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Anisotropic damage model for an inclined crack in thick plate and sensitivity study for its detection
International Journal of Solids and Structures 41 (2004) 4321–4336 www.elsevier.com/locate/ijsolstr
Anisotropic damage model for an inclined crack in thick plate and sensitivity study for its detection D. Wu, S.S. Law
*
Department of Civil and Structural Engineering, Hong Kong Polytechnic University, Yuk Choi Road, Kowloon, Hong Kong Received 12 January 2004; received in revised form 27 February 2004 Available online 12 April 2004
Abstract Many approaches on modeling of cracks in structural members have been reported in the literatures. However, most of them are explicitly developed for the purpose of studying the changes in static and dynamic responses of the structure due to the crack damage, which is a forward problem mathematically. The use of these models is inconvenient or even impossible for detecting damage in structures from vibration measurements, which is usually an inverse problem. An anisotropic damage model is proposed for a thick plate element with an inclined crack. The cracked plate element is represented by an effective plate element with anisotropic material properties expressed in terms of the virgin material stiffness and a vector of damage variables. A sensitivity-based model updating approach is developed incorporating the proposed anisotropic model and the estimated uniform load surface curvature from vibration measurements to identify the damage variables. The orientation of the crack is directly indicated by the geometric parameter and the damage severity can be estimated from other variables representing the relative stiffness reduction along the local axes. The validity of the methodology is demonstrated by numerical examples and experiment results. 2004 Elsevier Ltd. All rights reserved. Keywords: Damage detection; Crack model; Thick plate; Sensitivity method; Uniform load surface; Curvature; Vibration; Finite element; Chebyshev polynomial
1. Introduction The presence of a crack in a structural member introduces a local flexibility affecting its static behavior and vibration response. Many efforts have been devoted by engineers on the modeling of the crack-induced flexibility and investigating its effect on the dynamic characteristics of the damaged structure. Dimarogonas (1996) has summarized these works into three categories, namely: continuous model, discrete–continuous model and discrete models, i.e. finite element models. Historically the earliest method to model a fatigue crack in a beam is proposed by Hetenyi (1937) for determining the static deflection of beams with non-uniform cross-section, in which a crack is represented
*
Corresponding author. Tel.: +852-27666062; fax: +852-23346389. E-mail address: [email protected] (S.S. Law).
0020-7683/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijsolstr.2004.03.001
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by additional external equivalent loads. This method is further developed by Kirsmer (1944), Thomson (1949), and Petroski and Glazik (1980) to study the vibration response of the beams or shells with fatigue cracks. However the lack of explicit relation between the size of the crack and the magnitude of its external equivalent load makes it impossible to have the sensitivity analysis with the parameters of the structural members. Another approach related to the continuous model divides the cracked members into sub-domains by the crack lines. Special boundary conditions along the crack line are introduced (Stahl and Keer, 1972; Aggarwala and Ariel, 1981; Lee, 1992) to connect the sub-domains. The principal limitation of these continuous models is the fact that their partial differential-based mathematical derivation can provide sufficiently accurate results only in the case when the structural members have a very simple geometry. In the discrete–continuous models, a crack in a structural member is represented by additional springlike elements. A system with two parts of the undamaged member jointed with specific spring elements is created to model the cracked member. Many special compliance matrices for the spring elements were studied by the laws of fracture mechanics (Ostachowicz and Krawczak, 2001). These methods can successfully be used for modeling cracks in one-dimensional members. However, In the case of two or threedimensional members, this approach is practically inapplicable due to the fact that the large number of equations required for the boundary conditions of the connecting spring elements is not available. Real structures are more complicated than the geometrically simple ones described by the analytically continuous or discrete–continuous models. Researchers hence started to use discrete models to study the cracked structures, in which the finite element method (FEM) is the most popular and commonly utilized. From the published literature on FEM-based model of a cracked plate, one may find mainly three groups of methods. The simplest model represents the crack as a reduction in the elastic modulus of the element at the crack position (Cawley and Adams, 1979) or a reduction in the cross-sectional area of the element (Baschmid et al., 1984). These models have been successfully used in damage localization of plate-like structures (Cornwell et al., 1999; Li et al., 2002). However, as these methods studied the approximate crack size and location at the element level, a very fine finite element mesh is required to avoid large error. Another method models a crack by separating the nodes of finite elements along the crack line (Zastrau, 1985). To properly model the singular characteristics of the stress and strain fields around the crack tip, a very dense mesh of finite elements or singular-shaped isoparametric elements (Shen and Pierre, 1990) is used to cover the crack tip area. Obviously, this method can model the cracked structure very well, but is not suitable or feasible for damage identification. This is due to the low computation efficiency associated with the large number of finite elements. Also a finite element mesh has to be constructed for one suspected location of damage in the plate at one time. In the third group of methods, a rectangular plate element with an open and depth-through crack parallel to the plate boundary is modeled (Qian et al., 1991; Krawczuk, 1993; Krawczak and Ostachowicz, 1994). The stiffness matrix of a cracked element is written as K ¼ TF 1 T T where F is the matrix representing the sum of flexibility of the non-cracked plate and the additional flexibility due to the crack, and T is a transformation matrix. However, as the derived stiffness matrix cannot be explicitly parameterized in terms of the damage variable(s) to indicate the location, orientation and the extend of the crack, it is still difficult to incorporate this model into the inverse problem of structural damage identification. All the above-mentioned crack models were explicitly proposed for the purpose of studying the static behavior or dynamic response of the cracked structure, which is a forward problem in mathematical sense. Generally it is not suitable to use them for solving the inverse problem of damage identification. Therefore, a damage detection oriented crack model for plate-like structure is in need. The earliest crack model oriented to damage identification is probably presented by Lee et al. (1997), although their work originally focuses on fracture mechanics and aims to derive a damage evolution equation of an elliptical through
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micro-crack. By restricting the aspect ratio of the elliptical hole s ¼ 0, Lee et al. (2003) further studied the effective stiffness model of a thin plate with line micro-cracks, and then developed a model updating technique to identify the damage size and orientation by using the frequency response functions measured from the damaged plate. Numerical examples were simulated for demonstration. However, the validity of the theory still requires the support from experimental evidences. According to the authors’ understanding, Lee’s theory is derived from an infinite plate containing a central crack subjected to uniform stress load. Although the method can be approximately used for a micro-crack away from the plate boundary, it is no longer valid for the case of a macro-crack and corrections must be made to take into account the finite dimensions of the plate and different load patterns. Such correction has been demonstrated in the previous work by the authors on identifying edge-parallel cracks in a thin plate from vibration measurements (Wu and Law, submitted for publication). The objectives of this paper include: (1) to introduce a new effective stiffness model for thick plate elements with an inclined line-crack, in which the damage scalar in Lee et al. (1997) model is expanded and replaced by a vector of damage variables; (2) to develop a sensitivity-based model updating method incorporating the proposed anisotropic model and the estimated uniform load surface curvature (ULSC) from vibration measurements to identify the damage variables; (3) to carry out numerical simulation and experimental studies to verify the validity of the proposed methodology.
2. Anisotropic stiffness model for thick plate with an inclined crack The problem considered in this section is a thick plate with a non-propagating, open crack, which inclines with an angle n between the crack line and the plate boundary along the x-axis. It is assumed that the crack is depth-through but narrow such that it does not change the mass of the plate. A small rectangular plate element is shown in Fig. 1(a) containing the crack centrally. An effective stiffness model of the cracked element is proposed such that the modeled plate has equivalent dynamic characteristics as the real cracked plate. A crack induces local flexibility in the plate. According to the continuum elastic stiffness theory (Lee et al., 1997), the cracked plate element would exhibit orthotropic stiffness properties compatible with the orientation of the crack line. Furthermore, based on the observation from the fracture mechanics point of view that the high stress intensity at the crack edge along the major axis direction would effectively reduce the stiffness in the minor axis direction. A rational supposition can be made that the crack will mainly affect the flexural stiffness normal to the crack line, while contributing relatively little effect on the plate stiffness parallel to the crack. This is different from Lee’s model in that the stiffness reduction along the major axis is m2 times that along the minor axis. In order to verify this supposition, the cracked plate element is represented by an effective element of continuum anisotropic material with the major axis of the material parallel to the crack line and the minor axis normal to the crack. Thus, under the plane stress condition, the constitutive stress–strain relationship of the cracked element can be written as x
x ξ
≅
z
y
ξ
z
a
b
y
Fig. 1. Plate element with a through crack and the effective stiffness model of anisotropic material.
