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8/12/2019 Andrianov 2012 Composites Part B Engineering
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Buckling of fibers in fiber-reinforced composites
Igor V. Andrianov a, Alexander L. Kalamkarov b,⇑, Dieter Weichert a
a Institute of General Mechanics, RWTH Aachen University, Templergraben 64, Aachen D-52062, Germanyb Department of Mechanical Engineering, Dalhousie University, Halifax, Nova Scotia, Canada B3H 4R2
a r t i c l e i n f o
Article history:
Received 3 February 2011
Received in revised form 9 December 2011
Accepted 2 January 2012
Available online 25 January 2012
Keywords:
A. Fibers
B. Buckling
C. Micro-mechanics
Transversal buckling of fiber
a b s t r a c t
Elastic stability of fibers in fiber-reinforced composite materials subject to compressive loading is stud-
ied. The transversal buckling mode is considered, and two limiting cases, the dilute and non-dilutecomposites are analyzed. In the case of a non-dilute composite, the cylindrical model and the lubrication
approximation are applied. The original problem is reduced to a problem of stability of a rod on elastic
foundation. Through the solution of this problem a simple formula for the buckling load is obtained. In
the case of a dilute composite, the solution of a problem of stability of a compressed rod in elastic plane
is used. On the basis of the obtained solutions in two limiting cases the interpolation formulae are
derived. These formulae describe buckling of fiber in the fiber-reinforced composite for any value of
the fiber volume fraction. Comparison with known numerical and experimental results is carried out,
and the sufficient accuracy of the derived formulae is demonstrated.
2012 Elsevier Ltd. All rights reserved.
1. Introduction
One of important failure modes of the fiber-reinforced compos-
ite materials under the compressive loading is a loss of elastic sta-
bility of fibers, see, e.g., [1–6]. This phenomenon is studied in many
experimental investigations [2–5]. Basic conclusion from the
experimental investigations can be formulated as follows: if a
fiber-reinforced composite is compressed in the direction of fibers,
the most probable mechanism of failure is micro-buckling.
Theoretical studies of elastic stability of composites are often
based on some simplifying assumptions. In the investigation by
Rosen [7], see also Jones [8, Chapter 3.5.3], the buckling of fibers
was analyzed by considering the 2D problem for a two-layered
periodic composite in which the fibers and matrix were repre-
sented by the stiff and soft layers respectively. As it is mentioned
in [8], the 2D buckling model results should be upper bounds for
the original 3D fiber buckling problem, in which the fiber buckles
into a helix at a lower load then that corresponding to sinusoidal
buckling in the plane.
The buckling of fibers in elastic composite materials has been
studied by Parnes and Chiskis [9]. They modeled the composite
as a periodic two-layered material and analyzed the problem
by employing a mechanics of materials approach based on
Euler–Bernoulli theory of an infinite fiber layer embedded in an
elastic foundation matrix. The interaction between the fiber and
matrix layers was analyzed using the elasticity equations. A com-
prehensive list of references to various investigations of the present
subject canbe also foundin [9]. Aboudi and Gilat [10] used the anal-
ogybetween the governing equations for the analysis of buckling in
elastic structures and the elastodynamic equations of motionforthe
wave propagation. By employing this analogy, the exact and
approximate buckling stresses for the periodic layered materials
and for the continuous fiber-reinforced composites respectively
have been established. Guz and co-authors, see [4,5,11–13], used
solution of the problem in the form of series. In this approach the
original problem is reduced to the infinite systems of linear alge-
braic equations with their subsequent numerical solution. FEM
was also used in the number of publications, see e.g., [14–17].
Study of the Carbon nanotube-reinforced composites is of a
high importance. These materials have a very high stiffness and
strength. The major compressive failure mode of the Carbon
nanotube-reinforced composites is a loss of stability of the
embedded nanotubes. In [18] a failure theory for these materials
is developed on the basis of replacement of the nanotubes by the
infinitely long cylinders. Note that the assumption of infinitely
long fibers is not accurate. In this case a problem of buckling
should be considered assuming an infinite matrix and fibers of
a finite length, as it is assumed in the present paper.
