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Determination of elastic constants of materials by vibration testing
Composite Structures 49 (2000) 183±190
www.elsevier.com/locate/compstruct
Determination of elastic constants of materials by vibration testing Shun-Fa Hwang *, Chao-Shui Chang Department of Mechanical Engineering, National Yunlin University of Science & Technology, 123 University Road, Section 3, Touliu 640, Taiwan, ROC
Abstract Vibration testing, combining with numerical method, is a potential alternative approach for determining elastic constants of materials because of its nondestructive characteristic, single test, and producing average properties. In order to simplify the modeling processes and to reduce complicated derivation in the numerical method, the method of combining ®nite element analysis and optimum design is adopted to inversely determine the elastic constants in this work. Since both the ®nite element analysis and optimum design can be executed in a commercial program, the present method is easier to apply and very ¯exible. In this work, the present method is veri®ed by comparing with other methods, and it is applied to determine the elastic constants of aluminum and carbon/epoxy materials. The results indicate that the present method can obtain very accurate elastic constants for the both materials and for both the thin and thick plates. It is shown that the boundary conditions with all edges free are preferred, and the dierent dimensions and dierent stacking sequences have negligible eects on the determination of elastic constants of the materials. Ó 2000 Elsevier Science Ltd. All rights reserved. Keywords: Vibration testing; Elastic constant; Finite element analysis; Optimum design; Composite
1. Introduction Composites, which consist of two or more dierent materials, have the characteristics of high modulus/ weight and strength/weight ratios, excellent fatigue properties, and noncorroding behavior. These advantages encourage the extensive application of composites, for example, in sports and aerospace. The understanding of the mechanical behavior of composites is essential for the design and application. For a transversely isotropic composite, it needs ®ve elastic constants to describe the linear elastic stress±strain relationship. If the problem is two-dimensional, four constants are necessary. The most common method to determine these constants is static testing due to its simplicity. For composite materials, three types of specimens with dierent stacking sequences, which may be 008 , 0016 , and 450 2S , are required in this method [1]. In addition, boundary eects, nonuniform stress/strain ®elds, and localized data are its drawbacks that make this method not so attractive. An alternative approach for determining these constants is to combine the vibration testing and the nu*
Corresponding author. Tel.: +886-5-5342601-4143; fax: +886-55312062. E-mail address: [email protected] (S.-F. Hwang).
merical method. Some dynamic properties are obtained from vibration testing, and the elastic constants in a numerical model are updated such that the outputs from the numerical method ®t the results from vibration testing. By this approach, the elastic constants can be determined in a single test, and they represent the global properties of a structure panel, not just local values. Also this approach is a nondestructive method, which may be suitable for quality control. Because vibrating beam tests can provide only two of these constants, the vibration of plates is always considered. Thin plates have been focused to determine the four elastic constants. The Rayleigh±Ritz technique combined with a least-squares method is commonly chosen to model the dynamic behavior of a plate and to extract the elastic constants. Deobald and Gibson [2] invested a thin plate with dierent boundary conditions and discovered that a plate with all boundaries free can obtain better results than that with one or more ®xed edges. Ayorinde and Gibson [3] obtained the four independent elastic constants of a freely supported rectangular thin plate made from orthotropic materials with orthotropy ratio from one to thirteen. Lai and Lau [4] extended the approach to deal with a generally orthotropic plate. McIntyre and Woodhouse [5] identi®ed both elastic and damping constants of thin orthotropic plates by measuring and analyzing the low modes of
0263-8223/00/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 3 - 8 2 2 3 ( 9 9 ) 0 0 1 3 2 - 4
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vibration. De Visscher et al. [6,7] also obtained the stiness and damping properties of orthotropic composite plates by comparing experimental modal parameters and the corresponding results from a numerical model in combination with the modal strain energy method. Fallstrom et al. [8,9] used a real-time TV-holography system to obtain the modes of vibration. In addition to the Rayleigh±Ritz technique for modeling the vibration of plates, the superposition method was applied by Moussu and Nivoit [10] and ®nite element analysis was used by Fallstrom and Jonsson [8]. To determine all the ®ve independent elastic constants for transversely isotropic materials, thick plates are used for vibration testing. Since the eects of outplane shear modulus (or out-plane Poisson's ratio) are more evident in thick plates, one needs to include these eects in describing their vibration behavior. Frederiksen [11] used single-layer plate theories, including the eects of transverse shear, to analyze the vibration of thick symmetrically laminated rectangular plates with all edges free. Ayorinde [12] incorporated through-thethick shear and rotatory inertia in his method of obtaining the three-dimensional elastic constants of a completely free orthotropic plate from experimental plate vibration data. Frederiksen [13] identi®ed the elastic constants of thick orthotropic plates by the consideration of higher mode natural frequencies linked with a numerical model based on a higher-order shear deformation theory. Rayleigh's method [5,9,12,14,15], the Rayleigh±Ritz technique [2,3,11,13], and ®nite element analysis [8] are the three methods usually used for modelling the dynamic behavior of a plate. Rayleigh's method is easily implemented, but is too inaccurate. Finite element analysis is seldom used because it is time consuming. The Rayleigh±Ritz technique is commonly chosen, because it provides good accuracy and can be executed in a minicomputer. However, the accuracy and convergence of this technique are greatly dependent on the choice of admissible functions in the series representing the unknown functions in the displacement ®eld. If the boundary conditions or the shape of the plate is changed, admissible functions may be dierent such that it may be dicult to make a suitable choice. Furthermore, to obtain the natural frequencies of a plate by this technique, one needs to do complicated derivation and to do a lot of calculations. From these points, ®nite element analysis is a suitable choice to simplify the processes of modelling and to reduce the complicated derivation, since the analysis can be executed in a commercial code. That the ®nite element analysis is time consuming is trivial nowadays because of the high calculation capacity of personal computers. Also it can be extended to three-dimensional problems and can easily handle complex shapes and boundary conditions of plates. Hence, ®nite element
analysis is adopted to model the vibration of a plate in the present work. Instead of a least-squares method, the method of optimum design, which is also provided by a commercial ®nite element code, is chosen to ®nd the elastic constants. Thus this method is very helpful for technicians because they can obtain the elastic constants in a single program without considering the complicated derivation and calculation. 2. Experimental procedures Impulse technique [16,17] is adopted to do vibration testing, and modal analysis is used to extract the natural frequencies and mode shapes of plates tested. Since a full modal analysis is very time consuming, only natural frequencies are measured if the corresponding mode types are sure. The experimental set-up for impulse technique is shown in Fig. 1. The plate tested is put on a soft cotton pad to approximate the boundary conditions of all edges free. If ®xed boundary condition is considered, one edge of the plate is clamped on a ®xed heavy body by three C clamps. The impulse excitation is accomplished by a PCB impulse hammer with a built-in force transducer. The response is detected by an accelerometer (PCB A353B68), whose mass is less than 2g such that its eect is negligible. Both the input and output signals are converted to frequency domains by fast Fourier transformation in a two-channel signal analyzer, and then the frequency response functions are created. Totally, about 35 points are excited under this ®xed response technique. After that, those frequency response functions are exported to a modal analysis program, SMS STARModal [18], in a personal computer. A polynomial function is curve ®tted to the frequency response function data within a band under the assumption of single degree of freedom. After curve ®tting all frequency response functions, one can obtain the natural frequencies, damping ratios, and mode shapes by the SMS STARModal. The materials tested are aluminum and graphite/epoxy composites. To fabricate the carbon/epoxy plates, the bag molding process and a two-stage curing cycle [19] are used. 3. Plate vibration model A commercially available ®nite element code (ANSYS 5.3) is utilized to model the undamped free vibration of a plate. The natural frequencies and mode shapes of the plates are determined by modal analysis in the ANSYS program [20]. The whole domain is meshed by the shell elements (SHELL93 or SHELL99) that have eight nodes with six degrees of freedom at each node. SHELL93 is used in thin block materials, and SHELL99 is a special design for thin layered structures. Both ele-
S.-F. Hwang, C.-S. Chang / Composite Structures 49 (2000) 183±190
185
Fig. 1. Experimental set-up of vibration testing.