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r ¼ Ce;
or
8 9 2 C11 < r1 = r2 ¼ 4 C12 : ; 0 s12
C12 C22 0
38 9 0 < e1 = 0 5 e2 : ; C66 c12
ð1Þ
m12 E1 m21 E2 where the components C11 ¼ 1mE121 m21 , C22 ¼ 1mE122 m21 , C12 ¼ 1m ¼ 1m and C66 ¼ G12 , in which the terms 12 m21 12 m21 E, G and m denote the Young’s modulus, the shear modulus and the Poisson ratio respectively, and the subscripts 1, 2 define the local Cartesian coordinates with the 1-axis along the crack line and the 2-axis normal to the crack line. Comparing Eq. (1) with the constitutive stiffness of the intact material, which is expressed as 2 3 E=ð1 m2 Þ mE=ð1 m2 Þ 0 C0 ¼ 4 mE=ð1 m2 Þ E=ð1 m2 Þ 0 5 ð2Þ 0 0 G
one can introduce a vector of damage variables to relate the isotropic stiffness of the undamaged material to anisotropically reduced stiffness terms of the effective continuum model for the cracked plate element, as E1 ¼ Ea;
m21 ¼ m;
E2 ¼ Eb;
G12 ¼ Gv
ð3Þ
where a, b and v denote the stiffness scaling factors due to the crack. Considering the relation m12 =E2 ¼ m21 =E1 , one gets m12 ¼ m21
E2 b ¼m a E1
ð4Þ
Substituting (3) and (4) into (1), one obtains 3 2 Ea2 mEab 0 7 6 a bm2 a bm2 7 6 7 6 mEab Eab 7 C¼6 0 7 6 a bm2 a bm2 7 6 4 Ev 5 0 0 2ð1 þ mÞ
ð5Þ
If the crack line is not parallel to the plate boundary, and the term n represents the angle between the xaxis and the crack line (or the major axis of the effective anisotropic material). A transformation matrix can be defined as ¼ Tr r
and
e ¼ T Te
ð6Þ
T
T ¼ f rx ry sxy g , e ¼ ex ey cxy , and where r 2 3 sin2 n 2 sin n cos n cos2 n T ¼ 4 sin2 n cos2 n 2 sin n cos n 5 sin n cos n sin n cos n cos2 n sin2 n
ð7Þ
Substituting (6) into (1), we have ¼ TCT Te ¼ Ce r
ð8Þ
For a thick plate in which a plane normal to mid-plane before deformation does not remain normal to the mid-plane any longer after deformation, the effect of transverse shear deformation should be taken into account. Hence the constitutive equation of the effective continuum model of the cracked plate relating to the transverse shearing is expressed as
D. Wu, S.S. Law / International Journal of Solids and Structures 41 (2004) 4321–4336
s ¼ Dc;
or
s13 s23
¼
G13 0
0 G23
c13 c23
4325
ð9Þ
in which G13 ¼ Gd13 and G23 ¼ Gd23 , where the ratio factors d13 and d23 scale the reduced stiffness due to the crack against the transverse shear stresses in 1–3 plane and 2–3 plane respectively. Considering the inclined angle between the crack line and plate boundary, the shear constitutive relation can be written as s ¼ SDS Tc ¼ Dc where s ¼ f sxz
T
syz g and c ¼
ð10Þ
cxz
cyz
T
denoting the transverse shear stress and strain respectively, and cos n sin n the transformation matrix S ¼ . sin n cos n
3. Identification of the damage variables According to the above section, one needs a vector of six damage variables, p ¼ f a b v d13 d23 n g, to characterize a thick plate element with an inclined through crack. The purpose of this section is to identify the variables using vibration measurements from the undamaged and damaged plates. To achieve this, the ULSC, which is proposed by the authors as a local damage indicator (Wu and Law, in press), is adopted here as reference modal data instead of using modal frequencies or mode shapes. The reported method to estimate the ULSC from vibration measurements is briefly outlined below for clarity with the subsequent formulations of damage identification, followed by the sensitivity analysis of the ULSC with respect to the damage variables. Finally, a model updating method is developed based on the sensitivity formulations to identify the unknown variables. 3.1. Uniform load surface curvature (ULSC) For a linear structural system with n degrees-of-freedom, its flexibility matrix can be expressed by the superposition of mass normalized modes /r , where /Tr M/r ¼ 1 ðr ¼ 1; . . . ; nÞ, as (Berman and Flannelly, 1971) F ¼
n X /r /Tr x2r r¼1
ð11Þ
where xr is the rth natural frequency. Physically, each element of the flexibility matrix, fkl , can be interpreted as the displacement at the kth DOF due to a unit load at lth DOF. Thus the deflection vector of a plate under uniform unit load can be evaluated by u¼F L
ð12Þ
f1; . . . ; 1gT1 n
is the unit vector representing the uniform load acting on the plate. The deflection where L ¼ component uk can be expressed as uk ¼
n X l¼1
fkl ¼
n X n X /r ðkÞ/r ðlÞ x2r r¼1 l¼1
ð13Þ
This modal based formulation of the structural deflection vector under uniform load was defined as Uniform Load Surface (ULS) by Zhang and Aktan (1998). An approach to estimate the ULS curvature of plate structures using the Chebyshev polynomial approximation has been proposed by the authors (Wu and Law, in press). This approach is adopted here and outlined as follows.
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The ULS function of a plate can be modeled by the Chebyshev polynomial in two variables: uðx; yÞ ¼
N X M X i¼1
cij Ti ðxÞTj ðyÞ
ð14Þ
j¼1
where Ti ðxÞ, Tj ðyÞ are the first kind Chebyshev polynomials, and N , M are their orders. It is assumed that there are Q measuring points on the rectangular mesh of the plate. The ULS value, uðxp ; yq Þ, can be obtained at all the measuring points from Eq. (13). When substituting them into Eq. (14) separately, one has a system of Q equations written in matrix form as ½T ðxp ÞT ðyq Þ Q P fcij gP 1 ¼ fuðxp ; yq ÞgP 1
ð15Þ
where P ¼ N M. The coefficient vector fcij g can then be solved by the least-squares technique 1
T
T
fcij gP 1 ¼ ð½T ðxp ÞT ðyq Þ Q P ½T ðxP ÞT ðyq Þ Q P Þ ½T ðxp ÞT ðyq Þ Q P fuðxp ; yq ÞgQ 1
ð16Þ
By making use of the orthogonal property of the Chebyshev polynomial, the curvature of the ULS is obtained as the second derivatives of the Chebyshev polynomials in Eq. (14) as ucxx ¼
N X M X i¼1
ucxy ¼
j¼1
N X M X i¼1
cij
j¼1
oTi2 ðxÞ Tj ðyÞ; ox2
oT 2 ðxÞ cij i 2 ox
ucyy ¼
N X M X i¼1
cij Ti ðxÞ
j¼1
oTj2 ðyÞ oy 2
oTj2 ðyÞ ; oy 2 ð17Þ
3.2. Sensitivity of ULSC to generic damage variables Consider a damaged plate divided into ne rectangular elements. The characteristic equation for this undamped structural dynamic system can be expressed as K/r x2r M/r ¼ 0
ð18Þ
It is assumed that crack damage only causes a reduction in local stiffness, while the mass matrix remains unchanged. The stiffness and mass matrix of the damaged plate is expressed as K ¼ K0 þ DK ¼
ne X
f ðpe Þke ;
M ¼ M0
ð19Þ
e¼1
in which ke is the element stiffness matrix of the intact plate, and pe denotes a generic damage variable (or vector) in a specific function f which affects the elemental stiffness. M0 and M are the mass matrices of the intact and damaged states respectively. Taking the derivative of (18) with respect to the parameter pe and noting the relations in Eq. (19), we have o/r oxr oK 2 ðK xr MÞ ¼ 2xr M ð20Þ /r ope ope ope The sensitivity of natural frequency and mode shapes with respect to the parameter pe can be calculated using Nelson’s method (1976) as oxr 1 T oK ¼ / / ope 2xr r ope r
ð21Þ
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4327
and o/r ¼ gr þ cr /r ; ope
where cr ¼ gTr M/r
ð22Þ
in which gr is a particular solution of Eq. (20) by searching the component with the largest absolute value in /r and enforcing its derivative to zero. To find the sensitivity of the ULSC, oucðx;yÞ , one needs to calculate the sensitivity of the ULS at the ope measuring points first. Taking the derivative of Eq. (13) with respect to the parameter pe we get m X n ouðxi ; yj Þ ouk X 1 o/kr o/lr 2 oxr ¼ ¼ / / / / þ ð23Þ ope x2r ope lr ope kr x3r opk kr lr ope r¼1 l¼1 Substituting Eqs. (21) and (22) into (23), the sensitivities of ULS at the sensor grid points are obtained. Further taking the derivative of Eq. (16) with respect to the parameter pe and substituting Eq. (23), the sensitivity of the Chebyshev coefficients is obtained as ocij ouðxi ; yj Þ T 1 T ¼ ð½T ðxi ÞT ðyj Þ Q P ½T ðxi ÞT ðyj Þ Q P Þ ½T ðxi ÞT ðyj Þ Q P ð24Þ ope ope P 1 Q 1 Finally the sensitivity of ULS curvature with respect to the stiffness parameter pe can be derived from Eq. (17) and calculated as N X M N X M oTj2 ðyÞ oucxx ðx; yÞ X ocij oTi2 ðxÞ oucyy ðx; yÞ X ocij ¼ T ðyÞ; ¼ T ðxÞ ð25aÞ j i oy 2 ope ope ope ox2 ope i¼1 j¼1 i¼1 j¼1 and N X M oucxy ðx; yÞ X ocij oTi ðxÞ oTj ðyÞ ¼ ope oy ope ox i¼1 j¼1
