epoxy unidirectional laminates by the vibration testing of plates

Composites Science and Technology 59 (1999) 2015±2024

Determination of elastic constants of glass/epoxy unidirectional laminates by the vibration testing of plates A.K. Bledzki a, A. Kessler a, R. Rikards b,*, A. Chate b a Institut fuÈr Werksto€technik, UniversitaÈt-GH-Kassel, MuÈnchebergstr. 3, Kassel D-34125, Germany Institute of Computer Analysis of Structures, Riga Technical University, Kalku iela 1, LV-1658 Riga, Latvia

b

Received 9 November 1998; received in revised form 2 March 1999; accepted 9 April 1999

Abstract Identi®cation of elastic properties of unidirectional glass/epoxy laminates from the measured eigenfrequencies has been performed. The sti€ness of the laminates has been investigated by a mixed numerical/experimental method employing the vibration test of plates. Elastic constants of laminates have been determined by using an identi®cation procedure based on experiment design, the ®nite-element method and the response-surface approach. Elastic properties of laminates with two di€erent ®bre-surface treatments have been compared. It was found that only for the transverse elastic modulus is there a statistically signi®cant di€erence between the composites with good and poor ®bre/matrix adhesion. # 1999 Elsevier Science Ltd. All rights reserved. Keywords: Composites; B. Surface treatments; C. Laminates; C. Elastic properties

1. Introduction The transverse properties of unidirectional composite laminates such as sti€ness, strength, delamination fracture toughness, etc., depend on the ®bre/matrix adhesion quality. The in¯uence of interfacial e€ects on the mechanical properties have been studied in Refs. [1±6] and others. Di€erent experimental, analytical and numerical methods have been used in order to model the interface behaviour. Among the experimental methods mainly are used such as ®bre pull-out, fragmentation, transverse tension and shear and other destructive methods. Some non-destructive methods are discussed in [5] and other papers. One non-destructive method for studying the elastic properties of laminates is the vibration test [4,7], which is based on identi®cation of elastic constants from the measured natural frequencies. During recent years investigations for developing a new technique for material identi®cation, the so-called mixed numerical±experimental technique, have started [8±19]. The determination of sti€ness parameters for complex materials such as ®bre-reinforced composites is much more complicated than for isotropic materials since composites are anisotropic and non-homogeneous. * Corresponding author. Tel.: +371-708-9264; fax: +371-782-0094. E-mail address: [email protected] (R. Rikards)

Conventional methods for determining sti€ness parameters of the composite materials are based on direct measurement of strain ®elds. Boundary e€ects, samplesize dependencies and diculties in obtaining homogeneous stress and strain ®elds are some of the most serious problems. Because of this, indirect methods have recently received increasing attention. One such indirect method is based on measurements of the structure response and application of the numerical±experimental identi®cation technique. Mixed numerical±experimental methods are sensitive for model errors because the numerical model is always based on a series of hypotheses. If the real structure does not satisfy one or more of these hypotheses, the model of the structure is evidently not appropriate. Since the development of mixed numerical±experimental techniques for material identi®cation is aimed at obtaining a practical method which yields quick and reliable results, much research has been done in order to minimise these model errors [8,20,21]. In the meantime many di€erent approaches were produced for identi®cation of the physical parameters directly characterising structural behaviour (i.e. Young's modulus and density of the material). In [11] appropriate comparisons were made between actual eigenfrequencies of an existing structure and those obtained through the ®nite-element analysis. This led to

0266-3538/99/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved. PII: S0266-3538(99)00059-7

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A.K. Bledzki et al. / Composites Science and Technology 59 (1999) 2015±2024