ments can be applied to isotropic, orthotropic, or anisotropic materials. In order to perform a quicker and cheaper analysis, the direction perpendicular to the plane of the plate is assigned as master degrees of freedom that are signi®cant degrees of freedom characterizing the dynamic behavior of the structure. Because of the assignment of master degrees of freedom, a plate with all free boundary conditions, which is the same as in the experiments, can be easily handled in the ANSYS.
4. Optimum design A problem of optimum design can be stated as follows: Minimize
F F X
1
Subject to gi X 6 gi ;
i 1; 2; 3; . . . ; m1 ;
2
hi 6 hi X ;
i 1; 2; 3; . . . ; m2 ;
3
xi 6 xi 6 xi ;
i 1; 2; 3; . . . ; m3 ;
4
where m1 ; m2 and m3 are constants. Underbar and overbar represent lower and upper bounds, respectively. The objective function, F, as well as the state variables, gi and hi , may be linear or nonlinear functions of the design variables xi . The vector of design variables is indicated as
X x1 ; x2 ; x3 ; . . . ; xm3 :
5
The optimization module of the ANSYS program can be employed to determine the optimum design. A subproblem approximation method (SAM) is chosen to address the mathematical problem stated above. Since it requires only the values of the dependent variables (objective function and state variables) and not their derivatives, this method of optimization can be described as an advanced, zero-order method. The objective function is ®rst approximated by a fully quadratic representation with cross terms by means of leastsquares ®tting, and the state variables are handled in the similar manner except that only a quadratic ®t is used. Then, the constrained minimization problem is converted to an unconstrained problem using penalty functions. The search for a minimum of the approximated and penalized function (called the subproblem) is carried out by applying a sequential unconstrained minimization technique at each iteration. The ®nal step performed in each design iteration is the determination of the design variable vector to be used in the next iteration j 1. Vector X j1 is determined according to the following equation: ÿ 6 X j1 X b C X j ÿ X b ; where X b is the best design set and C is a constant chosen to vary between 0 and 1.
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mization method, and initial values and loops may aect the accuracy of the obtained elastic constants. Detail discussion of these eects has been done in the thesis of Chang [21]. According his recommendation under the consideration of accuracy and executing time, the ®nite element mesh along the in-plane dimensions is 10 ´ 10, which has total 100 shell elements in the mesh. Six vibration modes for objective function, state variables constrained to between 95% and 105% of their experimental values, and subproblem approximation method are suggested. Additionally, one can use average values to avoid the scatter in results due to dierent initial values, and do-loops may be not necessary. To verify this inverse determination method with the chosen parameters described above, an aluminum plate discussed in the paper of Deobald and Gibson [2] is considered. The dimension of the square aluminum plate is 25:4 cm 25:4 cm 0:316 cm, and the density is 2:77 g=cm3 . The Young's modulus of this isotropic plate is 72:4 GPa, the shear modulus 28 GPa, and the Poisson's ratio 0.33. Deobald and Gibson treated this plate as a transversely isotropic material and tried to obtain four elastic constants by a modal analysis/Rayleigh±Ritz technique. The natural frequencies obtained by impulse technique and modal analysis are shown in Table 1. In addition, it also shows the natural frequencies obtained by both the Rayleigh±Ritz technique and ®nite element method. In both methods the actual elastic constants has been used. Averagely the natural frequencies from these three methods are very close. However, there are still some frequencies cannot well match, and this may cause inherent errors in the methods to inversely determine elastic constants. Five dierent sets of initial values are put in the present inverse determination method. As shown in Table 2, Very small initial values are used in the ®rst set and very large initial values are used in the ®fth set. It is clear that dierent initial values still cause some scatters on the results. However, these results are all close enough to the correct ones and also better than the average values predicted by Deobald and Gibson. If one wants to minimize the scatters, average values from the results of dierent initial values are suggested to use. In the previous cases four elastic constant are sought. Since the aluminum plate is an isotropic material, two elastic constants are enough. If only two elastic constants are determined, the results are compared with those with four elastic constants in Table 3. Cases 1±3 in
5. Procedures to determine elastic constants To determine the elastic constants of a material by this method, one needs the natural frequencies of the plate made by this material. These frequencies obtained by vibration testing are used to de®ne the objective function. 2 n X fi ÿ fi ; 7 F fi i1 where fi is the natural frequency obtained by vibration testing, fi the natural frequency obtained by ®nite element analysis, and n the number of modes used in the object function. The square is chosen to make the objective function positive. The elastic constants are assigned as design variables and restrained to be positive. The number of design variable is dependent on the material and can be two, four, ®ve, or more. The state variables used to con®ne the domain for the design variables are chosen from the natural frequencies. The suitable elastic constants can be found, if the initial values of those constants are guessed. Using this set of initial values, the ®nite element analysis is executed, and the corresponding frequencies are obtained. In order to establish the relation between the objective function and the design variables, several sets of design variables and the corresponding frequencies will be automatically created and calculated. After that, the problem of optimum design is solved for the next best elastic constants. Once a new set of optimum design is generated, the approximate relationship between the objective function and the design variables is updated. This process is iterated until the conditions of convergence are satis®ed. The problem is said to be converged if the change in objective function or all design variables from the best feasible design to the current design is less than their respective tolerances or if the change in objective function or all design variables between the last two designs is less than their respective tolerances.
