A dynamic analysis approach for identifying the elastic properties of unstitched and stitched composite plates

Composite Structures 152 (2016) 959–968

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Composite Structures journal homepage: www.elsevier.com/locate/compstruct

A dynamic analysis approach for identifying the elastic properties of unstitched and stitched composite plates Nan Li ⇑, Mabrouk Ben Tahar, Zoheir Aboura, Kamel Khellil Sorbonne universités, Université de technologie de Compiègne, CNRS, UMR 7337 Roberval, Centre de recherche Royallieu, CS 60319, 60203 Compiègne cedex, France

a r t i c l e

i n f o

Article history: Received 30 September 2015 Revised 13 May 2016 Accepted 15 June 2016 Available online 16 June 2016 Keywords: Stitched composite Mechanical properties Vibration Finite element analysis (FEA)

a b s t r a c t The identification of elastic properties from the dynamic behavior of structure has long been an alternative method to mechanical test. This method can avoid the variability of properties due to extraction of the specimen and achieve better homogenization by involving the whole structure. The mode classification is proposed and integrated in the identification process in this paper and helps to reduce the calculation time. Firstly, the measured frequencies are provided by vibration test while the calculated frequencies are provided by a finite element model with discrete Kirchhoff plate element. Then a mode classification process is conducted to accelerate the identification. Every constant is identified by minimizing the difference between the calculated and the measured frequencies of the group that it dominates. This method has been applied to two types of composite plate: unstitched plate and dispersedly stitched plate which presents both macro and meso heterogeneity. The limited dimension along the width direction of the stitch makes the modulus inaccessible by mechanical test and thus this dynamic method turns to be indispensable. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction The composite materials are widely used in various industrial fields nowadays such as automotive, rail industry and aerospace. Composite materials will account for more than 50 percent (by mass) of the new generation of aircraft in the future. The wide utilization is mainly due to their high stiffness/mass ratio. However, the variety of constituents and molding technique of composite materials lead to a great diversity of their properties. On one hand, this characteristic brings us a certain degree of freedom in the optimization of their performance according to different areas of applications and desired functionality; on the other hand, it requires identification for each configuration of fabrication. The mechanical test is a traditional way to identify the elastic properties. For composite materials which are heterogeneous, the specimen is sometimes not big enough to achieve a satisfied level of homogenization. In addition, the exact values of some properties are difficult to obtain by classical mechanical test such as the in-plane shear modulus of a thin plate. To overcome the difficulties in the identification of elastic properties of composite materials, an alternative method of ⇑ Corresponding author. E-mail address: [email protected] (N. Li). http://dx.doi.org/10.1016/j.compstruct.2016.06.038 0263-8223/Ó 2016 Elsevier Ltd. All rights reserved.

identification from the vibration behavior has been proposed in the last two decades. The vibration behavior of a structure is controlled by the elastic properties (as the geometry and density are known in most cases). Inversely, the elastic properties can be determined by the vibration behavior via a sufficiently accurate model. Based on this principle, the identification is realized by minimizing the difference between the measured result of vibration behavior and a result calculated by a model. Compared to a traditional mechanical test, this approach presents several advantages:
For composite materials, the elastic properties are several independent constants. For example, 4 constants are needed to determine the stress-strain relation of an orthotropic plate under plane stress. With the help of this method, all the constants can be simultaneously identified (including the shear modulus).
Bigger and more representative specimen enables a more satisfied level of homogenization. In contrary to mechanical test, local imperfections of structure bring little influence to the result of this method.
The frequency dependence of elastic constants can be taken into consideration by this method.

