Theoretical Construction of Initial and Subsequent Yield Surfaces for Isotropic Strain-Hardening Elastoplastic Materials of the Differential Type
В рамках теории бесконечно малых деформаций предложен новый подход к теоретическому построению начальной и последующей поверхностей нагружения для некоторого класса простых по Ноллу начально твердых упрочняющихся упругопластических материалов дифференциального типа сложности 1. При этом поверхност...
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Інститут проблем міцності ім. Г.С. Писаренко НАН України
2010
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| Назва видання: | Проблемы прочности |
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| Цитувати: | Theoretical Construction of Initial and Subsequent Yield Surfaces for Isotropic Strain-Hardening Elastoplastic Materials of the Differential Type / P.P. Lepikhin // Проблемы прочности. — 2010. — № 5. — С. 165-173. — Бібліогр.: 14 назв. — англ. |
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irk-123456789-1120002017-01-17T03:03:14Z Theoretical Construction of Initial and Subsequent Yield Surfaces for Isotropic Strain-Hardening Elastoplastic Materials of the Differential Type Lepikhin, P.P. Научно-технический раздел В рамках теории бесконечно малых деформаций предложен новый подход к теоретическому построению начальной и последующей поверхностей нагружения для некоторого класса простых по Ноллу начально твердых упрочняющихся упругопластических материалов дифференциального типа сложности 1. При этом поверхности нагружения строятся без принятия каких-либо дополнительных предположений, для всего рассмотренного класса материалов начальная поверхность имеет форму Губера–Мизеса, а последующие поверхности в общем случае подвергаются расширению, смещению и изменению формы. В частных случаях уравнение последующей поверхности нагружения совпадает с уравнениями известных теорий пластичности с изотропно-кинематическим и кинематическим упрочнением. Установлено, что для начальных поверхностей нагружения и регулярных точек последующих поверхностей нагружения выполняется ассоциированный закон течения. При принятых определениях активного нагружения, разгрузки и нейтрального нагружения для начальной и гладких выпуклых последующих поверхностей нагружения выполняется постулат Друккера. Обосновано проведение опытов, необходимых для конкретизации полученных зависимостей. У рамках теорії малих деформацій запропоновано новий підхід до теоретичної побудови початкової і наступної поверхонь навантаження для деякого класу простих за Ноллом початково твердих зміцнюваних пружно-пластичних матеріалів диференціального типу складності 1. При цьому поверхні навантаження будуються без прийняття будь-яких додаткових припущень, для всього класу матеріалів, що розглядаються, початкова поверхня має форму Губера–Мізеса, а наступні поверхні в загальному випадку зазнають розширення, зміщення і зміни форми. В окремих випадках рівняння наступної поверхні навантаження збігається з рівнянням відомих теорій пластичності з ізотропно-кінематичним і кінематичним зміцненням. Установлено, що для початкових поверхонь навантаження і регулярних точок наступних поверхонь навантаження виконується асоційований закон текучості. При прийнятих визначеннях активного навантаження, розвантаження і нейтрального навантаження для початкової і гладких опуклих наступних поверхонь навантаження виконується постулат Друккера. Обґрунтовано проведення випробувань, що необхідні для конкретизації отриманих залежностей. 2010 Article Theoretical Construction of Initial and Subsequent Yield Surfaces for Isotropic Strain-Hardening Elastoplastic Materials of the Differential Type / P.P. Lepikhin // Проблемы прочности. — 2010. — № 5. — С. 165-173. — Бібліогр.: 14 назв. — англ. 0556-171X http://dspace.nbuv.gov.ua/handle/123456789/112000 539.37 en Проблемы прочности Інститут проблем міцності ім. Г.С. Писаренко НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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| topic |
