Dynamics Of Anisotropic Composite Cantilevers Weakened By Multiple Transverse Open Cracks

Engineering Fracture Mechanics 70 (2003) 105–123
www.elsevier.com/locate/engfracmech

Dynamics of anisotropic composite cantilevers weakened
by multiple transverse open cracks
Ohseop Song a, Tae-Wan Ha a, Liviu Librescu

b,*

a

Department of Mechanical Engineering, Chungnam National University, Taejon 305-764, South Korea
Department of Engineering Science and Mechanics, College of Engineering, Virginia Polytechnic Institute and State University,
Blacksburg, VA 24061-0219, USA

b

Received 7 May 2001; received in revised form 14 November 2001; accepted 15 January 2002

Abstract
In this paper an exact solution methodology, based on Laplace transform technique enabling one to analyze the
bending free vibration of cantilevered laminated composite beams weakened by multiple non-propagating part-through
surface cracks is presented. Toward determining the local flexibility characteristics induced by the individual cracks, the
concept of the massless rotational spring is applied. The governing equations of the composite beam with open cracks
as used in this paper have been derived via Hamilton’s variational principle in conjunction with Timoshenko’s beam
model. As a result, transverse shear and rotatory inertia effects are included in the model. The effects of various parameters such as the ply-angle, fiber volume fraction, crack number, position and depth on the beam free vibration are
highlighted. The extensive numerical results show that the existence of multiple cracks in anisotropic composite beams
affects the free vibration response in a more complex fashion than in the case of beam counterparts weakened by a
single crack. It should be mentioned that to the best of the authors’ knowledge, with the exception of the present study,
the problem of free vibration of composite beams weakened by multiple open cracks was not yet investigated.
Ó 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Cantilever composite beam; Shearable and unshearable; Multicracks; Vibration; Laplace transform method

1. Introduction
In recent years there is an increased use of fiber reinforced composites in weight-sensitive structures, such
as the aeronautical and aerospace constructions, helicopter, turbine blades, and there are all the reasons to
believe that this trend will continue and intensify in the years ahead. For such constructions, the issues
related with their structural integrity and strength degradation in the presence of cracks constitute vital
problems whose investigations present a considerable importance.

*

Corresponding author. Tel.: +1-540-231-5916; fax: +1-540-231-4574.
E-mail address: (L. Librescu).

0013-7944/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved.
PII: S 0 0 1 3 - 7 9 4 4 ( 0 2 ) 0 0 0 2 2 - X


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O. Song et al. / Engineering Fracture Mechanics 70 (2003) 105–123

It is well known in this context that the cracks appearing in a structure yield an increase of the vibrational
level, result in the reduction of their load carrying capacity, and can constitute the cause of catastrophic
failures. In order to prevent such highly detrimental events to occur, an early detection of the existence of
cracks is needed.
The existence of a crack results in a reduction of the local stiffness, and this additional flexibility alters
also the global dynamic structural response. In this sense, the obtained results have revealed that a good
prediction of changes in frequencies and mode shapes can contribute to the determination of the location
and size of cracks.

Due to the importance of this problem, a large number of research works addressing various issues
associated with the dynamic behavior of beams weakened by a single crack are available in the literature.
The reader is referred to the survey papers by Wauer [1], Dimarogonas [2], and Doebling and coworkers
[3,4], where ample information about the accomplishments and the literature in this field can be found.
These survey papers also reveal the extreme scarcity of results and methodologies enabling one to investigate the dynamics of beams weakened by multiple transverse open cracks. In this contexts, with the exception of a few papers (see in this respect Refs. [5–11], where the dynamic behavior of beams weakened by
two and multiple transverse surface open cracks was investigated), the specialized literature is quite void of
research works addressing the problem of free vibration of anisotropic shear deformable composite beams
exhibiting multiple surface open cracks.
All these research works, as well as the ones accomplished in Refs. [12–18] strongly substantiate the fact
that the use of vibrational characteristics can constitute a viable crack detection technique, and consequently, a reliable basis for structural health monitoring.
The present research addresses the problem of vibration of prismatic beams of length L, width b and
height h, weakened by a sequence of surface open cracks, of uniform but different depths ai , located at the
arbitrary positions li ði ¼ 1; nÞ along the beam span, measured from the section Z ¼ 0 perpendicular to the
beam longitudinal axis.
It is assumed that the material of the beam is orthotropic, its principal axes of orthotropy being rotated
in the plane XZ by an angle a considered to be positive when is measured from the X axis in the counterclockwise direction (see Fig. 1).
In the equations governing the transverse free vibration, the effects related with transverse shear flexibility and rotatory inertia are also included.

Fig. 1. Geometry of a composite beam with multiple cracks.


