Theoretical and Experimental Investigation of the Dissipated and Stored Energy Ratio in Iron under Quasi-Static and Cyclic Loading
The problem of energy storing (cold work accumulation) in metals was intensively investigated both theoretically and experimentally during all last century but a general theoretical conception of the process was not created. This work is devoted to an experimental investigation of energy dissipation...
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Інститут проблем міцності ім. Г.С. Писаренко НАН України
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| Zitieren: | Theoretical and Experimental Investigation of the Dissipated and Stored Energy Ratio in Iron under Quasi-Static and Cyclic Loading / O. Plekhov, S. Uvarov, O. Naimark // Проблемы прочности. — 2008. — № 1. — С. 101-104. — Бібліогр.: 5 назв. — англ. |
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irk-123456789-484482013-08-19T19:17:47Z Theoretical and Experimental Investigation of the Dissipated and Stored Energy Ratio in Iron under Quasi-Static and Cyclic Loading Plekhov, O. Uvarov, S. Naimark, O. Научно-технический раздел The problem of energy storing (cold work accumulation) in metals was intensively investigated both theoretically and experimentally during all last century but a general theoretical conception of the process was not created. This work is devoted to an experimental investigation of energy dissipation in metals under plastic deformation and to the development of a thermodynamic model to study the cold work accumulation under plastic deformation and failure. The proposed model is based on a statistical description of collective properties of mesoscopic defects and on dividing the plastic deformation into two parts (dissipative and structural). The structural plastic strain was considered as an independent thermodynamic variable that allowed us to determine the thermodynamic potential of the system. The derived constitutive relations were applied for numerical simulation of tensile and cyclic tests. The numerical results demonstrate a good agreement with experimental data. Экспериментально исследовано рассеяние энергии в металлах при пластическом де формировании и разработке термодинамической модели для изучения накопления наклепа при пластическом деформировании и разрушении. Предлагаемая модель основана на статистическом описании коллективных свойств мезоскопических дефектов и на разделении пластической деформации на две части (диссипативную и общую). Общая пластическая деформация рассматривалась как независимая термодинамическая переменная, что позволило определить термодинамический потенциал системы. Полученные определяющие соотношения применили для численного моделирования результатов испытаний на растяжение и циклических испытаний. Численные результаты хорошо согласуются с экспериментальными данными. 2008 Article Theoretical and Experimental Investigation of the Dissipated and Stored Energy Ratio in Iron under Quasi-Static and Cyclic Loading / O. Plekhov, S. Uvarov, O. Naimark // Проблемы прочности. — 2008. — № 1. — С. 101-104. — Бібліогр.: 5 назв. — англ. 0556-171X http://dspace.nbuv.gov.ua/handle/123456789/48448 539. 4 en Проблемы прочности Інститут проблем міцності ім. Г.С. Писаренко НАН України |
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Научно-технический раздел Научно-технический раздел Plekhov, O. Uvarov, S. Naimark, O. Theoretical and Experimental Investigation of the Dissipated and Stored Energy Ratio in Iron under Quasi-Static and Cyclic Loading Проблемы прочности |
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The problem of energy storing (cold work accumulation) in metals was intensively investigated both theoretically and experimentally during all last century but a general theoretical conception of the process was not created. This work is devoted to an experimental investigation of energy dissipation in metals under plastic deformation and to the development of a thermodynamic model to study the cold work accumulation under plastic deformation and failure. The proposed model is based on a statistical description of collective properties of mesoscopic defects and on dividing the plastic deformation into two parts (dissipative and structural). The structural plastic strain was considered as an independent thermodynamic variable that allowed us to determine the thermodynamic potential of the system. The derived constitutive relations were applied for numerical simulation of tensile and cyclic tests. The numerical results demonstrate a good agreement with experimental data. |