Научно-технический раздел Научно-технический раздел |
| spellingShingle |
Научно-технический раздел Научно-технический раздел Lepikhin, P.P. Theoretical Construction of Initial and Subsequent Yield Surfaces for Isotropic Strain-Hardening Elastoplastic Materials of the Differential Type Проблемы прочности |
| description |
В рамках теории бесконечно малых деформаций предложен новый подход к теоретическому
построению начальной и последующей поверхностей нагружения для некоторого класса
простых по Ноллу начально твердых упрочняющихся упругопластических материалов дифференциального типа сложности 1. При этом поверхности нагружения строятся без принятия
каких-либо дополнительных предположений, для всего рассмотренного класса материалов
начальная поверхность имеет форму Губера–Мизеса, а последующие поверхности в общем
случае подвергаются расширению, смещению и изменению формы. В частных случаях уравнение последующей поверхности нагружения совпадает с уравнениями известных теорий пластичности с изотропно-кинематическим и кинематическим упрочнением. Установлено, что
для начальных поверхностей нагружения и регулярных точек последующих поверхностей
нагружения выполняется ассоциированный закон течения. При принятых определениях активного нагружения, разгрузки и нейтрального нагружения для начальной и гладких выпуклых
последующих поверхностей нагружения выполняется постулат Друккера. Обосновано проведение опытов, необходимых для конкретизации полученных зависимостей. |
| format |
Article |
| author |
Lepikhin, P.P. |
| author_facet |
Lepikhin, P.P. |
| author_sort |
Lepikhin, P.P. |
| title |
Theoretical Construction of Initial and Subsequent Yield Surfaces for Isotropic Strain-Hardening Elastoplastic Materials of the Differential Type |
| title_short |
Theoretical Construction of Initial and Subsequent Yield Surfaces for Isotropic Strain-Hardening Elastoplastic Materials of the Differential Type |
| title_full |
Theoretical Construction of Initial and Subsequent Yield Surfaces for Isotropic Strain-Hardening Elastoplastic Materials of the Differential Type |
| title_fullStr |
Theoretical Construction of Initial and Subsequent Yield Surfaces for Isotropic Strain-Hardening Elastoplastic Materials of the Differential Type |
| title_full_unstemmed |
Theoretical Construction of Initial and Subsequent Yield Surfaces for Isotropic Strain-Hardening Elastoplastic Materials of the Differential Type |
| title_sort |
theoretical construction of initial and subsequent yield surfaces for isotropic strain-hardening elastoplastic materials of the differential type |
| publisher |
Інститут проблем міцності ім. Г.С. Писаренко НАН України |
| publishDate |
2010 |
| topic_facet |
Научно-технический раздел |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/112000 |
| citation_txt |
Theoretical Construction of Initial and Subsequent Yield Surfaces for Isotropic Strain-Hardening Elastoplastic Materials of the Differential Type / P.P. Lepikhin // Проблемы прочности. — 2010. — № 5. — С. 165-173. — Бібліогр.: 14 назв. — англ. |
| series |
Проблемы прочности |
| work_keys_str_mv |
AT lepikhinpp theoreticalconstructionofinitialandsubsequentyieldsurfacesforisotropicstrainhardeningelastoplasticmaterialsofthedifferentialtype |
| first_indexed |