Two types of buckling modes are commonly considered: the
shear and transversal buckling modes. In the first type of buckling
the fiber matrix layers exhibit in-phase deformation, see Fig. 1a,
whereas in the latter type the fiber and matrix layers exhibit
anti-phase deformation, see Fig. 1b [7,8]. Both buckling modes
are important from the practical point of view, however the trans-
versal buckling mode represents a particular interest in the cases of
dilute composites, see e.g., [3,8]. Therefore the analytical study of
transversal buckling of fibers is important problem.
1359-8368/$ - see front matter 2012 Elsevier Ltd. All rights reserved.doi:10.1016/j.compositesb.2012.01.055
⇑ Corresponding author. Tel.: +1 902 494 6072; fax: +1 902 423 6711.
E-mail address: [email protected] (A.L. Kalamkarov).
Composites: Part B 43 (2012) 2058–2062
Contents lists available at SciVerse ScienceDirect
Composites: Part B
j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c o m p o s i t e s b
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In the present paper we consider the transversal buckling of fi-
bers and we derive the simple interpolation formulae, suitable for
any value of the fiber volume fraction.
The paper is organized as follows. In Section 2, an extensional
matrix deformation for a non-dilute case for fibers of a square
cross-section is considered. The transversal buckling mode for
non-dilute case for fibers of a square cross-section is studied in
Section 3. In Section 4 the transversal buckling mode for the dilute
case for fibers of a square cross-section is analyzed, and the inter-
polation formula valid for any fiber volume fractions is derived.
The comparison of the obtained results with the published data
is carried out. It demonstrates the sufficient accuracy of the derived
formulae. In Section 5 the transversal buckling mode is studied for
the fibers of a circular cross-section. Section 6 concludes the paper.
2. Extensional matrix deformation for non-dilute case
Let us first consider a fiber-reinforced composite material with
the fibers of a square cross section distributed periodically with the
centers at the points of a simple square lattice, see Fig. 2. Compos-
ite material is assumed to be infinite in the direction of axis x3. It is
assumed that fibers are packed tightly in composite material, so
that h a, see Fig. 2.
Without loss of generality it is assumed that the buckling occurs
in the direction of x1-axis. The loading of a single fiber in the infi-
nite composite (or a finite, but clamped the lateral sides) is studied.
Direction of load is fixed and the problem is examined in the static
linear elastic formulation.
Due to the tight packing of fibers the matrix layer between the
fibers is thin, and it is possible to apply the lubrication approxima-
tion [19], to describe the deformation of matrix. Consider deforma-
tion in the direction x1 P 0. In view of the small thickness of layer
in the direction x1 P 0 the variation of stresses in this direction is
considerably larger than in the orthogonal direction, i.e.,
@ uðmÞ
@ x1
@ uðmÞ
@ x2
:
Therefore, the original equation of deformation
@ 2uðmÞ
@ x21
þ @ 2uðmÞ
@ x22
¼ 0
can be replaced by the following simplified equation:
@ 2uðmÞ
@ x21
¼ 0; ð1Þ
where u(m) is the displacement of matrix.
Let us now examine the loss of elastic stability of a single fiber.
We apply the cylindrical model, commonly used in the theory of
composites, see, e.g., [20]. Namely, we isolate a cell
a 2h 6 xi 6 a + 2h; i = 1, 2, and replace remaining composite
with the material with stiffness of fibers. The boundary conditions
are written as follows:
uðmÞ ¼ 0 for x1 ¼ aþ 2h: ð2ÞThe ideal bonding between fibers and matrix is assumed, and
therefore
uðmÞ ¼ w f for x1 ¼ a; ð3Þwhere w f is the displacement of fiber in the x1-direction.
Since the stiffness of fibers, as a rule, is substantially higher than
stiffness of matrix, we neglect the deformation of fibers.
The solution of the boundary-value problem (1)–(3) is the
following:
uðmÞ ¼ w f
2hðaþ 2h x1Þ: ð4Þ
3. The transversal buckling mode for non-dilute case
Stress in the matrix can be expressed as follows:
rðmÞ x1
¼ E ðmÞ
1 ðmðmÞÞ2eðmÞ x1
; ð5Þ
where eðmÞ
x1 ¼
@ uðmÞ
@ x1
, E (m) and m (m) are Young’s modulus and Poisson’s
ratio of the matrix material.
