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, Germany b 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 Avail able online 25 Januar y 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 transvers al buckl ing mode is consi dere d, and two limi ting cases, the dilute and non-dil ute composites 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 limitin g cases the interpolation formulae are deri ved. These formula e desc ribe buckling of fiber in the fiber-re infor ced composi te for any valu e ofthe 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- bil ity of fib ers, see, e.g ., [1–6]. This phenomeno n is studied in ma ny experim ental investigation s [2–5]. Basi c co nc lusion fr om the expe rimenta l inve stiga tions can be form ulat ed as foll ows: 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 simplif ying assump tion s. In the inve stiga tion 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 -lay ered peri odic comp osite in whi ch the fibe rs and matrix were rep re- 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] . The y mo deled the composi te as a per iodi c two -lay ered materi al and anal yze d the pro blem by emplo yin g a me cha nic s of ma terials appr oa ch based on Euler–Berno ulli theory of an infin ite fibe r laye r emb edd ed in an elastic foundation matrix. The interaction between the fiber and matrix layers was analyzed using the elasticity equations. A com- preh ensi ve list of refe rences to vari ous inve stiga tions of the pre sent sub jec t canbe als o foundin [9] . Aboudi and Gilat [10] used the anal- ogybetwee n the gov ern ing equ ati ons for the analy sis of buc kli ng in elas tic structures and the elas tody namic equations of mo tion forthe wa ve pr opag ati on. By empl oy ing thi s ana log y, the exact and approx imate buckling stresses for the period ic layered materials and for the continuous fiber-re inforced compos ites respective ly 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- brai c equations with their subsequ ent numeric al solu tion. FEM was also used in the number of publications, see e.g., [14–17]. Stud y of the Carb on nano tube -re info rced comp osite s is of a high impor tance. The se mat erials have a very high stiff ness and str ength. The ma jo r com pr ess ive fai lur e mo de of the Car bo n nan otu be- re inf or ced com po sites is a los s of sta bi lit y of the embedded nanotubes. In[18] a failure theory for these materials is developed on the basis of replacement of the nanotubes by the infin itely long cylin der s. Note that the assu mp tion of infin itely lon g fibers is not acc ur ate. In thi s cas e a pr ob lem of buckl ing shou ld be cons ider ed assumin g an infin ite mat rix and fibe rs ofa finite length, as it is assumed in the present paper. Two types of buck ling mo des are commo nly cons ide red : the shear and transversal buckling modes. In the first type of buckling the fiber matrix layers exhibit in-phase deformation, see Fig. 1a, wher eas in the lat ter typ e the fiber and ma trix lay er s ex hib it anti-phase deformatio n, see Fig . 1b [7,8]. Both bucklin g modes are important from the practical point of view, however the trans- ver sal buckling mode represents a particul ar interest in the cas es ofdilute composites, see e.g., [3,8]. Therefore the analytical study oftransversal 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 journal homepage: www.elsevier.com/locate/compositesb
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8/12/2019 Andrianov 2012 Composites Part B Engineering
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
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6. Conclusions
In the present paper the elastic stability of fiber in
[20] Tandon GP. Use of composite cylinder model as representative volume
element for unidirectional fiber composites. J Compos Mater
1995;29(3):388–409.
[21] Timoshenko SP, Gere JM. Theory of elastic stability. 2nd ed. Mineola,
NY: Dover; 2009.
[22] Rzhanitsin AR. Stability of elastic systems equilibrium. Moscow: GITTL; 1955.[23] Andrianov IV, Awrejcewicz J. New trends in asymptotic approaches:
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[24] GuzAN, DekretVA. On twomodels in thethree-dimensional theoryof stability
of composite materials. Int Appl Mech 2008;44(8):839–54.
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2062 I.V. Andrianov et al. / Composites: Part B 43 (2012) 2058–2062