Effect of Pores on Mechanical Behavior. Application to the Composite Material
The effects of microscopic pores, during the historical deformation on the behavior of ductile material are experimentally and numerically investigated. The material used in this study is constituted from the glass beads which are incorporated in the polyester resin. During moulding of these c...
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Інститут проблем міцності ім. Г.С. Писаренко НАН України
2002
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| Цитувати: | Effect of Pores on Mechanical Behavior. Application to the Composite Material / А. Sali, О. Fassi Fehri, S. Charif d ’Ouazzane, Z. Azari // Проблемы прочности. — 2002. — № 5. — С. 17-31. — Бібліогр.: 12 назв. — англ. |
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irk-123456789-468842016-04-15T13:34:30Z Effect of Pores on Mechanical Behavior. Application to the Composite Material Salih, A. Fassi Fehri, О. Charif d ’Ouazzane, S. Azari, Z. Научно-технический раздел The effects of microscopic pores, during the historical deformation on the behavior of ductile material are experimentally and numerically investigated. The material used in this study is constituted from the glass beads which are incorporated in the polyester resin. During moulding of these constituents, the pores remain in the material because of the resin viscosity. The experimental technique used for the characterization of the effect of pores is the tensile test. The constitutive equations of poroelastoplasticity are developed according to the micromechanical consideration. The material model developed by Gurson is extended to this composite material. The numerical results are obtained and compared with the experimental ones. Выполнены экспериментальное исследование и численное моделирование влияния микроскопических пор на поведение вязкого материала при деформировании с учетом истории нагружения. Объектом исследования служил композитный материал на основе полиэстер- ной резины с включениями из бисера. В процессе формовки указанных компонентов в материале образуются поры из-за вязкости резины. Для оценки эффекта пор использовали экспериментальную технологию испытаний на растяжение. В рамках микромеханического подхода получены основные уравнения пороупругопластичности. Проведено обобщение модели материала Гурсона на исследуемый композитный материал. Результаты расчетов сопоставлены с экспериментальными данными. Виконано експериментальні дослідження і числове моделювання впливу мікроскопічних пор на поведінку в ’язкого матеріалу при деформуванні з урахуванням історії навантаження. Об’єктом дослідження служив композитний матеріал на основі поліестерної гуми з вкрапленнями з бісеру. У процесі формовки указаних компонентів у матеріалі з ’являються пори внаслідок в ’язкості гуми. Для оцінки ефекту пор використовували експериментальну технологію випробувань на розтяг. У рамках мікромеханічного підходу отримано основні рівняння поропружнопластичності. Проведено узагальнення моделі матеріалу Гурсона на композитний матеріал, що досліджується. Розрахункові результати зіставляються з експериментальними. 2002 Article Effect of Pores on Mechanical Behavior. Application to the Composite Material / А. Sali, О. Fassi Fehri, S. Charif d ’Ouazzane, Z. Azari // Проблемы прочности. — 2002. — № 5. — С. 17-31. — Бібліогр.: 12 назв. — англ. 0556-171X http://dspace.nbuv.gov.ua/handle/123456789/46884 539.4 en Проблемы прочности Інститут проблем міцності ім. Г.С. Писаренко НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| language |
English |
| topic |
Научно-технический раздел Научно-технический раздел |
| spellingShingle |
Научно-технический раздел Научно-технический раздел Salih, A. Fassi Fehri, О. Charif d ’Ouazzane, S. Azari, Z. Effect of Pores on Mechanical Behavior. Application to the Composite Material Проблемы прочности |
| description |
The effects of microscopic pores, during the historical
deformation on the behavior of ductile
material are experimentally and numerically investigated.
The material used in this study is
constituted from the glass beads which are incorporated
in the polyester resin. During moulding
of these constituents, the pores remain in
the material because of the resin viscosity. The
experimental technique used for the characterization
of the effect of pores is the tensile test.
