Modelling of The Large Strains. The Message 3. About Theoretical Bases of Use of a Logarithmic Measure of Strains of Hencky

This paper represents the comprehensive data on theoretical foundations of applying logarithmic measure of Hencky deformation for deformation modelling with various types of large deformations: thermal, elastic, plastic and creep. We consider the properties of the logarithmic deformation (Hencky) for the fiber of materials. We use basic measures of stresses and the second law of thermodynamics as auxiliary information. Crucially, we define the law of elastic deformation (for an isotropic material) using Hencky strains – the stress measure corresponding to Hencky strains. We also uncover an equivalent formulation of the energy of internal forces. We show that components of the Hencky strains correspond to concepts of strains measure. As energy conjugating tensor stress components they have main components of Kirchhoff stress (stress of Noll). They are linked with key components of Euler-Cauchy “rotated” stress tensor solely through the scale factor. They allow using Hooke classical law to calculate the stress for unchanged isotropic material (metal).

Publication year: 
2012
Issue: 
6
УДК: 
539.3
С. 86—93. Бібліогр.: 15 назв.
References: 

1. Рудаков К.М., Добронравов О.А. Моделювання великих деформацій. Повідомлення 1. Мультиплікативний розклад при наявності чотирьох типів деформацій // Вісн. НТУУ “КПІ”. Сер. Машинобудування. – 2012. – № 64. – С. 7–12.
2. E.H. Lee, “Elastic–plastic deformations at finite strains”, J. Appl. Mech. (ASME), vol. 36, pp. 1–6, 1969.
3. Рудаков К.М., Яковлєв А.І. Моделювання великих деформацій. Повідомлення 2. Температурні деформації // Вісн. НТУУ “КПІ”. Сер. Машинобудування. – 2012. – № 65. – С. 10–18.
4. Жермен П. Курс механики сплошных сред. Общая теория: Пер. с фр. В.В. Федулова. – М.: Высш. шк., 1983. – 398 с.
5. Новожилов В.В. О формах связи между напряжениями и деформациями в первоначально изотропных неупругих телах (геометрическая сторона вопроса) // Прикл. матем. Механ. – 1963. – 27, вып. 5. – С. 794– 812.
6. Хилл Р. Об определяющих неравенствах для простых материалов // Механика. – 1969. – № 4 (116). – С. 94–118.
7. R. Hill, “Aspects of invariance in solid mechanics”, Adv. Appl. Mech., vol. 18, pp. 1–75, 1978.
8. Черных К.Ф. Нелинейная теория упругости в машиностроительных расчетах. – Л.: Машиностроение, Ленинград. отд-ние, 1986. – 336 с.
9. Коробейников С.Н. Нелинейное деформирование твердых тел. – Новосибирск: СО РАН, 2000. – 262 с.
10. L. Anand, “On H. Hencky’s approximate strain energy function for modeling deformations”, ASME J. Appl. Mech., vol. 46, pp. 78 82, 1979.
11. S.N. Atluri, “Alternate stress and conjugate strain measures, and mixed variational formulations involving rigid rotations, for computational analyses of finitely deformed solids, with application to plates and shells: I. Theory”, Comput. Struct., vol. 18, pp. 93–116, 1984.
12. A. Hoger, “The stress conjugate to logarithmic strain”, Int. J. Solids Struct., vol. 23, pp. 1645–1656, 1987.
13. H. Xiao at al., “Logarithmic strain, logarithmic spin and logarithmic rate”, Acta Mechanica, vol. 124(1–4), pp. 89–105, 1997.
14. A.L. Eterović, K-J. Bathe, “A hyperelastic-based large strain elasto–plastic constitutive formulation with combined isotropic-kinematic hardening using the logarithmic stress and strain measures”, Int. J. Num. Meth. Enging, vol. 30, pp. 1099–1114, 1990.
15. F.J. Montáns, K-J. Bathe, “Computational issues in large strain elasto-plasticity: an algorithm for mixed hardening and plastic spin”, Int. J. Num. Meth. Enging, vol. 63, pp. 159–196, 2005.

AttachmentSize
2012-6-13.pdf281.99 KB

Тематичні розділи журналу

,