| format |
Article |
| author |
Plekhov, O. Uvarov, S. Naimark, O. |
| author_facet |
Plekhov, O. Uvarov, S. Naimark, O. |
| author_sort |
Plekhov, O. |
| title |
Theoretical and Experimental Investigation of the Dissipated and Stored Energy Ratio in Iron under Quasi-Static and Cyclic Loading |
| title_short |
Theoretical and Experimental Investigation of the Dissipated and Stored Energy Ratio in Iron under Quasi-Static and Cyclic Loading |
| title_full |
Theoretical and Experimental Investigation of the Dissipated and Stored Energy Ratio in Iron under Quasi-Static and Cyclic Loading |
| title_fullStr |
Theoretical and Experimental Investigation of the Dissipated and Stored Energy Ratio in Iron under Quasi-Static and Cyclic Loading |
| title_full_unstemmed |
Theoretical and Experimental Investigation of the Dissipated and Stored Energy Ratio in Iron under Quasi-Static and Cyclic Loading |
| title_sort |
theoretical and experimental investigation of the dissipated and stored energy ratio in iron under quasi-static and cyclic loading |
| publisher |
Інститут проблем міцності ім. Г.С. Писаренко НАН України |
| publishDate |
2008 |
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Научно-технический раздел |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/48448 |
| citation_txt |
Theoretical and Experimental Investigation of the Dissipated and Stored Energy Ratio in Iron under Quasi-Static and Cyclic Loading / O. Plekhov, S. Uvarov, O. Naimark // Проблемы прочности. — 2008. — № 1. — С. 101-104. — Бібліогр.: 5 назв. — англ. |
| series |
Проблемы прочности |
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| first_indexed |
2025-07-04T08:57:39Z |
| last_indexed |
2025-07-04T08:57:39Z |
| _version_ |
1836706126537687040 |
| fulltext |
UDC 539. 4
T h e o r e t ic a l a n d E x p e r im e n t a l I n v e s t ig a t io n o f th e D is s ip a te d a n d S to r e d
E n e r g y R a t io in I r o n u n d e r Q u a s i-S t a t ic a n d C y c lic L o a d in g
O . P lek h o v ,1a S . U v a ro v ,1b and O . N a im a rk 1c
1 Institute o f Continuous Media Mechanics, RAS-Laboratory o f Physical Foundations o f Strength,
Russian Academy of Sciences, Perm, Russia
a poa@icmm.ru, b usv@icmm.ru, c naimark@icmm.ru
The problem o f energy storing (cold work accumulation) in metals was intensively investigated both
theoretically and experimentally during all last century but a general theoretical conception o f the
process was not created. This work is devoted to an experimental investigation o f energy dissipation
in metals under plastic deformation and to the development o f a thermodynamic model to study the
cold work accumulation under plastic deformation and failure. The proposed model is based on a
statistical description o f collective properties o f mesoscopic defects and on dividing the plastic
deformation into two parts (dissipative and structural). The structural plastic strain was considered
as an independent thermodynamic variable that allowed us to determine the thermodynamic potential
o f the system. The derived constitutive relations were applied fo r numerical simulation o f tensile
and cyclic tests. The numerical results demonstrate a good agreement with experimental data.
K e y w o rd s : cold work, energy dissipation, m esodefect evolution.
In trod u ction . The kinetics o f the microstructure o f m etallic materials has been the
subject o f m uch experim ental investigations. The available data indicate that the
deform ation o f m etals, especially plastic flow, is characterized by h igh d islocation activity
and specific m esodefect patterns. The evolution o f these structures, accom panied by
failure and rotation o f m esovolum es o f the material, leads to generation o f h igh internal
stresses and, as a consequence, to energy storage in a specim en. The problem o f the
dependence o f the storage energy Wst on the plastic w ork Wp = ~ : ( £ — ) is w id ely
covered in literature [1 -3 ] but their relation is still an open question. A generally accepted
assumption, W < 0.2 Wp , w hich is often justified b y citing the early study o f Taylor et al.
[4], is hardly applicable to m any m echanical processes [3].
The basic theoretical problem o f the m odels describing the energy balance under
plastic deform ation is determination o f n ew structure-sensitive parameters. The plastic
deform ation, conventionally considered to be such a parameter, cannot be interpreted as
an independent therm odynam ic variable.