O. Song et al. / Engineering Fracture Mechanics 70 (2003) 105–123

107

2. Governing equations
2.1. Shearable beam
The case of cantilevered prismatic beams weakened by an arbitrary sequence of part-through surface
open cracks of variable positions gi ð ‘i =LÞ, measured from the beam root Z ¼ 0, is considered (see Fig. 1).
In order to model the edge-notched structure, the entire beam can be conveniently divided into a number of

parts that are bordered by two consecutive cracks. In this context, St. Venant’s principle stipulating that the
stress field is influenced only in the region near to the crack is invoked. As was done in a number of papers
(see for example, Refs. [19–22]) also here, the discontinuity in the stiffness induced by the crack will be
modelled by a massless rotational spring of infinitesimal length whose stiffness is determined in accordance
with the principles of the Fracture Mechanics. As a result, the beam is converted to a continuous–discrete
model.
For the resulting system of differential equations, the boundary conditions should be prescribed at the
root section, g ð z=LÞ ¼ 0 (where the beam is assumed to be clamped), at the free section g ¼ 1, and at the
crack positions gi ð ‘i =LÞ:
Consequently, the governing equations obtained via Hamilton’s variational principle associated with
each continuous beam section, are given by for the case of shearable beams by:
a44 00
€ 0 ¼ 0 ði ¼ 1; n þ 1Þ;
ðU0ðiÞ þ h0yðiÞ Þ À Lb1 U
ð1aÞ
ðiÞ
L
a22 00
h À a44 ðU00 ðiÞ þ hyðiÞ Þ À b2 h€yðiÞ ¼ 0:
L2 yðiÞ
À À À À

ð1bÞ

The boundary conditions:
At the clamped root section, g ¼ 0:
U0ð1Þ ¼ hyð1Þ ¼ 0:

ð2a; bÞ

At the free edge, g ¼ 1:
U00 ðnþ1Þ þ hyðnþ1Þ ¼ 0

and

h0yðnþ1Þ ¼ 0;

ð2c; dÞ

At the crack location, g ¼ gi :
0
þ
h0yðiÞ ðgÀ
i Þ ¼ hyðiþ1Þ ðgi Þ;

ð2eÞ

0
À
þ
0
þ
hyðiÞ ðgÀ
i Þ þ U0ðiÞ ðgi Þ ¼ hyðiþ1Þ ðgi Þ þ U0ðiþ1Þ ðgi Þ;

ð2fÞ

þ
U0ðiÞ ðgÀ
i Þ ¼ U0ðiþ1Þ ðgi Þ;

ð2gÞ

À
KRi ½hyðiþ1Þ ðgþ
i Þ À hyðiÞ ðgi ފ ¼

a22 0
h
ðg Þ:
L yðiþ1Þ i

ð2hÞ

þ
Herein gÀ
i and gi identify the left and right sides of the cross-section g ¼ gi where the crack is located. The
conditions at the crack location supplied by Eqs. (2e)–(2h), express in succession the continuity of bending
moments, of shearing forces, of deflections and the jump of the rotation at the crack section, while KRi
denotes the stiffness of the rotational spring at section g ¼ gi . This quantity will be defined in the forthcoming developments.
In the previously displayed equations and boundary conditions, index i ð¼ 1; n þ 1Þ identifies the solutions associated with the various segments of the beam; U0 ðg; tÞ ð u0 ðz; tÞ=LÞ and hy ðz; tÞ denote the


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O. Song et al. / Engineering Fracture Mechanics 70 (2003) 105–123

dimensionless transverse deflection and elastic rotation of the normal about the y axis, respectively;
0
ðÁÞ  oðÁÞ=og; while a22 and a44 denote transverse bending and transverse shear stiffness, respectively.

Moreover, b1 and b2 denote reduced mass terms, while the term underscored by a dashed line stands for the
rotatory inertia effect.
2.2. Unshearable beams
In the case of the classical unshearable beam counterpart weakened by a sequence of cracks, the pertinent equations can be obtained from Eqs. (1a), (1b) and (2a)–(2h) in a straightforward way. To this end,
U00 ðiÞ þ hyðiÞ is eliminated in the two equations, a process that is followed by the consideration of
hyðiÞ ¼ ÀU00 ðiÞ . In this way one obtain the classical counterpart of Eqs. (1a), (1b) and (2a)–(2h) as:
Governing equation:
a22 0000
€ 0 ¼ 0:
€ 0 À b2 U
U þ b1 LU
0ðiÞ
ðiÞ
L3 0ðiÞ
À À À

ð3Þ

Boundary conditions:
At g ¼ 0:
U0ð1Þ ¼ U00 ð1Þ ¼ 0;

ð4a; bÞ

At g ¼ 1:
a22 000
€0
U
þ b2 U
¼ 0;

L2 0ðnþ1Þ À À 0Àðnþ1Þ
À
a22 00
U
¼ 0;
L 0ðnþ1Þ

ð4c; dÞ

and at the crack location g ¼ gi :
00
þ
U000ðiÞ ðgÀ
i Þ ¼ U0ðiþ1Þ ðgi Þ;