the identi®cation of parameters which can be used for the model calibration as well as for the detection of damaged zones in the structure. Numerical±experimental identi®cation methods are mainly used in structural applications [8±19]. For example, in [8] elastic properties of laminated composites have been identi®ed by using experimental eigenfrequencies. The sti€ness parameters were identi®ed from the measured natural frequencies of the laminated composite plate by direct minimisation of the identi®cation functional. A similar approach to identifying the sti€ness properties of the laminated composites was used in [18,19,22]. In [20] an improved plate model was used for identi®cation of elastic constants of the laminate. In [23] it was shown that the mixed numerical± experimental method can also be used for identi®cation of damping properties of polymeric composites. In the present study the numerical±experimental method for identi®cation of mechanical properties of laminated polymeric composites from the experimental results of the structure response has been further developed. The di€erence between the conventional [8,20] and the present approach is that, instead of direct minimisation of the identi®cation function, experimental design is used, by which response surfaces of the function to be minimised are obtained. The response surface approximations are obtained by using information on the behaviour of a structure in the reference points of the experiment design. Note that ®nite-element modelling of the structure is performed only in the reference points. The function to be minimised describes the di€erence between the measured and numerically calculated parameters of the response of structure. By minimising the function the identi®cation parameters were obtained. The method was employed to identify the elastic properties of the laminates from the measured eigenfrequencies of the plates. The main advantage of the present method is a signi®cant reduction of the computational e€ort. Previously this method was used for the solution of the optimum design problems [24±27]. In the present study the numerical±experimental method has been applied for identi®cation of elastic properties of glass/epoxy laminates with two di€erent ®bre-surface treatments. Experiments were performed on laminated plates and about 16 natural frequencies were measured. Elastic properties obtained from vibration tests have been compared with those from the transverse tensile test.

types of surface treatments. The ®rst type was treated by epoxy dispersion with aminosilane to promote good ®bre/matrix adhesion. More details about this ®bre treatment can be found in [28,29]. For this composite the notation EP is used. The second type was sized with polyethylene to prevent ®bre/matrix adhesion. For this composite the notation PE is used. A Ciba Geigy Ltd. Araldite epoxy system (LY556/917/DY070) with an ultimate tensile strain of 3.3% was used as the matrix material. Unidirectionally reinforced test samples were produced on a winding machine. The content of micropores was below 0.75% by volume. The glass ®bre-reinforced plates (see Fig. 1) were prepared with given geometric parameters. The total number of plies was eight in order to get a plate thickness of approximately 2 mm. The EP and PE laminates were produced by following the standard cure cycle recommended by Ciba Geigy Ltd. To improve the quality of the plates and to reduce the void content, the plates were placed in a vacuum before curing. Specimens were cut out with a diamond wheel and kept under standard conditions (23 C and 50% of relative humidity) until testing. All tests were performed in the same conditions. The ®bre content of the specimens was about 50‹1.5 vol%. The density was measured for each specimen. The specimen geometric parameters and density  for the unidirectionally (UD) reinforced PE composite plates are presented in Table 1. Similar data for the EP composite plates were given in [7]. 2.2. Vibration tests The eigenfrequencies of the test plates were measured by a real-time television (TV) holography. The samples having mean dimensions of 140 mm140 mm2 mm were hung upon two threads in order to simulate free± free boundary conditions. The sample was located in front of the holographic testing device. A piezoelectric resonator (sensor) was glued on one corner to excite the sample plate at increasing frequencies. The sensor is of

2. Experimental 2.1. Material Unidirectionally reinforced laminates were produced from E-glass ®bres (12 mm, 63 tex) with two di€erent

Fig. 1. Geometry of the unidirectionally reinforced plate.

A.K. Bledzki et al. / Composites Science and Technology 59 (1999) 2015±2024

circular shape with a diameter 25 mm located at the coordinates x ˆ a ÿ 12:5 mm and y ˆ b ÿ 12:5 mm (see Fig. 1). The mass of the sensor is ms ˆ 3:5 g. The plate is illuminated by laser light and imaged by CCD (charged couple device) array, resulting in a speckled image on the PC monitor. When the plate is deformed (excited), this interference pattern is slightly modi®ed. Digital substraction of two consecutive interference patterns yields a fringe pattern depicting the surface displacements of the plate. The nodal lines of the vibration modes can be easily identi®ed on the monitor in the form of white lines on the speckled image. The digital substraction of two consecutive pictures helps to minimise noises, such as rigid body motion of the hung plate. This measurement technique was described in more detail in [4,30]. Experiments were performed for all plates considered (see Table 1) and about 16 ¯exural eigenfrequencies were measured. In Table 2 the most frequently found experimental plate ¯exural frequencies for the PE composite are presented. The mode shapes were also recognized in the experiment. Experimentally measured frequencies for the EP composite were presented in [7]. Table 1 Geometric parameters and density of the PE composite plates 3

3

Sample

a (m)

b (m)

h  10 (m)

 (kg/m )