The constitutive equations of poroelastoplasticity
are developed according to the micromechanical
consideration. The material model developed
by Gurson is extended to this composite
material. The numerical results are obtained
and compared with the experimental ones. |
| format |
Article |
| author |
Salih, A. Fassi Fehri, О. Charif d ’Ouazzane, S. Azari, Z. |
| author_facet |
Salih, A. Fassi Fehri, О. Charif d ’Ouazzane, S. Azari, Z. |
| author_sort |
Salih, A. |
| title |
Effect of Pores on Mechanical Behavior. Application to the Composite Material |
| title_short |
Effect of Pores on Mechanical Behavior. Application to the Composite Material |
| title_full |
Effect of Pores on Mechanical Behavior. Application to the Composite Material |
| title_fullStr |
Effect of Pores on Mechanical Behavior. Application to the Composite Material |
| title_full_unstemmed |
Effect of Pores on Mechanical Behavior. Application to the Composite Material |
| title_sort |
effect of pores on mechanical behavior. application to the composite material |
| publisher |
Інститут проблем міцності ім. Г.С. Писаренко НАН України |
| publishDate |
2002 |
| topic_facet |
Научно-технический раздел |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/46884 |
| citation_txt |
Effect of Pores on Mechanical Behavior. Application to the Composite
Material / А. Sali, О. Fassi Fehri, S. Charif d ’Ouazzane, Z. Azari // Проблемы прочности. — 2002. — № 5. — С. 17-31. — Бібліогр.: 12 назв. — англ. |
| series |
Проблемы прочности |
| work_keys_str_mv |
AT saliha effectofporesonmechanicalbehaviorapplicationtothecompositematerial AT fassifehrio effectofporesonmechanicalbehaviorapplicationtothecompositematerial AT charifdouazzanes effectofporesonmechanicalbehaviorapplicationtothecompositematerial AT azariz effectofporesonmechanicalbehaviorapplicationtothecompositematerial |
| first_indexed |
2025-07-04T06:25:11Z |
| last_indexed |
2025-07-04T06:25:11Z |
| _version_ |
1836696534575480832 |
| fulltext |
UDC 539.4
Effect of Pores on Mechanical Behavior. Application to the Composite
Material
A. Salih,a O. Fassi Fehri,b S. C harif d ’Ouazzane,c and Z. A zarid
a Industrial Engineering Laboratory, Rabat Institute, Rabat, Maroc
b Laboratoire de Mécanique et Matériaux, Faculté des Sciences, Rabat, Maroc
c Laboratoire de Modélisation et Calcul en Mécanique, Rabat, Maroc
d Laboratoire de Fiabilité Mécanique, Université de Metz, Metz, France
УДК 539.4
Влияние пор на механическое поведение. Случай композитного
материала
А. Салиха, О. Фасси Фехри6, С. Ш ариф д’У аззанв, 3. А зариг
а Лаборатория промышленного строительства, Рабат, Марокко
6 Лаборатория механики и материалов, Рабат, Марокко
в Лаборатория моделирования и расчетов в механике, Рабат, Марокко
г Лаборатория механической надежности, Метц, Франция
Выполнены экспериментальное исследование и численное моделирование влияния микро
скопических пор на поведение вязкого материала при деформировании с учетом истории
нагружения. Объектом исследования служил композитный материал на основе полиэстер-
ной резины с включениями из бисера. В процессе формовки указанных компонентов в
материале образуются поры из-за вязкости резины. Для оценки эффекта пор использовали
экспериментальную технологию испытаний на растяжение. В рамках микромеханического
подхода получены основные уравнения пороупругопластичности. Проведено обобщение мо
дели материала Гурсона на исследуемый композитный материал. Результаты расчетов
сопоставлены с экспериментальными данными.
Introduction. The mechanical behavior of multiphase materials is related
not only to the properties of various components but also to the interfacial
adhesion between those constituents.
Generally, conventional composite materials contain pores introduced either
as by-products of the thermomechanical process or intentionally to control
microstructural features. During deformation, the inclusions may be debonded
from the matrix including the new pores. Consequently, those nucleating pores
grow by the plastic deformation.
There is considerable interest in the development of experimental [1] and
analytical [2] methods for the characterization of the pore nucleation and growth
process.
© A. SALIH, O. FASSI FEHRI, S. CHARIF D’OUAZZANE, Z. AZARI, 2002
ISSN 0556-I7IX. Проблемы прочности, 2002, № 5 17
A. Salih, O. Fassi Fehri, S. Charif D ’Ouazzane, Z. Azari
The best known experimental technique is tensile dilatometry (recently
reviewed by Meddad et al. [3]). In this method, the volume change in the material
is measured during the mechanical loading. Meddad et al. [3], who work with
bead-filled plastic, operate with a section of the tensile stress-strain curve and also
report that when tensile data are plotted using the compliance versus strain
presentation, the various stages of the stress-strain curve are easily distinguished.
