Dynamics Of Anisotropic Composite Cantilevers Weakened By Multiple Transverse Open Cracks

Dynamics of anisotropic composite cantilevers weakenedby multiple transverse open cracks Blacksburg, VA 24061-0219, USA Received 7 May 2001; received in revised form 14 November 2001; ac

Dynamics of anisotropic composite cantilevers weakened

by multiple transverse open cracks

Blacksburg, VA 24061-0219, USA 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 pa-rameters 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

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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 inves-tigate the dynamics of beams weakened by multiple transverse open cracks In this contexts, with the ex-ception 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 conse-quently, 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 liði 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 coun-terclockwise direction (see Fig 1)

In the equations governing the transverse free vibration, the effects related with transverse shear flexi-bility and rotatory inertia are also included

Fig 1 Geometry of a composite beam with multiple cracks.

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 gið ‘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, gð z=LÞ ¼ 0 (where the beam is assumed to be clamped), at the free section g ¼ 1, and at the crack positions gið ‘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

L ðU00

0 ðiÞþ h0yðiÞÞ  Lb1U€0

a22

L2h00y

ðiÞ a44ðU0

0ðiÞþ hy ðiÞÞ  b2h€yðiÞ¼ 0

   

The boundary conditions:

At the clamped root section, g¼ 0:

At the free edge, g¼ 1:

U00

ðnþ1Þþ hyðnþ1Þ ¼ 0 and h0y

At the crack location, g¼ gi:

h0y

ðiÞðg

iÞ ¼ h0yðiþ1Þðgþ

hyðiÞðgiÞ þ U0ðiÞ0 ðgiÞ ¼ hyðiþ1Þðgþi Þ þ U00ðiþ1Þðgþi Þ; ð2fÞ

U0ðiÞðg

iÞ ¼ U0ðiþ1Þðgþ

KRi½hy ðiþ1Þðgþ

i Þ  hy ðiÞðg

iÞ ¼a22

Herein g

i and gþ

i identify the left and right sides of the cross-section g¼ giwhere 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 KR i

denotes the stiffness of the rotational spring at section g¼ gi This quantity will be defined in the forth-coming developments

In the previously displayed equations and boundary conditions, index ið¼ 1; n þ 1Þ identifies the solu-tions associated with the various segments of the beam; Uðg; tÞ ð u ðz; tÞ=LÞ and hðz; tÞ denote the

dimensionless transverse deflection and elastic rotation of the normal about the y axis, respectively;

0

22 and a44 denote transverse bending and transverse shear stiffness, respectively Moreover, b1and b2denote 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 perti-nent equations can be obtained from Eqs (1a), (1b) and (2a)–(2h) in a straightforward way To this end,

U0

0 ðiÞþ hy ðiÞ is eliminated in the two equations, a process that is followed by the consideration of

hyðiÞ¼ U0

0 ðiÞ In this way one obtain the classical counterpart of Eqs (1a), (1b) and (2a)–(2h) as:

Governing equation:

a22

L3U00000

ðiÞþ b1L €U0ðiÞ b2U€0

0ðiÞ

  

Boundary conditions:

At g¼ 0:

U0ð1Þ¼ U0

At g¼ 1:

a22

L2U0000

ðnþ1Þþ b2U€0

0ðnþ1Þ

   

¼ 0;

a22

L U

00

0 ðnþ1Þ ¼ 0;

ð4c; dÞ

and at the crack location g¼ gi:

U000

ðiÞðg

iÞ ¼ U00

0 ðiþ1Þðgþ

a22

L2U0000

ðiÞðg

iÞ þ b2U€0

0 ðiÞðg

iÞ

    

¼a22

L2 U0000

ðiþ1Þðgþ

i Þ þ b2U€0

0 ðiþ1Þðgþ

i Þ

    

U0ðiÞðg

iÞ ¼ U0ðiþ1Þðgþ

KRiL U00

ðiþ1Þðgþ

iÞ

h

 U0

0 iðg

iÞi

¼ a22U000

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, g¼ 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 con-ditions would be free of such terms

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 Cijdue to the crack can be determined from Paris’ equations (see Ref [24]) as

Cij¼oui

oPj¼ o

2

oPioPj

Z

Af

where J is the energy release rate, Af is the area of the crack section, uiare 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 KI i, KII i and KIII i 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

Uc ¼

Z

Af

D1

XN i¼1

KI i

!2

2

XN i¼1

KII i

!2

þ D12

XN i¼1

KI i

XN i¼1

KII i

3

5 dAf

 Z

Af

JdAf

where Af is the area of the crack, and KIand KIIare 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

D1¼A22

2 Im

l1þ l2

l1l2

; D2¼A11

2 Imðl1þ l2Þ; D12¼ A11Imðl1l2Þ: ð7Þ

As concerns the elastic coefficients A22, A11, l1and l2these are supplied in Appendix A

The stress intensity factors KI and KII for the crack in a composite beam are expressed as

Kj¼ rj

ffiffiffiffiffiffi

pa

p

where rjis the stress in the each fracture mode, Fjða=hÞ is the correction factor for the finite specimen size and YjðnÞ 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

CðiÞmm¼o

2Uc

oP2

i

¼72pD1

hb2 T1þ 2pD2

hL2ð1  giÞ2T2þ

12pD12

where

T1¼

Z  i

0

½aiYI2ðfÞF2

IðaiÞ dai; T2¼

Z  i

0

½aiYII2ðfÞF2

IIðaiÞ dai;

