Construction of constitutive relationships for simple in Noll’s sense materials with viscoelastic-viscoplastic behavior
Within the class of simple in Noll’s sense materials, the media with viscoelastic-viscoplastic behavior have been singled out, whose arbitrary deformations and types of symmetry in properties have been expressed by general constitutive relationships, in which the long-term fading memories of the def...
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Інститут проблем міцності ім. Г.С. Писаренко НАН України
2009
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| Назва видання: | Проблемы прочности |
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| Цитувати: | Construction of constitutive relationships for simple in Noll’s sense materials with viscoelastic-viscoplastic behavior / P.P. Lepikhin // Проблемы прочности. — 2009. — № 1. — С. 6-12. — Бібліогр.: 18 назв. — англ. |
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irk-123456789-484812013-08-20T07:45:39Z Construction of constitutive relationships for simple in Noll’s sense materials with viscoelastic-viscoplastic behavior Lepikhin, P.P. Научно-технический раздел Within the class of simple in Noll’s sense materials, the media with viscoelastic-viscoplastic behavior have been singled out, whose arbitrary deformations and types of symmetry in properties have been expressed by general constitutive relationships, in which the long-term fading memories of the deformation and time histories take place, and the approaches to their specialization have been developed. В классе простых по Ноллу материалов выделены среды с вязкоупруго-вязкопластическим поведением, для произвольных деформаций и видов симметрии свойств которых построены общие определяющие соотношения с длительной затухающей памятью деформационной и временной историй и разработаны подходы к их специализации. 2009 Article Construction of constitutive relationships for simple in Noll’s sense materials with viscoelastic-viscoplastic behavior / P.P. Lepikhin // Проблемы прочности. — 2009. — № 1. — С. 6-12. — Бібліогр.: 18 назв. — англ. 0556-171X http://dspace.nbuv.gov.ua/handle/123456789/48481 539.4 en Проблемы прочности Інститут проблем міцності ім. Г.С. Писаренко НАН України |
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Научно-технический раздел Научно-технический раздел Lepikhin, P.P. Construction of constitutive relationships for simple in Noll’s sense materials with viscoelastic-viscoplastic behavior Проблемы прочности |
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Within the class of simple in Noll’s sense materials, the media with viscoelastic-viscoplastic behavior have been singled out, whose arbitrary deformations and types of symmetry in properties have been expressed by general constitutive relationships, in which the long-term fading memories of the deformation and time histories take place, and the approaches to their specialization have been developed. |
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Article |
| author |
Lepikhin, P.P. |
| author_facet |
Lepikhin, P.P. |
| author_sort |
Lepikhin, P.P. |
| title |
Construction of constitutive relationships for simple in Noll’s sense materials with viscoelastic-viscoplastic behavior |
| title_short |
Construction of constitutive relationships for simple in Noll’s sense materials with viscoelastic-viscoplastic behavior |
| title_full |
Construction of constitutive relationships for simple in Noll’s sense materials with viscoelastic-viscoplastic behavior |
| title_fullStr |
Construction of constitutive relationships for simple in Noll’s sense materials with viscoelastic-viscoplastic behavior |
| title_full_unstemmed |
Construction of constitutive relationships for simple in Noll’s sense materials with viscoelastic-viscoplastic behavior |
| title_sort |
construction of constitutive relationships for simple in noll’s sense materials with viscoelastic-viscoplastic behavior |
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Інститут проблем міцності ім. Г.С. Писаренко НАН України |
| publishDate |
2009 |
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Научно-технический раздел |
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http://dspace.nbuv.gov.ua/handle/123456789/48481 |
| citation_txt |
Construction of constitutive relationships for simple in Noll’s sense materials with viscoelastic-viscoplastic behavior / P.P. Lepikhin // Проблемы прочности. — 2009. — № 1. — С. 6-12. — Бібліогр.: 18 назв. — англ. |
| series |
Проблемы прочности |
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2025-07-04T09:00:22Z |
| last_indexed |
2025-07-04T09:00:22Z |
| _version_ |
1836706297603424256 |
| fulltext |
Scientif ic and Technical
Section
UDC 539.4
Construction of Constitutive Relationships for Simple in Noll’s Sense
Materials with Viscoelastic-Viscoplastic Behavior
P. P. L epikhin
Pisarenko Institute of Problems of Strength, National Academy of Sciences of Ukraine,
Kiev, Ukraine
Within the class o f simple in N oll’s sense materials, the media with viscoelastic-viscoplastic
behavior have been singled out, whose arbitrary deformations and types o f symmetry in properties
have been expressed by general constitutive relationships, in which the long-term fading memories
o f the deformation and time histories take place, and the approaches to their specialization have
been developed.