ð4eÞ

a22 000 À
€ 0 ðgÀ Þ ¼ a22 U 000 ðgþ Þ þ b2 U
€ 0 ðgþ Þ;
U ðg Þ þ b2 U
0ðiÞ
i
0ðiþ1Þ
i
0ðiþ1Þ
i
2
L2 0ðiÞ i
L

À À À À À
À À À À À

ð4fÞ

þ
U0ðiÞ ðgÀ
i Þ ¼ U0ðiþ1Þ ðgi Þ;

ð4gÞ

h
i
0
À
Þ
À
U
ðg
Þ
¼ a22 U000ðiþ1Þ ðgi Þ:
KRi L U00 ðiþ1Þ ðgþ
i
0i
i

ð4hÞ

The comparison of shearable (Eqs. (1a), (1b) and (2a)–(2h)), and of their unshearable counterpart Eq. (3)
and (4a)–(4h)) reveal that: (a) both governing equations systems feature the same order, namely four, and

as a result, in each of these cases two boundary conditions should be prescribed at each edge, g ¼ 0, 1, and
(b) in contrast to the shearable beam model, in the case of the unshearable beam counterpart, the rotatory
inertia terms are present in the boundary conditions at the free edge and at the crack location. However, as
is readily seen, when rotatory inertia terms are discarded, also in the shearable case, the boundary conditions would be free of such terms.


O. Song et al. / Engineering Fracture Mechanics 70 (2003) 105–123

109

3. Local flexibility of the beam induced by a transverse surface crack
3.1. General considerations
As is well known, a surface crack on a structural member introduces a local flexibility that is a function of
the crack length and depth, on material elastic constants and on the loading modes. The local flexibility
induced by a crack was studied within Griffith–Irwin theory (see Refs. [23,24]) who related the flexibility to
the stress intensity factor. The local flexibility coefficient Cij due to the crack can be determined from Paris’
equations (see Ref. [24]) as
Z
oui
o2
¼
J dAf ;
ð5Þ
Cij ¼
oPj oPi oPj Af
where J is the energy release rate, Af is the area of the crack section, ui are the additional displacements due
to the crack, and Pi are the corresponding loads.
The functional J was expressed in general form, in terms of stress intensity factors KIi , KIIi and KIIIi for
the three modes of fracture, where i denotes the independent forces acting on the beam (see Refs. [19–22]).
The additional strain energy of the beam due to the crack is

2
3
!2
!2
 Z

Z
N
N
N
N
X
X
X
X
4D1
KIi þ D2
KIIi þ D12 KIi
KIIi 5 dAf 
J dAf ;
Uc ¼
ð6Þ
Af

i¼1

i¼1

i¼1

i¼1

Af

where Af is the area of the crack, and KI and KII are the stress intensity factors for modes I and II of fracture
that result for every individual loading mode i, and the coefficients D1 , D2 and D12 are defined as


 22
 11
l1 þ l2
A
A
 11 Imðl1 l2 Þ:
Im
Imðl1 þ l2 Þ; D12 ¼ A
D1 ¼
ð7Þ
; D2 ¼
2
l1 l2
2
 22 , A
 11 , l1 and l2 these are supplied in Appendix A.
As concerns the elastic coefficients A
The stress intensity factors KI and KII for the crack in a composite beam are expressed as
pffiffiffiffiffiffi
Kj ¼ rj paFj ða=hÞYj ðnÞ ðj ¼ I; IIÞ;

ð8Þ

where rj is the stress in the each fracture mode, Fj ða=hÞ is the correction factor for the finite specimen size
and Yj ðnÞ is the correction factor for the anisotropic material (see Refs. [20,25,26]).
Replacement of (6)–(8) in (5), yields the additional flexibility of the composite beam weakened by a
transverse edge open crack of depth 
ai ð ai =hÞ; located at g ¼ gi along the beam span. Its expression is
o2 Uc 72pD1
2pD2
12pD12
T3 ;
¼
T1 þ
T þ
2 2
2
oPi2
hb2
hbLð1
À gi Þ
hL ð1 À gi Þ

ðiÞ
Cmm
¼

ð9Þ

where
T1 ¼

Z

ai


½
ai YI2 ðfÞFI2 ð
ai ފ d
ai ;

0

T3 ¼

Z

T2 ¼

Z

ai


½
ai YII2 ðfÞFII2 ðai ފ dai ;

0
ai


ð10Þ

½
ai YI ðfÞYII ðfÞFI ð
ai ÞFII ð
ai ފ d
ai :

0
ðiÞ

As a result, the local stiffness coefficient KR due to the crack is
ðiÞ

À1

ðiÞ
Þ :
KR ¼ ðCmm

ð11Þ


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O. Song et al. / Engineering Fracture Mechanics 70 (2003) 105–123

However, results not displayed here reveal that mode I is the predominant one, in the sense that contribution of fracture mode II to the predictions by mode I are lower than 0.1%. As a result, the subsequent
developments and numerical simulations are carried out within fracture mode I induced by a transverse
bending moment M.