T3¼

Z i

0

½aiYIðfÞYIIðfÞFIðaiÞFIIðaiÞ dai:

ð10Þ

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

However, results not displayed here reveal that mode I is the predominant one, in the sense that contri-bution 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

Uc¼

Z

A f

JdAf ¼

Z

A f

where KIMis the stress intensity factor for the crack opening mode, mode I, while D1is 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])

KIM¼ ð6M=bh2Þ ffiffiffiffiffiffi

pa

p

where the correction functions YIand FIM are expressed as (see Refs [24,25]):

YIðfÞ ¼ 1 þ 0:1ðf  1Þ  0:016ðf  1Þ2þ 0:002ðf  1Þ3; ð14aÞ and

FIMða=hÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tan c=c p cos c 0:923

h

þ 0:199ð1  sin cÞ4i

where

c¼ pa=2h and f¼

ffiffiffiffiffiffiffiffiffiffi

E1E2 p 2G12

 ffiffiffiffiffiffiffiffiffiffiffim

12m21

p

a being the crack depth As a result, the additional flexibility CI

MM of the composite beam weakened by a transverse edge open crack is expressed as

CMMI ¼72D1p

hb2

Z  0



aYI2ðfÞF2

and the local stiffness coefficient KI

R due to the crack is

KRI ¼ ðCI

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:

½U0ðjÞðg; tÞ; hyðjÞðg; tÞ ¼ ½ UjðgÞ; hjðgÞ eixt ðj ¼ 1; n þ 1Þ; ð17Þ where i¼ ffiffiffiffiffiffiffi

1

p

Replacement of Eq (17) into Eqs (1a) and (1b) yields:





h00j  f2ð Uj0þ hjÞ þ x2f3hj¼ 0; ð18bÞ where

f1¼ b1

a44L

2; f2¼a44

a22L

2; f3¼ b2

a22L

Similarly, the boundary conditions become:

At g¼ 0:



At g¼ 1:



Unþ10 þ hnþ1¼ 0; h0nþ1¼ 0; ð21a; bÞ and at g¼ gj:



Uj¼ Ujþ1; h0j¼ h0jþ1; ð21c; dÞ



hjþ Uj0¼ hjþ1þ Ujþ10 ;



hjþ1 hj¼ a22

KRjL



Applying (one-sided) Laplace transform to the governing equations associated with the various segments

of the beam, and using the boundary conditions at g¼ 0 one obtains

g11ðsÞ g12ðsÞ

g21ðsÞ g22ðsÞ

X1ðsÞ

Y1ðsÞ

¼ U

0

1ð0Þ



h01ð0Þ

" #

and

g11ðsÞ g12ðsÞ

g21ðsÞ g22ðsÞ

XjðsÞ

YjðsÞ

¼ FðgjÞ GðgjÞ

In these equations XjðsÞ and YjðsÞ stand for the Laplace transforms of UjðgÞ and hjðgÞ, respectively, i.e

½XjðsÞ; YjðsÞ ¼ L½ UjðgÞ; hjðgÞ ¼

Z 1 0

esg½ UjðgÞ; hjðgÞ dg; whileðj ¼ 1; n þ 1Þ; ð24Þ

g11¼ s2þ x2f1; g12¼ s; g21¼ f2s; g22¼ s2 f2þ x2f3; ð25a–eÞ

FðgjÞ ¼ s UjðgjÞ þ Uj0ðgjÞ þ hjðgjÞ;

GðgjÞ ¼ f2UjðgjÞ þ shjðgjÞ þ h0jðgjÞ;

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 g¼ 0 are used

Solving Eqs (17), (18a) and (18b) for XjðsÞ, and YjðsÞ, inverting these in the physical space as to obtain, respectively, UjðgÞ and hjðgÞ, and enforcing the boundary and continuity conditions at g ¼ 1 and g ¼ gj, respectively, the following eigenvalue problem expressed in matrix form is obtained

In Eq (21a,b), (21c,d) and (21e,f),

The entries of K contain the eigenfrequencies, while

fugTð4nþ2Þ1 f U10; h01; U2; h2; U20; h02; ; Ui; hi; Ui0; h0i; ; Unþ1; hnþ1; Unþ10 ; h0nþ1g; ð28Þ

is the eigenvector

The condition of non-triviality of fug, requires that det½K 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 L¼ 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

E¼ 2:1  1011 N/m2, Poisson’s ratio m¼ 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 a1¼ 2 mm located at l1¼ 0:12 m from the beam root, while Case 2 is associated with two cracks of depth and location, in their succession as: a1¼ 2 mm,

l1¼ 0:12 m and a2¼ 3 mm, l2¼ 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

w.r.t Ref [8]

Table 2

Comparison of natural frequencies for a beam with two crack, Case 2

w.r.t Ref [8]

5.2 Results for a composite shearable beam weakended by a single/multiple cracks of the same depth 5.2.1 Case A: Ef=Em¼ 100

The numerical illustrations are carried out for a cantilever beam featuring the same geometrical char-acteristics 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, re-spectively, are:

Em¼ 2:756 GPa; Ef ¼ 275:6 GPa;

mm¼ 0:33; mf ¼ 0:2;

Gm¼ 1:036 GPa; Gf ¼ 114:8 GPa;

qm¼ 1600 kg=m3; qf ¼ 1900 kg=m3:

ð29aÞ

In addition, consistent with Ref [22], the following geometrical beam characteristics are adopted:

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 a¼ 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

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 fun-damental 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 fre-quency 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 in-fluence 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

frequency.