PE01 PE02 PE03 PE04 PE05 PE06 Mean

0.1399 0.1401 0.1400 0.1400 0.1400 0.1401 0.1400

0.1401 0.1401 0.1399 0.1401 0.1404 0.1403 0.1401

2.095 2.072 2.060 2.070 2.010 2.060 2.061

1884 1889 1890 1900 1890 1900 1892

2017

3. Parameters of identi®cation and criterion The numerical±experimental method proposed for the identi®cation of the elastic constants from the vibration tests consists of the following stages. In the ®rst stage the physical experiments are performed. Also the parameters to be identi®ed, the domain of search and criterion containing experimental data are selected. In the second stage the ®nite-element method is used in order to model the response of the structure and calculations are performed in reference points of the variables to be identi®ed. The reference points are determined by using a method of experiment design. In the third stage the numerical data obtained by the ®nite-element solution in the reference points are used in order to determine simple functions (response surfaces) for a calculation of the structure response. In the fourth stage, on the basis of the simple models and experimentally measured values of the structure response, the identi®cation of the elastic properties is performed. For this a corresponding functional is minimised by using the method of nonlinear programming. 3.1. Parameters of identi®cation Identi®cation of parameters is performed from the experimentally measured eigenfrequencies. It is assumed that the plate dimensions (see Fig. 1), plate mass and the layer stacking sequence are known. The parameters to be identi®ed are ®ve elastic constants of a transversely isotropic plate [8]: . two Young's moduli, E1 , E2 ˆ E3 . shear moduli, G12 ˆ G13

Table 2 Experimental plate ¯exural frequencies fi (Hz) for the PE composite Mode

Mode shape

Specimens

Mean

i

m, n

PE01

PE02

PE03

PE04

PE05

PE06

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

1.1 2.0 2.1 0.2 1.2 3.0 2.2 3.1 3.2 0.3 4.0 4.1 2.3 4.2 3.3 5.0

164 270 415 493 587 742 891 ± ± ± 1470 1600 1750 2050 2240 2400

162 266 413 487 584 725 886 ± ± ± 1470 1600 1735 2036 ± 2395

167 270 420 485 590 740 920 ± 1340 ± 1470 1570 ± 2040 2200 2320

170 270 430 480 590 750 915 ± ± 1340 1480 1610 1760 2100 ± 2440

173 276 419 491 587 747 910 ± ± 1320 1480 1630 1760 ± ± 2450

170 270 410 470 570 720 910 ± ± 1290 1440 1560 ± ± ± 2310

168 270 418 484 585 737 905 ± 1340 1317 1468 1595 1751 2057 2220 2386

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where !~ 1 is the ®rst numerical eigenfrequency calculated with the prior selected value of a longitudinal Young's modulus E01 (initial guess value) of the layer [8].

. Poisson's ratio, 12 ˆ 13 . shear modulus, G23

E2 ˆ 2…1 ‡ 23 †

The plate is composed of unidirectionally reinforced layers with material axis 1±2±3, where 1 is the ®bre direction and 2, 3 are directions transverse to the ®bres. The unidirectional layer is assumed as homogeneous and transversely isotropic with respect to the ®bre direction. The parameters to be identi®ed can be expressed in terms of dimensionless constants [8]   E2 ; 2 ˆ 4 ÿ 4 E1   E2 G12 0 ; 3 ˆ 1 ‡ …1 ÿ 212 † ÿ 4 E1 E1   E2 G12 4 ˆ 1 ‡ …1 ‡ 612 † ÿ 4 0 ; E1 E1 5 ˆ 4…G23 ‡ G12 †

0 E1

…1†

E2 E1

…2†

The inverse relations of Eqs. (1) are as follows

4 ÿ 3 G23 25 ÿ 0:5…8 ÿ 2 ÿ 33 ÿ 4 † ; ˆ 8 ÿ 22 E1 80

…3†

Now the vector of parameters x to be identi®ed is de®ned through dimensionless quantities i x ˆ ‰x1 ; x2 ; x3 ; x4 Š ˆ ‰2 ; 3 ; 4 ; 5 Š

…4†

These parameters can be evaluated through the identi®cation procedure using the experimental eigenfrequencies of the laminated composite rectangular plate of constant thickness h, length a and width b (see Fig. 1). Let the experimental angular eigenfrequencies be I is the designated by !1 ; !2 ; :::; !I , where ÿ
number of measured eigenfrequencies fi i !~ i ˆ 2fi . The corresponding numerical eigenfrequencies fi …!~ i ˆ 2fi † for the set of material parameters i are represented by !~ 1 ; !~ 2 ; :::; !~ I . Let us consider the scaling parameter C which is chosen according to the relation [8] Cˆ

