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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: alex.kalamkarov@dal.ca (A.L. Kalamkarov).

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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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