6. Results and discussion 6.1. Veri®cation of the method In the present method, the ®nite element mesh, the number of modes, constraints on state variables, optiTable 1 Natural frequencies of aluminum plate Frequency
f1
f2
f3
f4
f5
f6
FEM (10 ´ 10) Experiment Rayleigh±Ritz
161.2 156.7 163.2
231.9 232.5 237.6
294.6 300.4 299.9
413.7 411.7 424.3
413.7 411.7 424.3
728.6 744.9 749.4
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187
Table 2 Elastic constant from dierent initial values for aluminum plate Initial values
a
E1 (GPa) E2 (Gpa) G12 (GPa) m12
0.96 0.90 0.65 0.35 Results
25 24 3 0.2
65 55 21 0.30
85 81 30 0.32
960 900 650 0.35
E1 (GPa) E2 (Gpa) G12 (GPa) m12
70.1 67.4 24.8 0.39
72.3 75.9 26.9 0.35
72.3 73.9 28.3 0.31
73.9 75.6 25.7 0.33
72.5 73.7 25.9 0.40
Average value
Papera value
Ref. value
72.2 ()0.3%) 73.3 (1.2%) 26.3 (6.1%) 0.356 (7.3%)
69.5 (4.0%) 69.9 (3.5%) 25.6 (8.6%) 0.361 (9.4%)
72.4 72.4 28.0 0.33
Averaged results from the paper of Deobald and Gibson [2].
Table 3 Two or four elastic constants obtained for aluminum plate Case
1
2
3
Ref. value
No. of constants
2
4
2
4
2
4
E1 (GPa) E2 (GPa) G12 (GPa) m12
72.9
71.6 65.3 24.7 0.40
75.5
72.1 70.0 28.0 0.28
75.5
76.7 72 25.4 0.32
0.30
0.34
this table represent dierent initial values used, and they are assigned the same values for both the cases of two and four elastic constants. It is evident that the results with only two elastic constants are very accurate and the executing time is much less. Hence, if the materials tested are sure to be isotropic, seeking only two elastic constants should be preferred. However, even though four elastic constants are sought, the obtained results are also very good. These four elastic constants approximately satisfy the following conditions for isotropic materials. E1 E2 ;
8
G12 E1 =2 1 m12 :
9
0.30
72.4 72.4 28.0 0.33
Therefore, the present inverse determination method with four elastic constants is applicable for both the isotropic and transversely isotropic materials. 6.2. Aluminum plates To verify the whole process of the present method, including the vibration testing and the inverse determination method, two aluminum plates are chosen. The dimensions and densities of these two plates are shown in Table 4. The F±F±F±F of A1 specimen represents all free boundary conditions in the plate, and the C±F±F±F of A2 specimen denotes one ®xed boundary condition and the other three edges free. The six natural
Table 4 Dimensions of aluminum and carbon/epoxy plates Length (cm)
Width (cm)
Thickness (cm)
Density (g=cm3 )
Aluminum A1 (F±F±F±F) A2 (C±F±F±F)
30 28
28 25
0.3 0.3
2.7 2.7
Carbon/epoxy B1 B2 C1 C2 D1 D2 E1 E2 E3
28.15 24.5 26 24.3 26 24.3 10.1 10.2 10
24.55 15.3 25.9 15.2 26 15.3 5.3 5.35 5.2
0.2 0.2 0.34 0.34 0.32 0.32 0.35 0.33 0.32
1.54 1.54 1.54 1.54 1.59 1.59 1.55 1.58 1.59
Stacking sequence
0 0 20 0020 0=908 =0 s 0=908 =0 s 0=908 s 0=90 0 8 s 0 20 0=908 =0 s 0=908 s
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Table 5 Natural frequencies of aluminum and carbon/epoxy plates f1
f2
f3
f4
f5
f6
Aluminum A1 (F±F±F±F) A2 (C±F±F±F)
114 39
167 91
218 237
289 261
300 337
497 545
Carbon/epoxy B1 B2 C1 C2 D1 D2
60 111 98 170 102 183
82 202 310 385 311 370
143 290 336 510 342 535
218 310 360 890 364 870
232 370 390 955 390 944
257 555 590 1030 612 1050
Table 6 Elastic constants of aluminum plate A1 A2 Ref. value
E1 (GPa)
E2 (GPa)
G12 (GPa)
m12
71.4 (3.5%) 68.8 ()0.3%) 63.2 ()8.5%) 67 ()2.9%) 69
64.9 ()5.9%)
24.67 ()4.9%)
70.5 (2.2%)
25.1 ()3.2%)
69
25.94
0.28 ()15%) 0.32 ()3%) 0.365 (10.6%) 0.32 ()3%) 0.33
frequencies for both types of boundary conditions obtained by vibration testing are shown in Table 5. From these frequencies, four or two elastic constants obtained by the present inverse determination method are presented in Table 6. The reference values in Table 6 are from metal handbook [22]. For both the A1 and A2 specimens, determining just two elastic constants is easier and more accurate than determining four elastic constants, even though the four constants predicted are good enough and approximately satis®ed the isotropic conditions. Comparing both the A1 and A2 cases, the results from A1 are superior to those from A2, especially in E1 , in which direction the A2 specimen is clamped. This clamped condition suppresses the response normal to the direction along the clamped edge such that the E1 value in A2 specimen is less. This also happens for the composite materials. Hence, free boundary conditions are recommended for determining elastic constants from vibration testing. 6.3. Carbon/epoxy plates From the above discussion, plates with all edge free are adopted to determine elastic constants for carbon/ epoxy composites. The dimensions, densities, and the stacking sequences of the specimens are listed in Table 4. The B type specimens are unidirectional 20-layer ®ber reinforced composite plates, which are 0020 . The stacking sequence of the C type specimens is =0 , and that for the D type specimens is 0=90 8 s 0=908 s . The dierence between these two symmetric types of specimens is that the type C specimens have one
more ply than the type D specimens. The B1 specimen has the dimensions approximating to square plate, and the B2 specimen is a rectangular plate. It is also similar in the dimensions for the type C and D specimens. All these specimens are considered as thin plates because the length/thickness values of the specimens are larger than about a value between 30 and 50. For type E specimens, this condition is not satis®ed, and they can be considered as thick plates. The dimensions and stacking sequences of E1, E2 and E3 are listed in Table 4. The natural frequencies obtained by vibration testing for carbon/epoxy thin plates are shown in Table 5, and the elastic constants determined by the present method are listed in Table 8. The reference values in Table 8 are from static testing. As shown in Table 8, the elastic constants from the approximately square plate and the rectangular plate for the type B, C, and D specimens do not have much dierence. Hence, the eects from the changes in plate dimensions upon the elastic constants determined by the present method can be neglected. Furthermore, the dierence in stacking sequences between the three types of specimens also cannot cause much variation in the elastic constants obtained by the present method. Even though the variation in natural frequencies from dierent response points in the type C and D specimens is reduced compared to that in the type B specimens, the elastic constants obtained show no clear dependence on the stacking sequences. For these six specimens, the determined elastic constants are in good agreement with the results from static testing, except for the major PoissonÕs ratios. The PoissonÕs ratios predicted by the present method are higher than those
