References
193
3. Quadratic invariants of the stress (D σ ) and strain (D ε ) deviators are proportional
τ 0 = Gγ 0 ,
or
σ ? = 3Gε ? .
(14.32)
The formulation of Hooke’s law in the form of theses 1–3 does not contain
coordinates and depends only on invariants of the stress and strain tensors. The
second advantage of such formulation is that it is possible to naturally make a bridge
between elasticity theory that is physically based on Hooke’s law and the simplest
so-called deformational theory of plasticity named as the theory of low elastic–
plastic strains by Ilyushin [3]. The following chapter describes the primary ratios of
this theory. Its applicability limits are also described.
References
1. P. Bridzhmen, Issledovaniya bol’shikh plasticheskikh deformatsii i razryva: Vliyanie vysokogo
gidrostaticheskogo davleniya na mekhanicheskie svoistva materialov (Studies of large plastic
deformations and impact of high hydrostatic pressure on the mechanical properties of
materials). (Izdatel’skaya gruppa URSS Publ., Moskow, 2010)
2. N. Davidenkov, Mekhanicheskie svoistva i ispytanie metallov (Mechanical properties and
testing of metals). (Oniks Publ., Moscow, 2012)
3. A. Il’yushin, Mekhanicheskie svoistva i ispytanie metallov (Mechanical properties and testing
of metals). (OGIZ Publ., Leningrad, Moscow, 1948)
4. Ispytatel’nye mashiny Schenck (Schenck test machines) (2014). https://schenck-rotec.de/
5. L. Kachanov, Fundamentals of Plasticity Theory (Dover, New York, 2004)
6. A. Lyav, Matematicheskaya teoriya uprugosti (Mathematical theory of elasticity). (ONTI
NKTP SSSR Publ., Moscow, Leningrad, 1935)
7. R. Mises, Mechanik der festen körper im plastischdeformablen zustand, in Göttinger:
Königlichen Gesellschaft (1913), pp. 582–592
8. V. Molotnikov, Mekhanika konstruktsii (Mechanics of structures). (SPb., Moscow, Krasnodar,
Lan’ Publ., 2012)
9. N. Muskhelishvili, Nekotorye osnovnye zadachi matematicheskoi teorii uprugosti (Some main
problems of mathematical theory elasticity). (Nauka Publ., Moscow, 1966)
193
3. Quadratic invariants of the stress (D σ ) and strain (D ε ) deviators are proportional
τ 0 = Gγ 0 ,
or
σ ? = 3Gε ? .
(14.32)
The formulation of Hooke’s law in the form of theses 1–3 does not contain
coordinates and depends only on invariants of the stress and strain tensors. The
second advantage of such formulation is that it is possible to naturally make a bridge
between elasticity theory that is physically based on Hooke’s law and the simplest
so-called deformational theory of plasticity named as the theory of low elastic–
plastic strains by Ilyushin [3]. The following chapter describes the primary ratios of
this theory. Its applicability limits are also described.
References
1. P. Bridzhmen, Issledovaniya bol’shikh plasticheskikh deformatsii i razryva: Vliyanie vysokogo
gidrostaticheskogo davleniya na mekhanicheskie svoistva materialov (Studies of large plastic
deformations and impact of high hydrostatic pressure on the mechanical properties of
materials). (Izdatel’skaya gruppa URSS Publ., Moskow, 2010)
2. N. Davidenkov, Mekhanicheskie svoistva i ispytanie metallov (Mechanical properties and
testing of metals). (Oniks Publ., Moscow, 2012)
3. A. Il’yushin, Mekhanicheskie svoistva i ispytanie metallov (Mechanical properties and testing
of metals). (OGIZ Publ., Leningrad, Moscow, 1948)
4. Ispytatel’nye mashiny Schenck (Schenck test machines) (2014). https://schenck-rotec.de/
5. L. Kachanov, Fundamentals of Plasticity Theory (Dover, New York, 2004)
6. A. Lyav, Matematicheskaya teoriya uprugosti (Mathematical theory of elasticity). (ONTI
NKTP SSSR Publ., Moscow, Leningrad, 1935)
7. R. Mises, Mechanik der festen körper im plastischdeformablen zustand, in Göttinger:
Königlichen Gesellschaft (1913), pp. 582–592
8. V. Molotnikov, Mekhanika konstruktsii (Mechanics of structures). (SPb., Moscow, Krasnodar,
Lan’ Publ., 2012)
9. N. Muskhelishvili, Nekotorye osnovnye zadachi matematicheskoi teorii uprugosti (Some main
problems of mathematical theory elasticity). (Nauka Publ., Moscow, 1966)