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The development of this identification method involves the evolution of model, the algorithm of optimization and the experimental instrument. The early work usually used an analytical model instead of a finite element model. Deobald and Gibson [1] used the Rayleigh–Ritz technique to model the vibration of rectangular orthotropic plate. Abrate and Perry [2] developed approximate expressions of the first six natural frequencies and used these expressions to provide initial values of the elastic properties for the Rayleigh–Ritz model. The finite element model has been introduced to this identification method in later works. A finite element model based on higher order assumed displacement field has been developed by Araujo [3] and a commercial finite element code has been adopted by Shun-Fa Hwang [4]. Another direction of evolution is the algorithm of optimization. Sol [5,6] considered the elastic properties as stochastic values and introduced a statistical optimization method into the identification process. His work has been extended to laminated plates [7] and sandwich beams [8]. Two different estimators, Bayesian estimator and minimum variance estimator, have been used and compared by Daghia [9] in the optimization process. Bledzki [10] and Rikards [11] have mentioned the utilization of experiment design which enables the construction of response surface based on only the reference points. A hybrid genetic algorithm has been used in the optimization by Shun-Fa Hwang [12]. As to the experimental instruments, evolution focuses on imposition of excitation and capture of response. For the excitation instruments, hammer [1], shaker and loudspeaker [13] are commonly used. The hammer-impact is a zero mass loading method and can give a broadband excitation to the structure in a short duration. However, the direction and magnitude of the excitation are difficult to control. As to the shaker, the frequency range excited and the level of force applied are easily controlled. In this case, attention should be paid to the influence brought by the excitation system (shaker, stinger and transducer) for light/flexible SUT (structure under test) [14,15]. The loudspeaker can avoid damage to fragile structure [16] as it is non-contact but the excitation imposed is distributed and it is difficult to measure the force transmitted to the structure accurately. As to the capture of response, the accelerometer is commonly used as contacted sensor[4]. The non-contact instruments such as laser vibrometer (single-point vibrometers, scanning vibrometers) [17,13,18] and microphone [19] can avoid adding mass to the system. Among these non-contact instruments, the measurement of microphone is indirect as it actually measures the pressure; sing-point vibrometer can only measure the vibration on single point while the scanning vibrometer can measure the vibration of plenty of points automatically. A couple of combinations of excitation and capture instruments are possible. The best combination may be the hammer and the scanning vibrameter. However, specific measuring setup is necessary. For example the auto–hammer to make the excitation reproducible proposed in [20]. In addition, the setup to reduce post-strike rigid body motion is also needed in the case of vibration under free-free boundary condition. In our case, the combination of scanning vibrometer/shaker is chosen although the influence of the excitation system should be taken care of. This combination is commonly used in laboratory and the setup is easy to install. In addition, the errors and uncertainties of the whole identification have also been studied by several researchers. Frederiksen has discussed various sources of errors in detail in [21]. Lauwagie [22] has studied the relation between the uncertainty of input parameters (for example the measured resonance frequencies) and the uncertainty of the output parameters. The researchers in this domain always try to identify all the parameters at the same time and have not tried to identify the parameters one by one. They are always bothered by the calculation time. For a plate with 4 elastic constants, the parameter space

is already four-dimensional. Although different algorithm of optimization can accelerate this process, the calculation time is still large. The classification of modes and identification of parameters one by one can help to solve this problem (the details will be presented in Sections 4 and 5). Each parameter is identified by a group of modes which it dominates. The dimension of the parameter space has been reduced to 1 and consequently the calculation time is reduced. The modes which is more complex (bending modes along two direction or torsion-bending mode) involve more than one parameter will be used to validate the robustness of this method. This approach is proved to be effective on both unstitched plate and dispersedly stitched plate which presents both macro and meso heterogeneity. According to the authors, the characterization of stitched composite by vibration test has not been presented in the literature. The assembly of composite structure by stitches is an interesting alternative to riveting and adhesion. Jegley [23] has shown an interesting contribution of the stitch to the buckling resistance of the stiffened structure assembled by stitches. Tan [24] has also demonstrated that the stitch can improve significantly the damage tolerance. Generally speaking, the employment of stitch is becoming widespread and brings composite materials an added-value which has been mentioned in many literatures, including but not limited to [25–28]. The mechanical characterization of stitched zone is very delicate. In fact, the zone of assembly (Fig. 1) always has very limited width (only includes several stitches). Consequently, it is very difficult to extract specimen for traditional mechanical characterization. In addition, the motif of stitch always makes the mission more complex, particularly the definition of representative elementary volume (REV). The determination of REV is in fact indispensable to mechanical tests. In the case of one-side stitching (Fig. 3), the extraction of specimen in the transverse direction is not possible due to the special motif of this technique. So it is necessary to find a new alternative for the characterization of the zones which include this type of stitch. The identification method from vibration test we present in this paper is one of these alternatives. 2. Finite element model The accuracy of the finite element model is a prerequisite of this method. For a thin plate, the Kirchhoff hypothesis is widely used in the literature. The discrete Kirchhoff elements started from Discrete Kirchhoff Triangular element (DKT) [29] and Discrete Kirchhoff Quadrilateral element (DKQ) [30] and have formed a big family [31]. Base on the DKQ element, a discrete Kirchhoff

Fig. 1. Stitches in the assembly of composite structure.