2025-07-08T03:10:01Z |
| last_indexed |
2025-07-08T03:10:01Z |
| _version_ |
1837046645829664768 |
| fulltext |
UDC 539.37
Theoretical Construction of Initial and Subsequent Yield Surfaces for
Isotropic Strain-Hardening Elastoplastic Materials of the Differential
Type
P. P. Lepikhin
Pisarenko Institute of Problems of Strength, National Academy of Sciences of Ukraine,
Kiev, Ukraine
ÓÄÊ 539.37
Òåîðåòè÷åñêîå ïîñòðîåíèå íà÷àëüíîé è ïîñëåäóþùèõ ïîâåðõíîñòåé
íàãðóæåíèÿ èçîòðîïíûõ óïðî÷íÿþùèõñÿ óïðóãîïëàñòè÷åñêèõ
ìàòåðèàëîâ äèôôåðåíöèàëüíîãî òèïà
Ï. Ï. Ëåïèõèí
Èíñòèòóò ïðîáëåì ïðî÷íîñòè èì. Ã. Ñ. Ïèñàðåíêî ÍÀÍ Óêðàèíû, Êèåâ, Óêðàèíà
 ðàìêàõ òåîðèè áåñêîíå÷íî ìàëûõ äåôîðìàöèé ïðåäëîæåí íîâûé ïîäõîä ê òåîðåòè÷åñêîìó
ïîñòðîåíèþ íà÷àëüíîé è ïîñëåäóþùåé ïîâåðõíîñòåé íàãðóæåíèÿ äëÿ íåêîòîðîãî êëàññà
ïðîñòûõ ïî Íîëëó íà÷àëüíî òâåðäûõ óïðî÷íÿþùèõñÿ óïðóãîïëàñòè÷åñêèõ ìàòåðèàëîâ äèôôå-
ðåíöèàëüíîãî òèïà ñëîæíîñòè 1. Ïðè ýòîì ïîâåðõíîñòè íàãðóæåíèÿ ñòðîÿòñÿ áåç ïðèíÿòèÿ
êàêèõ-ëèáî äîïîëíèòåëüíûõ ïðåäïîëîæåíèé, äëÿ âñåãî ðàññìîòðåííîãî êëàññà ìàòåðèàëîâ
íà÷àëüíàÿ ïîâåðõíîñòü èìååò ôîðìó Ãóáåðà–Ìèçåñà, à ïîñëåäóþùèå ïîâåðõíîñòè â îáùåì
ñëó÷àå ïîäâåðãàþòñÿ ðàñøèðåíèþ, ñìåùåíèþ è èçìåíåíèþ ôîðìû.  ÷àñòíûõ ñëó÷àÿõ óðàâíå-
íèå ïîñëåäóþùåé ïîâåðõíîñòè íàãðóæåíèÿ ñîâïàäàåò ñ óðàâíåíèÿìè èçâåñòíûõ òåîðèé ïëàñ-
òè÷íîñòè ñ èçîòðîïíî-êèíåìàòè÷åñêèì è êèíåìàòè÷åñêèì óïðî÷íåíèåì. Óñòàíîâëåíî, ÷òî
äëÿ íà÷àëüíûõ ïîâåðõíîñòåé íàãðóæåíèÿ è ðåãóëÿðíûõ òî÷åê ïîñëåäóþùèõ ïîâåðõíîñòåé
íàãðóæåíèÿ âûïîëíÿåòñÿ àññîöèèðîâàííûé çàêîí òå÷åíèÿ. Ïðè ïðèíÿòûõ îïðåäåëåíèÿõ àêòèâ-
íîãî íàãðóæåíèÿ, ðàçãðóçêè è íåéòðàëüíîãî íàãðóæåíèÿ äëÿ íà÷àëüíîé è ãëàäêèõ âûïóêëûõ
ïîñëåäóþùèõ ïîâåðõíîñòåé íàãðóæåíèÿ âûïîëíÿåòñÿ ïîñòóëàò Äðóêêåðà. Îáîñíîâàíî ïðîâå-
äåíèå îïûòîâ, íåîáõîäèìûõ äëÿ êîíêðåòèçàöèè ïîëó÷åííûõ çàâèñèìîñòåé.
Êëþ÷åâûå ñëîâà: ïðîñòîé ïî Íîëëó óïðî÷íÿþùèéñÿ óïðóãîïëàñòè÷åñêèé
ìàòåðèàë äèôôåðåíöèàëüíîãî òèïà ñëîæíîñòè 1, áåñêîíå÷íî ìàëûå äåôîð-
ìàöèè, èçîòðîïèÿ, ïîâåðõíîñòü íàãðóæåíèÿ.
In our earlier studies [1, 2], using rational continuum mechanics approaches, a
mathematical theory was developed for a strict construction and specialization of
constitutive equations for simple (in Noll’s sense) isotropic strain-hardening
elastoplastic materials of the differential type of complexity n as the most
important representatives of the materials with the infinitesimal memory of the
path shape (the path shape memory on an arbitrarily small interval of the “past”).
The strains were assumed to be finite. The hierarchy of the constitutive relations
was constructed according to the level of complexity of the material response to
deformation.
© P. P. LEPIKHIN, 2010
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2010, ¹ 5 165
In papers [1, 3], the method for specialization of the previously constructed
physical equations [1, 2] is developed within the theory of infinitesimal strains.
With this method, a number of constitutive relations are derived. For n�1, the
conditions for the existence of a loading surface are established.
In [3], a number of assumptions are made for the specialization of constitutive
relations.
Assumption 1. We assume that in the construction of the constitutive relations
for simple, in Noll’s sense, isotropic strain-hardening elastoplastic bodies of the
differential type of a different complexity, a specific character of the tensor space
structure related to the tensor product can be neglected.
Assumption 2. We assume that under the elastoplastic deformation, the stress
deviator is independent of the first invariants and the mean stress is independent of
the deviatoric components of the kinematic infinitesimal strain tensors.