S.-F. Hwang, C.-S. Chang / Composite Structures 49 (2000) 183±190
189
Table 7 Natural frequencies of thick carbon/epoxy plates Carbon/epoxy E1 E2 E3
f1
f2
f3
f4
f5
f6
f7
f8
f9
f10
1670 1230 1240
3480 2200 2170
3860 3300 3310
4400 5990 5350
5100 6880 6880
6300 7100 7030
8800 7530 7530
9900 8760 8910
10 500 11 480 11 340
11 000 11 580 11 710
Table 8 Elastic constants of carbon/epoxy plates B1 B2 C1 C2 D1 D2 Static test E1 E2 E3
E1 (GPa)
E2 (GPa)
G12 (GPa)
m12
G23 (GPa)
125.5 128.7 121.4 123.9 123.7 120.2 121.2 120 125 120
8.55 8.1 7.87 8.27 9.96 9.96 9.34 9.6 9.5 9.0
5.4 6.05 5.37 4.73 6.85 6.85 6.25 5.7 6.0 7.1
0.345 0.335 0.317 0.317 0.35 0.35 0.23 0.21 0.3 0.3
2.9 3.0 2.5
from static testing. This cannot be all attributed from the errors of the vibration testing and the present method, because it is also possible from the errors in the static testing. In order to determine the ®ve independent elastic constants for transversely isotropic materials, thick plates with dierent stacking sequences, which are E1, E2, and E3 specimens, are also invested. Since frequencies of higher modes are more aected by transverse shear deformation than those of lower modes, ten natural frequencies shown in Table 7 are identi®ed by vibration testing. The corresponding elastic constants predicted are shown in Table 8. Similarly, the elastic constants do not show clear dependence on the stacking sequence. The predicted in-plane elastic constants are also close to those from static testing and the thin plates. In addition, for thick plates one can also determine G23 , from which m23 can be calculated by the following equation: G23 E2 =2 1 m23 :
10
Therefore, the three-dimensional stress±strain relation can be established. Since there is no G23 values from static testing, it is dicult to judge the value from present method. However, one can reference the value of carbon/epoxy composites from the paper of Frederiksen [13], although his materials are dierent from the present materials. The G23 value of FrederiksenÕs material is 2.97 GPa, which was determined from fourteen natural frequencies, and the value of the present material is from 2.5 to 3.0 GPa. Hence, the G23 value predicted by the present method is reasonable, even though it is determined from only ten frequencies. Generally, the higher frequencies after ten are dicult to obtain under normal
testing instruments. Therefore, the present method is competitive with other methods in determining the ®ve independent elastic constants for thick plates. 7. Conclusion The combination method of ®nite element analysis and optimum design has been presented in this work to obtain the elastic constants of materials from their natural frequencies, which are measured by vibration testing. Under suitable selection of parameters in the present combination method, it is proved to be a fast and accurate method. The elastic constants of aluminum and carbon/epoxy composite materials are predicted by the present method. It is proved that the present method can obtain accurate elastic constants for both materials. For aluminum, it is also shown that the boundary conditions with all edges free can obtain better results, and seeking two elastic constants is a fast and accurate way for isotropic materials. For carbon/epoxy plates, the elastic constants have no clear dependency on the specimen dimensions and the stacking sequences. Furthermore, the present method can reasonably determine the out-of-plane shear modulus by using less number of higher frequencies. References [1] Carlsson LA, Pipes RB. Experimental characterization of advanced composite materials. New York: Prentice-Hall, 1987. [2] Deobald LR, Gibson RF. Determination of elastic constants of orthotropic plates by a modal analysis/Rayleigh±Ritz technique. J Sound Vib 1988;124:269±83.
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[3] Ayorinde EO, Gibson RF. Elastic constants of orthotropic composite materials using plate resonance frequencies classical lamination theory and an optimized three-mode Rayleigh formulation. Comp Eng 1993;3:395±407. [4] Lai TC, Lau TC. Determination of elastic constants of a generally orthotropic plate by modal analysis. Int J Anal Exp Modal Anal 1993;8:15±33. [5] McIntyre ME, Woodhouse J. On measuring the elastic and damping constants of orthotropic sheet materials. Acta Metallurgica 1988;36:1397±416. [6] DeVisscher J, Sol H, DeWilde WP, Vantomme J. Identi®cation of the damping properties of orthotropic composite materials using a mixed numerical experimental method. Appl Comp Mater 1997;4:13±33. [7] DeVisscher J, Sol H, Maton W, DeWilde WP, Vantomme J. Identi®cation of the temperature dependent complex moduli of composite materials using a mixed numerical experimental method. Proceedings of the 1998 Sixth International Conference on Computer Methods in Composite Materials, CADCOMP'98, Montreal, 1998;181±90. [8] Fallstrom KE, Jonsson M. A nondestructive method to determine material properties in anisotropic plates. Polymer Comp 1991;12:293±305. [9] Fallstrom KE. Determining material properties in anisotropic plates using Rayleigh's method. Polymer Comp 1991;12:306±14. [10] Moussu F, Nivoit M. Determination of elastic constants of orthotropic plates by a modal analysis/method of superposition. J Sound Vib 1993;165:149±63.