!21 ÿ
!~ 21 E01

 … x† ˆ

h i2 I !2i ÿ C‰!~ i …x†Š2 X !4i

iˆ2

…5†

…6†

It is seen that criterion (6) is a nonlinear function of the identi®cation parameters x. The identi®cation of the elastic constants x is performed on the basis of information obtained from the measurements of the I lowest frequencies. The identi®cation problem is formulated as follows min…x†

…7†

Subject to constraints g 1 … x† ˆ 4 ÿ  2 > 0 g 2 … x† ˆ

E2 4 ÿ 2 G12 8 ÿ 2 ÿ 33 ÿ 4 ; ˆ ˆ ; E1 4 E1 160 12 ˆ

The functional to be minimised describes deviation between the measured !i and numerically calculated !~ i …x† frequencies [8]

x

where 0 ˆ 1 ÿ 212

3.2. Identi®cation functional and minimization problem

or

E2 >0 E1

8 ÿ 2 ÿ 33 ÿ 4    # > 0 or 4 ÿ 3 2 4 ÿ 2 16 1 ÿ 8 ÿ 22 4

…8†

"

…9†

G12 >0 E1 25 ÿ 0:5…8 ÿ 2 ÿ 33 ÿ 4 † g 3 … x† ˆ "    # > 0 or 4 ÿ  3 2 4 ÿ 2 8 1ÿ 8 ÿ 22 4

…10†

G23 >0 E1 r 4 ÿ 3 4 ‡ > 0 or g4 …x† ˆ ÿ 4 ÿ 2 8 ÿ 22 r E1 ÿ j12 j > 0 E2

…11†

max ; min i 4i 4i

…12†

i ˆ 2; 3; 4; 5

and the lower min limits of the idenThe upper max i i ti®cation parameters will be chosen di€erent for each numerical example of identi®cation (see below). These values determine the so called domain of interest (domain of search). Constraints (8)±(11) denote conditions of a positive de®nite elasticity matrix.

A.K. Bledzki et al. / Composites Science and Technology 59 (1999) 2015±2024

The optimum values for the dimensionless material parameters i (i ˆ 2; 3; 4; 5) that satisfy Eq. (7) and constraints (8)±(12) were obtained by using a random search method [24]. The value of Young's modulus of the layer in the ®bre direction E1 can be easily evaluated, since C and 0 are known. The steps of this evaluation are shown below. 4. Finite-element solution The eigenvalue problem for harmonic vibrations of the plate can be represented by Ku ˆ !2 Mu

…13†

Here K is the sti€ness matrix of the plate, M is the mass matrix and u is the displacement vector. The eigenvalue relation (13) for the mode u1 which corresponds to the ®rst experimental eigenfrequency !1 can be written in an equivalent form placing E1 in evidence: E1 K u1 ˆ !21 Mu1

…14†

Here E1 K ˆ K is the sti€ness matrix. Taking into account relation (5) this equation can be written as CE01 K u1 ˆ C!~ 21 Mu1

…15†

hence E1 ˆ CE01

…16†

where E01 is the initial guess value given to the Young's modulus in the ®bre direction of the layer and E1 is the corresponding identi®ed mechanical property. After evaluation of the optimum value of x the remaining mechanical properties are calculated by inverse relations (3). The eigenvalue problem (13) was solved by the subspace iteration method [31] and using a triangular ®nite element of laminated thick plate with a shear correction [32]. In order to avoid `shear locking' a selective integration technique was applied. A 2222 regular mesh (968 ®nite elements) was considered in order to achieve the necessary accuracy for at least 16 ®rst eigenvalues of the laminated plate with FFFF (all edges free) boundary conditions. 5. Method of experiment design for identi®cation problems For direct identi®cation of the elastic properties of a material it is necessary to perform a multiple iterated ®nite element procedure. Such a direct procedure was