Assumption 3. We assume that the prehistory of variation of elastic
components of the strain tensor does not influence the stresses in the elastoplastic
material.
By the prehistory of variation of elastic components of the strain tensor one
should mean the whole history of variation of the elastic component of the strain
tensor except for its value at the end of the deformation process.
Assumptions 1 and 2 impose constraints on the material properties and, as
noted in [3], they are verified experimentally for a number of materials.
For strain-hardening elastoplastic materials of the differential type of
complexity 1, which obey assumptions 1–3, a constitutive relation is constructed
that is valid for the general case of active deformation and has the following form
[3]:
s e e
p
� �� �2 3 1� , (1)
where �2 and �3 depend on invariants
tr e( ) ,2 tr ee
p
( � ),1 (2)
s, e, and �e de d
p p p
1 � � are the stress, total strain, and plastic strain rate deviators,
respectively, e p is the plastic strain deviator, and � p is the arc length of the path
of the plastic strain deviator.
According to [3], equation (1) is a parametric representation of the loading
surface equation. To analyze in greater detail the loading surface properties (such
as closure, convexity, smoothness, the associated flow rule observance or violation,
etc.), the specialization of functions �2 and �3 – to a greater or lesser extent – is
required.
The purpose of this work is to develop an approach to the construction of the
initial and subsequent loading surfaces based on Eq. (1) without specializing
functions �2 and �3 .
As shown by the analysis, it follows from the general properties of the
initially solid strain-hardening elastoplastic material (hereinafter referred to as the
strain-hardening elastoplastic material) [4] that the path (history of variation) of the
166 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2010, ¹ 5
P. P. Lepikhin
plastic strain during active deformation fully determines the stresses at the end of
the deformation process. Based on the aforementioned, a conclusion can be made
that assumption 3 stems from the general properties of strain-hardening elastoplastic
materials and can be reformulated as follows: in setting the history of variation of
plastic strains, the prehistory of variation of elastic components of the strain tensor
can be neglected in determining the stresses in the strain-hardening elastoplastic
material. Moreover, in constructing constitutive relations for such materials we can
admit that, in case of setting the history of variation of plastic strains, the history of
variation of elastic components of the strain tensor can be neglected in determining
the stresses in the strain-hardening elastoplastic material.
If the last corollary that proceeds from the general properties of the strain-
hardening elastoplastic materials considered proves to be valid, Eq. (1) takes the
form:
s e e
p p p p
� �� � � ,� �2 3 1 (3)
where coefficients ��2
p
and ��3
p
are certain functions of invariants
tr e p( ) ,2 tr e ep p
( � ).1 (4)
As follows from [3], Eq. (3) can be written in the vector space. In this case,
the stress, plastic strain and plastic strain rate deviators can be substituted by
relevant vectors.
Let us write the vector representation of Eq. (3) in the Il’yushin space [5]
� � � � �
� � �� �2 3 1
p p p p
E E , (5)
where
�
� ,
�
E p , and
� �
E dE d
p p p
1 � � are the stress, plastic strain, and plastic strain
rate vectors, respectively,
�
�2
p
and
�
�3
p
, in view of (4), depend on the modulus of
the plastic strain vector and the angle between the plastic strain vector and the
plastic strain “rate” vector. At every point of the active deformation path, coefficients
��2
p
and ��3
p
correspond to coefficients
�
�2
p
and
�
�3
p
, respectively.
At the beginning of the active deformation process, the plastic strain is equal
to zero. In this case, Eq. (3) takes the form
s e
p p
� � � ,�30 1 (6)
where, in view of (4), ��30
p
is the positive quantity that does not depend on the type
of the stress (strain) state.
Taking into account the fact that the stress deviator in the active process of
uniaxial tension at the beginning of plastic deformation equals to sT 0 , where sT 0
is the stress deviator at the yield point under uniaxial tension, equation (6) can be
rewritten as
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2010, ¹ 5 167
Theoretical Construction of Initial and Subsequent Yield Surfaces ...
s e s
p p
T� �� � .�30 1 0 (7)
Let us bring Eq. (7) to the invariant form. To do so, let us square it, take the
trace of the obtained equation, multiply all its members by 3/2 for convenience,
and write it in the coordinate form as follows:
3
2
3
2
3
230
2
0
2
0 0
s s s sij ij
p
ij ij TT T
� � �( � ) ,� � (8)
where �T 0 is the initial yield strength under uniaxial tension. In writing Eq. (8),
we took into account the fact that �e
p
1 is a normalized deviator.