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N. Li et al. / Composite Structures 152 (2016) 959–968

quadrilateral element with in-plane displacement is proposed (Fig. 2).

(b) at the mid-nodes:

bsk þ w0 ;sk ¼ 0 k ¼ 5; 6; 7; 8

2.1. Displacement field and strain field

where s represents the co-ordinate along the element boundary.

Development of this element starts from a traditional Reissner/ Mindlin displacement field with in-plane displacement. The displacement components of an arbitrary point of the plate are:

uðx; y; zÞ

v ðx; y; zÞ

¼ u0 ðx; yÞ þ zbx ðx; yÞ ¼

v 0 ðx; yÞ þ zby ðx; yÞ

ð1Þ

wðx; y; zÞ ¼ w0 ðx; yÞ; where u0 and v 0 are in-plane displacements of the middle surface, w0 is transverse displacement, bx and by are the rotations of the normal to the undeformed middle surface in the x-z and y-z planes respectively. The strain field is:

8 9 9 @bx @u0 > > > > > > > > 8 9 > > > > > > @x @x > > > e > > > > < x> = < = = @b @ v 0 y p fe g ¼ ey ¼ þz ; @y @y > > > > > > : > > > > > > > exy ; > > > > > > > > > @by > @b > > : @u0 þ @ v 0 > > ; > : xþ ; @y @x @y @x 8 9 @w0 > > >   > b þ < y =  h c @y e ¼ yz ¼ ; > > cxz > : bx þ @w0 > ; @x 8 > > > > > > <

ð2Þ

ð3Þ

For a thin plate, the transverse shear strain can be neglected. In the continuous Kirchhoff theory, the shear strain eh is imposed to be zero on the whole domain. This assumption required C 1 continuity for bx and by . The discrete Kirchhoff relaxes a little this strict assumption and impose eh ¼ 0 only on some discrete points of the boundary of an element. Considering a quadrilateral element with 8 nodes, the Kirchhoff assumptions are introduced

2.2. Shape function bx and by are firstly expressed in terms of a 8-node Serendipity element [32] shape functions and transitory nodal variables at corner and mid-nodes. The nodal variables at mid-nodes can be related to corner nodes by Eqs. (4) and (5) and the following other considerations: w0 ;s varies quadratically along the element side and its value at mid-node w0 ;sk can be related to the nodal values of w0 and w0 ;s ; bn varies linearly along the sides. Then the midnodes nodal variables no longer exist. bx and by can be expressed just in terms of variables at the corner nodes w0i ; bxi and byi . As to transverse displacement w0 , no interpolation function is needed to develop the stiffness matrix. However, the interpolation function is necessary to develop the mass matrix. w0 is expressed quadratically over the element in our work. The explicit expression of shape functions of w0 ; bxi and byi can be found in [30]. The in-plane displacement u0 and v 0 can be expressed independently in terms of the nodal in-plane DOF through a linear interpolation of Q4 element:

Npi ¼

bxi þ w0;xi byi þ w0;yi

)

¼

  0 0

i ¼ 1; 2; 3; 4

ð4Þ

1 ð1 þ n1 nÞð1 þ gi gÞ: 4

ð6Þ

2.3. Stress–strain relation If the x–y–z coordinate system is the same as the material orthotropic coordinate system, the stress–strain relation of an orthotropic plate is:

8 9 > > < rx = > :

(a) at the corner nodes:

(

ð5Þ

2

Q 11

Q 12

0

ry ¼ 6 4 Q 12 Q 22 0 > rxy ; 0 0 Q 66

38 > < ex 7 e 5 y > :c

xy

9 > = > ;

;

ð7Þ

where

E1 E2 ; Q 22 ¼ ; E2 2 E2 1  m12 1  m212 E1 E1 ¼ m12 Q 22 ; Q 66 ¼ G12 :

Q 11 ¼ Q 12

ð8Þ

E1 and E2 are Young’s modulus along warp and weft direction respectively. G12 is shear modulus and m12 is Poisson’s ratio. 2.4. Stiffness matrix and mass matrix The stiffness matrix and the mass matrix can be derived from the strain energy and the kinetic energy respectively:

U

K

Fig. 2. Quadrilateral element with in-plane DOF.