Naim ark et al. [5] developed an original m ethod for describing the damage kinetics
using a statistical description o f a m esodefect ensemble. Based on the statistical description
o f the problem , it is possib le to determine characteristic responses o f solids w ith defects
and to w ork out an appropriate constitutive m odel. We use this approach to define
therm odynam ic internal variables and to obtain nonlinear kinetic equations that describe
the energy balance in m etals under plastic deformation. The plastic deform ation is divided
into tw o parts (“pure” plastic deform ation and “structural” or “potential” deform ation),
and on ly one part (the structural deform ation) is interpreted as an independent thermo
dynam ic variable. The obtained kinetic equations are used to describe the thermal
behavior o f m etals (for exam ple, o f pure iron) subjected to tensile and cyclic tests. We
present num erical sim ulation that incorporates our m odel and show s that theoretical
predictions and experim ental results are in good quantitative agreement.
T h erm od yn am ic M od el. A general therm odynam ic process obeys the m om entum
balance equation and the first and second law s o f therm odynam ics. In the case o f sm all
deform ations, these equations involve the fo llow in g therm odynam ic quantities: strain and
© O. PLEK H O V , S. U V A R O V , O. N A IM A R K , 2008
ISSN 0556-171X. Проблемы прочности, 2008, № 1 101
mailto:poa@icmm.ru
mailto:usv@icmm.ru
mailto:naimark@icmm.ru
O. Plekhov, S. Uvarov, and O. Naimark
stress tensors £ and ~ , heat supply r, and specific H elm holtz free energy F . A ssum ing
the fo llow ing kinem atic relation for the material under study £ = £ e + £ p + p + i ( T — T ' )
w here ~ e is the elastic strain tensor, ~ p is the plastic strain tensor (related to the defect
m otion), ~ is the defect-induced strain tensor (structural part o f the plastic deform ation),
i is the tensor o f the thermal expansion coefficient, and T ' is the reference temperature,
w e can write the fo llow ing equation for the solid temperature evolution:
c T = Q e + Q p + r + V T , ( 1)
w here Q e = TF~t :~ e is heating due to the th erm oelastic e ffect, Q p = d : e p +
(o — Fp-): J j+ T F ~t : ~ represents the inelastic contribution to the heating, c is the
specific heat capacity, and Fx denotes the derivative o f F w ith respect to x.
A nalysis o f the inelastic contribution to the heating g ives the fo llow ing relation for
the stored energy rate:
—TF~t + F p
Wst = ~ ■ p ■ . (2 )
o : ( £ p + p )
To so lve Eq. (2), one should determine the structural plastic deform ation p.
The structural parameters associated w ith typical m esoscop ic defects (microcracks,
m icroshears) were introduced in [5] as the derivatives o f the d islocation density tensor.
Those defects are described by sym m etric tensors o f the form s = sv v for microcracks
and s = 1 2 s ( v l + l v ) for microshears. Here v is the unit vector normal to the base o f a
m icrocrack or the microshear slip plane, l is the unit vector in the direction o f shear, and
s is the m icrocrack volum e or the shear intensity for microshear. The average o f the
“m icroscopic” tensor s g ives the m acroscopic tensor o f the m icrocrack or the microshear
density p = n{S) , where n is the defect concentration.
Statistical description o f the microcrack (m icroshear) ensem ble w as developed in
terms o f the solution o f the Fokker-Plank equation in the phase space o f the possible
states o f the m icroscopic variable s linking the size s and the v , l orientation m odes.
The obtained solution allow ed the definition o f the part o f the free energy caused by
defects F ~. The “equilibrium” correlation o f the defect density tensor and the applied
stress is g iven by the form ula (for a one-dim ensional case) dFp ( o , p ) / dp = 0. The
solution to the latter relationship depends on a new structural-scaling parameter (5. This
parameter indicates the scale distribution o f the defect density tensor in a specific volum e
and plays the role o f the second structural variable related to the m ultiscale nature o f
damage accumulation.
Finally, assum ing linear relations betw een therm odynam ic forces and fluxes, w e can
obtain the fo llow ing constitutive equations:
5 = L6 F d .