ð25bÞ
3.3. Sensitivity of effective stiffness to crack damage variables To formulate the stiffness matrix of the proposed effective continuum model of the cracked plate element, one needs to express the internal strains in terms of the nodal variables, which include the transverse deflection w and the rotations of the mid-plane about the x- and y-axes, i.e. hx and hy , respectively. These rotation variables are formed from the bending and transverse shear deformations as ow ow cxz ; hy ¼ cyz hx ¼ ð26Þ ox oy where cxz and cyz are the deformed angles arising from the transverse shear stress. The in-plane displacements are given as u ¼ zhx ðx; yÞ;
v ¼ zhy ðx; yÞ
ð27Þ
Taking the three displacement variables of each node independently, the displacement field of the plate element can be written in terms of the nodal displacements as nn nn nn X X X Hi ðx; yÞðhx Þi ; hy ¼ Hi ðx; yÞðhy Þi ; w ¼ Hi ðx; yÞwi ð28Þ hx ¼ i¼1
i¼1
i¼1
where Hi ðx; yÞ is the isoparametric shape function for the ith node and nn denotes the number of nodes of the plate element. Considering the relations in Eq. (27), the in-plane strains can be expressed in terms of the nodal variables as
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3
2
where
2
o 07 6 ox 7 6 6 o7 7 u ¼ z6 0 7 6 ox 7 v 6 5 4 o o ox oy
o 8 9 6 ox < ex = 6 6 ey 0 ¼6 :c ; 6 6 xy 4 o oy 2
oH1 6 ox 6 6 Bb ¼ 6 6 0 6 4 oH1 oy
0 oH1 oy oH1 ox
0
oHnn ox
0
0
0
oHnn oy
3 0 78 9 7 < hx = 7 07 7: hy ; ¼ zBb d 7 w 5 0
0 o oy o ox
ð29Þ
3 07 7 7 07 7 7 5 0
0 oHnn oy oHnn ox
T and d ¼ f hx;1 hy;1 w1 hx;nn hy;nn wnn g . From Eq. (26) the transverse shearing strains is expressed in terms of the nodal displacements as 2 3 o 8 9 1 0 < hx = 6 cxz ox 7 7 h y ¼ Bs d ð30Þ ¼6 4 o 5: ; cyz 0 1 w oy
where
2
6 H1 ½Bs ¼ 6 4 0
0 H1
oH1 ox oH1 oy
Hnn
0
0
Hnn
3 oHnn ox 7 7 oHnn 5 oy
According to the Reissner–Mindlin theory, one can formulate the stiffness matrix of the effective model of the cracked plate element by Z Z Z Z h3 ke ¼ BTb TCT T Bb dA þ jh BTs SDS T Bs dA ð31Þ 12 A A where h denotes the thickness of the plate, A denotes the plane area of the plate element, and Bb , Bs are the strain–displacement relationship matrices for the bending and transverse shear strains respectively, j is the shear energy correction factor of 5/6. Thus, the sensitivity of the elemental stiffness matrix with respect to the crack variables can be obtained as follows: 3 2 Eaða 2bm2 Þ Eb2 m3 07 6 ZZ 7 6 ða bm2 Þ2 ða bm2 Þ2 oke h3 7 T 6 2 2 ¼ BTb T 6 Eb2 m3 ð32aÞ 7T Bb dA Eb m 6 oa 12 07 A 2 5 4 ða bm2 Þ2 2 ða bm Þ 0 2
Ea2 m2 6 ZZ 6 ða bm2 Þ2 oke h3 T 6 ¼ Bb T 6 Ea2 m 6 ob 12 A 4 ða bm2 Þ2 0
0
0 3
Ea2 m ða bm2 Þ Ea2 ða
2
2 bm2 Þ
0
07 7 7 T 7T Bb dA 07 5 0
ð32bÞ
D. Wu, S.S. Law / International Journal of Solids and Structures 41 (2004) 4321–4336
oke h3 ¼ ov 12
Z Z
6
A
oke ¼ lh od13 oke h3 ¼ on 12
2
Z Z
0
0 BTb T 6 4 0
0 0
0 7 T 0 7T Bb dA 5 E 0 2ð1 þ mÞ
2
E BTs S 4 2ð1 þ mÞ A 0
Z Z
BTb
A
4329
3
oC Bb dA þ jt on
3 05 T S Bs dA; 0 Z Z
BTs A
ð32cÞ
oke ¼ lh od23
3 0 5S T Bs dA E BTs S 4 0 A 2ð1 þ mÞ
Z Z
2
oD Bs dA on
0
ð32dÞ
ð32eÞ
Substituting Eq. (32) into (20) and (21) to replace the sensitivity of stiffness matrix with respect to a generic oK parameter op , the sensitivity of ULSC to the crack variables is now ready to be calculated. e 3.4. Model updating procedure Consider an unknown damage case with single or multiple cracks producing changes in the vectors of T damage variables for all the suspected elements, written as Dpi ¼ f Dai Dbi Dvi Dd13;i Dd23;i Dni g ði ¼ 1; . . . ; neÞ, where ne denotes the number of suspected elements in the plate. With the small damage assumption, the damaged plate is still considered as a linear structural system, and the damage-induced changes in the ULSC of the plate can be expressed by the first order Taylor series approximation Ducxx ðxi ; yj Þ ¼ ucDxx ucxx ¼
oucxx oucxx oucxx Dp1 þ Dp2 þ þ Dpne op1 op2 opne
ð33Þ
Similar relations can be obtained for the curvature changes Ducyy ðxi ; yj Þ and Ducxy ðxi ; yj Þ. Provided the ULS curvature and its sensitivity are estimated at Q measuring points, one can repeatedly obtain the above relations for each point, and rearrange them in a matrix form to have 3 2 oucxx ðx1 ; y1 Þ oucxx ðx1 ; y1 Þ oucxx ðx1 ; y1 Þ 7 6 op1 op2 opne 9 8 7 6 7 6 > > Ducxx ðx1 ; y1 Þ > . . . . > 7 6 > > .. .. .. .. > > .. 7 6 > > > > 9 8 7 6 oucxx ðxQ ; yQ Þ oucxx ðxQ ; yQ Þ > > . > > oucxx ðxQ ; yQ Þ 7 Dp 6 > > > > > > 1 > > > 7> 6 Duc ðx ; y Þ > > xx Q Q > op1 op2 opne > > > > 7> 6 = = < < Dp 2 7 6 . .. .. .. .. 7 6 . ð34aÞ ¼ . . 7> . > > 6 . . . . > > > 7> 6 . > > > > > > > Ducyy ðxQ ; yQ Þ > 6 oucyy ðxQ ; yQ Þ oucyy ðxQ ; yQ Þ oucyy ðxQ ; yQ Þ 7> > > > 7: Dpne ; > 6 > > > > 7 6 op op op > > . 1 2 ne > > 7 6 . > > . > > 7 6 . . . . > > ; : 7 6 .. .. .. .. 7 6 Ducxy ðxQ ; yQ Þ 5 4 oucxy ðxQ ; yQ Þ oucxy ðxQ ; yQ Þ oucxy ðxQ ; yQ Þ op1 op2 opne or U fDpg ¼ fDucg
ð34bÞ
It should be noted that Eq. (34) comprises 6ne unknown variables and a total of neq ¼ 3 Q linear equations. To ensure a unique over-determined solution to be found, the condition neq P 6ne is required. The Singular Value Decomposition (SVD) with error-truncation technique (Ren et al., 2000) is adopted to
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solve the linearised equations. To include the non-linear effects from large magnitude damage or interaction between the variables, the equations are solved by an iterative algorithm as follows. n
e Step 1. Initialize the angle variable n0 ji¼1 ¼ 0 and the other variables to unity. Step 2. At the beginning of each iteration, construct the ULSC sensitivity matrix Uk for the kth iteration, and calculate the curvature change, fDucgk , due to damage. Step 3. Solve the increment of damage variables as, fDpgk ¼ Uk fDucgk , where Uk is the generalized inverse ne of Uk from SVD technique. Each solution is divided into two stages: (1) fix the angle variable nk ji¼1 and update the other variables; (2) fix the other variables and update the angle variable. ne Step 4. Normalize all the variables in fDpgk except the angle variable nk ji¼1 by fDpgk ¼ fDpgk k such that ne the condition lmin 6 Dbk ji¼1 6 lpos is satisfied, where k is a scale factor and lmin , lpos are the convergence limits as shown below. Variable bk is chosen as a criterion because it has the largest range of variation other than the angle variable nk . e e e Step 5. Evaluate the new angle variable as nk jni¼1 ¼ nk1 jni¼1 þ Dnk jni¼1 and the other variables as fpgk ¼
fpgk1 ð1 þ fDpgk Þ. If w1 ¼ jfDpgk j < w1 or w2 ¼
jfDucgkþ1 j jfDucgk j
< w2 , the solution is considered con-
verged. Otherwise go to Step 2 until the condition is satisfied. lpos lmin The scale factor k in Step 4 is calculated as k ¼ min minðDb ; , in which the range ½lmin ; lpos ne ne j Þ maxðDb j Þ k k i¼1
i¼1
is set to compromise the convergence speed and solution stability. A wider range could speed up the solution convergence, but for some cases, it may lead to a divergent solution, while a narrower range would have an opposite effect. For all the cases in this paper, lmin ¼ 0:5, lpos ¼ 0:25 and w1 ¼ w2 ¼ 0:001. 4. Numerical examples An aluminium plate with free boundary condition is studied in this section to validate the proposed method. The plate has the dimension of 400 mm · 300 mm · 15 mm. The whole intact plate is divided into 4 · 3 Mindlin/Reissner plate elements. An inclined through crack is then assumed to occur in one element with a length of 100 mm and an angle n ¼ 45 as shown in Fig. 2. The finite element mesh of the crack is modeled using an open-source finite element toolbox, OPENFEM, integrated with MATLAB. The natural frequencies and corresponding mode shapes of the plate at the 20 finite element nodes are ‘measured’ y
x
Fig. 2. Finite element model of the thick plate with an inclined crack.
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Table 1 Identified parameters of the inclined crack with the thick plate model Element no.
