasticity,” Proc. R. Soc. A 1: 20170607 (2017).
L. Anand, “A Constitutive Model for Compressible Elastomeric Solids,” Comp. Mech. 18, 339–355 (1996).
H. Xiao and L. S. Chen, “Hencky’s Elasticity Model and Linear Stress-Strain Relations in Isotropic Finite Hyperelasticity,” Acta Mech. 157, 51–60 (2002).
M. Latorre and F. J. Montans, “On the Interpretation of the Logarithmic Strain Tensor in an Arbitrary System of Representation,” Int. J. Sol. Struct. 51, 1507–1515 (2014).
S. N. Korobeinikov and A. A. Oleinikov, “Lagrangian Formulation of Hencky’s Hyperelastic Material,” Dal’nevost. Mat. Zh. 11 (2), 155–180 (2011).
A. A. Markin and M. Yu. Sokolova, Thermomechanics of Elastoplastic Deformation (Fizmatlit, Moscow, 2013) [in Russian].
A. V. Muravlev, “On a Representation of an Elastic Potential in A. A. Il’yushin’s Generalized Strain Space,” Izv. Ros. Akad. Nauk Mekh. Tv. Tela, No. 1, 99–102 (2011) [Mech. Sol. (Engl. Transl.) 46, (1), 77–79 (2011)].
K. Brugger, “Thermodynamic Definition of Higher Order Elastic Coefficients,” Phys. Rev. 133 (6A), A1611 (1964).
R. Hill, “On Constitutive Inequalities for Simple Materials—I,” J. Mech. Phys. Sol. 16 (4), 229–242 (1968).
M. Yu. Sokolova and D. V. Khristich, “The Symmetry of the Thermoelastic Properties of Quasicrystals,” Prikl. Mat. Mekh. 78 (5), 728–734 (2014) [J. Appl. Math. Mech. (Engl. Transl. 78 (5), 524–528 (2014)].
Y. Astapov, D. Khristich, A. Markin, and M. Sokolova, “The Construction of Nonlinear Elasticity Tensors for Crystals and Quasicrystals,” Int. J. Appl. Mech. 9 (6), 1750080-1–1750080-15 (2017).
R. A. Toupin and B. Bernstein “Sound Waves in Deformed Perfectly Elastic Materials. Acoustoelastic Effect,” J. Acoust. Soc. Am. 33 (2), 216–225 (1961).
Yu. I. Sirotin and M. P. Shaskol’skaya, Foundations of Crystal Physics (Nauka, Moscow, 1975) [in Russian].
G. L. Brovko, Constitutive Relations of Continuum Mechanics. The Development of Mathematical Apparatus and the Foundations of the General Theory (Nauka, Moscow, 2017) [in Russian].
M. O. Glagoleva, A. A. Markin, N. M. Matchenko, and A. A. Treshchev, “Properties of Isotropic Elastic Materials,” Izv. Tuls. Univ. Ser. Mat. Mekh. Inf. 4 (2), 15–19 (1998).
V. F. Astapov, A. A. Markin and M. Yu. Sokolova, “Determination of the Elastic Properties of Materials from Experiments for Solid Cylinders,” Izv. Ros. Akad. Nauk Mekh. Tv. Tela, No. 1, 104–111 (2002) [Mech. Sol. (Engl. Transl.) 37 (1), 85–91 (2002)].
GOST R 53568-2009. Non-Destructive Testing. Evaluation of the Third Order Elasticity Modulus by Ultrasound Method. General Requirements (Standartinform, Moscow, 2010) [in Russian].
Acknowledgements
This work was supported by a grant from the Russian Foundation for Basic Research (project No. 18-31-20053).
Author information
Authors and Affiliations
Corresponding author
Additional information
Russian Text © The Author(s), 2019, published in Izvestiya Akademii Nauk, Mekhanika Tverdogo Tela, 2019, No. 6, pp. 68–75.
About this article
Cite this article
Markin, A.A., Sokolova, M.Y. Variant of Nonlinear Elasticity Relations. Mech. Solids 54, 1182–1188 (2019). https://doi.org/10.3103/S0025654419080089
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S0025654419080089