2019

used by Mota Soares et al. [8] and also in [19]. On each iterative stage the eigenvalue problem for the linear system was solved by the ®nite element method or by the Ritz method and the nonlinear programming algorithm applied in order to minimise directly the identi®cation functional (6). Such a procedure requires very large computational e€orts. Instead of direct minimisation of the criterion (6), a method based on experiment design can be used. The selection of points in the domain of interest where the response must be evaluated is commonly called design of experiments. The choice of experiment design can have a large in¯uence on the accuracy of the approximation and the cost of constructing the response surface [33]. A commonly used experiment design is the Doptimality criterion [34]. Other experiment design methods require an a priori knowledge of the functional form of the response. This knowledge might not be available. The method of experiment design in which the a priori knowledge of the functional form of the response is not necessary was proposed in [24]. This method previously was used in the solution of optimisation problems for sandwich and laminated composite structures [25±27]. The identi®cation and optimum design problems are very similar. In both cases the solution can be obtained by minimisation of the function considered. The method of experiment design for the solution of the identi®cation problems was outlined in [7]. Further, as an example the method is shown for identi®cation of elastic constants for PE composite by the use of the mean values of plate ¯exural frequencies (see last column in Table 2). The mean values of the geometric parameters and density of the plates are taken from Table 1. Firstly the experiment design with four variables (n=4) and 35 reference points is selected. The second step is selection of the domain of interest. The upper and lower limits of the identi®cation parameters are taken as follows 5 GPa4E2 440 GPa; 5 GPa4G23 430 GPa;

5 GPa4G12 430 GPa; 0:2412 40:4

…17†

The initial guess value of Young's modulus is taken E01 =40 GPa, which is related to E1 by expression (16). After selection of the domain of interest the experiment design is employed. The matrix ij of the experiment design for n ˆ 4 and k ˆ 35 is used and the parameters of identi®cation in the reference points of the domain of interest (17) are calculated by the expression ‡ x…j i† ˆ xmin j

ÿ
1  max xj ÿ xmin Bij ÿ 1 j kÿ1

Here i ˆ 1; 2; :::; k and j ˆ 1; 2; :::; n.

…18†

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A.K. Bledzki et al. / Composites Science and Technology 59 (1999) 2015±2024

Further, in these reference points (18) the ®nite-element solution of the eigenvalue problem (13) is performed and at least 16 ®rst numerical frequencies of the PE composite are obtained for each reference point. A similar procedure was repeated also for the other plates (PE01±PE06). 6. Approximation of the response surface Techniques from experiment design and responsesurface methodology [35] are used to build the approximate models from the data in the reference points. Information on the behaviour of an object can be obtained from the computer solution in the reference points of the experiment design. The information can be represented as a data table, where the response function y…x† of the object is to be in relationship with the variables x. In our case there are four identi®cation variables representing the elastic constants of the material. The goal is, by using the data only in the reference points [in our case these data are obtained by the ®nite-element solution of the eigenvalue problem (13) in the reference points], to obtain the relation y…x† in the mathematical form or the so-called response surface. Here such mathematical models (response surfaces) have been obtained for the ®rst 16 eigenfrequencies of the laminated plate. The existing methods of regression analysis are based on the principle that the equation form is known a priori, and the problem is to ®nd coecients of the equation. However, in most cases the form of the equation must also be determined. Such a method was proposed earlier [24±27] to obtain a simple mathematical model for the structural optimisation problems. This method was brie¯y outlined and applied to the identi®cation problems in [7]. Further, the procedure to get the approximation y…x† for the ®rst natural frequency f1 of the PE composite plate is shown. In order to obtain the equation of regression, the resint program [24] was employed. In this program the perspective functions are selected by using the least-squares estimation. Then a step-by-step reduction procedure of the number of terms in the model is applied. The diagram of reduction of terms in the model for the ®rst frequency is shown in Fig. 2. It is seen that the ®rst break in the diagram corresponds to the regression equation with eight terms. The second break corresponds to the expression with four terms. This means that the model with eight terms should be chosen. On eliminating the eighth term, the correlation with the data of numerical experiment decreases more in comparison with the previous step of reduction. In the expression with four terms, only three parameters of identi®cation are presented. Thus, the simple model (response surface) for the ®rst frequency of the PE

Fig. 2. Diagram of the term reduction for the function f1 (x).

composite plate (mean values of geometric parameters and experimentally measured frequencies) is given by expression with eight terms: f1 …x† ˆ 158:5 ÿ 4:3z1 ÿ 15:44z2 ÿ 8:14z3 ‡ 0:234z4 ÿ 0:567z22 ÿ 1:43z2 z3 ÿ 0:454z1 z2