Analytically, there are significant developments of pore nucleation and
growth process; we list those developed by Gurson [4, 5] on the ductile solid
material containing pores. Our work parallels that of Chu and Needleman [6],
who analyzed numerically the effects of pore nucleation in the biaxially stretched
sheets.
In this work, we present the extension of the famous Gurson model [4, 5] to
the porous composite material. The introduction of the pore pressure and the
current porosity to that model allows to describe the material behavior. Based on
the Bridgman [7] approach, we have also developed a new model of the
nucleation porosity.
The theory of isotropic poroelastopasticity is introduced in section 1. The
Hooke’s relations and micromechanical constitutive constants are briefly listed in
order to facilitate the engineering applications. A detailed description of the
poroelastic theory was introduced by Biot [8]. An extended review of the
development of the isotropic theory can be found in Detournay and Cheng [9].
Considering the same formality of continuous media, Coussy [10] presented the
mechanical behavior of porous media according to the micromechanical analysis.
In section 2, a brief description of the experimental method is given. Using
the dilatometry theory, the terminology used and the technique for obtaining the
experimental data are based on [11]. In glass bead-filled unsatured polyester (UP),
the global behavior is nonlinear. This behavior is characterized by the stress and
pore pressure methods. The interfacial debonding will be suggested by the current
porosity. The porous material model that has been developed by Gurson and the
applications of the Euler integration to that model are also given in section 3. A
comparison of the numerical results with those obtained from the experimental
study close this paper.
1. Poroelastoplastic Constitutive Relations. Let us consider a geometrical
domain Q constituted by a fluid phase (pores) and solid (matrix) one, having in
its initial configuration, the voluminal fraction p o and (1 — p o ), respectively. In
the case of poroelastoplastic configuration, we make an elementary dump defined
by do increments; we, respectively, measure the reversible (dee , d t e) and
irreversible (dep , d t p ) increments of strain and the variation of fluid content,
such that
where de and dt, are, respectively, a differential change in the total strain and the
total variation of fluid content.
de = de e + de p , (1)
d t= d t e + d t p , (2)
18 ISSN 0556-171X. npoôëeMbi npounocmu, 2002, N 5
Effect o f Pores on Mechanical Behavior. Application
In the framework of the hypothesis of infinitesimal transformations and for
the case of linear poroelasticity, the changes in the stress o and pore pressure p,
which represents the internal forces in the material, are defined in the following
relations:
0 ij ~ C ijkle k l, (3)
p = M (Ç e - a e ekk), (4)
where Cjki is the modulus tensor
Cjki = { K j - 3 Gj ^ j ô kl + 2 G j ô ikÔj i . (5)
The superscript y = u (resp. y = d) denotes undrained (resp. drained) quantities.
The undrained modulus tensor C j ki satisfies the following relation:
j = Cfm + « 2 M I j (6)
where Igki is a fourth order identity tensor, a is the Biot’s coefficient, and M is
a modulus associated with the combined fluid/solid compressibility.
a = \-
K r
K ,
and M = K r
K r
K ■Ф\
m j
K
K
- i
(7)
For the isotropic case, the flow function includes not only the first and the
second invariants of the stress tensor but also pore pressure and state variables.
This flow function is illustrated by the following formula:
F ( a h , a q , p , H a ) = 0, (8)
where H a (a = 1,2, ..., n) is a set of values. The function F is defined whenever:
F < 0 the response is purely poroelastic and F = 0 which is the consistence
condition represent the current yield surface; o H and a , which are,
respectively, the hydrostatic and the equivalent stress, are defined by the
following relations:
a H
1
= 3 a I
and a = л / д а ss , (9)
where I is the second order identity tensor and s is the stress deviator.
ISSN 0556-171X. Проблемы прочности, 2002, N2 5 19
A. Salih, O. Fassi Fehri, S. Charif D ’Ouazzane, Z. Azari
With the above notation, the stress tensor can be written as
2
O — OH I +--- OqU. (10)
The flow rules are written as
dep = dX
1 dg , 1 dg N
----- n + ---------1
dO q 3 dO h
and d(bp = dX\ —
\à p ,
(11)
where dX is a positive potential scalar, d ÿ p is a differential change in the plastic
porosity, and g = g (o H , o q , p , H a ) is the yield potential. The first flow rule
gives:
1 pde p = 3 de H I + de P n , (12)
where
' dg X
de H = dX\
y do
dep = dX
/ a N dg
H \ dO q )
and n = ------
2 O q
in which de qP and de H are related by the ‘dilatancy’ relation [10]
de H
~dep
(13)
where d is the dilatational factor.