1a
1b
1v
1 d13
1 d23
n
1 2 3 4 5 6 7 8 9 10 11 12
0.0185 0.0069 0.0182 0.0096 0.0268 0.0315 0.0129 0.0137 0.0006 0.0047 0.0134 0.0185
0.0038 0.0277 0.0222 0.0053 0.0122 0.1486 0.0481 0.0275 0.0123 0.0268 0.0017 0.0009
0.0244 0.0003 0.0042 0.0061 0.006 0.0883 0.0121 0.0082 0.0056 0.0005 0.0224 0.0134
0.0073 0.014 0.0126 0.0254 0.0058 0.0266 0.0060 0.0202 0.0252 0.0006 0.0205 0.0114
0.025 0.0151 0.0213 0.0129 0.0391 0.1075 0.0057 0.0058 0.0025 0.0091 0.0163 0.0045
1.37 0.45 2.64 )4.61 6.12 44.57 )4.33 )1.42 7.81 16.24 2.19 )5.54
through eigenvalue analysis of the finite element model. The ULSC are then estimated from the modal data via the procedure described in Eqs. (13)–(17), and input into a MATLAB program to calculate their sensitivities. According to the proposed crack model for a thick plate, there are six unknown parameters to model each suspected cracked element, and totally there are 6ne ¼ 6 12 unknowns for all the suspected elements of the whole plate. To satisfy the over-determined condition for Eq. (34), the ULS curvatures are also estimated at the center node of each element besides the 20 finite element nodes so that the number of linear equations 3Q equals 96. All the parameters for the initial plate model are set to unity and the Poisson ratio m is fixed at 0.3 in calculating the initial curvature sensitivities from Eq. (25). The final damaged indices after seven iterations are listed in Table 1, in which the location of the damage is clearly detected and the inclined angle of the crack is accurately predicted. The relative flexural and shearing stiffness b and d23 due to the crack show clearly a larger reduction than the others, while a and d13 remain almost unchanged. This finding is consistent with the authors’ supposition that a crack mainly affects its stiffness normal to the crack line.
5. Experiment verification Three aluminium plate specimens with the same dimensions of 600 mm · 500 mm · 3 mm are named as plate A, B and C. Fig. 3 shows the experimental set-up for testing. A rectangular mesh of 7 · 6 measuring points is outlined on each plate. The intact plate is suspended from a rigid frame by two steel wires of 0.5 mm in diameter and 0.75 m in length to simulate the free boundary condition. An impulsive signal was applied on the plate by hitting with a dynamic hammer Model B&K 8202 at each measuring point, and the vibration response of the plate due to the impulse is collected by an accelerometer Model B&K 4370 as shown in the figure. Both the input and output signals are amplified and input into a commercial modal testing and analysis system DASP2000. The natural frequencies and corresponding mode shapes of the plate at the rectangular mesh are then extracted through a Multi-Input-Single-Output transfer function analysis. An artificial crack is then cut in each specimen as shown in Fig. 4(a). To verify the proposed crack model and the identification method with cases of different crack lengths and orientations, a scheme of crack cutting is formulated and listed in Table 2. The crack in the first state is 0.5 mm wide, and it is 0.3 mm wide for the other two states. After each crack cutting exercise, the above hammer test is repeated to obtain the modal data of the plate. The first five natural frequencies of the plate specimens in different damage states are listed in Table 3.
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Fig. 3. Diagram of the experimental system.
Fig. 4. (a) Artificial cracks (State I). (b) Finite element meshes to model the cracked plates A, B and C.
As the thickness of the test plates are very small relative to the other two dimension of the plates, the transverse shear deformations are neglected so that the vector of damage variables consists of only four components for each plate element, i.e. pi ¼ f ai bi vi ni gT ði ¼ 1; . . . ; neÞ. The initial finite element model of the intact plate with one 9-nodes plate element and 26 number of 4-nodes elements is shown in Fig. 5, with the parameter vectors for all the elements initialized to unity. The crack identification procedure is adopted here with an over-determined equation system characterized by a sensitivity matrix of ULS
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Table 2 Scheme of artificial crack cutting for the three specimens Crack information
Plate A
Plate B
Plate C
Center (x; y) Orientation (n)
(200; 200) 45
(200; 200) 60
(200; 200) 90
Length (mm) State I State II State III
80 140 200
80 120 160
80 120 160
Table 3 Natural frequencies (Hz) of the plate specimens in different crack states Mode no.
1
2
3
4
5
Plate A Intact State I State II State III
33.854 33.714 33.421 32.876
44.796 44.675 44.454 44.043
69.250 68.655 67.541 65.941
83.015 82.965 82.861 82.484
94.795 94.810 94.739 94.497
Plate B Intact State I State II State III
33.966 33.817 33.666 33.402
45.035 44.772 44.446 43.976
68.703 68.348 67.931 67.342
82.829 82.782 82.709 82.582
94.750 94.772 94.685 94.508
Plate C Intact State I State II State III
31.379 31.372 31.372 31.361
41.578 41.228 40.812 40.133
65.859 65.571 65.238 64.843
78.105 78.040 77.991 77.870
88.263 88.245 88.194 88.028
curvature sized 3Q 4ne, where Q ¼ 7 6 and ne ¼ 27. The ULS curvatures for both the test plate and initial finite element model are estimated from the modal data of the first five modes. The identified crack parameters for each specimen with crack states are plotted in Fig. 6, where the x-axis gives the pffiffidifferent ffi relative crack length defined by c= 2a, in which c denotes the length of the crack and a denotes the size of the square element. Besides testing the three plate specimens with different crack states, three refined finite element model is also constructed by OPENFEM to model the cracked aluminium plates, as shown in Fig. 4(b), to study the relationship between the proposed crack parameters and the crack length and orientation. Each model has initially a crack length of 40 mm, and then the crack is lengthened in steps of 20 mm each and the finite element mesh is modified. After each step of extending the crack, ULS curvatures are estimated from the ‘measured’ modal data of the plate, and then they are used as references to solve the unknown vector Dp. The resulting crack parameters are shown separately in Fig. 6(a)–(c) with different symbols as denoted in legends, and they are curve-fitted as shown. It is seen that the test results are consistent with the numerical results, and that both sets of result exhibit the same trend of changing with the crack propagation. Parameter a decays very slowly and changes little, which means the stiffness reduction in the direction of crack extension is limited. Parameter b sharply drops with the extension of the crack indicating remarkable reduction in the stiffness normal to the crack line. There are two phases for v, which represents the in-plane twisting stiffness of the cracked plate element, in
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y
x Fig. 5. Simplified finite element model of the plate with a inclined crack.
Fig. 6. Identified crack model parameters from experiment and finite element model.
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40
30
20
Magnitude (dB)
10
0
-10
-20
-30
Proposed model Refined model Test
-40
-50 0
10
20
30
40
50
60
70
80
90
100
Frequency (Hz)
Fig. 7. The transfer acceleration FRF for Plate A (State I).
the cases when the crack angle n > 45. Firstly the parameter drops evenly when the relative crack length is less than 0.43 of the element dimension, and then the twisting stiffness degrades abruptly as the crack propagating towards the element edges. The magnitude of the transfer acceleration FRF between the excitation at the node denoted by (see Fig. 3) and the output at the sensor location, for the test specimen A (State I), the refined model in Fig. 4(b) and the proposed model respectively, are plotted in Fig. 7. It can be seen that both the refined finite element model and the proposed model can predict accurately the dynamic response of the real plate.
6. Conclusions An effective stiffness continuum model of a cracked plate for the non-destructive fault detection from vibration measurements is proposed. The characterizing damage variables are identified using a model updating approach based on the sensitivity analysis of the ULSC of the plate. The proposed methodology uses a baseline finite element model along with modal test data. The ULS curvatures of the undamaged and damaged plates are estimated from the measurements using the Chebyshev polynomial approximation. An analytical sensitivity matrix of the measured ULSC with respect to the crack variables is then derived from the baseline model. Combining the obtained sensitivity matrix with the measured ULSC changes due to the crack, the unknown crack parameters can be solved simultaneously through a gradient-based model updating method. Both numerical examples and experiment results demonstrate that the proposed model predicts the real cracked plate very well and the identification methods are effective for localizing the crack in the plate.
Acknowledgements The work described in this paper was supported by a grant from the Hong Kong Polytechnic University Research Funding Project No. G-W092.