K e y w o r d s : constitutive relationships, viscoelastic-viscoplastic materials, long-term
fading memory.
The m ajor problem o f the mechanics o f a deformable solid body is the
development o f methods for constructing physically grounded, m athematically
rigorous constitutive relationships that allow not only describing, but also
predicting, at various levels o f accuracy, the behavior o f a w ide class o f materials
existing in nature in a broad range o f variation o f the conditions o f their
deformation.
Despite the achievements in using the phenom enological approach to the
construction o f constitutive relationships and a large num ber o f proposed models
[1- 10], at present, this approach does not allow for a complete solution to the
above problem, particularly when applied to arbitrary deformations and types of
symmetry in the properties o f viscoelastic-viscoplastic m aterials.
The fact that the theory o f simple in N oll’s sense materials (hereinafter
referred to as simple materials, media, continua), is still general enough to include
practically all known purely m echanical phenom enological models o f m aterial
deformation that are governed by the principle o f a specimen, and the success in
constructing constitutive relationships for simple elastic, viscoelastic, and elasto-
plastic continua by the methods o f rational continuum mechanics [11, 12] testify
to a great potential for using this approach in developing constitutive relationships
for viscoelastic-viscoplastic media.
In this paper, the m edia w ith viscoelastic-viscoplastic behavior have been
distinguished w ithin the class o f simple materials [11], whose arbitrary
deformations and types o f symmetry in properties are expressed by constructed
general constitutive relationships in which the long-term fading memories o f the
© P. P. LEPIKHIN, 2009
6 ISSN 0556-171X. Проблемы прочности, 2009, № 1
Construction of Constitutive Relationships
deformation and time histories take place, and the approaches to their specialization
have been developed.
Let us single out simple viscoelastic-viscoplastic materials by postulating the
following basic properties:
(i) stresses depend on the path shape in the tensor strain space (deformation
history) and on the history o f traversing this deformation history in time (time
history);
(ii) the time history m em ory o f the materials w ithin the active and passive
deformation fades in time;
(iii) the independent o f time m em ory o f the deformation history within the
active deformation fades along the length o f the path in the tensor strain space;
(iv) the total strains can in some way be divided into elastic and plastic
components;
(v) a certain yield criterion is true;
(vi) a certain law o f yielding is fulfilled.
Hereafter, in considering scalar or tensor functions p at the present moment
and in the past, it w ill be convenient to characterize the past m om ent t' by the
positive value s = t — t' [11], where t is the present mom ent o f time. The history
o f the function p up to the mom ent t will be defined by p t , its value being
p t (s):
p t = p t ( s) = p ( t — s).
Here t is fixed and s > 0. For every t, the history o f p t is defined over [0, °°).
We describe the behavior o f a viscoelastic-viscoplastic m aterial by a general
constitutive relationship for a simple m aterial [11]:
o R = G (C t ), ( 1)
R R Twhere o is the Cauchy stress tensor, o is defined by o = R o R , R is the
rotation tensor in m ultiplicative decomposition F = RU = V R o f the deformation
gradient F, U and V are the right and left stretch tensors o f the deformation,
T trespectively, R is the transpose o f R , C is the history o f the right C auchy-
Green tensor, and G denotes a mapping o f histories C t onto symmetric tensors.
Proceeding from the first key property o f viscoelastic-viscoplastic materials
and using the data from [13, 14], Eq. (1) can be presented as
o R = G (C t ) = G (C *; £ t ), (2)
where C * is the deformation history o f the o f the right Cauchy-Green tensor, * is
the arc length along the strain path determined according to [15], and * t is the
time history o f traversing C * or simply the time history.
Later throughout this text we shall consider the processes o f deformation as
those starting at a certain reference mom ent o f time t 0 from an unstressed and
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P. P. Lepikhin
unstrained reference configuration к 0 assuming that the active process begins
with the onset o f the deformation process, unloading is absent, and C £ and £ 1
are smooth continuous param eter functions differentiated as m any times as
necessary.
Let us assume that viscoelastic-viscoplastic materials have a long-term
fading in time m em ory (hereinafter referred to as the fading memory), and this
mem ory represents a property that can be mathem atically expressed using the
function o f the simple m aterial response.