Using Eqs. (1) and (2), relations (3) and (4) can be incrementally rewritten as
o t+dt = C ̂ e £t+dt = C ̂ ( e e + de e ) = o e - C ̂ de p , (14)
p t+dt — M £ f + % e - a (eH t + d eH )]— p e - M [% p - adeH ], (15)
where
o f — C Y (e f + de) (16)
and
p e — M [£f + d t - a ( e Ht + de h )], (17)
where t is the time at the start of the increment and t + dt is the time at the end
of the increment.
20 ISSN 0556-171X. npoôëeubi npounocmu, 2002, N 5
Effect o f Pores on Mechanical Behavior. Application
Using the Eqs. (5) and (12), we can rewrite the Eqs. (14) and (15) as
a t+dt = a e - K y d£H 1 - 2Gy d£Pn , (18)
Pt+dt = p e - M (d ^ P - ad£h ). (19)
Finally, the poroplasticity model is achieved by describing the evolution of
the state variables
d H a = h a (d£ p , dip p , a , p , H P ) (20)
as far as the rate independent materials h a remain homogenous of degree one in
d£p and dp p .
2. Application.
2.1. Experim ental. Material. The material used in this study is constituted
by the glass beads (type E) incorporated in the polyester (UP) resin. The average
bead diameter is about 350 ^m . During the moulding of these constituents, the
pores remain in the composite because of the resin viscosity. The bead volume
fraction f and the initial porosity f o are obtained by the statistical method
which is described in [11]. Using an Versamet-2 optical microscope, with
magnification X1600, the material was visualized (see Fig. 1).
Fig. 1. Microscopic section of the material (resin in clear, beads in white, pores in black).
Technical Test. In order to determine the experimental data, we adopted the
tensile test technique. This method is specified in [11]. The values of the
mechanical characteristics are obtained in the undrained test, in which the
variation of the fluid content is zero (£ = 0). A tensile test machine with the
cross-head speed of 1 mm/mn was used. The strain data is collected by using two
tensometers: axial tensometer and transverse tensometer (gauge length was
10 mm). The strains £ and the stresses a were simultaneously recorded.
The stress a versus strain £ curves for various initial porosity p 0 are
shown in Fig. 2. It shows that the stress decreases with the porosity content. The
behavior of these materials is nonlinear. The most plausible reason on this
ISSN 0556-171X. npoôëeubi npounocmu, 2002, N2 5 21
A. Salih, O. Fassi Fehri, S. Charif D ’Ouazzane, Z. Azari
nonlinearity is the debonding process. The effect of treatment of the glass beads
interface appears negligible. It is difficult to separate the contribution of
debonding and the behavior of the matrix because almost all the glass beads have
became debonded from the matrix.
o ,
7.0
6.0
5.0
4.0
3.0
2.0
1.0
o a.i a.2 as a4 os 07 e, %
Fig. 2. Stress o vs nominal strain e for several values of ^ 0.
2.2. Continuum Model of Porous M aterial. In this section, the material
model developed by Gurson [4, 5] for a porous ductile material is applied to the
material studied. Based on an analysis of single spherical pore in a shell,
including several simplified approximations (for example, the change of pore
chap is neglected so that the yield function remains effectively isotropic), Gurson
[4] proposed the following approximate form for the yield surface of periodically
porous solid containing a volume fraction 0 of pores:
I
F =
o
+ 2q10 cosh| 3 q 2 o H
2 o
(1+ q 3 0 2) = 0, (21)
2
where o m is the equivalent stress representing the actual microscopic stress state
in the matrix material and 0 is the current porosity. The parameters q i , q 2 , and
q 3 (where q 3 = q1) were introduced by Tvergaard [12] in an attempt to make the
predictions of Gurson’s equations. This assumption agrees with numerical studies
of materials containing pores periodically distributed in the matrix. For qi =
= q 2 = q 3 = 1, the function (21) was obtained by Gurson for the spherical pores.
The yield surface given by the equation (21) becomes that of von Mises when
0 = 0. Whenever the pore volume fraction is non-zero, there is an effect of the
hydrostatic stress on the plastic flow.
The microscopic equivalent plastic strain (e P) is assumed to be governed by
the equivalent plastic work
22 ISSN 0556-171X. npo6n.eubi npounocmu, 2002, № 5
Effect o f Pores on Mechanical Behavior. Application
ode p = ( 1 - 0 0 )o md e qp (22)
or, equivalently
ode p
d e P = ---------------- . (23)
q ( 1 - 0 o ) o m 1 J
The change in pore volume fraction during an increment of deformation is
partly due to growth of the existing pores and partly due to nucleation of now
pores by cracking or interfacial decohesion of inclusions or precipitate particles.
