О СПОСОБНОСТИ НЕЛИНЕЙНОГО ОПРЕДЕЛЯЮЩЕГО СООТНОШЕНИЯ РАБОТНОВА ДЛЯ НАСЛЕДСТВЕННЫХ МАТЕРИАЛОВ МОДЕЛИРОВАТЬ ДИАГРАММЫ ДЕФОРМИРОВАНИЯ С ПАДАЮЩИМ УЧАСТКОМ
Ключевые слова:
вязкоупругость, физическая нелинейность, диаграммы деформирования, положительная скоростная чувствительность, равновесная диаграмма деформирования, деформационное разупрочнение
Аннотация
Исследуется физически нелинейное определяющее соотношение Ю.Н. Работнова с двумя произвольными материальными функциями для изотропных реономных материалов с целью определения комплекса моделируемых реологических эффектов, границ и индикаторов его области применимости. Выведено уравнение семейства диаграмм деформирования с постоянными скоростями, порождаемых этим соотношением в условиях одноосного квазистатического нагружения, аналитически изучены их общие качественные свойства, зависимости от скорости деформирования и характеристик материальных функций. Обнаруженные свойства сопоставлены с типичными свойствами диаграмм испытаний вязкоупругопластичных материалов и с общими свойствами диаграмм деформирования, порождаемых линейным интегральным соотношением вязкоупругости Больцмана - Вольтерры с произвольной функцией ползучести, которое соотношение Работнова обобщает в одноосном случае; указаны унаследованные свойства и свойства, приобретенные вследствие введения второй материальной функции. В частности, доказано, что соотношение Работнова моделирует только положительную скоростную чувствительность, а при стремлении скорости нагружения к нулю или бесконечности семейство диаграмм деформирования соотношения Работнова сходится к предельным кривым (мгновенного или равновесного деформирования). Получены достаточные условия монотонного возрастания диаграмм деформирования и наличия у них максимума. Показано, что определяющее соотношение Работнова может качественно описывать разупрочнение материала, то есть диаграммы деформирования с точкой максимума и ниспадающей ветвью, наблюдаемые в квазистатических испытаниях многих материалов. Способностью описывать деформационное разупрочнение соотношение Работнова кардинально отличается от линейного определяющего соотношения вязкоупругости Больцмана - Вольтерры, которое порождает только возрастающие и выпуклые вверх диаграммы деформирования.
Литература
1. Khokhlov A.V. Kharakternye osobennosti semeystv krivykh deformirovaniya lineynykh modeley vyazkouprugosti [Specific features of stress-strain curves at constant stress rate or strain rate yielding from linear viscoelasticity]. Problemy prochnosti i plastichnosti [Problems of Strength and Plasticity]. 2015. Vol. 77. No 2. P. 139-154 (In Russian).
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3. Khokhlov A.V. Analiz obshchikh svoystv krivykh polzuchesti pri tsiklicheskikh stupenchatykh nagruzheniyakh, porozhdaemykh lineynoy teoriey nasledstvennosti [Analysis of creep curves produced by the linear viscoelasticity theory under cyclic stepwise loadings]. Vestnik Samarskogo gosudarstvennogo tekhnicheskogo universiteta. Seriya Fiziko-matematicheskie nauki [Journal Samara State Tech. Univ. Ser. Phys. Math. Sci.]. 2017. Vol. 21. No 2. P. 326-361. DOI: 10.14498/vsgtu1533 (In Russian).
4. Khokhlov A.V. Dvustoronniye otsenki dlya funktsii relaksatsii lineynoy teorii nasledstvennosti cherez krivyye relaksatsii pri ramp-deformirovanii i metodiki yeye identifikatsii [Тwo-sided bounds for relaxation modulus in the linear viscoelasticity via relaxation curves at ramp strain histories and identification techniques]. Izvestiya Rossiyskoy Akademii Nauk. Mekhanika tverdogo tela [Mechanics of Solids]. 2018. No 3. P. 81-104. DOI: 10.7868/S0572329918030108 (In Russian).