Eqs. (4) and (5) yield
rðmÞ x1
¼ E ðmÞ
2½1 ðmðmÞÞ2w f
h : ð6Þ
Tension force from the matrix to the fiber from two sides,
neglecting the lateral forces, can be expressed as follows:
T ðmÞ ¼ 2a
h
E ðmÞ
½1 ðmðmÞÞ2w f : ð7Þ
Otherwords, it is assumed that there will be similar tension of
matrix from the opposite side without debonding.
Equation of the elastic stability of fiber loaded by the compres-
sive stress r0 can be written as follows:
E f 2a3
3
d4w f
dx4
3
þ 2ar0
d2w f
dx2
3
þ E ðmÞ
½1 ðmðmÞÞ2hw f ¼ 0 ð8Þ
It can be seen that Eq. (8) coincides with the equation of the
elastic stability of a rod on an elastic foundation. It is natural to as-
sume that many waves are formed in this case when the buckling
occurs, and that is observed in the experiments [6]. The wave-
length is then determined from the formula [21]:
Lb ¼ a
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2hE f ½1 ðmðmÞÞ2
3aE ðmÞ4
s : ð9Þ
Therefore, the formula for the buckling stress can be calculated
from the following expression [21]:
Fig. 1. (a) Shear buckling mode, and (b) transversal buckling mode.
Fig. 2. Cross section of fiber-reinforced composite material.
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r0 ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2aE f E ðmÞ
q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3h½1 ðmðmÞÞ2q : ð10Þ
For the transversal buckling mode the following formula for
buckling stress was proposed by Rosen, see [7,8, Chapter 3.5.3]:
r0 ¼ 2c ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficE f E
ðmÞ
3ð1 c Þs ; ð11Þ
where c is the volume fraction of inclusions, i.e.,
c ¼ a2
ðaþ hÞ2 :
As we consider the case h a, we get c 1 2h/a. Discarding the
members of the order h/a, the Eq. (11) yields
r0 ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2aE f E ðmÞ
q ffiffiffiffiffiffi
3hp : ð12Þ
The formula (12) differs from the above derived Eq. (10) only by
a Poisson’s ratio of matrix material which enters into Eq. (10) and
is absent in Eq. (12). The presence of the Poisson’s ratio m
(m)
in Eq.(10) shows that this formula takes into account the mechanics of
deformation more accurately. Note that as it is shown in [9], the
formula (11) gives good results precisely for large concentrations
of rigid inclusions.
4. Transversal buckling mode for the dilute case: Matching of
limiting solutions
For the case of dilute composite, the following expression for
the buckling stress is derived in [9]:
r0 ¼ a
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE f ðE ðmÞÞ23
q ; where aðmðmÞÞ ¼ 3ð1 mðmÞÞ
ð1 þ mðmÞÞð3 4mðmÞÞ 2=3
:
ð13ÞNote that in this case results for the shear and transversal
buckling modes practically coincide.
Since a(0) = 1,and a(0.35) = 0.97, Eq. (13) can be written as
follows:
r0 ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE f ðE ðmÞÞ23
q : ð14Þ
Note that the following qualitatively similar formula is obtained
in [22]:
r0 ¼ 0:52
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE f ðE ðmÞÞ23
q :
Since we have obtained the expressions for the transversal
buckling load in both limiting cases of small and large values of
the fiber volume fraction, it is possible to match them, and to de-rive the interpolation formula. Note that it is more convenient to
match the limiting expressions for the transversal buckling strains
e0, i.e., the following formulae:
e0 ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE ðmÞ
E f
!2
3
v uut for c ! 0; ð15Þ
e0 ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2E ðmÞ ffiffiffic
p
3E f ð1 ffiffiffic
p Þ½1 ðmðmÞÞ2
v uut for c ! 1: ð16Þ
We apply the multiplicative matching [23], and use the volume
fraction of inclusions c as the matching parameter. The following
interpolation formula for the transversal buckling strain is derivedas a result:
e0 ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE ðmÞ
E f
!2
3
v uut 1 þ 2
ffiffiffiffiffiffiffiffiE f
E ðmÞ6
s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1
3½1 ðmðmÞÞ2ð1 c Þ
s c k
" #: ð17Þ
Note that the value of the parameter k in Eq. (17) allows to
achieve the best approximation of formulae (15) and (16) in the
limiting cases. Calculations show that the best approximation is
achieved for k ¼ 1:2.