Z 1 heifrgdX 2 X Z 1 ¼ hei½C fegdX 2 X Z 1 T T ¼ hqe i½B ½T  ½Hk ½T ½BdetJdndgfqe g; 2 A

ð9Þ

Z 1 2 x q huifugdX 2 X Z 1 ¼ x2 qhqe i ½Bu T ½Hu ½Bu detJdndgfqe g 2 A

ð10Þ

¼

¼

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N. Li et al. / Composite Structures 152 (2016) 959–968

Fig. 3. One side stitching.

where

Z ½H k  ¼

t 2

½FðzÞT ½C ½FðzÞdz

2t

Z ½H u  ¼

t 2

2t

½F u ðzÞT ½F u ðzÞdz

2

3 1 0 0 0 z 0 0 0 4 ½FðzÞ ¼ 0 0 0 1 0 0 0 z 5; 0 1 1 0 0 z z 0 2

3 1 0 0 z 0 4 ½F u ðzÞ ¼ 0 1 0 0 z 5 0 0 1 0 0 2

j

60 6 22 ½T  ¼ 6 4 022 022

022 j 022 022

022 022 j 022

3 022 022 7 7 7 022 5 j

 hqe i ¼ u1 u2 u3 u4 v 1 v 2 v 3 v 4 w01 bx1 by1 w02 bx2 by2 w03 bx3 by3 w04 bx4 by4

Fig. 4. Distribution of stitches.



3. Experiments 3.1. Fabrication We fabricated two plates: one is unstitched composite plate and the other is stitched composite plate. Both plates are made of 10 plies of woven 5 harness satin carbon fiber fabric and epoxy resin and molded by Vacuum Assisted Resin Infusion Molding technique. After molding, the plates were kept at 80 °C for 24 h. For the stitched plate, a technique called one-side stitching (Fig. 3) was employed to stitch the dry fabrics by a robot KUKA. This technique uses two needles and a single thread. One needle which inclines 45° wears the thread and penetrates the dry fabrics; the other needle which is actually a hook will pick up the thread and takes it to another side. Each stitch is 25 mm in width. Instead of being completely covered by the stitches, the stitched plate was stitched dispersedly in order to concentrate on the property of the stitch itself and avoid involving the property between stitches. Detailed distribution of the stitches is shown in Fig. 4.

3.2. Vibration test In the vibration test, the test plate is suspended with two thin strings in order to approximate the free-free boundary conditions (Fig. 5(a)). In order to guarantee a good reflection of laser, the plate is painted white. We have chosen a combination of scanning vibrometer and shaker which are available in the our laboratory. The scanning vibrometer used is Polytec PSV-400 1D laser vibrometer. The excitation is provided by a Brüel & Kjær Type 4824 shaker driven by Brüel & Kjær Type 2732 power amplifier. The connection between the shaker and the structure is shown in Fig. 5(b). The shaker armature is connected to the a Brüel & Kjær 8001 Impedance head with a stinger made of steel (1.25 mm in radius and 70 mm in length); while the impedance head is connected to the test plate. The plate is drilled and tapped with thread and a perfect connection is guaranteed. A part of the impedance head (part below force gauge) and its connection to the plate brings an additional mass of 3.16 g (according to the supplier of the instrument) to the plate under test (4.27 kg). The result of the vibration test will not be perturbed as this additional mass is neglectable and the test frequency band that we are interested is far below the first bending resonance frequency of the stinger (1032 Hz).

N. Li et al. / Composite Structures 152 (2016) 959–968

963

Fig. 5. The setup of the vibration test.