5. Khokhlov A.V. Analiz svoystv krivykh polzuchesti s proizvol'noy nachal'noy stadiyey nagruzheniya, porozhdayemykh lineynoy teoriyey nasledstvennosti [Analysis of properties of creep curves generated by the linear viscoelasticity theory under arbitrary loading programs at initial stage]. Vestnik Samarskogo gosudarstvennogo tekhnicheskogo universiteta. Seriya Fiziko-matematicheskie nauki [Journal Samara State Tech. Univ. Ser. Phys. Math. Sci.]. 2018. No 1. P. 65-95. DOI: 10.14498/vsgtu1543. (In Russian).
6. Khokhlov A.V. Krivyye polzuchesti i relaksatsii nelineynogo opredelyayushchego sootnosheniya Yu.N. Rabotnova dlya vyazkouprugoplastichnykh materialov [Creep and relaxation curves produced by the Rabotnov nonlinear constitutive relation for viscoelastoplastic materials]. Problemy prochnosti i plastichnosti [Problems of Strength and Plasticity]. 2016. Vol. 78. No 4. P. 452-466 (In Russian).
7. Khokhlov A.V. Analiz obshchikh svoystv krivykh polzuchesti pri stupenchatom nagruzhenii, porozhdaemykh nelineynym sootnosheniem Rabotnova dlya vyazkouprugoplastichnykh materialov [Analysis of general properties of creep curves generated by the Rabotnov nonlinear hereditary relation under multi-step loadings]. Vestnik Moskovskogo gosudarstvennogo tekhnicheskogo universiteta im. N.E. Baumana. Ser. Estestvennye nauki [Herald of the Bauman Moscow State Technical University. Series Natural Sciences]. 2017. No 3. P. 93-123. DOI: 10.18698/1812-3368-2017-3-93-123 (In Russian).
8. Khokhlov A.V. Analysis of properties of ramp stress relaxation curves produced by the Rabotnov non-linear hereditary theory. Mechanics of Composite Materials. 2018. Vol. 54. No 4. P. 473-486. DOI: 10.1007/s11029-018-9757-1.
9. Khokhlov A.V. Asymptotic behavior of creep curves in the Rabotnov nonlinear heredity theory under piecewise constant loadings and memory decay conditions. Moscow University Mechanics Bulletin. 2017. Vol. 72. No 5. P. 103-107.
10. Khokhlov A.V. Modelirovanie zavisimosti krivykh polzuchesti pri rastyazhenii i koeffitsienta Puassona reonomnykh materialov ot gidrostaticheskogo davleniya s pomoshch'yu nelineyno-nasledstvennogo sootnosheniya Rabotnova [Simulation of hydrostatic pressure influence on creep curves and Poisson's ratio of rheonomic materials under tension using the Rabotnov non-linear hereditary relation]. Mekhanika kompozitsionnykh materialov i konstruktsiy [Journal on Composite Mechanics and Design]. 2018. Vol. 24. No 3. P. 407-436 (In Russian).
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14. Rabotnov Yu.N., Papernik L.K., Stepanychev E.I. Application of the nonlinear theory of heredity to the description of time effects in polymeric material. Polimer Mechanics. 1971. Vol. 7. No 1. P. 63-73.
15. Dergunov N.N., Papernik L.Kh., Rabotnov Yu.N. Analysis of behavior of graphite on the basis of nonlinear heredity theory. Journal of Applied Mechanics and Technical Physics. 1971. Vol. 12. No 2. P. 235-240. DOI: 10.1007/BF00850695.
16. Rabotnov Yu.N., Papernik L.Kh., Stepanychev E.I. Nelineynaya polzuchest' stekloplastika ТS8/3-250 [Non-linear creep of fibreglass reinforced plastic ТS 8/3-250]. Mekhanika polymerov [Polymer Mechanics]. 1971. No 3. P. 391-397 (In Russian).
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18. Rabotnov Yu.N., Suvorova Yu.V. О zakone deformirovaniya metallov pri odnoosnom nagruzhenii [On a law of metals deforming under uniaxial loading]. Izvestiya AN SSSR. Mekhanika tverdogo tela [Mechanics of Solids]. 1972. No 4. P. 41-54 (In Russian).
19. Rabotnov Yu.N., Papernik L.K., Stepanychev E.I. Description of creep of composition materials under tension and compression. Polymer Mechanics. 1973. Vol. 9. No 5. P. 690-695. https://doi.org/10.1007/BF00856259.