Accordingly, we write
d0 = d0 g + d0 n . (24)
The growth of pores is related to the change of the total volume as
d0 g = ( 1 - 0 o )de h ■ (25)
Suggestion o f Chu and Needleman [6]. The second phase particles are the
primary source of internal cavitations (pores). Pores initiate either by cracking of
the inclusions or by decohesion of the inclusion matrix interface. Assuming that
plastic strain controls nucleation process, as suggested by Chu and Needleman [6]
we get
d0 n = A d e PI. (26)
Parameter A depends in same complicated way on the distribution of the
inclusions and the mean equivalent plastic strain for nucleation e n ;
A f N A = ------1= exp
s N ĵ 2n
e „ - e N
\
s N
(27)
where s N is the standard deviation of the distribution and f N is determined so
that the total volume fraction nucleated is consistent with the volume fraction of
inclusions.
New M odel o f the Nucleation Porosity. The model of porous ductile material
to be studied here is defined by a periodical arrangement of the spherical
inclusions (beads) in the elastic-plastic media, as shown in Fig. 3. Referring to the
microscopic scale, we also denote by dQ m and dQ f the initial and the current
elementary domain of the matrix material and dQ f is the volume occupied at
time t by the nucleating inclusions initially located in dQ 0, so that dQ 0 =
= dQ m U dQ 0 and dQ t = dQ f U dQ f .
At time t the total volume strain of the material is irreversible and is sum of
the constituents which are related to:
ISSN 0556-171X. npoôëeubi npounocmu, 2002, N2 5 23
A. Salih, O. Fassi Fehri, S. Charif D ’Ouazzane, Z. Azari
(i) growth of the available pores;
(ii) decohesion of matrix-bead interface;
(iii) formation of new pores.
The second constituent is ignored in this study.
Here, we suppose that the pores grow as the spherical cavities once they
nucleated. Based on an analysis of a elementary volume which contains a simple
spherical inclusion in a matrix media, we consider the equivalent plastic strain in
the material (ep ) which controls the nucleation of new pores as such [7]
e p = 2 l ” ( t ,
(28)
where R o and R t are, respectively, the initial and the current radii of the
inclusion. Equation (28) is validated in two cases:
1) we assume that the behavior of the matrix material is elastic. In this case
the plastic strain is controlled by the pore volume strain;
2) the matrix is assumed to satisfy the incompressibility condition but,
because of the existence of pores, the macroscopic response does not. This remark
has been explicitly used by several authors (see for example [5] and [6]).
r t
0
o
Î
O
2Rt
2R 0
dQ 0 O , dQ t
t
Q OO
^
o
Fig. 3. Geometrical model of porous material.
Furthermore the material elementary volumes are defined as
dQ o = d X id X 2 d X 3 and dQ t = dx idx 2 dx 3 , (29)
where X t and x t (i= 1, 2, 3) are the initial and the current positions so that
d x i = d X i + d d X i .
Let P be the deformation gradient of the material so that J = det P. It can be
shown that
dQ t = JdQ 0 (30)
24 ISSN 0556-171X. npoôëeubi npounocmu, 2002, N2 5
Effect o f Pores on Mechanical Behavior. Application
and
J = 1 + £ , (31)
where J is the Jacobian of the transformation and £ is the volume strain of the
material. Using Eqs. (29), (30), and (31) the volume strain £ is expressed as
H i +
ô d X 1
d X
1 +
i / dX
1 +
ô d X 3
d X
- 1, (32)
ôdX 1 ô d X 2 ô d X 3
where £ L = and £T = ^ z — = ^ ; — = _ v£ l represent, respectively, the
dX i dX 2 dX 3
longitudinal and the transverse strains and v is the Poison’s ratio. In the plastic
phase (or when the elastic phase is neglected) the volume strain is defined as
(33)
This equation is equally valuated in the parfait plasticity case.
By using Eqs. (30) and (31), relationships (28) can be rewritten as
1 d&b_ dO f
\ d Q 0 dO о
-( £ +1) (34)
ep з 1п( ^ ( l + 1)
(35)
and f N
dO 0
dO о
are, respectively, the volume fractions of
nucleation pores and of the inclusions that are susceptible to be cracked or
nucleated.