3.2. Flexibility of cantilevered composite notched beams corresponding to fracture mode I
Within these conditions, the strain energy of the beam with a crack area Af is
Z
Z
2
Uc ¼
J dAf ¼
D1 KIM
dAf ;
Af

ð12Þ

Af

where KIM is the stress intensity factor for the crack opening mode, mode I, while D1 is expressed by the first
term of Eq. (7).
For slender beams featuring L=h P 4; KIM can be expressed as (see Refs. [19,22,25,26])
pffiffiffiffiffiffi
ð13Þ
KIM ¼ ð6M=bh2 Þ paYI ðfÞFIM ða=hÞ;
where the correction functions YI and FIM are expressed as (see Refs. [24,25]):
YI ðfÞ ¼ 1 þ 0:1ðf À 1Þ À 0:016ðf À 1Þ2 þ 0:002ðf À 1Þ3 ;

ð14aÞ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h
i
tan c=c
4

0:923 þ 0:199ð1 À sin cÞ ;
FIM ða=hÞ ¼
cos c

ð14bÞ

and

where
c ¼ pa=2h

and

f¼

pffiffiffiffiffiffiffiffiffiffi
E1 E2 pffiffiffiffiffiffiffiffiffiffiffi
À m12 m21 ;
2G12

ð14cÞ

I
a being the crack depth. As a result, the additional flexibility CMM
of the composite beam weakened by a
transverse edge open crack is expressed as
Z
72D1 p a 2
I
2


CMM
¼
ð
aÞ d
a;
ð15Þ
aYI ðfÞFIM
hb2
0

and the local stiffness coefficient KRI due to the crack is
À1

I
KRI ¼ ðCMM
Þ :

ð16Þ

where the index M indicates that the respective flexibility/stiffness coefficient corresponds to the beam acted
by the bending moment M.

4. Solution methodology
Laplace transform method is used to solve exactly the free vibration problem of cantilever beams
weakened by multiple surface cracks. Assuming synchronous motion, we represent the displacement
quantities associated with the various parts of the beam as:
 j ðgÞ; hj ðgފeixt
½U0ðjÞ ðg; tÞ; hyðjÞ ðg; tފ ¼ ½U
pffiffiffiffiffiffiffi

where i ¼ À1.

ðj ¼ 1; n þ 1Þ;

ð17Þ


O. Song et al. / Engineering Fracture Mechanics 70 (2003) 105–123

111

Replacement of Eq. (17) into Eqs. (1a) and (1b) yields:
 j ¼ 0;
 00 þ h0 þ x2 f1 U
U
j
j

ð18aÞ

 0 þ hj Þ þ x2 f3 hj ¼ 0;
h00j À f2 ðU
j

ð18bÞ

where
f1 ¼

b1 2

L;
a44

f2 ¼

a44 2
L;
a22

f3 ¼

b2 2
L:
a22

ð19a–cÞ

Similarly, the boundary conditions become:
At g ¼ 0:
 1 ¼ h1 ¼ 0;
U

ð20a; bÞ

At g ¼ 1:
 0 þ hnþ1 ¼ 0;
U
nþ1

h0nþ1 ¼ 0;

ð21a; bÞ

and at g ¼ gj :
h0j ¼ h0jþ1 ;

ð21c; dÞ

 0 ¼ hjþ1 þ U
0 ;
hj þ U
j
jþ1
a
22
0
hjþ1 À hj ¼
h :
KRj L jþ1

ð21e; fÞ

 jþ1 ;
j ¼ U
U

Applying (one-sided) Laplace transform to the governing equations associated with the various segments
of the beam, and using the boundary conditions at g ¼ 0 one obtains
#



 " 0
 ð0Þ
g11 ðsÞ g12 ðsÞ X1 ðsÞ
U
1
¼ 0
for j ¼ 1;
ð22Þ
g21 ðsÞ g22 ðsÞ Y1 ðsÞ
h1 ð0Þ
and


g11 ðsÞ g12 ðsÞ
g21 ðsÞ g22 ðsÞ




 
F ðgj Þ
Xj ðsÞ
¼
Gðgj Þ
Yj ðsÞ

for j ¼ 2; n:

ð23Þ

 j ðgÞ and hj ðgÞ, respectively, i.e.
In these equations Xj ðsÞ and Yj ðsÞ stand for the Laplace transforms of U
Z 1
 j ðgÞ; hj ðgފ ¼
 j ðgÞ; hj ðgފ dg; while ðj ¼ 1; n þ 1Þ;
½Xj ðsÞ; Yj ðsފ ¼ L½U
eÀsg ½U
ð24Þ
0

g11 ¼ s2 þ x2 f1 ;

g12 ¼ s;

g21 ¼ Àf2 s;

g22 ¼ s2 À f2 þ x2 f3 ;