4. Methodology The finite element model generates the calculated frequencies and mode shapes. Meanwhile, measured resonance frequencies and mode shapes can be obtained by vibration test. By observing both the measured and calculated modes shapes, the modes can be divided into several groups. Each group has its dominating parameter, which can be verified by sensitivity analysis of parameters. Within one group, the dominating parameter is updated in each iteration in order to minimize the difference between the calculated frequencies and the measured frequencies. At the same time, the Modal Assurance Criterion (MAC) is calculated to guarantee the consistency of corresponding modes. An optimized result is obtained when the difference of frequencies is small enough. The modes which is more complex (bending modes along two direction or torsion-bending mode) and involves more than one parameter will be used to validate the whole identification process and

guarantee the quality of the identification. The whole identification process is shown in Fig. 6. Among the four elastic constants of orthotropic plate under plane stress, the Poisson’s ratio varies in a very narrow interval. Its influence on the natural frequencies can be neglected in most cases (except for some specials cases which have already been excluded [33]). Thus the Poisson’s ratio is fixed to a value obtained by mechanical test in the whole identification. For unstitched plate, the Poisson’s ratio is 0:110:028 ; for stitches, it is 0:040:002 . The identification process will focus on the other three elastic constants. 5. First application—the unstitched plate 5.1. Identification based on correlation experiment-calculation The identification is realized thanks to the correlation between the result of vibration test (Section 3.2) and the result of the finite

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N. Li et al. / Composite Structures 152 (2016) 959–968

Fig. 6. The flowchart of the identification.

element model (Section 2). The result of vibration test is listed in Table 3 (f Mea ). The identification is divided into three steps: (1) Categorization of modes according to different behaviors; (2) Determination of the parameter which dominates a group; (3) Identification of the value of the dominating parameter. Through a simple observation of modes shapes coming from vibration test, the modes can be divided into the following groups: (a) torsional modes; (b) bending modes along direction X; (c) bending modes along direction Y; (d) bending mode in two direction (X, Y); (e) torsion-bending mode. The representative of each family is presented in Fig. 7. The evidences can also be found by comparing the result of vibration test with analytical solution. The bending equation of an orthotropic plate is given as [34]:

D11

@4w @4w @4w @2w þ 2ðD12 þ 2D66 Þ 2 2 þ D22 4 þ qh 2 ¼ 0 4 @x @x @y @y @t

(a) torsion mode

ð11Þ

where 3

3

D11 ¼

h E1 h E2 ; D22 ¼ 12 1  m21 m12 12 1  m21 m12

D12 ¼

h m12 E2 h ; D66 ¼ G12 12 1  m21 m12 12 3

3

As the plate is orthotropic (material coordinate system is the same as the global coordinate system E1 ¼ Ex ; E2 ¼ Ey ; G12 ¼ Gxy ; m12 ¼ mxy ) and rectangular, the solution can be written as a production of two variables, one depends only on x and the other depends only on y.

wðx; yÞ ¼ XðxÞYðyÞeiwt

ð12Þ

Under free-free boundary conditions, there are pure bending modes along just one direction. For the bending modes along direction X, the solution doesn’t depend on y. This is to say YðyÞ = C. So the solution can be written as:

wðx; yÞ ¼ CXðxÞeiwt

(b) bending modes along di-(c) bending modes along direction X

rection Y

(d) bending mode in two di- (e) torsion-bending mode rections Fig. 7. Categorization of modes.

ð13Þ

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N. Li et al. / Composite Structures 152 (2016) 959–968 Table 1 Confirmation by analytical solution.

Table 3 The obtained frequency match using the identified values.

Bending modes along direction X N° of modes

Frequencies [Hz]

2 6 10 16

34.6875 92.8125 180.3125 296.5625

2pf i ðbi LÞ2 9.7416066 9.4556855 9.3706532 9.3233358

Table 2 Analysis of sensibility of parameters in element finite model (group b).