20. Rabotnov Yu.N. Elementy nasledstvennoi mekhaniki tverdykh tel [Fundamentals of Hereditary Solid Mechanics]. Moscow. Nauka Publ. 1977. 384 p. (In Russian).
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22. Osokin A.E., Suvorova Yu.V. Nonlinear governing equation of a hereditary medium and methodology of determining its parameters. Journal of Applied Mathematics and Mechanics. 1978. Vol. 42. No 6. P. 1214-1222. DOI: 10.1016/0021-8928(78)90072-2.
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2. Khokhlov A.V. Kachestvennyy analiz obshchikh svoystv teoreticheskikh krivykh lineynogo opredelyayushchego sootnosheniya vyazkouprugosti [The qualitative analysis of theoretic curves generated by linear viscoelasticity constitutive equation]. Nauka i obrazovanie: nauchnoe izdanie MGTU im. N.E. Baumana [Science and Education: Scientific Publication of BMSTU]. 2016. No 5. P. 187-245.DOI: 10.7463/0516.0840650. URL: http://technomagelpub.elpub.ru/jour/article/view/869 (In Russian).
3. Khokhlov A.V. Analiz obshchikh svoystv krivykh polzuchesti pri tsiklicheskikh stupenchatykh nagruzheniyakh, porozhdaemykh lineynoy teoriey nasledstvennosti [Analysis of creep curves produced by the linear viscoelasticity theory under cyclic stepwise loadings]. Vestnik Samarskogo gosudarstvennogo tekhnicheskogo universiteta. Seriya Fiziko-matematicheskie nauki [Journal Samara State Tech. Univ. Ser. Phys. Math. Sci.]. 2017. Vol. 21. No 2. P. 326-361. DOI: 10.14498/vsgtu1533 (In Russian).
4. Khokhlov A.V. Dvustoronniye otsenki dlya funktsii relaksatsii lineynoy teorii nasledstvennosti cherez krivyye relaksatsii pri ramp-deformirovanii i metodiki yeye identifikatsii [Тwo-sided bounds for relaxation modulus in the linear viscoelasticity via relaxation curves at ramp strain histories and identification techniques]. Izvestiya Rossiyskoy Akademii Nauk. Mekhanika tverdogo tela [Mechanics of Solids]. 2018. No 3. P. 81-104. DOI: 10.7868/S0572329918030108 (In Russian).
5. Khokhlov A.V. Analiz svoystv krivykh polzuchesti s proizvol'noy nachal'noy stadiyey nagruzheniya, porozhdayemykh lineynoy teoriyey nasledstvennosti [Analysis of properties of creep curves generated by the linear viscoelasticity theory under arbitrary loading programs at initial stage]. Vestnik Samarskogo gosudarstvennogo tekhnicheskogo universiteta. Seriya Fiziko-matematicheskie nauki [Journal Samara State Tech. Univ. Ser. Phys. Math. Sci.]. 2018. No 1. P. 65-95. DOI: 10.14498/vsgtu1543. (In Russian).
6. Khokhlov A.V. Krivyye polzuchesti i relaksatsii nelineynogo opredelyayushchego sootnosheniya Yu.N. Rabotnova dlya vyazkouprugoplastichnykh materialov [Creep and relaxation curves produced by the Rabotnov nonlinear constitutive relation for viscoelastoplastic materials]. Problemy prochnosti i plastichnosti [Problems of Strength and Plasticity]. 2016. Vol. 78. No 4. P. 452-466 (In Russian).
7. Khokhlov A.V. Analiz obshchikh svoystv krivykh polzuchesti pri stupenchatom nagruzhenii, porozhdaemykh nelineynym sootnosheniem Rabotnova dlya vyazkouprugoplastichnykh materialov [Analysis of general properties of creep curves generated by the Rabotnov nonlinear hereditary relation under multi-step loadings]. Vestnik Moskovskogo gosudarstvennogo tekhnicheskogo universiteta im. N.E. Baumana. Ser. Estestvennye nauki [Herald of the Bauman Moscow State Technical University. Series Natural Sciences]. 2017. No 3. P. 93-123. DOI: 10.18698/1812-3368-2017-3-93-123 (In Russian).