Having taken the history o f C £ in (2) to be constant, we vary £ t .F o r this
family o f the deformation processes, Eq. (2) takes the following form:
о R = G ( £ t ). (3)
Basing on relationship (3), consider the difference betw een the static
response and all other responses. Just as f l designates the history up to the
m om ent t o f the arbitrary function f over (—те, + те), we designate the history
o f the constant function f , whose value always equals to a, by a c :
a c (s) = a, 0 < s <те. (4)
Thus, £( t ) c represents a constant history (or a history constant) corresponding
to the current value £( t ) o f the arc length £ along the strain path for point in
reference configuration X in the history C £ . I n order to enable consideration of
the static case outlined here, ju st as it was done by the author o f [11], we assume
that i f £ t is the history belonging to the domain D 1 o f the response G
definition, then for each s over [0 , те) the constant history ( £ t (s ) ) c also belongs
to D 1. The value G (£ ( t ) c ) o f the response G represents the stresses
corresponding to being at rest in the state obtained from к 0 during deformation
along the path C £ whose arc length equals to £( t ).
In an elastoplastic material, particularly w ith a fixed C £ history, the stresses
are always static for all £ t in D 1 [13]:
о R = 0 R = G (C t ) = G ( £ t ) = G (C £ ) = G ( £( t ) c ) = g( £( t )), (5)
where о s is the value o f the static stress.
The m ain idea o f the fading m em ory in a viscoelastic-viscoplastic material is
that when the history £ t is close to the constant history £( t ) c, the stresses G (£ t )
are close to the static stresses. In other words, a small deviation from the constant
history £( t ) c induces the stresses, which are only slightly different from those in
an elastoplastic m aterial that correspond to £( t ) c. We specify the notions of
“smallness” and “closeness” with the help o f topology. W hen the topologies are
8 ISSN 0556-171X. Проблемы прочности, 2009, N 1
Construction of Constitutive Relationships
introduced, we can speak o f continuity in precise terms and formulate a precise
and general axiom o f continuity as an essential condition for the fading memory:
the response G is continuous in each constant history £ ( t ) c in D ^.
Just as it was done by the author o f [11], we consider real functions that are
summable w ith respect to some Lebesque-Stieltjes measure i on a real line R
and assume that the following relationship is true [11]
where | • | and || • || are, respectively, the norm and the semi-norm.
The measure i is generated by a real non-decreasing function i in a
well-known manner
i ( s - 0) = i ( s X I {[a,b)} = i ( b ) - i ( a ) (7)
for all real values o f a and b. We consider only those histories that represent the
functions set over [0 , oo) and assume [11] that the past only makes a finite
contribution to the semi-norms o f the bounded histories. Let us call the measure
I an obliviating measure, i f [11]
i ( s ) = 0 at s < 0, lim i ( s ) = M < o . (8)
s^ 00
This implies that transferring any interval o f the line infinitely far in the past
reduces its measure to zero:
lim i{ [ a + c , b + c )} = 0. (9)
We call semi-norm (6) calculated from the measure satisfying condition (8)
m history memory, which corresponds to this measure. The collection o f m
histories, for which the semi-norm 11 m|| is finite, forms a functional space, which
is a subspace o f the space o f all the histories m easurable w ith respect to i . This
subspace is called here the space o f histories with finite memory. It includes all
the bounded measurable histories and, in particular, all constant histories £( t ) c.
Just as the author o f [11] did it, we assume that a certain obliviating measure i
has been established once and for all keeping in m ind that our results depend on
the choice o f this measure. Suppose that the definition domain D 1 o f the
response G from (3) is a connected subset o f the space o f histories w ith a finite
mem ory with respect to i .
Consider the materials, which satisfy the axiom o f continuity for the
topology obtained on the basis o f the obliviating measure, and give the following
definition.
D e fin itio n . A viscoelastic-viscoplastic m aterial has a weak fading m em ory if
it satisfies the axiom o f continuity, w ith the discontinuity determined using the
obliviating measure:
ISSN 0556-171X. npodxeMbi npounocmu, 2009, N9 1 9
P. P. Lepikhin
o R = G (£ t ) = g (£ ( t)) + o(1) for | | | t - £ ( t ) c || —0. (10)
Thus, on condition that the m em ory o f the difference o f the history £ t and
constant history £( t ) c is rather small, the stresses are close to those in an
elastoplastic m aterial corresponding to £ ( t ).