Relation (8) corresponds to the equation of the initial loading surface, which
can be represented as
f s sin ij ij ij T i T( ) ,� � � �� � � � �
3
2
00
2 2
0
2
(9)
where � ij and � i ij ijs s� ( )3 2 are the Cauchy stress tensor and the stress
intensity, respectively.
Under assumptions adopted in the construction of Eq. (9), the initial loading
surface in the three-dimensional space of principal stresses represents a circular
cylinder (the Huber–Mises cylinder) – a regular (smooth), continuous, convex
surface with the radius equal to 2 3 0�T .
Given the validity of Eq. (9), we can show [6] that
�
��
�
�
f f
s
s
in
ij
in
ij
ij� �3 , and
�
��
�
�
�
f
d
f
s
ds
in
ij
ij
in
ij
ij� . (10)
Based on Eqs. (6), (8), and (9), we write in the coordinate form
de
d
sij
p
p
i
ij�
3
2
�
�
. (11)
In view of (10), Eq. (11) expresses the flow rule associated to (9)
de d
f
d
f
s
d
sij
p
ij ij
p
i
ij� � �
�
��
�
�
�
�
3
2
, (12)
where, as follows from (12), d
d p
i
�
�
�
1
6
0.
Using the concepts of the active loading, unloading and neutral loading [7]
and taking into account (10), we obtain for the initial loading surface (9) under
active loading, unloading and neutral loading, respectively [6]
168 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2010, ¹ 5
P. P. Lepikhin
�
�
f
s
ds f d
in
ij
ij in
�
0 0 0, , , (13)
�
�
f
s
ds f d
in
ij
ij in� � �0 0 0, , , (14)
�
�
f
s
ds f d
in
ij
ij in� � �0 0 0, , . (15)
Given the validity of Eqs. (12)–(15), the Drucker’s postulate is fulfilled for
the initial Huber–Mises loading surface.
In constructing subsequent loading surfaces when plastic strains are not equal
to zero, we go back to Eq. (3), whose coefficients depend on invariants (4), and
which has form (5) in the Il’yushin vector space.
Let us introduce the concept of the active stress deviator [8, 9], by which one
should mean the component of the stress deviator decomposition in the direction of
the line tangent to the plastic strain deviator path. Then we can write from Eq. (3)
s s e ea p p p p
� � �� � � ,� �2 3 1 (16)
where sa is the active stress deviator.
It follows from Eq. (16) that the direction tensors, strain mode angles and
Lode–Nadai parameters of the active stress deviator and �e
p
1 in the arbitrary
active-deformation process, within the limits of validity of relation (3), coincide.
Since the �e
p
1 deviator is normalized, then sa p
� � .�3
By squaring (16), taking the trace of th obtained equation, multiplying its
right- and left-hand sides by 3/2 for convenience, we can write it in the coordinate
form
3
2
3
2
3
22 2 3s s s e s eij
a
ij
a
ij
p
ij
p
ij
p
ij
p p
� � � �( � )( � ) ( � )� � � 2 . (17)
Equation (17) represents the equation of the subsequent loading surface,
which can be rewritten as follows:
f s sij i
a p
ij
a
ij
a p
( ) ( ) ( � ) ( � ) ,� � � �� � � � �2
3
2
3
23
2
3
2
3
2
0 (18)
where � i
a
ij
a
ij
as s� ( )3 2 is the active stress intensity.
In the case of validity of Eq. (18), using the approach described in [6], we can
show that
�
��
�
�
�
f f
s
s e s
ij ij
ij
p
ij
p
ij
a� � � �3 32( ) , and
�
��
�
�
�
f
d
f
s
ds
ij
ij
ij
ij� . (19)
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2010, ¹ 5 169
Theoretical Construction of Initial and Subsequent Yield Surfaces ...
As follows from Eqs. (16)–(18) in the coordinate form,
de
d
sij
p
p
i
a ij
a�
3
2
�
�
. (20)
In view of (19), Eq. (20) expresses the flow rule associated to (18)
de d
f
d
f
s
d
sij
p
ij ij
p
i
a ij
a� � �
�
��
�
�
�
�
3
2
, (21)
where, as follows from (21), d
d p
i
a
�
�
�
1
6
0.