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References Aggarwala, B.D., Ariel, P.D., 1981. Vibration and bending of a cracked plate. Engineering Transactions 29, 295–310. Baschmid, A., Dizua, G., Pizzigoni, B., 1984. The influence of unbalance on cracked rotors. Vibration of Rotating Machinery, London, England, pp. 193–198. Berman, A., Flannelly, W.G., 1971. Theory of incomplete models of dynamic structures. AIAA Journal 9, 1481–1487. Cawley, P., Adams, R.D., 1979. The location of defects in structures from measurement of natural frequencies. Journal of Strain Analysis 14, 49–57. Cornwell, P., Doebling, S.W., Farrar, C.R., 1999. Application of the strain energy damage detection method to plate-like structures. Journal of Sound and Vibration 224 (2), 359–374. Dimarogonas, A.D., 1996. Vibration of cracked structures: a state of the art review. Engineering Fracture Mechanics 55 (5), 831–857. Hetenyi, M., 1937. Deflection of beams of varying cross section. ASME, Journal of Applied Mechanics 4, A49–A52. Kirsmer, P.G., 1944. The effect of discontinuities on the natural frequency of beams. ASTM Proceedings 44, 897–900. Krawczuk, M., 1993. A rectangular plate finite element with an open crack. Computers and Structures 46, 487–493. Krawczak, M., Ostachowicz, W., 1994. A finite plate element for dynamic analysis of a crack plate. Computer Methods in Applied Mechanics and Engineering 115, 67–78. Lee, H.P., 1992. Fundamental frequencies of annular plates with internal cracks. Computers and Structures 43, 1085–1089. Lee, U., Lesieutre, G.A., Fang, L., 1997. Anisotropic damage mechanics based on strain energy equivalence and equivalent elliptical microcracks. International Journal of Solids and Structures 34, 4377–4397. Lee, U., Cho, K.K., Shin, J., 2003. Identification of orthotropic damages within a thin uniform plate. International Journal of Solids and Structures 40, 2195–2213. Li, Y.Y., Cheng, L., Yam, L.H., Wong, W.O., 2002. Identification of damage locations for plate-like structures using damage sensitive indices: strain modal approach. Computers and Structures 80, 1881–1894. Nelson, R.B., 1976. Simplified calculations of eigenvector derivatives. AIAA Journal 14, 1201–1205. Ostachowicz, W., Krawczak, M., 2001. On modeling of structural stiffness loss due to damage. Damage Assessment of Structures, Key Engineering Materials 204–205, 185–200. Petroski, H.J., Glazik, J.L., 1980. The response of cracked cylindrical shells. ASME, Journal of Applied Mechanics 44, 444–446. Qian, G.L, Gu, S.N., Jiang, J.S., 1991. A finite element model of cracked plates and application to vibration problems. Computers and Structures 39, 483–487. Ren, W.X., De Roeck, D., Harik, I.E., 2000. Singular value decomposition based truncation algorithm in structural damage identification through modal data. In: Proceedings of 14th Engineering Mechanics Conference, pp. 21–24. Shen, M.H.H., Pierre, C., 1990. Natural modes of Bernoulli–Euler beams with symmetric cracks. Journal of Sound and Vibration 138, 115–134. Stahl, B., Keer, L.M., 1972. Vibration and stability of cracked rectangular plates. Journal of Solids and Structures 8, 69–91. Thomson, W.J., 1949. Vibration of slender bars with discontinuities in stiffness. ASME, Journal of Applied Mechanics 17, 361–367. Wu, D., Law, S.S. Damage localization in plate structures from uniform load surface curvature. Journal of Sound and Vibration, in press (doi:10.1016/j.jsv.2003.07.040). Wu, D., Law, S.S. Crack identification in thin plates with anisotropic damage model and vibration measurements. Journal of Applied Mechanics, ASME, submitted for publication. Zastrau, B., 1985. Vibration of cracked structures. Archives of Mechanics 37, 743–781. Zhang, Z., Aktan, A.E., 1998. Application of modal flexibility and its derivatives in structural identification. Research in Nondestructive Evaluation 10 (1), 43–61.
Anisotropic damage model for an inclined crack in thick plate and sensitivity study for its detection D. Wu, S.S. Law
*
Department of Civil and Structural Engineering, Hong Kong Polytechnic University, Yuk Choi Road, Kowloon, Hong Kong Received 12 January 2004; received in revised form 27 February 2004 Available online 12 April 2004
Abstract Many approaches on modeling of cracks in structural members have been reported in the literatures. However, most of them are explicitly developed for the purpose of studying the changes in static and dynamic responses of the structure due to the crack damage, which is a forward problem mathematically. The use of these models is inconvenient or even impossible for detecting damage in structures from vibration measurements, which is usually an inverse problem. An anisotropic damage model is proposed for a thick plate element with an inclined crack. The cracked plate element is represented by an effective plate element with anisotropic material properties expressed in terms of the virgin material stiffness and a vector of damage variables. A sensitivity-based model updating approach is developed incorporating the proposed anisotropic model and the estimated uniform load surface curvature from vibration measurements to identify the damage variables. The orientation of the crack is directly indicated by the geometric parameter and the damage severity can be estimated from other variables representing the relative stiffness reduction along the local axes. The validity of the methodology is demonstrated by numerical examples and experiment results. 2004 Elsevier Ltd. All rights reserved. Keywords: Damage detection; Crack model; Thick plate; Sensitivity method; Uniform load surface; Curvature; Vibration; Finite element; Chebyshev polynomial
1. Introduction The presence of a crack in a structural member introduces a local flexibility affecting its static behavior and vibration response. Many efforts have been devoted by engineers on the modeling of the crack-induced flexibility and investigating its effect on the dynamic characteristics of the damaged structure. Dimarogonas (1996) has summarized these works into three categories, namely: continuous model, discrete–continuous model and discrete models, i.e. finite element models. Historically the earliest method to model a fatigue crack in a beam is proposed by Hetenyi (1937) for determining the static deflection of beams with non-uniform cross-section, in which a crack is represented
*
Corresponding author. Tel.: +852-27666062; fax: +852-23346389. E-mail address: [email protected] (S.S. Law).
0020-7683/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijsolstr.2004.03.001
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by additional external equivalent loads. This method is further developed by Kirsmer (1944), Thomson (1949), and Petroski and Glazik (1980) to study the vibration response of the beams or shells with fatigue cracks. However the lack of explicit relation between the size of the crack and the magnitude of its external equivalent load makes it impossible to have the sensitivity analysis with the parameters of the structural members. Another approach related to the continuous model divides the cracked members into sub-domains by the crack lines. Special boundary conditions along the crack line are introduced (Stahl and Keer, 1972; Aggarwala and Ariel, 1981; Lee, 1992) to connect the sub-domains. The principal limitation of these continuous models is the fact that their partial differential-based mathematical derivation can provide sufficiently accurate results only in the case when the structural members have a very simple geometry. In the discrete–continuous models, a crack in a structural member is represented by additional springlike elements. A system with two parts of the undamaged member jointed with specific spring elements is created to model the cracked member. Many special compliance matrices for the spring elements were studied by the laws of fracture mechanics (Ostachowicz and Krawczak, 2001). These methods can successfully be used for modeling cracks in one-dimensional members. However, In the case of two or threedimensional members, this approach is practically inapplicable due to the fact that the large number of equations required for the boundary conditions of the connecting spring elements is not available. Real structures are more complicated than the geometrically simple ones described by the analytically continuous or discrete–continuous models. Researchers hence started to use discrete models to study the cracked structures, in which the finite element method (FEM) is the most popular and commonly utilized. From the published literature on FEM-based model of a cracked plate, one may find mainly three groups of methods. The simplest model represents the crack as a reduction in the elastic modulus of the element at the crack position (Cawley and Adams, 1979) or a reduction in the cross-sectional area of the element (Baschmid et al., 1984). These models have been successfully used in damage localization of plate-like structures (Cornwell et al., 1999; Li et al., 2002). However, as these methods studied the approximate crack size and location at the element level, a very fine finite element mesh is required to avoid large error. Another method models a crack by separating the nodes of finite elements along the crack line (Zastrau, 1985). To properly model the singular characteristics of the stress and strain fields around the crack tip, a very dense mesh of finite elements or singular-shaped isoparametric elements (Shen and Pierre, 1990) is used to cover the crack tip area. Obviously, this method can model the cracked structure very well, but is not suitable or feasible for damage identification. This is due to the low computation efficiency associated with the large number of finite elements. Also a finite element mesh has to be constructed for one suspected location of damage in the plate at one time. In the third group of methods, a rectangular plate element with an open and depth-through crack parallel to the plate boundary is modeled (Qian et al., 1991; Krawczuk, 1993; Krawczak and Ostachowicz, 1994). The stiffness matrix of a cracked element is written as K ¼ TF 1 T T where F is the matrix representing the sum of flexibility of the non-cracked plate and the additional flexibility due to the crack, and T is a transformation matrix. However, as the derived stiffness matrix cannot be explicitly parameterized in terms of the damage variable(s) to indicate the location, orientation and the extend of the crack, it is still difficult to incorporate this model into the inverse problem of structural damage identification. All the above-mentioned crack models were explicitly proposed for the purpose of studying the static behavior or dynamic response of the cracked structure, which is a forward problem in mathematical sense. Generally it is not suitable to use them for solving the inverse problem of damage identification. Therefore, a damage detection oriented crack model for plate-like structure is in need. The earliest crack model oriented to damage identification is probably presented by Lee et al. (1997), although their work originally focuses on fracture mechanics and aims to derive a damage evolution equation of an elliptical through
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micro-crack. By restricting the aspect ratio of the elliptical hole s ¼ 0, Lee et al. (2003) further studied the effective stiffness model of a thin plate with line micro-cracks, and then developed a model updating technique to identify the damage size and orientation by using the frequency response functions measured from the damaged plate. Numerical examples were simulated for demonstration. However, the validity of the theory still requires the support from experimental evidences. According to the authors’ understanding, Lee’s theory is derived from an infinite plate containing a central crack subjected to uniform stress load. Although the method can be approximately used for a micro-crack away from the plate boundary, it is no longer valid for the case of a macro-crack and corrections must be made to take into account the finite dimensions of the plate and different load patterns. Such correction has been demonstrated in the previous work by the authors on identifying edge-parallel cracks in a thin plate from vibration measurements (Wu and Law, submitted for publication). The objectives of this paper include: (1) to introduce a new effective stiffness model for thick plate elements with an inclined line-crack, in which the damage scalar in Lee et al. (1997) model is expanded and replaced by a vector of damage variables; (2) to develop a sensitivity-based model updating method incorporating the proposed anisotropic model and the estimated uniform load surface curvature (ULSC) from vibration measurements to identify the damage variables; (3) to carry out numerical simulation and experimental studies to verify the validity of the proposed methodology.