In particular, the remainder term in (10) identically equals to zero in an
elastoplastic material. That is why, for (10) to hold true, the obliviating measure
should be such that
| |£ t - £ ( t ) c ||= 0. ( 11)
Inversely, if, according to this definition o f the memory, relationship (10) is
true w ith the remainder term equal to zero, the m aterial is elastoplastic. The
function P, w hich defines a obliviating measure o f this kind, is a single jum p at
s = 0.
Each time we assume the m aterial to have a weak fading memory, we choose
some function y3.
D e fin itio n [1 1 ] . Let £ t be a time history. Then the time history £ defined
by
£ t = \ £ ( t ° ) , s =£t - t ° ,
{t°} [£ t0( s - ( t - 10 )), s > t - 10 ,
is called the static continuation o f the given time history.
Using the notion o f static continuation, the phenom enon o f stress relaxation
in a viscoelastic-viscoplastic m aterial is m odeled based on the assertion that, if
some neighborhood o f the particle X has been m aintained in the state o f rest for
quite a long time, the stresses in X approach the value they would have had if
this neighborhood had always been in the state o f rest.
S tr e s s R e la x a tio n T h eo rem . For any fixed m om ent t and any history £ t in
D 1, the history o f static continuation £ also belongs to D 1, and the limit
G (£ {t0} ) at 10 — - w exists and represents static stresses corresponding to £ ( t ):
lim (G (£ {t0})) = G (£ ( t ) c ) = g (£ ( t )). (12)
t0 ~— —w v ’
A similar theorem for viscoelastic materials was proved in [11] w ith some
limitations. Analysis has shown that this proving is also true for the above case.
We take the history m em ory £ t in the form proposed by Coleman and Noll
[11]:
w
|| £ 11|2 = B | £ ( t ) |2+ f k (s) | £ t (s ) |2 d s , (13)
where B is a positive constant. We call the function k a obliviator or an
influence function.
0
10 ISSN 0556-171X. npodxeMbi npounocmu, 2009, N9 1
Construction of Constitutive Relationships
Similarly to the way it was done in [11], we construct approximations that
are higher than (10). To this end, assume that the principle o f the fading memory
o f the nth order is as follows: for static history £( t ) c , the response G is n times
Frechet-differentiable. Then
n
a R = G (£ t ) = g (£ (t)) G i ( £ t — £ (t ) c ) + o ( | |£ t — £ ( t ) c ||n ), (14)
i =1
where G j- are the bounded homogeneous polynomial mappings o f the ith degree
dependent on the variable £( t ) at a fixed C £ . In the Frechets expansion, the
above m apping is replaced by the sum o f simpler mappings with an error tending
to zero faster than the nth degree o f the m em ory 11 £ t — £ ( t) c 11 o f the difference
between the true history £ t and the corresponding constant history £( t ) c .
The viscoelastic-viscoplastic m aterials considered herein exhibit the long
term fading memories o f the deformation and tim e histories on the active
deformation. These two types o f the fading m em ory are independent and are
governed by different laws o f fading. W ith a constant strain value and varying
time, the material considered shows the long-term m em ory o f the time history
fading in time, whereas the long-term fading m em ory o f the deformation history
is absent. During passive deformation, the material has the fading m em ory alone
(viscoelastic behavior).
I f we assume that the material has the fading m em ory o f the first order, then
Eq. (14) approximates the deviations from the stresses in an elastoplastic material
w ith the help o f the bounded linear functional. The collection o f all the histories
w ith the finite m em ory forms the Hilbert space, and, according to the F rechet-
Riss theorem , each bounded linear functional in the H ilbert space admits
presentation in the form o f a scalar product. Assum e that the fading m em ory of
the Colem an-N oll type is being considered, then, according to (13), we obtain
a R = g (£ (t )) + f h (s)K (£ ( t ), s )[£ t (s) — £ (t )]d s + o ( ||£ t — £ (t ) c ||) , (15)
0
where the kernel K is the second-rank tensor such that
f |K (£ ( t), s ) |2 d s < x .
0
I f we truncate the correction term, we obtain a relationship independent of
the reference system, w hich can be used at large deformations for describing the
behavior o f a viscoelastic-viscoplastic material with a fading m em ory and
arbitrary symmetry o f properties.
Constitutive relationships for simple hardening elastoplastic materials with a
long-term fading m em ory o f the deformation history based on Eq. (5) were
constructed elsewhere [16-18].
ISSN 0556-171X. npodxeMbi npounocmu, 2009, N 1 11
P. P. Lepikhin
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Received 11. 06. 2008
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