Thus, the rate of the nucleated porosity dp n is obtained by the following
formula
exp
( p
2 ô
de P = Sde P , (36)
where d, which is defined by the Eq. (13), is the dilatational factor of the
material. To introduce it, let us consider the elementary experience that only the
£ q phydrostatic strain £ H and the distortion у = ^ = are not null. If £ H and £ P
л/3
correspond to the plastic contributions, the factor d is defined by the relationship:
ISSN 0556-171X. Проблемы прочности, 2002, N2 5 25
A. Salih, O. Fassi Fehri, S. Charif D ’Ouazzane, Z. Azari
e Hô = — ,e p
e q
where, in the isotropic conditions, we get
e H = ( 1 - 2v)e L and e p = (1+ v )e l . (37)
Finally, using the above remarks and Eq. (37), we find
, 1 - 2v
ô ^ - ------ . (38)
1+ v
3. Num erical Integration. Using the equations given in Section 1, and
enforcing the consistency condition (dF = 0):
By posing
dF dF m dF
dF = — do + ------ do + — dà = 0. (39)
do d0 m дф v ’
y = 2 я м 2ф sinh(1 q 2 o)\, (40)
p = q1cosh|2 q 2 ® )- q 3ф , (41)
ü ( (ü + у )
(42)B =
1- ф ,
d o m
H = — B 2 - o mA<pB - 3o m ( 1 - ф 0)у<р, (43)
d e q
and
where
de у H
C = ----- = 1 + - +
de pq ü Е В (1 -ф 0 )
d e pq ü + y
de С (1 -ф 0)
(44)
(45)
and after some algebraic manipulations, we find that the equations describing the
poroelastoplastic problem can be written as
26 ISSN 0556-171X. Проблемы прочности, 2002, N 5
Effect o f Pores on Mechanical Behavior. Application
do H
~de~ CB(1 —é 0 ) , (46)
dp _ 3aM y
de Cm
(47)
d é 3y A( m + y )
~dë~ C m ( 1 - é o) + C ( 1 - é 0) , (48)
where o and £ are, respectively, the true macroscopic axial stress and
logarithmic strain. The set of nonlinear equations (46), (47) and (48) is integrated
using the Euler method with equal strain increments of 1/1000 of the yield strain
in order to ensure high accuracy. The results of this ‘exact’ solution are compared
with those experimentally obtained.
From this comparison, let us consider the behavior of the composite material
with an initial porosity at p 0 = 0.2. The uniaxial stress-strain curve of the matrix
is represented in the following way:
0
EE m
1 „ m _mO — Ov
K y /
for o o
(49)
for o m > o Y ,
m
where £ m is the logarithmic strain, o m is the true stress, o ^ is the uniaxial
yield stress. The coefficients K y and M y are, respectively, the plastic strength
and the hardening exponent. The elastic and plastic properties of the matrix are
specified by E j a m = 4000, v m = 0.4, K y = 1100 MPa, and M y = 1.33,
where E m is the Young modulus and v m is Poisson’s ratio of the matrix
material. The Tvergaard constants (q1, q 2 , and q 3) are equal to 1.
The plastic strain that controls the nucleation process is described by the
volume fraction of nucleated or cracked beads f N = (1 — p 0 ) f 0, where f 0 is
the bead volume fraction corresponding at p 0 = 0. The value of f 0 can be
estimated in most cases to 0.52. Since the debonding process begins from the
nominal strain at 0.03%, we have assigned this value to the £ n , and the standard
deviation is about s n = 0.06.
The simulation test is assumed to be undrained. We suppose that the fluid
enclosed in pores is a perfect gas. Consequently, for an isothermal transformation
the fluid’s bulk modulus is equal to the initial pore pressure K f = 0.1 MPa.
The Biot’s coefficients a and M are calculated by the Eq. (7) in which the
drained bulk modulus is obtained by the following formula:
ISSN 0556-171X. npoôëeubi npounocmu, 2002, N2 5 27
A. Salih, O. Fassi Fehri, S. Charif D ’Ouazzane, Z. Azari
K d = Ku
Km
K Ф(
u j
Km
K f У
1 -
Ku
K
+ 0 ,
m
1 -
f
(50)
In the studied range of axial strain (0-0.6%), the stress curve (o /o r ), the rate
of pore pressure (£p = A p /p 0) and the rate of the current porosity
(£0 = A 0 /0 0) are, respectively, plotted in Figs. 4, 5, and 6. These figures show
that the numerical results (exact solutions) agree very well with those practically
obtained.
o /o y %
Fig. 4. Uniaxial stress-strain curve.