[11] Frederiksen PS. Single-layer plate theories applied to the ¯exural vibration of completely free thick laminates. J Sound Vib 1995;186:743±59. [12] Ayorinde EO. Elastic constants of thick orthotropic composite plates. J Comp Mater 1995;29:1025±39. [13] Frederiksen PS. Experimental procedure and results for the identi®cation of elastic constants of thick orthotropic plates. J Comp Mater 1997;31:360±82. [14] Dickinson SM. The buckling and frequency of ¯exural vibration of rectangular isotropic and orthotropic plates using Rayleigh's method. J Sound Vib 1978;61:1±8. [15] Kim CS, Dickinson SM. Improved approximate expressions for the natural frequencies of isotropic and orthotropic rectangular plates. J Sound Vib 1985;103:142±9. [16] Edwins DJ. Modal testing: theory and practice. Research Studies Press, 1986. [17] McConnell KG. Vibration testing. New York: Wiley, 1995. [18] The STAR reference manual. Structural Measurement Systems, 1994. [19] Mallick PK. Fiber-reinforced composites: materials manufacturing and design. New York: Marcel Dekker, 1993. [20] The ANSYS user's manual. Swanson Analysis Systems Inc, 1995. [21] Chang CS. The Determination of elastic constants of materials by a vibration testing and optimum design method, M.S. thesis. Department of Mechanical Engineering, National Yunlin University of Science & Technology. [22] Metals handbook. Vol. 2 ± Properties and selection: nonferrous alloys and pure metals, ninth ed. American Society for Metals, 1979.
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Determination of elastic constants of materials by vibration testing Shun-Fa Hwang *, Chao-Shui Chang Department of Mechanical Engineering, National Yunlin University of Science & Technology, 123 University Road, Section 3, Touliu 640, Taiwan, ROC
Abstract Vibration testing, combining with numerical method, is a potential alternative approach for determining elastic constants of materials because of its nondestructive characteristic, single test, and producing average properties. In order to simplify the modeling processes and to reduce complicated derivation in the numerical method, the method of combining ®nite element analysis and optimum design is adopted to inversely determine the elastic constants in this work. Since both the ®nite element analysis and optimum design can be executed in a commercial program, the present method is easier to apply and very ¯exible. In this work, the present method is veri®ed by comparing with other methods, and it is applied to determine the elastic constants of aluminum and carbon/epoxy materials. The results indicate that the present method can obtain very accurate elastic constants for the both materials and for both the thin and thick plates. It is shown that the boundary conditions with all edges free are preferred, and the dierent dimensions and dierent stacking sequences have negligible eects on the determination of elastic constants of the materials. Ó 2000 Elsevier Science Ltd. All rights reserved. Keywords: Vibration testing; Elastic constant; Finite element analysis; Optimum design; Composite
1. Introduction Composites, which consist of two or more dierent materials, have the characteristics of high modulus/ weight and strength/weight ratios, excellent fatigue properties, and noncorroding behavior. These advantages encourage the extensive application of composites, for example, in sports and aerospace. The understanding of the mechanical behavior of composites is essential for the design and application. For a transversely isotropic composite, it needs ®ve elastic constants to describe the linear elastic stress±strain relationship. If the problem is two-dimensional, four constants are necessary. The most common method to determine these constants is static testing due to its simplicity. For composite materials, three types of specimens with dierent stacking sequences, which may be 008 , 0016 , and 450 2S , are required in this method [1]. In addition, boundary eects, nonuniform stress/strain ®elds, and localized data are its drawbacks that make this method not so attractive. An alternative approach for determining these constants is to combine the vibration testing and the nu*
Corresponding author. Tel.: +886-5-5342601-4143; fax: +886-55312062. E-mail address: [email protected] (S.-F. Hwang).
merical method. Some dynamic properties are obtained from vibration testing, and the elastic constants in a numerical model are updated such that the outputs from the numerical method ®t the results from vibration testing. By this approach, the elastic constants can be determined in a single test, and they represent the global properties of a structure panel, not just local values. Also this approach is a nondestructive method, which may be suitable for quality control. Because vibrating beam tests can provide only two of these constants, the vibration of plates is always considered. Thin plates have been focused to determine the four elastic constants. The Rayleigh±Ritz technique combined with a least-squares method is commonly chosen to model the dynamic behavior of a plate and to extract the elastic constants. Deobald and Gibson [2] invested a thin plate with dierent boundary conditions and discovered that a plate with all boundaries free can obtain better results than that with one or more ®xed edges. Ayorinde and Gibson [3] obtained the four independent elastic constants of a freely supported rectangular thin plate made from orthotropic materials with orthotropy ratio from one to thirteen. Lai and Lau [4] extended the approach to deal with a generally orthotropic plate. McIntyre and Woodhouse [5] identi®ed both elastic and damping constants of thin orthotropic plates by measuring and analyzing the low modes of
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vibration. De Visscher et al. [6,7] also obtained the stiness and damping properties of orthotropic composite plates by comparing experimental modal parameters and the corresponding results from a numerical model in combination with the modal strain energy method. Fallstrom et al. [8,9] used a real-time TV-holography system to obtain the modes of vibration. In addition to the Rayleigh±Ritz technique for modeling the vibration of plates, the superposition method was applied by Moussu and Nivoit [10] and ®nite element analysis was used by Fallstrom and Jonsson [8]. To determine all the ®ve independent elastic constants for transversely isotropic materials, thick plates are used for vibration testing. Since the eects of outplane shear modulus (or out-plane Poisson's ratio) are more evident in thick plates, one needs to include these eects in describing their vibration behavior. Frederiksen [11] used single-layer plate theories, including the eects of transverse shear, to analyze the vibration of thick symmetrically laminated rectangular plates with all edges free. Ayorinde [12] incorporated through-thethick shear and rotatory inertia in his method of obtaining the three-dimensional elastic constants of a completely free orthotropic plate from experimental plate vibration data. Frederiksen [13] identi®ed the elastic constants of thick orthotropic plates by the consideration of higher mode natural frequencies linked with a numerical model based on a higher-order shear deformation theory. Rayleigh's method [5,9,12,14,15], the Rayleigh±Ritz technique [2,3,11,13], and ®nite element analysis [8] are the three methods usually used for modelling the dynamic behavior of a plate. Rayleigh's method is easily implemented, but is too inaccurate. Finite element analysis is seldom used because it is time consuming. The Rayleigh±Ritz technique is commonly chosen, because it provides good accuracy and can be executed in a minicomputer. However, the accuracy and convergence of this technique are greatly dependent on the choice of admissible functions in the series representing the unknown functions in the displacement ®eld. If the boundary conditions or the shape of the plate is changed, admissible functions may be dierent such that it may be dicult to make a suitable choice. Furthermore, to obtain the natural frequencies of a plate by this technique, one needs to do complicated derivation and to do a lot of calculations. From these points, ®nite element analysis is a suitable choice to simplify the processes of modelling and to reduce the complicated derivation, since the analysis can be executed in a commercial code. That the ®nite element analysis is time consuming is trivial nowadays because of the high calculation capacity of personal computers. Also it can be extended to three-dimensional problems and can easily handle complex shapes and boundary conditions of plates. Hence, ®nite element