2. Anisotropic stiffness model for thick plate with an inclined crack The problem considered in this section is a thick plate with a non-propagating, open crack, which inclines with an angle n between the crack line and the plate boundary along the x-axis. It is assumed that the crack is depth-through but narrow such that it does not change the mass of the plate. A small rectangular plate element is shown in Fig. 1(a) containing the crack centrally. An effective stiffness model of the cracked element is proposed such that the modeled plate has equivalent dynamic characteristics as the real cracked plate. A crack induces local flexibility in the plate. According to the continuum elastic stiffness theory (Lee et al., 1997), the cracked plate element would exhibit orthotropic stiffness properties compatible with the orientation of the crack line. Furthermore, based on the observation from the fracture mechanics point of view that the high stress intensity at the crack edge along the major axis direction would effectively reduce the stiffness in the minor axis direction. A rational supposition can be made that the crack will mainly affect the flexural stiffness normal to the crack line, while contributing relatively little effect on the plate stiffness parallel to the crack. This is different from Lee’s model in that the stiffness reduction along the major axis is m2 times that along the minor axis. In order to verify this supposition, the cracked plate element is represented by an effective element of continuum anisotropic material with the major axis of the material parallel to the crack line and the minor axis normal to the crack. Thus, under the plane stress condition, the constitutive stress–strain relationship of the cracked element can be written as x
x ξ
≅
z
y
ξ
z
a
b
y
Fig. 1. Plate element with a through crack and the effective stiffness model of anisotropic material.
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r ¼ Ce;
or
8 9 2 C11 < r1 = r2 ¼ 4 C12 : ; 0 s12
C12 C22 0
38 9 0 < e1 = 0 5 e2 : ; C66 c12
ð1Þ
m12 E1 m21 E2 where the components C11 ¼ 1mE121 m21 , C22 ¼ 1mE122 m21 , C12 ¼ 1m ¼ 1m and C66 ¼ G12 , in which the terms 12 m21 12 m21 E, G and m denote the Young’s modulus, the shear modulus and the Poisson ratio respectively, and the subscripts 1, 2 define the local Cartesian coordinates with the 1-axis along the crack line and the 2-axis normal to the crack line. Comparing Eq. (1) with the constitutive stiffness of the intact material, which is expressed as 2 3 E=ð1 m2 Þ mE=ð1 m2 Þ 0 C0 ¼ 4 mE=ð1 m2 Þ E=ð1 m2 Þ 0 5 ð2Þ 0 0 G
one can introduce a vector of damage variables to relate the isotropic stiffness of the undamaged material to anisotropically reduced stiffness terms of the effective continuum model for the cracked plate element, as E1 ¼ Ea;
m21 ¼ m;
E2 ¼ Eb;
G12 ¼ Gv
ð3Þ
where a, b and v denote the stiffness scaling factors due to the crack. Considering the relation m12 =E2 ¼ m21 =E1 , one gets m12 ¼ m21
E2 b ¼m a E1
ð4Þ
Substituting (3) and (4) into (1), one obtains 3 2 Ea2 mEab 0 7 6 a bm2 a bm2 7 6 7 6 mEab Eab 7 C¼6 0 7 6 a bm2 a bm2 7 6 4 Ev 5 0 0 2ð1 þ mÞ
ð5Þ
If the crack line is not parallel to the plate boundary, and the term n represents the angle between the xaxis and the crack line (or the major axis of the effective anisotropic material). A transformation matrix can be defined as ¼ Tr r
and
e ¼ T Te
ð6Þ
T
T ¼ f rx ry sxy g , e ¼ ex ey cxy , and where r 2 3 sin2 n 2 sin n cos n cos2 n T ¼ 4 sin2 n cos2 n 2 sin n cos n 5 sin n cos n sin n cos n cos2 n sin2 n
ð7Þ
Substituting (6) into (1), we have ¼ TCT Te ¼ Ce r
ð8Þ
For a thick plate in which a plane normal to mid-plane before deformation does not remain normal to the mid-plane any longer after deformation, the effect of transverse shear deformation should be taken into account. Hence the constitutive equation of the effective continuum model of the cracked plate relating to the transverse shearing is expressed as
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s ¼ Dc;
or
s13 s23
¼
G13 0
0 G23
c13 c23
4325
ð9Þ
in which G13 ¼ Gd13 and G23 ¼ Gd23 , where the ratio factors d13 and d23 scale the reduced stiffness due to the crack against the transverse shear stresses in 1–3 plane and 2–3 plane respectively. Considering the inclined angle between the crack line and plate boundary, the shear constitutive relation can be written as s ¼ SDS Tc ¼ Dc where s ¼ f sxz
T
syz g and c ¼
ð10Þ
cxz
cyz
T
denoting the transverse shear stress and strain respectively, and cos n sin n the transformation matrix S ¼ . sin n cos n
3. Identification of the damage variables According to the above section, one needs a vector of six damage variables, p ¼ f a b v d13 d23 n g, to characterize a thick plate element with an inclined through crack. The purpose of this section is to identify the variables using vibration measurements from the undamaged and damaged plates. To achieve this, the ULSC, which is proposed by the authors as a local damage indicator (Wu and Law, in press), is adopted here as reference modal data instead of using modal frequencies or mode shapes. The reported method to estimate the ULSC from vibration measurements is briefly outlined below for clarity with the subsequent formulations of damage identification, followed by the sensitivity analysis of the ULSC with respect to the damage variables. Finally, a model updating method is developed based on the sensitivity formulations to identify the unknown variables. 3.1. Uniform load surface curvature (ULSC) For a linear structural system with n degrees-of-freedom, its flexibility matrix can be expressed by the superposition of mass normalized modes /r , where /Tr M/r ¼ 1 ðr ¼ 1; . . . ; nÞ, as (Berman and Flannelly, 1971) F ¼
n X /r /Tr x2r r¼1
ð11Þ
where xr is the rth natural frequency. Physically, each element of the flexibility matrix, fkl , can be interpreted as the displacement at the kth DOF due to a unit load at lth DOF. Thus the deflection vector of a plate under uniform unit load can be evaluated by u¼F L
ð12Þ
f1; . . . ; 1gT1 n
is the unit vector representing the uniform load acting on the plate. The deflection where L ¼ component uk can be expressed as uk ¼
n X l¼1
fkl ¼
n X n X /r ðkÞ/r ðlÞ x2r r¼1 l¼1
ð13Þ
This modal based formulation of the structural deflection vector under uniform load was defined as Uniform Load Surface (ULS) by Zhang and Aktan (1998). An approach to estimate the ULS curvature of plate structures using the Chebyshev polynomial approximation has been proposed by the authors (Wu and Law, in press). This approach is adopted here and outlined as follows.
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The ULS function of a plate can be modeled by the Chebyshev polynomial in two variables: uðx; yÞ ¼
N X M X i¼1
cij Ti ðxÞTj ðyÞ
ð14Þ
j¼1
where Ti ðxÞ, Tj ðyÞ are the first kind Chebyshev polynomials, and N , M are their orders. It is assumed that there are Q measuring points on the rectangular mesh of the plate. The ULS value, uðxp ; yq Þ, can be obtained at all the measuring points from Eq. (13). When substituting them into Eq. (14) separately, one has a system of Q equations written in matrix form as ½T ðxp ÞT ðyq Þ Q P fcij gP 1 ¼ fuðxp ; yq ÞgP 1
ð15Þ
where P ¼ N M. The coefficient vector fcij g can then be solved by the least-squares technique 1
T
T
fcij gP 1 ¼ ð½T ðxp ÞT ðyq Þ Q P ½T ðxP ÞT ðyq Þ Q P Þ ½T ðxp ÞT ðyq Þ Q P fuðxp ; yq ÞgQ 1
ð16Þ
By making use of the orthogonal property of the Chebyshev polynomial, the curvature of the ULS is obtained as the second derivatives of the Chebyshev polynomials in Eq. (14) as ucxx ¼
N X M X i¼1
ucxy ¼
j¼1
N X M X i¼1
cij
j¼1
oTi2 ðxÞ Tj ðyÞ; ox2
oT 2 ðxÞ cij i 2 ox
ucyy ¼
N X M X i¼1
cij Ti ðxÞ
j¼1
oTj2 ðyÞ oy 2
oTj2 ðyÞ ; oy 2 ð17Þ
3.2. Sensitivity of ULSC to generic damage variables Consider a damaged plate divided into ne rectangular elements. The characteristic equation for this undamped structural dynamic system can be expressed as K/r x2r M/r ¼ 0
ð18Þ
It is assumed that crack damage only causes a reduction in local stiffness, while the mass matrix remains unchanged. The stiffness and mass matrix of the damaged plate is expressed as K ¼ K0 þ DK ¼
ne X
f ðpe Þke ;
M ¼ M0
ð19Þ
e¼1
in which ke is the element stiffness matrix of the intact plate, and pe denotes a generic damage variable (or vector) in a specific function f which affects the elemental stiffness. M0 and M are the mass matrices of the intact and damaged states respectively. Taking the derivative of (18) with respect to the parameter pe and noting the relations in Eq. (19), we have o/r oxr oK 2 ðK xr MÞ ¼ 2xr M ð20Þ /r ope ope ope The sensitivity of natural frequency and mode shapes with respect to the parameter pe can be calculated using Nelson’s method (1976) as oxr 1 T oK ¼ / / ope 2xr r ope r
ð21Þ
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and o/r ¼ gr þ cr /r ; ope
where cr ¼ gTr M/r
ð22Þ