£0, %
Fig. 5. Pressure rate vs nominal strain e.
28
Fig. 6. Porosity rate vs nominal strain e.
ISSN 0556-171X. Проблемы прочности, 2002, № 5
Effect o f Pores on Mechanical Behavior. Application
The behavior of porous composite material is described by the stress and the
pore pressure. The interfacial debonding process is characterized by the current
porosity p . The pores are assumed to grow as spherical cavities once they are
nucleated. This is an appropriate idealization, in which nucleation takes place by
debonding process.
The porosity influences significantly the stiffness and the characteristics of
the material. Therefore, the behavior can be divided into three distinct stages:
I. Up to a deformation £ = 0.03%, the behavior of this material is linear
elastic with an initial modulus E 0 = 5800 MPa and initial Poisson’s ratio
v 0 = 0.34. The rate of the pore pressure (£p) and the current porosity are zero. It
is worth noting that between 0 and 0.03%, the plastic strain is zero. The ‘loading-
unloading’ test illustrated this phenomena.
II. In the range of 0.03% and 0.12% the curves o (£), £p(£), and £p(£) are
nonlinear. This stage corresponds to the progressive debonding of the beads from
the matrix. The value of the nucleation porosity p N is higher than that of the
growing porosity p g . When the porosity is not initially the case, the microbeads
can sometimes be nucleated during straining in the second phase. The yield stress
o y , at which the debonding process is initiated, decreases with the increasing
pores concentration.
III. At £ = 0.12%, the o (£) curve starts to deviate from nonlinearity to
linearity. The porosity increases exponentially. After the glass beads are debonded
from the matrix, it creates pores which grow by plastic deformation. This
deformation of the pores created an additional porosity. In this stage, the behavior
of porous composite material is similar to that of porous resin.
0 N % 0 N , %
0.7
/ : 1.0
y rf-0 .1 :
0.6 / 0.9
/ ■
/ 0.8 Y :
0.5 y
0.7 J O .S=0.15 *jT ' V ‘
0.4 Jf 0.6
7 Y
0.5 / y « t 6 =0.2 :
0.3 y Yjt 0.4 V y -j* «
0.2 / 0.3 / JT •
0 ( hu • Vvdk’liun
A « X l'inpnxJ m><kk'l ■ 0.2 jr * uiw»mp nui “
v. 1
V 0.1
(unp nul ;
0 O.S 0.? 0.3 0.« 0.5 0.® % 3,1 «,.< 3.3 0,4 0.5 C.S
Fig. 7. Nucleation porosity p N vs strain £ Fig. 8. Proposed model of p N for compressible
for Chu-Needleman’s and proposed models. and incompressible material.
Having found the expression (36) for the new nucleation porosity model, the
critical point of cracked beads into a shear band is pronounced in the behavior.
This point is located at the nominal strain 0.03%. Figure 7 shows that the
comparison, between the results of this model and those obtained by Chu and
Needleman is conformed. This comparison between Chu and Needleman’s
ISSN 0556-171X. npoôëeubi npounocmu, 2002, N 5 29
A. Salih, O. Fassi Fehri, S. Charif D ’Ouazzane, Z. Azari
parameters at <p 0 = 0.2, £ N = 0.03%, and s N = 0.06 and the new model’s
coefficients f n = 0.416. The parameters £ and d are, respectively, obtained by
the Eqs. (32) and (38). For the Poisson’s ratio 0.3 < v < 0.4 the dilatational factor
is defined between 0.3 < v < 0.15.
In view of this, the transformations related to the integration of Eqs. (39)-
(48) yield to the formula that predicts a monotonous increase in ^ n with
increasing strain, while the Chu-Needleman relation asserts that this value is not
monotonic and even change rater at £ p > £ n , but because the value taken in this
study is very small, this phenomena no longer appears.
The effects of the volume strain of the matrix and the dilatational factor are
illustrated in Fig. 8, which shows that the volume strain appears to be negligible.
Therefore the study is similar to compressible or incompressible material, but the
effect of the dilatational factor is significant.
Conclusions. In many situations, pores remain in the composite material
during its moulding because of the viscosity of the resin. The pores modify not
only the mechanism of load transfer between the matrix and the inclusions, but
also the macroscopic characteristic. These pores are the origin of the stress
concentration.
The problem analyzed in this study is the influence of the pores on the
mechanical behavior.
