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 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 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: librescu@vt.edu (L Librescu) 0013-7944/02/$ - see front matter Ó 2002 Elsevier Science Ltd All rights reserved PII: S 0 - 4 ( ) 0 2 - X 106 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 , located at the arbitrary positions li ði ¼ 1; nÞ along the beam span, measured from the section Z ¼ 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 Geometry of a composite beam with multiple cracks O Song et al / Engineering Fracture Mechanics 70 (2003) 105–123 107 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 (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 € ¼ 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 and h0yðnþ1Þ ¼ 0; ð2c; dÞ At the crack location, g ¼ gi : þ h0yðiÞ ðgÀ i Þ ¼ hyðiþ1Þ ðgi Þ; ð2eÞ À þ 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 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 108 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; 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 € ¼ 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 € ðgÀ Þ ¼ a22 U 000 ðgþ Þ þ b2 U € ðgþ Þ; U ðg Þ þ b2 U 0ðiÞ i 0ðiþ1Þ i 0ðiþ1Þ i 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 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 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 !2  Z  Z N N N N X X X X 4D1 KIi þ D2 KIIi þ D12 KIi KIIi 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 l1 l2  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  ð =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 oPi2 hb2 hbLð1 À gi Þ hL ð1 À gi Þ ðiÞ Cmm ¼ ð9Þ where T1 ¼ Z  ½ YI2 ðfÞFI2 ð ފ d ; T3 ¼ Z T2 ¼ Z  ½ YII2 ðfÞFII2 ðai ފ dai ;  ð10Þ ½ YI ðfÞYII ðfÞFI ð ÞFII ð ފ d : ðiÞ As a result, the local stiffness coefficient KR due to the crack is ðiÞ À1 ðiÞ Þ : KR ¼ ðCmm ð11Þ 110 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 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 þ 0:1ðf À 1Þ À 0:016ðf À 1Þ2 þ 0:002ðf À 1Þ3 ; ð14aÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i tan c=c 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 I  CMM ¼ ð aÞ d a; ð15Þ aYI ðfÞFIM hb2 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 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  þ hj Þ þ x2 f3 hj ¼ 0; h00j À f2 ðU j ð18bÞ where f1 ¼ b1 L; a44 f2 ¼ a44 L; a22 f3 ¼ b2 L: a22 ð19a–cÞ Similarly, the boundary conditions become: At g ¼ 0:  ¼ h1 ¼ 0; U ð20a; bÞ At g ¼ 1:  þ hnþ1 ¼ 0; U nþ1 h0nþ1 ¼ 0; ð21a; bÞ and at g ¼ gj : h0j ¼ h0jþ1 ; ð21c; dÞ  ¼ hjþ1 þ U 0 ; hj þ U j jþ1 a 22 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 one obtains #    "  ð0Þ g11 ðsÞ g12 ðsÞ X1 ðsÞ U 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  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 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  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 are used 112 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 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  ; h0 ; U  ; h2 ; U  ; h0 ; ; U  i ; hi ; U  ; h0 ; ; U  nþ1 ; hnþ1 ; U  ; h0 g; fugTð4nþ2ÞÂ1  fU i nþ1 i nþ1 ð28Þ is the eigenvector The condition of non-triviality of fug, requires that det½KŠ ¼ 0, wherefrom, the eigenfrequencies are obtained 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  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 and 2, are considered here Case is associated with a single crack of depth a1 ¼ mm located at l1 ¼ 0:12 m from the beam root, while Case is associated with two cracks of depth and location, in their succession as: a1 ¼ mm, l1 ¼ 0:12 m and a2 ¼ mm, l2 ¼ 0:4 m The results of the comparisons are summarized in Tables and The results reveal an excellent agreement of predictions Table Comparison of natural frequencies for a beam with one crack, Case 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 Comparison of natural frequencies for a beam with two crack, Case 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 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 =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 (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 (a) Variation of the first natural frequency vs ply-angle for the case of three cracks distributed differently, as indicated (   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 (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 =hÞ ¼ 0:2 and volume fraction of fibers, vf ¼ 0:5 116 O Song et al / Engineering Fracture Mechanics 70 (2003) 105–123 Fig (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 < 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 O Song et al / Engineering Fracture Mechanics 70 (2003) 105–123 117 Fig (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 118 O Song et al / Engineering Fracture Mechanics 70 (2003) 105–123 Fig (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 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 O Song et al / Engineering Fracture Mechanics 70 (2003) 105–123 119 Fig (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 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 120 O Song et al / Engineering Fracture Mechanics 70 (2003) 105–123 Fig (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) 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 O Song et al / Engineering Fracture Mechanics 70 (2003) 105–123 121 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, and of the or ij ¼ Aij and in addition A  16 ¼ A  26 ¼ 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 E22 A11 ¼ À m12 ð1 À m223 Þ ; A22 ¼ E11 E22 E11 m12 ðA:2Þ ð1 þ m23 Þ; A66 ¼ ; A12 ¼ À G12 E11 1 ; A55 ¼ : A44 ¼ G23 G13 122 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 and correspond to the direction of fibers and to that normal to the fiber directions, respectively, while index 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]) References [1] Wauer J On the dynamics of cracked rotors: a literature survey Appl Mech Rev 1990;43:13–7 [2] Dimarogonas AD 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