Using the concepts of the active loading, unloading and neutral loading for
regular points of the loading surface, we derive that for the active loading,
unloading and neutral loading the following equations are true, respectively
�
�
f
s
ds f d
ij
ij
�
0 0 0, , , (22)
�
�
f
s
ds f d
ij
ij � � �0 0 0, , , (23)
�
�
f
s
ds f d
ij
ij � � �0 0 0, , . (24)
Given the validity of Eqs. (21)–(24), the Drucker’s postulate is fulfilled for
regular points of the subsequent loading surface of convex shape.
Following the procedure described in [6], we can show that, for the initial and
subsequent loading surfaces represented by (9) and (21), the plastic change of
volume equals to zero, i.e., the material is plastically incompressible.
Let us point out that no other assumption was taken into account in the
construction of Eqs. (9) and (18) based on Eq. (1).
We will call ��2
p
ij
p
e and ��3
p
in (17) the evolution parameters of the subsequent
loading surface. In view of (4) and the vector representation of the deviatoric
space, ��2
p
and ��3
p
depend on the plastic strain deviator (vector) modulus, angle
between the plastic strain vector and the plastic strain “rate” vector. Equation (17)
allows modeling the translation, expansion and distortion of the subsequent loading
surface in the active deformation process. Here, the distortion of the subsequent
loading surface is described by the angle between the plastic strain vector and the
plastic strain rate vector. As we know from [6, 10], this kind of evolution of the
subsequent loading surface and the initial surface of the Huber–Mises type are
characteristic of a number of elastoplastic materials.
170 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2010, ¹ 5
P. P. Lepikhin
Let us note that the plastic strain deviator modulus does not describe the effect
caused by the path shape of the plastic strain deviator on ��2
p
and ��3
p
in Eq. (17).
Therefore, these coefficients can be specified based on uniaxial tests with the
additional construction of subsequent loading surfaces for every plastic strain
deviator modulus within the studied plastic strain range in accordance with the
technique presented in [10], for instance.
Let us consider some special cases of Eq. (17).
1) We assume that for a certain class of materials considered here coefficient
��2
p
does not depend on the angle between the plastic strain vector and the plastic
strain “rate” vector. At the same time, a distortion of the subsequent loading
surface can be described by the evolution parameter ��3
p
that depends on the above
angle.
2) We assume that ��2
p
and ��3
p
do not depend on the angle between the
plastic strain vector and the plastic strain “rate” vector. Then the equation of the
subsequent loading surface takes the Huber–Mises form, ��2
p
and ��3
p
are
determined by the plastic strain deviator modulus. In this case, the equation of the
subsequent loading surface can be specified based on uniaxial tests performed in
accordance with the technique presented in [6], which was offered for the
Kadashevich–Novozhilov theory of plasticity with isotropic-kinematic hardening
[8]. In [11], the translation of the subsequent loading surface (��2
p
ij
p
e ) is modeled by
the dependence of the plastic strain deviator coefficient on the plastic strain
deviator modulus.
3) In special cases such as 3 2 3 0�� �
p
T� �const and � ,�2
p
c� �const Eq. (17)
corresponds to the equation of the subsequent loading surface from the Ishlinsky–
Prager theory [12, 13].
The equation of the initial loading surface for special classes of materials
considered here will not change.
The main distinction of the proposed approach from the well-known approaches
[6–9, 11–14, and others] is that the loading surface type is not postulated, as well
as the constitutive equation based on which it is constructed. The initial and
subsequent loading surfaces, as well as the expression for the plastic strain
increment deviator, are constructed strictly on the basis of the earlier obtained
constitutive relation that is valid for the active loading processes.