ð25a–eÞ

 j ðgj Þ þ U
 0 ðgj Þ þ hj ðgj Þ;
F ðgj Þ ¼ sU
j
 j ðgj Þ þ shj ðgj Þ þ h0 ðgj Þ;
Gðgj Þ ¼ Àf2 U
j
s being the Laplace transform variable, while L stands for Laplace transform operator.
In the process of applying Laplace transformation to Eqs. 13, (14a)–(14c), (15), and (16), the two

boundary conditions at g ¼ 0 are used.


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O. Song et al. / Engineering Fracture Mechanics 70 (2003) 105–123

Solving Eqs. (17), (18a) and (18b) for Xj ðsÞ, and Yj ðsÞ, inverting these in the physical space as to obtain,
 j ðgÞ and hj ðgÞ, and enforcing the boundary and continuity conditions at g ¼ 1 and g ¼ gj ,
respectively, U
respectively, the following eigenvalue problem expressed in matrix form is obtained
½KŠfug ¼ 0:

ð26Þ

In Eq. (21a,b), (21c,d) and (21e,f),
½Kij Š ¼ ½KŠð4nþ2ÞÂð4nþ2Þ :

ð27Þ

The entries of K contain the eigenfrequencies, while
 0 ; h0 ; U
 2 ; h2 ; U
 0 ; h0 ; . . . ; U
 i ; hi ; U
 0 ; h0 ; . . . ; U
 nþ1 ; hnþ1 ; U
 0 ; h0 g;
fugTð4nþ2ÞÂ1  fU
1

2
i
nþ1
1
2
i
nþ1

ð28Þ

is the eigenvector.
The condition of non-triviality of fug, requires that det½KŠ ¼ 0, wherefrom, the eigenfrequencies are
obtained.

5. Numerical simulations
5.1. Validation of the present approach
It is important first, to validate the present approach of the problem, by comparing the actual predictions
with the ones obtained in the literature via other methods. To this end, the case considered in Ref. [8] will be
adopted here.
It consists of a cantilevered beam of L ¼ 0:8 m and square cross-section, b ¼ h ¼ 0:02 m, modelled within
Euler–Bernoulli theory. The material properties corresponds to an isotropic material of Young’s modulus
E ¼ 2:1 Â 1011 N/m2 , Poisson’s ratio m ¼ 0:35, its material density being q ¼ 7800 kg/m3 . Two scenarios
addressed in Ref. [8] referred here to as Cases 1 and 2, are considered here.
Case 1 is associated with a single crack of depth a1 ¼ 2 mm located at l1 ¼ 0:12 m from the beam root,
while Case 2 is associated with two cracks of depth and location, in their succession as: a1 ¼ 2 mm,
l1 ¼ 0:12 m and a2 ¼ 3 mm, l2 ¼ 0:4 m.
The results of the comparisons are summarized in Tables 1 and 2.
The results reveal an excellent agreement of predictions.
Table 1
Comparison of natural frequencies for a beam with one crack, Case 1

Natural frequency (Hz)

Ref. [8]

Present paper

Percentage difference
w.r.t. Ref. [8]

x1
x2
x3

26.1231
164.0921
459.6028

26.1015
163.5959
456.3634

0.0827
0.3024
0.7048

Table 2
Comparison of natural frequencies for a beam with two crack, Case 2
Natural frequency (Hz)

Ref. [8]

Present paper

Percentage difference
w.r.t. Ref. [8]

x1
x2
x3

26.0954
163.3221
459.6011

26.0694
162.7112
456.3611

0.0996
0.3740
0.7049


O. Song et al. / Engineering Fracture Mechanics 70 (2003) 105–123

113

5.2. Results for a composite shearable beam weakended by a single/multiple cracks of the same depth
5.2.1. Case A: Ef =Em ¼ 100
The numerical illustrations are carried out for a cantilever beam featuring the same geometrical characteristics and material properties as the ones considered in Ref. [22]. Due to the complexity of the problem,

acquirement of a closed form solution is precluded.
As supplied in Ref. [27], the properties of graphite–fiber reinforced polyimide materials used in the
present numerical simulations, in terms of those of fibers and matrix, identified by the index f and m, respectively, are:
Em ¼ 2:756 GPa; Ef ¼ 275:6 GPa;
mm ¼ 0:33; mf ¼ 0:2;
Gm ¼ 1:036 GPa;
qm ¼ 1600 kg=m3 ;

Gf ¼ 114:8 GPa;

ð29aÞ

qf ¼ 1900 kg=m3 :

In addition, consistent with Ref. [22], the following geometrical beam characteristics are adopted:
L ¼ 1 m;

h ¼ 0:025 m;

b ¼ 0:05 m:

ð29bÞ

In Fig. 2(a)–(c) the variation of the first three natural frequencies are represented in succession as a function
of the unicrack position and ply-angle. In all these cases the crack depths is ai ð ai =hÞ ¼ 0:2 and volume
fraction of fibers is vf ¼ 0:5: The results displayed in these three-dimensional (3-D) graphs reveal that,
corresponding to the ply-angle a ¼ 0, the natural frequencies are the lowest ones and are insensitive to the

Fig. 2. (a) 3-D plot depicting the variation of the first natural frequency as a function of the dimensionless crack position and ply-angle
(vf ¼ 0:5). (b) The counterpart of (a) for the second natural frequency. (c) The counterpart of (a) for the third natural frequency.

114

O. Song et al. / Engineering Fracture Mechanics 70 (2003) 105–123

variation of the crack location. However, the increase of the ply-angle is accompanied by a significant
increase of natural frequencies. In this sense, when the crack is located closer to the beam root, the fundamental eigenfrequency is much lower than in the case of the crack located toward the beam tip.
As concerns the implications of the position of the unicrack coupled with that of the ply-angle on the
second and third eigenfrequencies, these appear more complex than in the case of the fundamental frequency. In this sense, for specific locations of the crack, decreases of the eigenfrequencies are occurring. As
it will be seen later, the largest decreases of natural frequencies are experienced when the crack is located at
positions of maximum curvature of the respective mode shapes.
On the other hand, when the crack is located at points of minimum curvature of mode shapes, the influence of the crack upon the natural frequencies is much smaller. In this context, one can say that in
contrast to the trend of variation of the first natural frequency as a function of the crack position, for the
second and third natural frequencies their variations depend strongly on how close the crack is to the nodal
or antinodal points of the respective mode shape. These conclusions are in perfect agreement to those
outlined by Krawczuk and Ostachowicz [22].
This conclusion is enforced further, in the case of multicracks, (see Fig. 3(a)–(c) that should be considered
in conjunction with Fig. 4(a)–(c) that display the corresponding eigenmodes. Indeed, in these figures the
case of three cracks located according to the scenarios labelled as E, F and G, are considered. For the first

Fig. 3. (a) Variation of the first natural frequency vs. ply-angle for the case of three cracks distributed differently, as indicated
(
ai  
a ¼ 0:2, vf ¼ 0:5). (b) The counterpart of (a) for the second natural frequency. (c) The counterpart of (a) for the third natural
frequency.


O. Song et al. / Engineering Fracture Mechanics 70 (2003) 105–123

115

Fig. 4. (a) Variation of the normalized first mode shape that corresponds to the case in Fig. 3(a), and a ¼ 90°. (b) Variation of the
normalized second mode shape that corresponds to the case in Fig. 3(b), and a ¼ 90°. (c) Variation of the normalized third mode shape
that corresponds to the case in Fig. 3(c), and a ¼ 90°.

natural frequency, it becomes clear that when the cracks are remote from the root section, its value continuously increases. In this sense, it is readily seen that the fundamental frequency of case G is larger than in
cases E and F, while the fundamental frequency corresponding to case F is smaller than that corresponding
to case G, and larger than that of case E.
However, a change from this rule intervenes for the second and third modal frequencies, for the cases E,
F, and G. In this sense, the results reveal, for example, that for the case E, the second frequency is not the
lowest one, but the one associated with case F, while in the case of the third modal frequency, the minimum
one is that associated with case G. This change in trend can easily be explained by examining the variation
of the corresponding mode shapes.
In this sense, from Fig. 4(b) is readily seen that for case F, in the region of the location of the three cracks,
the maximum curvature of the corresponding mode shape is experienced. The same conclusion can be
obtained when examining the third mode shape associated with case G, that exhibits the minimum natural
frequency.
In Fig. 5(a)–(c) the first three natural bending frequencies are displayed in succession, as a function of the
ply-angle, number of cracks and crack location gi . In all these cases the crack depth is ai ð ai =hÞ ¼ 0:2 and
volume fraction of fibers, vf ¼ 0:5.


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Fig. 5. (a) Variation of the first natural frequency vs. ply-angle for n ¼ 2–5 cracks of equal depth (ai  a ¼ 0:2). The case of the noncracked beam is also displayed. (b) The counterpart of (a) for the second natural frequency. (c) The counterpart of (a) for the third
natural frequency.