Number of mode

E1 = 47.872 E2 = 45.143 G12 = 5.818

E1 = 47.872 E2 =40 G12 = 3

Gap%

2 6 10 16

33.66 92.58 180.39 297.67

33.65 92.39 180.22 297.34

0.04 0.20 0.10 0.11

We substitute Eq. (13) into Eq. (11) and we obtain:

@ 4 X qhx2  X¼0 @x4 D11

ð14Þ

N

f Mea

f Cal

Error %

Group

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

15,3125 34,6875 43,7500 70,9375 75,0000 92,8125 94,0625 100,3125 140,6250 180,3125 186,5625 193,4375 195,9375 210,9375 220,0000

0,24 1,37 0,69 1,32 0,45 1,26 1,82 1,53 2,43 1,69 0,80 0,35 0,16 1,02 2,63

a b d c d b d e d b d c e d d

16 17

296,5625 302,5000

2,02 2,46

b e

18

330,9375

2,70

d

19 20 21 22 23

378,4375 391,8750 404,0625 417,5000 439,6875

15,2757 34,2127 44,0504 69,9978 74,6617 93,9830 95,7767 101,8474 144,0481 183,3513 188,0524 194,1083 196,2436 213,0786 225,7899 248,8336* 302,5403 309,9337 315,8702* 339,8644 375,8196* 379,9624 393,1522 417,2004 423,1458 451,4924

0,40 0,33 3,25 1,35 2,68

d d e d b

Max error Min error Average error

3,25 1,37 1,14

The characteristic equation of Eq. (14) is:

chðbLx Þ cosðbLx Þ ¼ 1

ð15Þ *

where

sffiffiffiffiffiffiffiffiffiffiffiffiffi 2 4 qhx b¼ D11

ð16Þ

These modes are not found in experiment.

Table 4 The identified value of the elastic constants.

Finally,

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u uðb L Þ4 D xi ¼ t i x 4 11 qhLx which makes

2pf i ðbi Lx Þ2

Identified value

E1 [Gpa]

E2 [Gpa]

G12 [Gpa]

49.5

47.5

3.2

ð17Þ ¼

qffiffiffiffiffiffiffiffi D11

qhL4x

the same constants for all the bending

modes along X. If the frequencies of bending modes we have recognized conform to this condition, they are bending modes along the same direction. In Table 1, the same constant is found for the bending group along X and the classification is further confirmed. E1 h h As m12 m21  1; D11 ¼ 12  12 E1 . Eq. (17) can simplified as 1m21 m12 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 h2 xi ¼ ðbqi LLx4Þ 12 E1 , which means the bending modes along X is dom3

3

x

inated by E1 . An analysis of sensitivity of the parameters in the finite element model can also confirm this statement (Table 2). Two groups of parameters with the same E1 but different E2 and G12 are injected into the finite element model but the gap of frequencies caused by this difference is small enough to be neglected. So the bending group along direction X is dominated by E1 . The same process can be applied to the other groups (a, c). So the bending group along direction Y is dominated by E2 ; the torsional group is controlled by G12 . The bending mode along two direction (d) is controlled by two parameters (E1 and E2 ). The torsion-bending mode (e) is more complicated and may be controlled by all the parameters. Then, the parameters are identified by minimizing the gap between the experimental frequencies and the calculated frequencies of the group they dominate (G12 identified by group a, E1 by group b and E2 by group c). The group d and group e are not involved in the identification of each parameter but act as a validation. The group d is used to validate the identification of E1 and E2

while the group e is used to validate the identification of all the parameters. 5.2. Result and discussion With the method introduced in Section 5.1, the differences between the calculated frequencies and the measured frequencies diminish to an acceptable level after several iterations (Table 3). We can see that the average difference of all the 23 modes is only 1.14%. The identified constants are listed in Table 4. 6. Second application —the stitched plate 6.1. Identification based on correlation experiment-calculation The stitched plate contains three domains (Fig. 8): the unstitched part (III), the stitches along the direction X (II) and the stitches along the direction Y (I). Compared to other parts, the intersections of the stitches along two directions are very small and are considered to have the same properties as the stitch along X. As to the unstitched part, a thin layer of resin was added on the surface after molding to make the plate smoother and flatter. For the vibrometer, the measuring quality can be improved if the plate is smooth and flat. In this case, the property of the unstitched zone is considered as the homogenization of the thin resin and the optimized result of the unstitched plate. The property of the resin is E ¼ 3000 Mpa;m ¼ 0:3 while the property of the unstitched plate is listed in Table 4. The homogenized property is calculated by