analysis is adopted to model the vibration of a plate in the present work. Instead of a least-squares method, the method of optimum design, which is also provided by a commercial ®nite element code, is chosen to ®nd the elastic constants. Thus this method is very helpful for technicians because they can obtain the elastic constants in a single program without considering the complicated derivation and calculation. 2. Experimental procedures Impulse technique [16,17] is adopted to do vibration testing, and modal analysis is used to extract the natural frequencies and mode shapes of plates tested. Since a full modal analysis is very time consuming, only natural frequencies are measured if the corresponding mode types are sure. The experimental set-up for impulse technique is shown in Fig. 1. The plate tested is put on a soft cotton pad to approximate the boundary conditions of all edges free. If ®xed boundary condition is considered, one edge of the plate is clamped on a ®xed heavy body by three C clamps. The impulse excitation is accomplished by a PCB impulse hammer with a built-in force transducer. The response is detected by an accelerometer (PCB A353B68), whose mass is less than 2g such that its eect is negligible. Both the input and output signals are converted to frequency domains by fast Fourier transformation in a two-channel signal analyzer, and then the frequency response functions are created. Totally, about 35 points are excited under this ®xed response technique. After that, those frequency response functions are exported to a modal analysis program, SMS STARModal [18], in a personal computer. A polynomial function is curve ®tted to the frequency response function data within a band under the assumption of single degree of freedom. After curve ®tting all frequency response functions, one can obtain the natural frequencies, damping ratios, and mode shapes by the SMS STARModal. The materials tested are aluminum and graphite/epoxy composites. To fabricate the carbon/epoxy plates, the bag molding process and a two-stage curing cycle [19] are used. 3. Plate vibration model A commercially available ®nite element code (ANSYS 5.3) is utilized to model the undamped free vibration of a plate. The natural frequencies and mode shapes of the plates are determined by modal analysis in the ANSYS program [20]. The whole domain is meshed by the shell elements (SHELL93 or SHELL99) that have eight nodes with six degrees of freedom at each node. SHELL93 is used in thin block materials, and SHELL99 is a special design for thin layered structures. Both ele-
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185
Fig. 1. Experimental set-up of vibration testing.
ments can be applied to isotropic, orthotropic, or anisotropic materials. In order to perform a quicker and cheaper analysis, the direction perpendicular to the plane of the plate is assigned as master degrees of freedom that are signi®cant degrees of freedom characterizing the dynamic behavior of the structure. Because of the assignment of master degrees of freedom, a plate with all free boundary conditions, which is the same as in the experiments, can be easily handled in the ANSYS.
4. Optimum design A problem of optimum design can be stated as follows: Minimize
F F X
1
Subject to gi X 6 gi ;
i 1; 2; 3; . . . ; m1 ;
2
hi 6 hi X ;
i 1; 2; 3; . . . ; m2 ;
3
xi 6 xi 6 xi ;
i 1; 2; 3; . . . ; m3 ;
4
where m1 ; m2 and m3 are constants. Underbar and overbar represent lower and upper bounds, respectively. The objective function, F, as well as the state variables, gi and hi , may be linear or nonlinear functions of the design variables xi . The vector of design variables is indicated as
X x1 ; x2 ; x3 ; . . . ; xm3 :
5
The optimization module of the ANSYS program can be employed to determine the optimum design. A subproblem approximation method (SAM) is chosen to address the mathematical problem stated above. Since it requires only the values of the dependent variables (objective function and state variables) and not their derivatives, this method of optimization can be described as an advanced, zero-order method. The objective function is ®rst approximated by a fully quadratic representation with cross terms by means of leastsquares ®tting, and the state variables are handled in the similar manner except that only a quadratic ®t is used. Then, the constrained minimization problem is converted to an unconstrained problem using penalty functions. The search for a minimum of the approximated and penalized function (called the subproblem) is carried out by applying a sequential unconstrained minimization technique at each iteration. The ®nal step performed in each design iteration is the determination of the design variable vector to be used in the next iteration j 1. Vector X j1 is determined according to the following equation: ÿ 6 X j1 X b C X j ÿ X b ; where X b is the best design set and C is a constant chosen to vary between 0 and 1.
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mization method, and initial values and loops may aect the accuracy of the obtained elastic constants. Detail discussion of these eects has been done in the thesis of Chang [21]. According his recommendation under the consideration of accuracy and executing time, the ®nite element mesh along the in-plane dimensions is 10 ´ 10, which has total 100 shell elements in the mesh. Six vibration modes for objective function, state variables constrained to between 95% and 105% of their experimental values, and subproblem approximation method are suggested. Additionally, one can use average values to avoid the scatter in results due to dierent initial values, and do-loops may be not necessary. To verify this inverse determination method with the chosen parameters described above, an aluminum plate discussed in the paper of Deobald and Gibson [2] is considered. The dimension of the square aluminum plate is 25:4 cm 25:4 cm 0:316 cm, and the density is 2:77 g=cm3 . The Young's modulus of this isotropic plate is 72:4 GPa, the shear modulus 28 GPa, and the Poisson's ratio 0.33. Deobald and Gibson treated this plate as a transversely isotropic material and tried to obtain four elastic constants by a modal analysis/Rayleigh±Ritz technique. The natural frequencies obtained by impulse technique and modal analysis are shown in Table 1. In addition, it also shows the natural frequencies obtained by both the Rayleigh±Ritz technique and ®nite element method. In both methods the actual elastic constants has been used. Averagely the natural frequencies from these three methods are very close. However, there are still some frequencies cannot well match, and this may cause inherent errors in the methods to inversely determine elastic constants. Five dierent sets of initial values are put in the present inverse determination method. As shown in Table 2, Very small initial values are used in the ®rst set and very large initial values are used in the ®fth set. It is clear that dierent initial values still cause some scatters on the results. However, these results are all close enough to the correct ones and also better than the average values predicted by Deobald and Gibson. If one wants to minimize the scatters, average values from the results of dierent initial values are suggested to use. In the previous cases four elastic constant are sought. Since the aluminum plate is an isotropic material, two elastic constants are enough. If only two elastic constants are determined, the results are compared with those with four elastic constants in Table 3. Cases 1±3 in
5. Procedures to determine elastic constants To determine the elastic constants of a material by this method, one needs the natural frequencies of the plate made by this material. These frequencies obtained by vibration testing are used to de®ne the objective function. 2 n X fi ÿ fi ; 7 F fi i1 where fi is the natural frequency obtained by vibration testing, fi the natural frequency obtained by ®nite element analysis, and n the number of modes used in the object function. The square is chosen to make the objective function positive. The elastic constants are assigned as design variables and restrained to be positive. The number of design variable is dependent on the material and can be two, four, ®ve, or more. The state variables used to con®ne the domain for the design variables are chosen from the natural frequencies. The suitable elastic constants can be found, if the initial values of those constants are guessed. Using this set of initial values, the ®nite element analysis is executed, and the corresponding frequencies are obtained. In order to establish the relation between the objective function and the design variables, several sets of design variables and the corresponding frequencies will be automatically created and calculated. After that, the problem of optimum design is solved for the next best elastic constants. Once a new set of optimum design is generated, the approximate relationship between the objective function and the design variables is updated. This process is iterated until the conditions of convergence are satis®ed. The problem is said to be converged if the change in objective function or all design variables from the best feasible design to the current design is less than their respective tolerances or if the change in objective function or all design variables between the last two designs is less than their respective tolerances.