Within the theory of infinitesimal strains, a new approach is proposed for the
theoretical construction of the initial and subsequent loading surfaces for simple in
Noll’s sense initially solid strain-hardening materials of the differential type of
complexity 1. The loading surfaces are constructed without making additional
assumptions. For the whole class of materials considered, the initial surface has the
Huber–Mises form, the subsequent surface undergoes expansion, translation and
distortion in the general case. In special cases, the equation of the subsequent
loading surface corresponds to the equations of the well-known theories of
plasticity with the isotropic-kinematic and kinematic hardening. It is established
that the associated flow rule is fulfilled for the initial loading surfaces and regular
points of the subsequent loading surfaces. With the accepted definitions of the
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2010, ¹ 5 171
Theoretical Construction of Initial and Subsequent Yield Surfaces ...
active loading, unloading and neutral loading, the Drucker postulate is fulfilled
for the initial surface and subsequent surfaces of smooth convex shape. The
experiments necessary for the concrete definition of the derived dependences are
verified.
Ð å ç þ ì å
Ó ðàìêàõ òåî𳿠ìàëèõ äåôîðìàö³é çàïðîïîíîâàíî íîâèé ï³äõ³ä äî òåîðå-
òè÷íî¿ ïîáóäîâè ïî÷àòêîâî¿ ³ íàñòóïíî¿ ïîâåðõîíü íàâàíòàæåííÿ äëÿ äåÿêîãî
êëàñó ïðîñòèõ çà Íîëëîì ïî÷àòêîâî òâåðäèõ çì³öíþâàíèõ ïðóæíî-ïëàñòè÷-
íèõ ìàòåð³àë³â äèôåðåíö³àëüíîãî òèïó ñêëàäíîñò³ 1. Ïðè öüîìó ïîâåðõí³
íàâàíòàæåííÿ áóäóþòüñÿ áåç ïðèéíÿòòÿ áóäü-ÿêèõ äîäàòêîâèõ ïðèïóùåíü,
äëÿ âñüîãî êëàñó ìàòåð³àë³â, ùî ðîçãëÿäàþòüñÿ, ïî÷àòêîâà ïîâåðõíÿ ìàº
ôîðìó Ãóáåðà–̳çåñà, à íàñòóïí³ ïîâåðõí³ â çàãàëüíîìó âèïàäêó çàçíàþòü
ðîçøèðåííÿ, çì³ùåííÿ ³ çì³íè ôîðìè.  îêðåìèõ âèïàäêàõ ð³âíÿííÿ íàñòóï-
íî¿ ïîâåðõí³ íàâàíòàæåííÿ çá³ãàºòüñÿ ç ð³âíÿííÿì â³äîìèõ òåîð³é ïëàñòè÷-
íîñò³ ç ³çîòðîïíî-ê³íåìàòè÷íèì ³ ê³íåìàòè÷íèì çì³öíåííÿì. Óñòàíîâëåíî, ùî
äëÿ ïî÷àòêîâèõ ïîâåðõîíü íàâàíòàæåííÿ ³ ðåãóëÿðíèõ òî÷îê íàñòóïíèõ ïî-
âåðõîíü íàâàíòàæåííÿ âèêîíóºòüñÿ àñîö³éîâàíèé çàêîí òåêó÷îñò³. Ïðè ïðèé-
íÿòèõ âèçíà÷åííÿõ àêòèâíîãî íàâàíòàæåííÿ, ðîçâàíòàæåííÿ ³ íåéòðàëüíîãî
íàâàíòàæåííÿ äëÿ ïî÷àòêîâî¿ ³ ãëàäêèõ îïóêëèõ íàñòóïíèõ ïîâåðõîíü íàâàí-
òàæåííÿ âèêîíóºòüñÿ ïîñòóëàò Äðóêêåðà. Îá´ðóíòîâàíî ïðîâåäåííÿ âèïðî-
áóâàíü, ùî íåîáõ³äí³ äëÿ êîíêðåòèçàö³¿ îòðèìàíèõ çàëåæíîñòåé.
1. P. P. Lepikhin, Structure of Constitutive Relations for Viscoelastic-Visco-
plastic State of Materials [in Russian], Author’s Abstract of the Doctor
Degree Thesis (Phys. & Math. Sci.), Kiev (1997).
2. P. P. Lepikhin, “The construction of constitutive relations for isotropic
strain-hardening elastoplastic materials of the differential type of complexity
n. Part 1. Finite strains,” Strength Mater., 41, No. 2, 135–146 (2009).
3. P. P. Lepikhin, “The construction of constitutive relations for isotropic
strain-hardening elastoplastic materials of the differential type of complexity
n. Part 2. Infinitesimal strains,” Strength Mater., 41, No. 4, 363–376 (2009).