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Table 5 The obtained frequency match using the identified values. N

f Mea

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 *

15 34 43 72 77 91 96 98 141 176 185 194 199 212 220 247

Error %

f Cal 15,41 34,08 43,98 69,82 74,41 93,55 95,38 100,98 142,70 180,34 185,42 193,03 195,35 211,62 223,22 246,84

2,73 0,22 2,27 3,03 3,36 2,80 0,64 3,04 1,21 2,47 0,23 0,50 1,84 0,18 1,46 0,06

Group a b d c d b d e d b d c e d d e

N 17 18 19 20 21 22 23 24 25 26

f Mea

f Cal

Error %

299,04 304,96 312,86 336,03 368,39 371,92 386,72 413,54 419,09

0,32

*

280 300 303* 322* 336 379 387 398 409 425

Max error Min error Average Error

0,01 2,80 3,90 2,83 1,11 1,39

Group – b – – d c d d e d

3.04 3,90 0.14

These modes shapes are not very clear.

Table 6 The identified value of the elastic constants.

Stitch along X

Stitch along Y

EIIx

EIIy

GIIxy

36.177

38.5

3

EIy

EIx

GIIxy

34.716

40.121

3

the classic theory of laminate. As all the modes of the plate we obtained both by vibration test and calculation are bending or torsion modes, we are interested in the bending/torsion part of the Pk¼2 1 3 3 stiffness matrix of the laminate D ¼ k¼1 ðhk  hk1 ÞQ k (Q k and 3 hk are the stiffness matrix and thickness of the kth layer). Then we can imagine a homogeneous plate who has the same thickness and same matrix D as this laminate and calculate its engineering constant inversely. We find that the property is equal to 80% of the optimized result of the unstitched plate. The property of the unstitched domain is kept unchanged during the identification process of the stitched plate. With regard to the stitched part, several assumptions are proposed between the elastic constants. Given that the ratio of aniso

III tropy of the unstitched plate is a EExIII ¼ a , we can assume the y

ratio of EIIx and EIy is also a because their difference is due to the anisotropy of the basic fabric. The moduli of the other direction EIx and EIIy conform with the same relation. The above assumptions lead to

EIIx EIy

¼ a;

EIx EIIy

¼ a:

ð18Þ

Based on Eq. (18), one relation between the four moduli can be established. The shear moduli of the stitches along the two directions are assumed to be equal:

GIxy

¼

GIIxy :

ð19Þ

Eqs. (18) and (19) are respected during the identification process. The other parts of the identification are the same as the unstitched plate. 6.2. Result and discussion The identification result of the stitched plate is summarized in Table 5, Table 6 and Fig. 9. The result of vibration test is listed in Table 5 (f Mea ). The differences of all the modes are within 5% and

Fig. 8. The partition of different domains.

the average difference is only 0.14%. The identification quality is also guaranteed by some modes at high frequencies which have complex mode shapes and sensitive to the stitch. In Fig. 9, the stitch can be seen in the mode shapes of vibration test. We can see that the stitches play an important role in the modes at high frequencies. The experiment-calculation differences of these modes are small. For example, the 25th mode at 409 Hz is approximated by the finite element model with a difference of only 1.11%. In Table 6, the fact that EIIy > EIIx et EIx > EIy can be explained by the geometry of the stitch. As shown in Fig. 3(a), the stitch left one thread along the direction in width every step. These threads became rigid bars under the embracement of resin. Unlike the thread along direction in length which can only reinforce two edges of the stitch, these threads reinforce the whole surface. 7. Conclusion An identification approach of elastic properties of composite plate from vibration test is presented in this work. This approach has been applied to two types of composite plate. For each plate, an acceptable level of difference is achieved between the calculated and measured frequencies of more than 20 modes. The shear modulus and the Young’s modulus along the width of the stitch which are difficult to obtain via mechanical test are also successfully identified by this method. In addition, to the authors’ knowledge, this method has been applied to a stitched plate for the first time. This plate is heterogeneous not only on the scale of the stitch

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Fig. 9. The experimental (first row) and corresponding calculated (second row) mode shapes of several modes.

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