8. Khokhlov A.V. Analysis of properties of ramp stress relaxation curves produced by the Rabotnov non-linear hereditary theory. Mechanics of Composite Materials. 2018. Vol. 54. No 4. P. 473-486. DOI: 10.1007/s11029-018-9757-1.
9. Khokhlov A.V. Asymptotic behavior of creep curves in the Rabotnov nonlinear heredity theory under piecewise constant loadings and memory decay conditions. Moscow University Mechanics Bulletin. 2017. Vol. 72. No 5. P. 103-107.
10. Khokhlov A.V. Modelirovanie zavisimosti krivykh polzuchesti pri rastyazhenii i koeffitsienta Puassona reonomnykh materialov ot gidrostaticheskogo davleniya s pomoshch'yu nelineyno-nasledstvennogo sootnosheniya Rabotnova [Simulation of hydrostatic pressure influence on creep curves and Poisson's ratio of rheonomic materials under tension using the Rabotnov non-linear hereditary relation]. Mekhanika kompozitsionnykh materialov i konstruktsiy [Journal on Composite Mechanics and Design]. 2018. Vol. 24. No 3. P. 407-436 (In Russian).
11. Rabotnov Yu.N. Ravnovesiye uprugoy sredy s posledeystviyem [Equilibrium of elastic medium with heredity]. Prikladnaya matematika i mekhanika [Applied Mathematics and Mechanics]. 1948. Vol. 12. No 1. P. 53-62 (In Russian).
12. Namestnikov V.S., Rabotnov Yu.N. O nasledstvennykh teoriyakh polzuchesti [On the hereditary theories of creep]. Prikladnaya mekhanika i tekhnicheskaya fizika [Journal of Applied Mechanics and Technical Physics]. 1961. Vol. 2. No 4. P. 148-150 (In Russian).
13. Rabotnov Yu.N. Creep Problems in Structural Members. Amsterdam, London, North-Holland Publ. Co. 1969. 822 p.
14. Rabotnov Yu.N., Papernik L.K., Stepanychev E.I. Application of the nonlinear theory of heredity to the description of time effects in polymeric material. Polimer Mechanics. 1971. Vol. 7. No 1. P. 63-73.
15. Dergunov N.N., Papernik L.Kh., Rabotnov Yu.N. Analysis of behavior of graphite on the basis of nonlinear heredity theory. Journal of Applied Mechanics and Technical Physics. 1971. Vol. 12. No 2. P. 235-240. DOI: 10.1007/BF00850695.
16. Rabotnov Yu.N., Papernik L.Kh., Stepanychev E.I. Nelineynaya polzuchest' stekloplastika ТS8/3-250 [Non-linear creep of fibreglass reinforced plastic ТS 8/3-250]. Mekhanika polymerov [Polymer Mechanics]. 1971. No 3. P. 391-397 (In Russian).
17. Rabotnov Yu.N., Papernik L.Kh., Stepanychev E.I. O svyazi kharakteristik polzuchesti stekloplastikov s krivoi mgnovennogo deformirovaniya [On connection between fiberglass creep behavior and momentary deforming curve]. Mekhanika polimerov [Polymer Mechanics]. 1971. No 4. P. 624-628 (In Russian).
18. Rabotnov Yu.N., Suvorova Yu.V. О zakone deformirovaniya metallov pri odnoosnom nagruzhenii [On a law of metals deforming under uniaxial loading]. Izvestiya AN SSSR. Mekhanika tverdogo tela [Mechanics of Solids]. 1972. No 4. P. 41-54 (In Russian).
19. Rabotnov Yu.N., Papernik L.K., Stepanychev E.I. Description of creep of composition materials under tension and compression. Polymer Mechanics. 1973. Vol. 9. No 5. P. 690-695. https://doi.org/10.1007/BF00856259.
20. Rabotnov Yu.N. Elementy nasledstvennoi mekhaniki tverdykh tel [Fundamentals of Hereditary Solid Mechanics]. Moscow. Nauka Publ. 1977. 384 p. (In Russian).
21. Suvorova Yu.V. Nelineynye effekty pri deformirovanii nasledstvennykh sred [Nonlinear effects in case of deformation of hereditary medium]. Mekhanika polimerov [Polymer Mechanics]. 1977. No 6. P. 976-980 (In Russian).
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2018-12-15
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