Numerical results from Eq. (17) are shown in Fig. 3 (green1
curves). For the comparison, the results from [9] are shown inFig. 3 by diamonds, and blue curves show the results of the follow-
ing formula obtained by Rosen [7,8, Chapter 3.5.3]:
e0 ¼ 2c
ffiffiffiffiffiffiffiffiE ðmÞ
E f
s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic
3ð1 c Þr
: ð18Þ
It is seen from Fig. 3 that the presently derived Eq. (17) gives
close results with those from [9], especially in the cases of smaller
fiber volume fractions c , while formula (18) underestimates the
transversal buckling strains e0 .
Thus, the presently derived formula (17) gives good results in
dilute case. Whereas Rosen’s formula (18) leads to not-accurate re-
sults in dilute case. At the same time, as shown above, in the non-
dilute case the formulae (17) and (18) practically coincide. And
both of them give good approximation of the exact results in
non-dilute case [8,9].
5. The transversal buckling mode for fibers of a circular cross-
section
Consider now the elastic stability of fibers of a circular cross-
section in the fiber-reinforced composite material, see Fig. 4.
In the case of tight packing of fibers the lubrication approxima-
tion [19] can be applied. In this approach the unit cell with curvi-
linear boundaries of inclusion, see Fig. 5a is replaced by a much
simpler problem for a strip shown in Fig. 5b.
Fig. 3. Comparison of results: green (upper) curves show results of presently
derived Eq. (17) for k ¼ 1:2; blue (lower) curves show results of Rosen’s formula
(18) [7,8, Chapter 3.5.3]; and diamonds show results from [9]. The upper curves
correspond to E f /E (m) = 100, and the lower curves correspond to E f /E (m) = 1000.
1 For interpretation of color in Fig. 3, the reader is referred to the web version of this article.
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As a result the buckling stress in the case of axial compression
for the tightly packed fibers of the circular cross section with a ra-
dius R can be determined from the following formula:
r0
¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2RE f E
ðmÞ
q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3h½1 ðmðmÞÞ
2q ;
ð19
Þ
where R is the radius of the fiber, and h is the minimum distance
between the neighboring fibers.
It should be noted that the formula (19) is less accurate than the
above obtained formula (10) because in the case of circular fibers
the matrix occupies larger area than in the case of fibers of a square
cross section.
The following interpolation formula for the transversal buckling
strain in the case of circular fibers is derived as a result:
e0 ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE ðmÞ
E f
!2
3
v uut 1 þ 2
ffiffiffiffiffiffiffiffiE f
E ðmÞ6
s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1
3½1 ðmðmÞÞ2ð1 c =c maxÞ
s ðc c maxÞk
" #:
ð20Þ
6. Conclusions
In the present paper the elastic stability of fiber in
fiber-reinforcedcompositematerials subjectto compressiveloading
is considered, and, in particular, the transversal buckling mode is
analyzed. The interpolation formula (19) (in the case of fibers of a
rectangular cross-section) and the formula (20) (in the case offibers
of a circular cross-section) are derived to determine the bucklingloads for any value of the fiber volume fraction. Comparison with
some known numerical and experimental results is carried out,
andthe sufficient accuracyof the derivedformulae is demonstrated.
In particular, it is shown that the presently derived formula (17)
gives accurate results in dilute case. Whereas Rosen’s formula (18)
leads to non-accurate results in dilute case. At the same time, for
non-dilute case the formulae (17) and (18) practically coincide.
And both of them give good approximation of the exact results in
non-dilute case.
The obtained results can be also used for the evaluation of the
elastic stability of laminated composite materials.
Note that in the analysis of stability of composites reinforced
with the short Carbon nanotubes the results for the critical
buckling load obtained using the assumption of infinitely long fi-bers, see [24,25], are not accurate. In this case a problem of elastic
stability should be considered assuming an infinite matrix and the
fibers with the finite length, as it is done in the present paper. It is
also of interest to investigate the influence of boundary conditions
for a composite specimen with the finite dimensions, see, e.g., [2].
Acknowledgments
The present work was supported by the German Research Foun-
dation (Deutsche Forschungsgemeinschaft), Grant # WE 736/30-1
(I.V. Andrianov); and by the Natural Sciences and Engineering
Research Council of Canada, NSERC (A.L. Kalamkarov).
Authors thank Prof. A.I. Manevitch for his suggestions and com-
ments related to the obtained results.
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