in which gr is a particular solution of Eq. (20) by searching the component with the largest absolute value in /r and enforcing its derivative to zero. To find the sensitivity of the ULSC, oucðx;yÞ , one needs to calculate the sensitivity of the ULS at the ope measuring points first. Taking the derivative of Eq. (13) with respect to the parameter pe we get m X n ouðxi ; yj Þ ouk X 1 o/kr o/lr 2 oxr ¼ ¼ / / / / þ ð23Þ ope x2r ope lr ope kr x3r opk kr lr ope r¼1 l¼1 Substituting Eqs. (21) and (22) into (23), the sensitivities of ULS at the sensor grid points are obtained. Further taking the derivative of Eq. (16) with respect to the parameter pe and substituting Eq. (23), the sensitivity of the Chebyshev coefficients is obtained as ocij ouðxi ; yj Þ T 1 T ¼ ð½T ðxi ÞT ðyj Þ Q P ½T ðxi ÞT ðyj Þ Q P Þ ½T ðxi ÞT ðyj Þ Q P ð24Þ ope ope P 1 Q 1 Finally the sensitivity of ULS curvature with respect to the stiffness parameter pe can be derived from Eq. (17) and calculated as N X M N X M oTj2 ðyÞ oucxx ðx; yÞ X ocij oTi2 ðxÞ oucyy ðx; yÞ X ocij ¼ T ðyÞ; ¼ T ðxÞ ð25aÞ j i oy 2 ope ope ope ox2 ope i¼1 j¼1 i¼1 j¼1 and N X M oucxy ðx; yÞ X ocij oTi ðxÞ oTj ðyÞ ¼ ope oy ope ox i¼1 j¼1
ð25bÞ
3.3. Sensitivity of effective stiffness to crack damage variables To formulate the stiffness matrix of the proposed effective continuum model of the cracked plate element, one needs to express the internal strains in terms of the nodal variables, which include the transverse deflection w and the rotations of the mid-plane about the x- and y-axes, i.e. hx and hy , respectively. These rotation variables are formed from the bending and transverse shear deformations as ow ow cxz ; hy ¼ cyz hx ¼ ð26Þ ox oy where cxz and cyz are the deformed angles arising from the transverse shear stress. The in-plane displacements are given as u ¼ zhx ðx; yÞ;
v ¼ zhy ðx; yÞ
ð27Þ
Taking the three displacement variables of each node independently, the displacement field of the plate element can be written in terms of the nodal displacements as nn nn nn X X X Hi ðx; yÞðhx Þi ; hy ¼ Hi ðx; yÞðhy Þi ; w ¼ Hi ðx; yÞwi ð28Þ hx ¼ i¼1
i¼1
i¼1
where Hi ðx; yÞ is the isoparametric shape function for the ith node and nn denotes the number of nodes of the plate element. Considering the relations in Eq. (27), the in-plane strains can be expressed in terms of the nodal variables as
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3
2
where
2
o 07 6 ox 7 6 6 o7 7 u ¼ z6 0 7 6 ox 7 v 6 5 4 o o ox oy
o 8 9 6 ox < ex = 6 6 ey 0 ¼6 :c ; 6 6 xy 4 o oy 2
oH1 6 ox 6 6 Bb ¼ 6 6 0 6 4 oH1 oy
0 oH1 oy oH1 ox
0
oHnn ox
0
0
0
oHnn oy
3 0 78 9 7 < hx = 7 07 7: hy ; ¼ zBb d 7 w 5 0
0 o oy o ox
ð29Þ
3 07 7 7 07 7 7 5 0
0 oHnn oy oHnn ox
T and d ¼ f hx;1 hy;1 w1 hx;nn hy;nn wnn g . From Eq. (26) the transverse shearing strains is expressed in terms of the nodal displacements as 2 3 o 8 9 1 0 < hx = 6 cxz ox 7 7 h y ¼ Bs d ð30Þ ¼6 4 o 5: ; cyz 0 1 w oy
where
2
6 H1 ½Bs ¼ 6 4 0
0 H1
oH1 ox oH1 oy
Hnn
0
0
Hnn
3 oHnn ox 7 7 oHnn 5 oy
According to the Reissner–Mindlin theory, one can formulate the stiffness matrix of the effective model of the cracked plate element by Z Z Z Z h3 ke ¼ BTb TCT T Bb dA þ jh BTs SDS T Bs dA ð31Þ 12 A A where h denotes the thickness of the plate, A denotes the plane area of the plate element, and Bb , Bs are the strain–displacement relationship matrices for the bending and transverse shear strains respectively, j is the shear energy correction factor of 5/6. Thus, the sensitivity of the elemental stiffness matrix with respect to the crack variables can be obtained as follows: 3 2 Eaða 2bm2 Þ Eb2 m3 07 6 ZZ 7 6 ða bm2 Þ2 ða bm2 Þ2 oke h3 7 T 6 2 2 ¼ BTb T 6 Eb2 m3 ð32aÞ 7T Bb dA Eb m 6 oa 12 07 A 2 5 4 ða bm2 Þ2 2 ða bm Þ 0 2
Ea2 m2 6 ZZ 6 ða bm2 Þ2 oke h3 T 6 ¼ Bb T 6 Ea2 m 6 ob 12 A 4 ða bm2 Þ2 0
0
0 3
Ea2 m ða bm2 Þ Ea2 ða
2
2 bm2 Þ
0
07 7 7 T 7T Bb dA 07 5 0
ð32bÞ
D. Wu, S.S. Law / International Journal of Solids and Structures 41 (2004) 4321–4336
oke h3 ¼ ov 12
Z Z
6
A
oke ¼ lh od13 oke h3 ¼ on 12
2
Z Z
0
0 BTb T 6 4 0
0 0
0 7 T 0 7T Bb dA 5 E 0 2ð1 þ mÞ
2
E BTs S 4 2ð1 þ mÞ A 0
Z Z
BTb
A
4329
3
oC Bb dA þ jt on
3 05 T S Bs dA; 0 Z Z
BTs A
ð32cÞ
oke ¼ lh od23
3 0 5S T Bs dA E BTs S 4 0 A 2ð1 þ mÞ
Z Z
2
oD Bs dA on
0
ð32dÞ
ð32eÞ
Substituting Eq. (32) into (20) and (21) to replace the sensitivity of stiffness matrix with respect to a generic oK parameter op , the sensitivity of ULSC to the crack variables is now ready to be calculated. e 3.4. Model updating procedure Consider an unknown damage case with single or multiple cracks producing changes in the vectors of T damage variables for all the suspected elements, written as Dpi ¼ f Dai Dbi Dvi Dd13;i Dd23;i Dni g ði ¼ 1; . . . ; neÞ, where ne denotes the number of suspected elements in the plate. With the small damage assumption, the damaged plate is still considered as a linear structural system, and the damage-induced changes in the ULSC of the plate can be expressed by the first order Taylor series approximation Ducxx ðxi ; yj Þ ¼ ucDxx ucxx ¼
oucxx oucxx oucxx Dp1 þ Dp2 þ þ Dpne op1 op2 opne
ð33Þ
Similar relations can be obtained for the curvature changes Ducyy ðxi ; yj Þ and Ducxy ðxi ; yj Þ. Provided the ULS curvature and its sensitivity are estimated at Q measuring points, one can repeatedly obtain the above relations for each point, and rearrange them in a matrix form to have 3 2 oucxx ðx1 ; y1 Þ oucxx ðx1 ; y1 Þ oucxx ðx1 ; y1 Þ 7 6 op1 op2 opne 9 8 7 6 7 6 > > Ducxx ðx1 ; y1 Þ > . . . . > 7 6 > > .. .. .. .. > > .. 7 6 > > > > 9 8 7 6 oucxx ðxQ ; yQ Þ oucxx ðxQ ; yQ Þ > > . > > oucxx ðxQ ; yQ Þ 7 Dp 6 > > > > > > 1 > > > 7> 6 Duc ðx ; y Þ > > xx Q Q > op1 op2 opne > > > > 7> 6 = = < < Dp 2 7 6 . .. .. .. .. 7 6 . ð34aÞ ¼ . . 7> . > > 6 . . . . > > > 7> 6 . > > > > > > > Ducyy ðxQ ; yQ Þ > 6 oucyy ðxQ ; yQ Þ oucyy ðxQ ; yQ Þ oucyy ðxQ ; yQ Þ 7> > > > 7: Dpne ; > 6 > > > > 7 6 op op op > > . 1 2 ne > > 7 6 . > > . > > 7 6 . . . . > > ; : 7 6 .. .. .. .. 7 6 Ducxy ðxQ ; yQ Þ 5 4 oucxy ðxQ ; yQ Þ oucxy ðxQ ; yQ Þ oucxy ðxQ ; yQ Þ op1 op2 opne or U fDpg ¼ fDucg
ð34bÞ
It should be noted that Eq. (34) comprises 6ne unknown variables and a total of neq ¼ 3 Q linear equations. To ensure a unique over-determined solution to be found, the condition neq P 6ne is required. The Singular Value Decomposition (SVD) with error-truncation technique (Ren et al., 2000) is adopted to
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solve the linearised equations. To include the non-linear effects from large magnitude damage or interaction between the variables, the equations are solved by an iterative algorithm as follows. n
e Step 1. Initialize the angle variable n0 ji¼1 ¼ 0 and the other variables to unity. Step 2. At the beginning of each iteration, construct the ULSC sensitivity matrix Uk for the kth iteration, and calculate the curvature change, fDucgk , due to damage. Step 3. Solve the increment of damage variables as, fDpgk ¼ Uk fDucgk , where Uk is the generalized inverse ne of Uk from SVD technique. Each solution is divided into two stages: (1) fix the angle variable nk ji¼1 and update the other variables; (2) fix the other variables and update the angle variable. ne Step 4. Normalize all the variables in fDpgk except the angle variable nk ji¼1 by fDpgk ¼ fDpgk k such that ne the condition lmin 6 Dbk ji¼1 6 lpos is satisfied, where k is a scale factor and lmin , lpos are the convergence limits as shown below. Variable bk is chosen as a criterion because it has the largest range of variation other than the angle variable nk . e e e Step 5. Evaluate the new angle variable as nk jni¼1 ¼ nk1 jni¼1 þ Dnk jni¼1 and the other variables as fpgk ¼
fpgk1 ð1 þ fDpgk Þ. If w1 ¼ jfDpgk j < w1 or w2 ¼
jfDucgkþ1 j jfDucgk j
< w2 , the solution is considered con-
verged. Otherwise go to Step 2 until the condition is satisfied. lpos lmin The scale factor k in Step 4 is calculated as k ¼ min minðDb ; , in which the range ½lmin ; lpos ne ne j Þ maxðDb j Þ k k i¼1
i¼1
is set to compromise the convergence speed and solution stability. A wider range could speed up the solution convergence, but for some cases, it may lead to a divergent solution, while a narrower range would have an opposite effect. For all the cases in this paper, lmin ¼ 0:5, lpos ¼ 0:25 and w1 ¼ w2 ¼ 0:001. 4. Numerical examples An aluminium plate with free boundary condition is studied in this section to validate the proposed method. The plate has the dimension of 400 mm · 300 mm · 15 mm. The whole intact plate is divided into 4 · 3 Mindlin/Reissner plate elements. An inclined through crack is then assumed to occur in one element with a length of 100 mm and an angle n ¼ 45 as shown in Fig. 2. The finite element mesh of the crack is modeled using an open-source finite element toolbox, OPENFEM, integrated with MATLAB. The natural frequencies and corresponding mode shapes of the plate at the 20 finite element nodes are ‘measured’ y
x
Fig. 2. Finite element model of the thick plate with an inclined crack.