…19†

where the normalised variables zi are introduced: z1 ˆ ÿ28:5 ‡ 10x1 ; z3 ˆ ÿ6:07 ‡ 4:43x3 ;

z2 ˆ ÿ5:84 ‡ 8:67x2 ; z4 ˆ ÿ3:39 ‡ 3:83x4

…20†

Here the identi®cation parameters x are related to the elastic constants of the layer by Eqs. (1)±(3). Similar models (response surface) to expression (19) were also obtained for the other frequencies and for other plate specimens (PE01±PE06). Note that there are no general rules for the procedure of reduction of terms in the model (response-surface function) and it is necessary to acquire some experience to obtain appropriate function. Another possibility for building a model is using engineering knowledge of the true functional form of the response [36]. 7. Results of identi®cation for the PE composite Identi®cation of the elastic constants of the PE composite from the measured eigenfrequencies was performed in two di€erent ways. In the ®rst approach the identi®cation was carried out for each sample (PE01± PE06). In the second approach the mean values of the measured frequencies (see Table 2) were taken for the identi®cation. Such a method can be realised since the

A.K. Bledzki et al. / Composites Science and Technology 59 (1999) 2015±2024

geometric parameters and density of the samples (see Table 1) are almost the same for all specimens and the scatter of the measured frequencies is rather small (see Table 2). However, since frequencies are very sensitive to thickness h and density  of the plate, in the case of larger scatter of the measurements the ®rst approach must be used, i.e. the identi®cation of the elastic constants should be performed for each specimen. For the plate specimen with a sensor there are some di€erences in calculation of the mass matrix M in Eq. (13) in comparison with the plate without sensor. In order to represent more accurately the inertia forces of the plate, the mass of sensor ms should be taken into account. In the ®nite-element modelling it is assumed that the ®nite elements where the sensor is located have the same thickness h as the plate, but for these ®nite elements an equivalent density eqv is calculated: eqv ˆ  ‡

2021

points and the function (response surface) obtained by the approximation have been selected. Such selection is necessary in order to minimise the approximation errors. The vibration modes selected for identi®cation are given in Table 4. In the last row (Mean) the vibration modes selected for identi®cation from the mean values (the second approach) are presented. Veri®cation of the results was performed by the ®niteelement method (FEM) and through the independent experiments. For the ®nite-element analysis the elastic constants obtained by the identi®cation procedure were used [see Table 3, the mean values …2† ]. Results are shown in Table 5. Residuals were calculated by the expression
i ˆ

f FEM …x † ÿ f exp i i  100 f exp i

…21†

ms Fs h

Here Fs is the area of the sensor. Results of identi®cation are presented in Table 3. Utilising the ®rst way the elastic constants were identi®ed for each specimen using the data from Tables 1 and 2, and then the mean values …1† and standard deviations s…1† were calculated. In the second approach at the beginning the mean values of frequencies (see Table 2) and geometric parameters (see Table 1) were calculated. These mean values were used as input data and then the identi®cation procedure was applied. The results obtained by identi®cation utilising the second way are also given in Table 3 and denoted by …2† . In Table 3 it is seen that results for the in-plane elastic constants E1 ; E2 ; G12 and 12 obtained by both approaches are in good agreement. The transverse shear modulus G23 can not be reliable determined from the measured frequencies since the plates were too thin (h=a=1/70) for identi®cation of this property. In this case thick plates should be used. It should be noted that the number of frequencies which were selected for the identi®cation was di€erent for di€erent specimens. For the identi®cation only the modes with a high values of the coecient of correlation (correlation higher than 98%) between the data obtained by the ®nite-element solution in the reference

7.1. Comparison with independent tests It is of interest to compare the elastic constants obtained from the vibration test through the identi®cation (see Table 3) with the values obtained by the independent tension test. The same PE composite was tested in static tension transverse to the ®bres [4]. The composite also was tested in the ®bre direction. Results obtained by independent tests are compared in Table 6. For the tension test in the ®bre direction, six specimens were tested. In the tension test transverse to the ®bre direction, eight specimens were tested. The data reduction was performed according to DIN 53457 and DIN 29971, i.e. the modulus E1 was calculated as a slope between the points for the strain level 0.05 and 0.25%, and the modulus E2 was calculated as a slope between the points for the stress at 10 and at 50% of the maximum stress [4]. The sample mean and standard deviations are presented. In order to evaluate the di€erences between the values obtained by independent tests the methods of statistics should be applied. The t-test [38] for comparing mean values of two normal distributions (variances being equal) was employed. Assuming the two mean values of the test groups 1 (vibration test) and 2 (tension test)