6. Results and discussion 6.1. Veri®cation of the method In the present method, the ®nite element mesh, the number of modes, constraints on state variables, optiTable 1 Natural frequencies of aluminum plate Frequency
f1
f2
f3
f4
f5
f6
FEM (10 ´ 10) Experiment Rayleigh±Ritz
161.2 156.7 163.2
231.9 232.5 237.6
294.6 300.4 299.9
413.7 411.7 424.3
413.7 411.7 424.3
728.6 744.9 749.4
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187
Table 2 Elastic constant from dierent initial values for aluminum plate Initial values
a
E1 (GPa) E2 (Gpa) G12 (GPa) m12
0.96 0.90 0.65 0.35 Results
25 24 3 0.2
65 55 21 0.30
85 81 30 0.32
960 900 650 0.35
E1 (GPa) E2 (Gpa) G12 (GPa) m12
70.1 67.4 24.8 0.39
72.3 75.9 26.9 0.35
72.3 73.9 28.3 0.31
73.9 75.6 25.7 0.33
72.5 73.7 25.9 0.40
Average value
Papera value
Ref. value
72.2 ()0.3%) 73.3 (1.2%) 26.3 (6.1%) 0.356 (7.3%)
69.5 (4.0%) 69.9 (3.5%) 25.6 (8.6%) 0.361 (9.4%)
72.4 72.4 28.0 0.33
Averaged results from the paper of Deobald and Gibson [2].
Table 3 Two or four elastic constants obtained for aluminum plate Case
1
2
3
Ref. value
No. of constants
2
4
2
4
2
4
E1 (GPa) E2 (GPa) G12 (GPa) m12
72.9
71.6 65.3 24.7 0.40
75.5
72.1 70.0 28.0 0.28
75.5
76.7 72 25.4 0.32
0.30
0.34
this table represent dierent initial values used, and they are assigned the same values for both the cases of two and four elastic constants. It is evident that the results with only two elastic constants are very accurate and the executing time is much less. Hence, if the materials tested are sure to be isotropic, seeking only two elastic constants should be preferred. However, even though four elastic constants are sought, the obtained results are also very good. These four elastic constants approximately satisfy the following conditions for isotropic materials. E1 E2 ;
8
G12 E1 =2 1 m12 :
9
0.30
72.4 72.4 28.0 0.33
Therefore, the present inverse determination method with four elastic constants is applicable for both the isotropic and transversely isotropic materials. 6.2. Aluminum plates To verify the whole process of the present method, including the vibration testing and the inverse determination method, two aluminum plates are chosen. The dimensions and densities of these two plates are shown in Table 4. The F±F±F±F of A1 specimen represents all free boundary conditions in the plate, and the C±F±F±F of A2 specimen denotes one ®xed boundary condition and the other three edges free. The six natural
Table 4 Dimensions of aluminum and carbon/epoxy plates Length (cm)
Width (cm)
Thickness (cm)
Density (g=cm3 )
Aluminum A1 (F±F±F±F) A2 (C±F±F±F)
30 28
28 25
0.3 0.3
2.7 2.7
Carbon/epoxy B1 B2 C1 C2 D1 D2 E1 E2 E3
28.15 24.5 26 24.3 26 24.3 10.1 10.2 10
24.55 15.3 25.9 15.2 26 15.3 5.3 5.35 5.2
0.2 0.2 0.34 0.34 0.32 0.32 0.35 0.33 0.32
1.54 1.54 1.54 1.54 1.59 1.59 1.55 1.58 1.59
Stacking sequence
0 0 20 0020 0=908 =0 s 0=908 =0 s 0=908 s 0=90 0 8 s 0 20 0=908 =0 s 0=908 s
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Table 5 Natural frequencies of aluminum and carbon/epoxy plates f1
f2
f3
f4
f5
f6
Aluminum A1 (F±F±F±F) A2 (C±F±F±F)
114 39
167 91
218 237
289 261
300 337
497 545
Carbon/epoxy B1 B2 C1 C2 D1 D2
60 111 98 170 102 183
82 202 310 385 311 370
143 290 336 510 342 535
218 310 360 890 364 870
232 370 390 955 390 944
257 555 590 1030 612 1050
Table 6 Elastic constants of aluminum plate A1 A2 Ref. value
E1 (GPa)
E2 (GPa)
G12 (GPa)
m12
71.4 (3.5%) 68.8 ()0.3%) 63.2 ()8.5%) 67 ()2.9%) 69
64.9 ()5.9%)
24.67 ()4.9%)
70.5 (2.2%)
25.1 ()3.2%)
69
25.94
0.28 ()15%) 0.32 ()3%) 0.365 (10.6%) 0.32 ()3%) 0.33
frequencies for both types of boundary conditions obtained by vibration testing are shown in Table 5. From these frequencies, four or two elastic constants obtained by the present inverse determination method are presented in Table 6. The reference values in Table 6 are from metal handbook [22]. For both the A1 and A2 specimens, determining just two elastic constants is easier and more accurate than determining four elastic constants, even though the four constants predicted are good enough and approximately satis®ed the isotropic conditions. Comparing both the A1 and A2 cases, the results from A1 are superior to those from A2, especially in E1 , in which direction the A2 specimen is clamped. This clamped condition suppresses the response normal to the direction along the clamped edge such that the E1 value in A2 specimen is less. This also happens for the composite materials. Hence, free boundary conditions are recommended for determining elastic constants from vibration testing. 6.3. Carbon/epoxy plates From the above discussion, plates with all edge free are adopted to determine elastic constants for carbon/ epoxy composites. The dimensions, densities, and the stacking sequences of the specimens are listed in Table 4. The B type specimens are unidirectional 20-layer ®ber reinforced composite plates, which are 0020 . The stacking sequence of the C type specimens is =0 , and that for the D type specimens is 0=90 8 s 0=908 s . The dierence between these two symmetric types of specimens is that the type C specimens have one