4. M. Lucchesi, D. R. Owen, and P. Podio-Guidugli, “Materials with elastic
range: A theory with a view toward applications. Pt. 3,” Arch. Rat. Mech.
Analysis, 117, 53–96 (1992).
5. A. A. Il’yushin, “On the fundamentals of the general theory of plasticity,” in:
Problems of the Plasticity Theory [in Russian], Izd. AN SSSR, Moscow
(1961), pp. 3–29.
6. N. N. Malinin, Applied Theory of Plasticity [in Russian], Mashinostroenie,
Moscow (1975).
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[in Russian], Nauka, Moscow (1971).
8. Yu. I. Kadashevich and V. V. Novozhilov, “Theory of plasticity taking into
account residual microstresses,” Appl. Math. Mech., 22, No. 1, 78–89 (1958).
172 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2010, ¹ 5
P. P. Lepikhin
9. D. Kolarov, A. Baltov, and N. Boncheva, Mechanics of Plastic Media [in
Russian], Mir, Moscow (1979).
10. A. S. Khan, R. Kazmi, A. Pandey, and T. Stoughton, Evolution of subsequent
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account the Bauschinger effect,” Dokl. AN SSSR, 117, No. 4, 586–588 (1957).
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14. H.-C. Wu, H.-K. Hong, and J.-K Lu, “ An endochronic theory accounted for
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Received 01. 02. 2010
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2010, ¹ 5 173
Theoretical Construction of Initial and Subsequent Yield Surfaces ...
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/NOR <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>
/PTB <FEFF005500740069006c0069007a006500200065007300730061007300200063006f006e00660069006700750072006100e700f50065007300200064006500200066006f0072006d00610020006100200063007200690061007200200064006f00630075006d0065006e0074006f0073002000410064006f0062006500200050004400460020007000610072006100200069006d0070007200650073007300f5006500730020006400650020007100750061006c0069006400610064006500200065006d00200069006d00700072006500730073006f0072006100730020006400650073006b0074006f00700020006500200064006900730070006f00730069007400690076006f0073002000640065002000700072006f00760061002e0020004f007300200064006f00630075006d0065006e0074006f00730020005000440046002000630072006900610064006f007300200070006f00640065006d0020007300650072002000610062006500720074006f007300200063006f006d0020006f0020004100630072006f006200610074002000650020006f002000410064006f00620065002000520065006100640065007200200035002e0030002000650020007600650072007300f50065007300200070006f00730074006500720069006f007200650073002e>
/SUO <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>
/SVE <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>
/ENU (Use these settings to create Adobe PDF documents for quality printing on desktop printers and proofers. Created PDF documents can be opened with Acrobat and Adobe Reader 5.0 and later.)
>>
/Namespace [
(Adobe)
(Common)
(1.0)
]
/OtherNamespaces [
<<
/AsReaderSpreads false
/CropImagesToFrames true
/ErrorControl /WarnAndContinue
/FlattenerIgnoreSpreadOverrides false
/IncludeGuidesGrids false
/IncludeNonPrinting false
/IncludeSlug false
/Namespace [
(Adobe)
(InDesign)
(4.0)
]
/OmitPlacedBitmaps false
/OmitPlacedEPS false
/OmitPlacedPDF false
/SimulateOverprint /Legacy
>>
<<
/AddBleedMarks false
/AddColorBars false
/AddCropMarks false
/AddPageInfo false
/AddRegMarks false
/ConvertColors /NoConversion
/DestinationProfileName ()
/DestinationProfileSelector /NA
/Downsample16BitImages true
/FlattenerPreset <<
/PresetSelector /MediumResolution
>>
/FormElements false
/GenerateStructure true
/IncludeBookmarks false
/IncludeHyperlinks false
/IncludeInteractive false
/IncludeLayers false
/IncludeProfiles true
/MultimediaHandling /UseObjectSettings
/Namespace [
(Adobe)
(CreativeSuite)
(2.0)
]
/PDFXOutputIntentProfileSelector /NA
/PreserveEditing true
/UntaggedCMYKHandling /LeaveUntagged
/UntaggedRGBHandling /LeaveUntagged
/UseDocumentBleed false
>>
]
>> setdistillerparams
<<
/HWResolution [2400 2400]
/PageSize [612.000 792.000]
>> setpagedevice
|