O n e-D im en sion a l T ensile L oad ing . The experim entally investigated material was
annealed iron. During all tests, an infrared camera (CEDIP Jade III M W R) w as used to
record the temperature field evolution on the specim en surface. The m ain technical
102 ISSN 0556-171X. npo6n.eMH npounocmu, 2008, N9 1
Theoretical and Experimental Investigation
characteristics o f the camera are as follow s: spectral range 3 -5 ,«m, m axim um picture size
320X 240 p ixels, m axim um framing rate 500 Hz, N E T D < 25 m K at 300 K, and digital
conversion 14 bits. To increase the surface em issiv ity properties, the specim en surface
w as painted black (mat paint) after polishing.
The experim entally obtained temperature fie ld w as num erically processed to
determ ine the space-tim e evolution o f the heat sources and the stored energy value. The
typical results are presented in Fig 1.
Fig. 1. Space-time evolution o f the heat sources (in W/m3) in the longitudinal specimen section (a),
stress-strain, mean temperature variation, and stored energy rate curves (b) obtained during the test.
(The stress and temperature difference were normalized by the corresponding maximum values.
Maximum stress is 27 MPa, maximum temperature difference is 4 °C.)
To simulate the elastic-plastic transition accom panied b y the heat w ave propagation,
the system o f equations (1) and (3) together w ith the m om entum balance equation was
num erically solved. Figure 2 presents the experim entally and num erically determined
space-tim e evolution o f the heat sources in iron under elastic-plastic transition.
Fig. 2. Experimentally (a) and numerically (b) obtained space-time evolution of the heat sources in
the longitudinal specimen section under elastic-plastic transition.
C yclic L oad in g . The developed m odel allow s us to simulate temperature evolution
under cyclic loading. The m ain feature o f cyclic loading is the creation o f m any different
dislocation structures during the test. In terms o f the m odel, it m eans that both structure-
sensitive parameters p (deform ation caused by defect initiation) and (5 (describing the
arrangement o f d islocation ensem ble) vary and interact during the test. Figure 3 presents
the temperature evolution in iron during the numerical test. We obtain three (experimentally
observed) stages o f temperature evolution during the fatigue test: (I) initial temperature
increase, (II) constant temperature region, (III) abrupt temperature increase before failure.
ISSN 0556-171X. npodxeMbi npounocmu, 2008, N 1 103
O. Plekhov, S. Uvarov, and O. Naimark
a 2 4 6 8 10 12 14 16 18 20
lime (c)
Fig. 3. Temperature evolution under cyclic loading during a numerical test.
C on clu sions. Experimental and theoretical investigations o f the energy storage and
dissipation in m etals a llow us to propose a therm odynam ic m odel to describe the energy
balance under plastic deformation o f metals. The key point o f the m odel is the presentation
o f plastic deform ation in terms o f tw o variables: plastic strain tensor (related to the
dissipation effect) ~ p and defect-induced strain tensor (related to the stored energy) ~.
This m akes it possib le to consider the structure-related part o f plastic deform ation as a
therm odynam ic variable and to formulate a corresponding nonequilibrium therm odynamic
potential (free energy). The developed approach has been successfu lly applied to the
sim ulation o f nonlinear thermal effects observed under quasi-static and cyclic loading o f
iron.
Acknowledgments. The authors thank the laboratory LAMEFIP ENSAM (personally Dr. T. Palen-
Luc and Dr. N. Saintier) for the help in the experimental investigations. The work was partly
supported by the grants o f RFBR (05-08-33652, 07-08-96001, 07-05-96019).
1. M. B. Bever, D. L. Hilt, and A. L. Titchener, Prog. Mater. Sci., 17, 1 (1973).
2. R. Kapoor and S. Nemat-Nasser, Mech. Mater., 27, 1 (1998).
3. P. Rosakis, A. J. Rosakis, G. Ravichandran, and J. Hodowany, J. Mech. Phys. Solids, 48, 581
(2000).
4. W. S. Farren and G. I. Taylor, Proc. Roy. Soc. London, A107, 422 (1925).
5. O. Naimark, M. Davydova, O. Plekhov, and S. Uvarov, Phys. Mesomech, 2, 43 (1999).
Received 28. 06. 2007
104 ISSN 0556-171X. npo6neMbi nponuocmu, 2008, № 1
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