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Table 1 Identified parameters of the inclined crack with the thick plate model Element no.
1a
1b
1v
1 d13
1 d23
n
1 2 3 4 5 6 7 8 9 10 11 12
0.0185 0.0069 0.0182 0.0096 0.0268 0.0315 0.0129 0.0137 0.0006 0.0047 0.0134 0.0185
0.0038 0.0277 0.0222 0.0053 0.0122 0.1486 0.0481 0.0275 0.0123 0.0268 0.0017 0.0009
0.0244 0.0003 0.0042 0.0061 0.006 0.0883 0.0121 0.0082 0.0056 0.0005 0.0224 0.0134
0.0073 0.014 0.0126 0.0254 0.0058 0.0266 0.0060 0.0202 0.0252 0.0006 0.0205 0.0114
0.025 0.0151 0.0213 0.0129 0.0391 0.1075 0.0057 0.0058 0.0025 0.0091 0.0163 0.0045
1.37 0.45 2.64 )4.61 6.12 44.57 )4.33 )1.42 7.81 16.24 2.19 )5.54
through eigenvalue analysis of the finite element model. The ULSC are then estimated from the modal data via the procedure described in Eqs. (13)–(17), and input into a MATLAB program to calculate their sensitivities. According to the proposed crack model for a thick plate, there are six unknown parameters to model each suspected cracked element, and totally there are 6ne ¼ 6 12 unknowns for all the suspected elements of the whole plate. To satisfy the over-determined condition for Eq. (34), the ULS curvatures are also estimated at the center node of each element besides the 20 finite element nodes so that the number of linear equations 3Q equals 96. All the parameters for the initial plate model are set to unity and the Poisson ratio m is fixed at 0.3 in calculating the initial curvature sensitivities from Eq. (25). The final damaged indices after seven iterations are listed in Table 1, in which the location of the damage is clearly detected and the inclined angle of the crack is accurately predicted. The relative flexural and shearing stiffness b and d23 due to the crack show clearly a larger reduction than the others, while a and d13 remain almost unchanged. This finding is consistent with the authors’ supposition that a crack mainly affects its stiffness normal to the crack line.
5. Experiment verification Three aluminium plate specimens with the same dimensions of 600 mm · 500 mm · 3 mm are named as plate A, B and C. Fig. 3 shows the experimental set-up for testing. A rectangular mesh of 7 · 6 measuring points is outlined on each plate. The intact plate is suspended from a rigid frame by two steel wires of 0.5 mm in diameter and 0.75 m in length to simulate the free boundary condition. An impulsive signal was applied on the plate by hitting with a dynamic hammer Model B&K 8202 at each measuring point, and the vibration response of the plate due to the impulse is collected by an accelerometer Model B&K 4370 as shown in the figure. Both the input and output signals are amplified and input into a commercial modal testing and analysis system DASP2000. The natural frequencies and corresponding mode shapes of the plate at the rectangular mesh are then extracted through a Multi-Input-Single-Output transfer function analysis. An artificial crack is then cut in each specimen as shown in Fig. 4(a). To verify the proposed crack model and the identification method with cases of different crack lengths and orientations, a scheme of crack cutting is formulated and listed in Table 2. The crack in the first state is 0.5 mm wide, and it is 0.3 mm wide for the other two states. After each crack cutting exercise, the above hammer test is repeated to obtain the modal data of the plate. The first five natural frequencies of the plate specimens in different damage states are listed in Table 3.
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Fig. 3. Diagram of the experimental system.
Fig. 4. (a) Artificial cracks (State I). (b) Finite element meshes to model the cracked plates A, B and C.
As the thickness of the test plates are very small relative to the other two dimension of the plates, the transverse shear deformations are neglected so that the vector of damage variables consists of only four components for each plate element, i.e. pi ¼ f ai bi vi ni gT ði ¼ 1; . . . ; neÞ. The initial finite element model of the intact plate with one 9-nodes plate element and 26 number of 4-nodes elements is shown in Fig. 5, with the parameter vectors for all the elements initialized to unity. The crack identification procedure is adopted here with an over-determined equation system characterized by a sensitivity matrix of ULS
D. Wu, S.S. Law / International Journal of Solids and Structures 41 (2004) 4321–4336
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Table 2 Scheme of artificial crack cutting for the three specimens Crack information
Plate A
Plate B
Plate C
Center (x; y) Orientation (n)
(200; 200) 45
(200; 200) 60
(200; 200) 90
Length (mm) State I State II State III
80 140 200
80 120 160
80 120 160
Table 3 Natural frequencies (Hz) of the plate specimens in different crack states Mode no.
1
2
3
4
5
Plate A Intact State I State II State III
33.854 33.714 33.421 32.876
44.796 44.675 44.454 44.043
69.250 68.655 67.541 65.941
83.015 82.965 82.861 82.484
94.795 94.810 94.739 94.497
Plate B Intact State I State II State III
33.966 33.817 33.666 33.402
45.035 44.772 44.446 43.976
68.703 68.348 67.931 67.342
82.829 82.782 82.709 82.582
94.750 94.772 94.685 94.508
Plate C Intact State I State II State III
31.379 31.372 31.372 31.361
41.578 41.228 40.812 40.133
65.859 65.571 65.238 64.843
78.105 78.040 77.991 77.870
88.263 88.245 88.194 88.028
curvature sized 3Q 4ne, where Q ¼ 7 6 and ne ¼ 27. The ULS curvatures for both the test plate and initial finite element model are estimated from the modal data of the first five modes. The identified crack parameters for each specimen with crack states are plotted in Fig. 6, where the x-axis gives the pffiffidifferent ffi relative crack length defined by c= 2a, in which c denotes the length of the crack and a denotes the size of the square element. Besides testing the three plate specimens with different crack states, three refined finite element model is also constructed by OPENFEM to model the cracked aluminium plates, as shown in Fig. 4(b), to study the relationship between the proposed crack parameters and the crack length and orientation. Each model has initially a crack length of 40 mm, and then the crack is lengthened in steps of 20 mm each and the finite element mesh is modified. After each step of extending the crack, ULS curvatures are estimated from the ‘measured’ modal data of the plate, and then they are used as references to solve the unknown vector Dp. The resulting crack parameters are shown separately in Fig. 6(a)–(c) with different symbols as denoted in legends, and they are curve-fitted as shown. It is seen that the test results are consistent with the numerical results, and that both sets of result exhibit the same trend of changing with the crack propagation. Parameter a decays very slowly and changes little, which means the stiffness reduction in the direction of crack extension is limited. Parameter b sharply drops with the extension of the crack indicating remarkable reduction in the stiffness normal to the crack line. There are two phases for v, which represents the in-plane twisting stiffness of the cracked plate element, in
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y
x Fig. 5. Simplified finite element model of the plate with a inclined crack.
Fig. 6. Identified crack model parameters from experiment and finite element model.
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4335
40
30
20
Magnitude (dB)
10
0
-10
-20
-30
Proposed model Refined model Test
-40
-50 0
10
20
30
40
50
60
70
80
90
100
Frequency (Hz)
Fig. 7. The transfer acceleration FRF for Plate A (State I).
the cases when the crack angle n > 45. Firstly the parameter drops evenly when the relative crack length is less than 0.43 of the element dimension, and then the twisting stiffness degrades abruptly as the crack propagating towards the element edges. The magnitude of the transfer acceleration FRF between the excitation at the node denoted by (see Fig. 3) and the output at the sensor location, for the test specimen A (State I), the refined model in Fig. 4(b) and the proposed model respectively, are plotted in Fig. 7. It can be seen that both the refined finite element model and the proposed model can predict accurately the dynamic response of the real plate.
6. Conclusions An effective stiffness continuum model of a cracked plate for the non-destructive fault detection from vibration measurements is proposed. The characterizing damage variables are identified using a model updating approach based on the sensitivity analysis of the ULSC of the plate. The proposed methodology uses a baseline finite element model along with modal test data. The ULS curvatures of the undamaged and damaged plates are estimated from the measurements using the Chebyshev polynomial approximation. An analytical sensitivity matrix of the measured ULSC with respect to the crack variables is then derived from the baseline model. Combining the obtained sensitivity matrix with the measured ULSC changes due to the crack, the unknown crack parameters can be solved simultaneously through a gradient-based model updating method. Both numerical examples and experiment results demonstrate that the proposed model predicts the real cracked plate very well and the identification methods are effective for localizing the crack in the plate.
Acknowledgements The work described in this paper was supported by a grant from the Hong Kong Polytechnic University Research Funding Project No. G-W092.
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