On one hand, the poroelastoplastic formulation is developed in the isotropic
case, according to the micromechanical consideration. The Gurson’s material, in
which we have introduced the new porosity model, is extended to the porous
composite material. The application of the Euler integration method to the
Gurson’s model is discussed.
On the other hand, we have studied experimentally a material constituted
from the microbeads of glass (E) which are incorporated in a polyester resin (UP).
From this study we deduce that the pores influence not only the
characteristics of the material, but is also the nonlinearity of the behavior that has
three distinct stages:
1). Linear poroelastic stage, where Young’s modulus and Poisson’s ratio are
constant. In this stage, the current porosity and the pore pressure are neglected.
2). Debonding stage, in which the behavior deviates from linearity to
nonlinearity. The debonding process is describing by the new model of the
nucleation porosity. In this stage, the porosity increases exponentially.
3). Growing stage, in which the slope of the behavior decreases with the
pores content. In this stage, that follows the complete debonding, the load of the
bearing section is reduced by the area occupied by the glass beads.
We have also found that in the proposed model the effect of the
incompressibility of the material appears to be negligible, but the dilatational
factor influences significantly the nucleation porosity. This model consists in the
replacement of the Chu-Needleman relation by a new one, which is more
easy-to-use and more adapted to the micromechanics of this phenomena.
However, the refinements proposed here cover mostly the expansion of an
isolated spherical pore under uniaxial tensile loading conditions of a porous
structure.
30 ISSN 0556-171X. npo6n.eubi npounocmu, 2002, № 5
Effect o f Pores on Mechanical Behavior. Application
Р е з ю м е
Виконано експериментальні дослідження і числове моделювання впливу
мікроскопічних пор на поведінку в ’язкого матеріалу при деформуванні з
урахуванням історії навантаження. Об’єктом дослідження служив компо
зитний матеріал на основі поліестерної гуми з вкрапленнями з бісеру. У
процесі формовки указаних компонентів у матеріалі з ’являються пори вна
слідок в ’язкості гуми. Для оцінки ефекту пор використовували експери
ментальну технологію випробувань на розтяг. У рамках мікромеханічного
підходу отримано основні рівняння поропружнопластичності. Проведено
узагальнення моделі матеріалу Гурсона на композитний матеріал, що до
сліджується. Розрахункові результати зіставляються з експериментальними.
1. J. M. Jalinier, J. L. Salsmann, and B. Baudelet, “Strain hardening and
damage,” in: M echanical Behavior o f M aterials, University of Cambridge
(1979).
2. A. Needleman, and N. Triantafyllidis, “Void growth and local necking in
biaxially stretched sheets,” J. Eng. Math. Tech., 100, 164 (1978).
3. A. Meddad, S. Fellah, M. Pinard, and B. Fisa, “Filler-matrix debonding
monitored by tensile test,” CRSP, 1-5, E. Poly. de Montreal (1994).
4. A. L. Gurson, Plastic Flow and Fracture Behavior o f D uctile M aterials
Incorporating Void Nucleation, Growth, and Interaction, Ph.D. Thesis,
Brown University (1975).
5. A. L. Gurson, “Continuum theory o f ductile rupture by void nucleation
and growth: Part I. Yield criteria and flow rules for porous ductile materials,”
J. Eng. Math. Tech., 99, 2-15 (1977).
6. C. C. Chu and A. Needleman, “Void nucleation effects in biaxially stretched
sheets,” J. Eng. Math. Tech., 102, 249-256 (1980).
7. P. W. Bridgeman, Studies in Large Plastic Flow and Fracture, University
Cambridge, Massachusetts, (1964).
8. M. A. Biot, “General theory of three-dimensional consolidation,” J. Appl.
P h ys , 12, 155-164 (1941).
9. A. H. D. Cheng, “Material coefficients of anisotropic poroelasticity,” Int. J.
Rock. Mech. Min. S c i , 34, 199-205 (1997).
10. O. Caussy, “Mécanique des milieux poreux,” Technip, Paris (1991), pp. 89
217.
11. “Plastiques renforcés au verre tex tile ,” in: F ra n ce A sso c ia tio n o f
Normalisation, 2nd ed., AFNOR (1982), pp. 338-351, 396-399.
12. V. Tvergaard, “Influence of voids on shear band instabilities under plane
strain,” Int. J. Fract., 17, 389-407 (1981).
Received 03. 12. 2001
ISSN 0556-171X. Проблеми прочности, 2002, № 5 31
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