In these figures, the cases of the uncracked beam, of two (n ¼ 2ðAÞ), three (n ¼ 3ðBÞ), four (n ¼ 4ðCÞ)
and five (n ¼ 5ðDÞ) cracks have been displayed. The location of the cracks gi , is identified by the numbers
that associate the letters A, B, C and D. The results displayed in Fig. 5(a)–(c) reveal that in the range of plyangles 0 < a K 30°, for the relatively small crack depth considered here, the fundamental frequency is
practically insensitive to the number of cracks and their location. In addition, with the increase of the plyangle, for a > 30°, a significant increase of the fundamental frequency is experienced. However, a relatively
low sensitivity to the location of cracks is manifested.
The results also reveal that for ab30°, the fundamental frequency of the beam without cracks does not
differ too much of that corresponding to the cracked beam. However, with the increase of the ply-angle, a
big decay of the eigenfrequency corresponding to the cracked beam as compared to that of the uncracked
one is experienced.
In Fig. 6(a)–(c) there are depicted the variations of the first three normalized bending mode shapes for the
cases of the uncracked beam, as well as those of beams weakened by one until five cracks. In the case of the
unicrack (case labelled as U), its location is at g1 ¼ 0:1.
In addition to the positions of the cracks, the natural frequencies associated to each of the considered
cases are also supplied. The displayed results reveal that the increase of the number of cracks results in a


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117

Fig. 6. (a) Variation of the first normalized bending mode shape as a function of the number of cracks (a ¼ 0:2, a ¼ 90°).
(b) Counterpart of (a) for the second normalized mode shape. (c) Counterpart of (a) for the third normalized mode shape.

decrease of eigenfrequencies, decrease that is exacerbated by the increase of the mode number. From these
plots the conclusions outlined in connection with the trend of variation of eigenfrequencies as a function of
the location of the crack, as resulting from Fig. 2(b) and (c), can be restated also in this case.
Indeed from these figures, the maximum curvature of the second mode shape occurs at g ffi 0:5, while for
the third mode shape at g ffi 0:3 and 0.7. As revealed in Fig. 2(b) and (c), for a location of the crack in these
sections, the most severe decay of the natural frequencies is occurring.
In Fig. 7(a)–(c) there are depictions of the variation of the first three natural frequencies, in succession, as

a function of the fiber volume fraction and of the number of cracks, including the case of the uncracked
beam. For all these cases, the depth of the cracks ai  a ¼ 0:2 and the ply-angle a ¼ 90°. It clearly appears
from these plots the dramatic effect played by the increase of the fiber volume fraction on the eigenfrequencies.
In Fig. 8(a)–(c) there are presented the effects of the crack depth considered in conjunction with that of
the fiber volume fraction on the natural frequencies, in the case of the existence of five cracks (Case D) and
of a fixed ply-angle, a ¼ 90°.
The results emerging from these plots reveal that the increase of the crack depth yielding a decay of the
natural frequency, can hardly be compensated by the increase of the fiber volume fraction.


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Fig. 7. (a) Variation of the fundamental eigenfrequency vs. fiber volume fraction for various number of cracks (a ¼ 0:2, a ¼ 90°).
(b) The counterpart of (a) for the second natural frequency. (c) The counterpart of (a) for the third natural frequency.

5.2.2. Case B: Ef =Em < 100
In order to get an understanding on the effects of the ratio Ef =Em on natural frequencies and natural
modes, in addition to the previous case, that of the beam whose constituent material is E-glass/epoxy is
considered.
For this case Ef ¼ 69 GPa, Em ¼ 3:8 GPa, Ef =Em ¼ 18:2, mf ¼ 0:2, mm ¼ 0:36. For the purpose of capturing the effects of the ratio Ef =Em the remaining geometrical and physical characteristies are considered to
be the same to those listed in (29b).
In this context, Fig. 9(a)–(c) display in succession, the variation of the first three natural frequencies, as a
function of the ply-angle a, for the three cases involving the different locations of the three cracks, labelled
as E, F and G. The comparison of Fig. 9(a)–(c) with their counterparts, Fig. 3(a)–(c), obtained for
Ef =Em ¼ 100 is of interest. As it clearly appears, in the case Ef =Em ¼ 18:2, the frequencies are much lower
than their counterparts that are experienced when Ef =Em ¼ 100. It can also be remarked that in the range
35° J a° J 0 in the former case, the effect of the location of cracks on natural frequencies is more prominent than in the latter.
However, for a J 35°, the influence of the ply-angle on natural frequencies corresponding to the three

crack location scenarios, E, F and G, is more evident in the case Ef =Em ¼ 100 than for Ef =Em ¼ 18:2.


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119

Fig. 8. (a) Variation of the fundamental eigenfrequency vs. fiber volume fraction for different crack depths (n ¼ 5ðDÞ, a ¼ 90°). (b) The
counterpart of (a) for the second natural frequency. (c) The counterpart of (a) for the third natural frequency.