Table 3 Elastic constants of the PE composite obtained by identi®cation Property

PE01

PE02

PE03

PE04

PE05

PE06

…1†  s…1†

…2†

E1 (GPa) E2 (GPa) G12 (GPa) G23 (GPa) 12

38.69 11.80 4.65 4.90 0.263

40.12 11.80 4.67 7.19 0.213

39.80 12.16 5.01 1.31 0.137

38.75 12.30 5.19 2.78 0.315

39.57 12.67 5.73 7.71 0.358

36.16 11.99 5.26 ± 0.246

38.84‹1.44 12.12‹0.33 5.09‹0.41 4.78‹2.76 0.255‹0.077

39.15 12.16 5.11 2.06 0.269

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A.K. Bledzki et al. / Composites Science and Technology 59 (1999) 2015±2024

Table 4 Vibration modes used for identi®cation Plate

PE01 PE02 PE03 PE04 PE05 PE06 Mean

Mode i 1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

+ + + + + + +

+ + + + ÿ + +

+ + + + + ÿ +

ÿ ÿ ÿ ÿ + + +

+ + + + + + +

+ + + + ÿ + +

+ + ÿ ÿ ÿ ÿ +

ÿ ÿ ÿ ÿ ÿ ÿ ÿ

ÿ ÿ ÿ ÿ ÿ ÿ ÿ

ÿ ÿ ÿ ÿ ÿ ÿ ÿ

ÿ ÿ ÿ ÿ ÿ ÿ ÿ

+ + + + + + +

+ + ÿ + + ÿ +

+ + + + ÿ ÿ +

ÿ ÿ ÿ ÿ ÿ ÿ +

ÿ ÿ ÿ ÿ ÿ ÿ ÿ

Table 5 Flexural frequencies fi (Hz) (i ˆ 1; 2; :::; 16) and residuals
i for PE composite Mode i

FEM

Exp.


i (%)

Used in identi®cation

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

168 270.5 418 477 585 737 905 ± 1340 1317 1468 1595 1751 2057 2220 2386

167.9 266.4 430.7 484.5 588.3 740.7 887.6 919.6 1382 1320 1449 1588 1737 2073 2232 2337

0.06 1.37 3.04 0.10 0.56 0.50 1.31 ± 3.13 0.23 1.29 0.44 0.80 0.78 0.54 2.06

+ + + + + + + ÿ ÿ ÿ ÿ + + + + ÿ

Table 6 Elastic constants for Pe composite obtained by independent tests Property

Tension test [4]

Vibration test

E1 (GPa) E2 (GPa)

39.50‹1.18 8.47‹1.76

38.84‹1.44 12.12‹0.33

are equal, then the sample means x1 and x2 will not di€er signi®cantly at some con®dence level if j x2 ÿ x 1 j s   2 …n1 ÿ 1†s1 ‡ …n2 ÿ 1†s22 1 1 ‡ n1 ‡ n2 ÿ 2 n1 n2 < t=2; n1 ‡ n2 ÿ 2

…22†

Here, s1 , s2 are the standard deviations and n1 , n2 are the number of samples in the specimen groups 1 and 2, respectively. On the right hand side of Eq. (22) t=2; n1 ‡ n2 ÿ 2 is the Student's coecient for the signi®cance