Moreover, the results not displayed here reveal that when Ef =Em ¼ 18:2, and for a ¼ 0, the effects of the
location of cracks on natural modes is less visible than when a ¼ 90°, and, in both these cases, the effect of
crack location is much lighter than in their counterpart cases, obtained for Ef =Em ¼ 100.
5.2.3. Case C: influence of the crack depth
In all previous cases, an unique crack depth was considered. Herein, four scenarios that involve various
crack depths of the beam experiencing three cracks located invariably at gi ¼ 0:1; 0.3 and 0.5 are compared.
In Fig. 10(a)–(c) there is displayed the variation of the first three natural frequencies as a function of the
ply-angle a, while in Fig. 11(a)–(c), there are presented the natural mode counterparts, depicted for a fixed
value of the ply-angle, a ¼ 90°.
The material and geometrical characteristics of the beam as considered in these simulations correspond
to those listed in (29a) and (29b). The crack positions and depths, for each of the considered four scenarios
are identified by the sequence gi ð
ai Þ.
From these plots, it clearly appears that the lowest fundamental frequency is experienced when the
deepest cracks are located closer to the beam root, and increase when the deepest cracks are more remote
from the beam root. It is also clearly seen that in the case of the worst scenario, yielding the smallest
fundamental frequency, the use of the tailoring technique via the variation of the ply-angle as to increase
the fundamental frequency should be used with caution.
However, the trend of variation of higher mode natural frequencies with that of crack depth and their
location does not follow that one manifested by the fundamental frequency. The more complex variation

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Fig. 9. (a) Counterpart of Fig. 3(a) for the case of E-glass/epoxy composite beam (Ef =Em ¼ 18:2Þ, mf ¼ 0:5. In the remaining, all the
beam characteristies are similar to those in Fig. 3(a). (b) The counterpart of Fig. 3(b) for E-glass/epoxy composite beam. (c) The
counterpart of Fig. 3(c) for E-glass/epoxy composite beam.

trend experienced by the higher mode natural frequencies can be explained by examining the variation of
the associated mode shapes, in the sense already explained in the case of the multiple cracks featuring the
same depth. For the sake of completion, the associated mode shapes determined for a ¼ 90° are supplied in
Fig. 11(a)–(c).

6. Conclusions
An analytical methodology enabling one to investigate the free vibration characteristics of anisotropic
beams weakened by a sequence of surface open cracks was presented.
Results that reflect the implications of multiple cracks of equal or different depths on vibration characteristics of composite beams encompassing a number of non-classical effects, such as transverse shear,
anisotropy, and rotatory inertia have been displayed, and pertinent conclusions on the implications of
location of cracks and depth, fiber volume fraction, number of cracks, ply-angle of the material on the
eigenvibration characteristics have been outlined. It should be emphasized that to the best of the authors’
knowledge this paper represents the first work addressing, in a so broad context, the issue of vibration of
anisotropic shearable beams featuring multiple surface cracks.


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121

Fig. 10. (a) Variation of the first natural frequency vs. ply-angle for the case of three cracks featuring various depths. In the inset, gi
identifies the crack locations while (ai ) their depth. The beam characteristies are the ones supplied in (29a) and (29b). (b) The
counterpart of Fig. 10(a) for the second natural frequency. (c) The counterpart of Fig. 10(a) for the third natural frequency.

Appendix A. Expressions of the quantities intervening in Eq. (7)
l1 and l2 are two non-conjugate roots of the characteristic equations selected to correspond to those
roots with positive imaginary parts (see Refs. [28,29])
 16 l3 þ ð2A
 12 þ A
 66 Þl2 À 2A
 26 l þ A
 22 ¼ 0:
 11 l4 À 2A
A

ðA:1Þ

In the case when the geometrical axes X–Z coincide with the material principal axes, 1 and 2 of the or ij ¼ Aij and in addition A
 16 ¼ A
 26 ¼ 0.
thotropic material i.e. when the ply-angle a ¼ 0, A
The constants Aij are related to the mechanical properties of the material by


1
1
2 E22
A11 ¼
1 À m12
ð1 À m223 Þ

; A22 ¼
E11
E22
E11
m12
1
ðA:2Þ
ð1 þ m23 Þ; A66 ¼
;
A12 ¼ À
G12
E11
1
1
; A55 ¼
:
A44 ¼
G23
G13


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O. Song et al. / Engineering Fracture Mechanics 70 (2003) 105–123

Fig. 11. (a) First natural bending mode shape, associated to the case in Fig. 10(a) as a function of the crack depth (a ¼ 90°). (b) Second
natural bending mode shape. (c) Third natural bending mode shape.

In these equations the index 1 and 2 correspond to the direction of fibers and to that normal to the fiber
directions, respectively, while index 3 to a direction perpendicular to the plane of the fibers.

On the other hand, for given mechanical characteristics of the fiber and matrix and given volume fraction
of fibers, the equivalent material properties of the composite i.e., the Young’s shear moduli and Poisson’s
ratios intervening in Eq. (A.2) can be determined. In terms of these equations the expressions of Aij for any
value of the ply-angle can be determined (see for example Refs. [27,30]).

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