level  on n1 ‡ n2 ÿ 2 degrees of freedom. These values can be obtained from the t-tables [38]. Using the t-test for comparing the vibration test results for E1 (group 1, number of samples n1 ˆ 6) and the tension test results (group 2, number of samples n2 ˆ 6) the expression on the left hand side of Eq. (22) is 0:868 < t0:025;10 ˆ 2:228. For calculations the 95% con®dence level was assumed, i.e. signi®cance level  ˆ 0:05. Therefore, the results of t-test shows that for E1 there is no statistically signi®cant di€erence between the values obtained by the tension and vibration tests (the t-test passes). Utilising the t-test for comparing the vibration test results for E2 (group 1, number of samples n1 ˆ 6) and the tension test results (group 2, number of samples n2 ˆ 8) according Eq. (22) the t-test fails, i.e. the expression on the left hand side of Eq. (22) is 11:62 > t0:025;12 ˆ 2:179. So, the t-test shows that there is a statistically signi®cant di€erence between the values obtained by the transverse tension and vibration test (the t-test fails). For the transverse modulus E2 the values obtained in the vibration test are higher. The reason is that in the vibration test a dynamic (storage) modulus is measured. Since the epoxy matrix is a viscoelastic material the storage modulus could be higher than the static one. The in¯uence of the viscoelastic behaviour is more signi®cant in the direction transverse to the ®bres. Since the glass ®bres can be assumed as an elastic material the in¯uence of the viscoelastic behaviour of the matrix in the ®bre direction is smaller. Other reason of di€erences is that in the static tensile test after few loading steps due to poor ®bre/matrix adhesion a break in the stress/strain curve can be observed (see Fig. 3). For this test the modulus E2 was calculated as a slope between the points for the stress at 10 and at 50% of the maximum stress. Due to the nonlinearity in the stress/strain diagram the value of E2 calculated in such a way is rather a secant modulus than modulus of elasticity. For the material in which ®bre/ matrix debonding occurs in the ®rst few steps of loading it is dicult to measure the transverse modulus of elasticity in the static test. On the other hand for the

A.K. Bledzki et al. / Composites Science and Technology 59 (1999) 2015±2024

2023

Table 7 Comparison of di€erent properties

Fig. 3. Typical stress/strain diagram in transverse tension for PE composite.

Property

Test

EP composite

PE composite

t-test (22)

E1 (GPa) E1 (GPa) E2 (GPa) E2 (GPa) G12 (GPa)

Vibration Tension [4] Vibration Tension [4] Vibration

39.03‹1.14 38.45‹1.62 12.69‹0.20 10.35‹0.69 5.33‹0.23

38.84‹1.44 39.50‹1.18 12.12‹0.33 8.47‹1.76 5.09‹0.41

Pass Pass Fail Fail Pass

compared (see Table 7). For the all elastic constants statistics are available and according to Eq. (22) the ttest has been performed. Results of the t-test are presented in the last column. Comparing the composites PE and EP it can be concluded that for both the vibration and the tension test there is no statistically signi®cant di€erence between the values E1 (the t-test passes). The same can be concluded for the inplane shear modulus G12 . At the same time for the sti€ness property transverse to the ®bre direction E2 for both the vibration and the transverse tension test there is a statistically signi®cant di€erence between the composites PE and EP (the t-test fails). The transverse sti€ness for the PE composite with poor ®bre/matrix adhesion is lower. 9. Conclusions

Fig. 4. Typical stress/strain diagram in transverse tension for EP composite.

material with good ®bre/matrix adhesion (EP composite) the stress/strain curve is nearly linear till ultimate stress (see Fig. 4) and the value of E2 can be measured in the static test [4]. For the EP composite the E2 modulus obtained in the vibration test [7] is also about 20% higher than modulus obtained in the static tensile test. This could be due to the in¯uence of viscoelasticity in the dynamic test. 8. Comparison of elastic properties for PE and EP composites In order to compare the laminates with good ®bre/ matrix adhesion (EP composite) and poor ®bre/matrix adhesion (PE composite), di€erent material properties can be used. For example, in [4] the transverse strength and in [37] the interlaminar fracture toughness properties have been compared for the same composites considered here. In the present paper elastic properties are

The main advantage of the present identi®cation method is signi®cant reduction (about 50±100 times) of the computational e€orts in calculating the numerical frequencies, which participate in the functional to be minimised. Another advantage of the identi®cation method used is that all elastic constants are determined only from one vibration test by using a plate sample. Thus, the material is not destroyed by cutting samples in order to determine di€erent elastic constants. Comparing elastic properties for the composites with good (EP composite) and poor (PE composite) ®bre/matrix adhesion it was shown that the transverse sti€ness for the PE composite is lower. The t-test shows that there is a statistically signi®cant di€erence for E2 values obtained in the vibration and the static tension test. Acknowledgements The investigations concerning the development of the identi®cation method for elastic properties of laminated composites were sponsored through Grant 96.0504. The authors are pleased to acknowledge the ®nancial support by the Latvian Council of Science. Thanks are also due to German Federal Ministry of Science and Technology (Grant WTZ LET010.97) and Latvian Ministry of Education and Science (Grant 6292/98) for their generous support of this paper.

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