more ply than the type D specimens. The B1 specimen has the dimensions approximating to square plate, and the B2 specimen is a rectangular plate. It is also similar in the dimensions for the type C and D specimens. All these specimens are considered as thin plates because the length/thickness values of the specimens are larger than about a value between 30 and 50. For type E specimens, this condition is not satis®ed, and they can be considered as thick plates. The dimensions and stacking sequences of E1, E2 and E3 are listed in Table 4. The natural frequencies obtained by vibration testing for carbon/epoxy thin plates are shown in Table 5, and the elastic constants determined by the present method are listed in Table 8. The reference values in Table 8 are from static testing. As shown in Table 8, the elastic constants from the approximately square plate and the rectangular plate for the type B, C, and D specimens do not have much dierence. Hence, the eects from the changes in plate dimensions upon the elastic constants determined by the present method can be neglected. Furthermore, the dierence in stacking sequences between the three types of specimens also cannot cause much variation in the elastic constants obtained by the present method. Even though the variation in natural frequencies from dierent response points in the type C and D specimens is reduced compared to that in the type B specimens, the elastic constants obtained show no clear dependence on the stacking sequences. For these six specimens, the determined elastic constants are in good agreement with the results from static testing, except for the major PoissonÕs ratios. The PoissonÕs ratios predicted by the present method are higher than those
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189
Table 7 Natural frequencies of thick carbon/epoxy plates Carbon/epoxy E1 E2 E3
f1
f2
f3
f4
f5
f6
f7
f8
f9
f10
1670 1230 1240
3480 2200 2170
3860 3300 3310
4400 5990 5350
5100 6880 6880
6300 7100 7030
8800 7530 7530
9900 8760 8910
10 500 11 480 11 340
11 000 11 580 11 710
Table 8 Elastic constants of carbon/epoxy plates B1 B2 C1 C2 D1 D2 Static test E1 E2 E3
E1 (GPa)
E2 (GPa)
G12 (GPa)
m12
G23 (GPa)
125.5 128.7 121.4 123.9 123.7 120.2 121.2 120 125 120
8.55 8.1 7.87 8.27 9.96 9.96 9.34 9.6 9.5 9.0
5.4 6.05 5.37 4.73 6.85 6.85 6.25 5.7 6.0 7.1
0.345 0.335 0.317 0.317 0.35 0.35 0.23 0.21 0.3 0.3
2.9 3.0 2.5
from static testing. This cannot be all attributed from the errors of the vibration testing and the present method, because it is also possible from the errors in the static testing. In order to determine the ®ve independent elastic constants for transversely isotropic materials, thick plates with dierent stacking sequences, which are E1, E2, and E3 specimens, are also invested. Since frequencies of higher modes are more aected by transverse shear deformation than those of lower modes, ten natural frequencies shown in Table 7 are identi®ed by vibration testing. The corresponding elastic constants predicted are shown in Table 8. Similarly, the elastic constants do not show clear dependence on the stacking sequence. The predicted in-plane elastic constants are also close to those from static testing and the thin plates. In addition, for thick plates one can also determine G23 , from which m23 can be calculated by the following equation: G23 E2 =2 1 m23 :
10
Therefore, the three-dimensional stress±strain relation can be established. Since there is no G23 values from static testing, it is dicult to judge the value from present method. However, one can reference the value of carbon/epoxy composites from the paper of Frederiksen [13], although his materials are dierent from the present materials. The G23 value of FrederiksenÕs material is 2.97 GPa, which was determined from fourteen natural frequencies, and the value of the present material is from 2.5 to 3.0 GPa. Hence, the G23 value predicted by the present method is reasonable, even though it is determined from only ten frequencies. Generally, the higher frequencies after ten are dicult to obtain under normal
testing instruments. Therefore, the present method is competitive with other methods in determining the ®ve independent elastic constants for thick plates. 7. Conclusion The combination method of ®nite element analysis and optimum design has been presented in this work to obtain the elastic constants of materials from their natural frequencies, which are measured by vibration testing. Under suitable selection of parameters in the present combination method, it is proved to be a fast and accurate method. The elastic constants of aluminum and carbon/epoxy composite materials are predicted by the present method. It is proved that the present method can obtain accurate elastic constants for both materials. For aluminum, it is also shown that the boundary conditions with all edges free can obtain better results, and seeking two elastic constants is a fast and accurate way for isotropic materials. For carbon/epoxy plates, the elastic constants have no clear dependency on the specimen dimensions and the stacking sequences. Furthermore, the present method can reasonably determine the out-of-plane shear modulus by using less number of higher frequencies. References [1] Carlsson LA, Pipes RB. Experimental characterization of advanced composite materials. New York: Prentice-Hall, 1987. [2] Deobald LR, Gibson RF. Determination of elastic constants of orthotropic plates by a modal analysis/Rayleigh±Ritz technique. J Sound Vib 1988;124:269±83.
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[3] Ayorinde EO, Gibson RF. Elastic constants of orthotropic composite materials using plate resonance frequencies classical lamination theory and an optimized three-mode Rayleigh formulation. Comp Eng 1993;3:395±407. [4] Lai TC, Lau TC. Determination of elastic constants of a generally orthotropic plate by modal analysis. Int J Anal Exp Modal Anal 1993;8:15±33. [5] McIntyre ME, Woodhouse J. On measuring the elastic and damping constants of orthotropic sheet materials. Acta Metallurgica 1988;36:1397±416. [6] DeVisscher J, Sol H, DeWilde WP, Vantomme J. Identi®cation of the damping properties of orthotropic composite materials using a mixed numerical experimental method. Appl Comp Mater 1997;4:13±33. [7] DeVisscher J, Sol H, Maton W, DeWilde WP, Vantomme J. Identi®cation of the temperature dependent complex moduli of composite materials using a mixed numerical experimental method. Proceedings of the 1998 Sixth International Conference on Computer Methods in Composite Materials, CADCOMP'98, Montreal, 1998;181±90. [8] Fallstrom KE, Jonsson M. A nondestructive method to determine material properties in anisotropic plates. Polymer Comp 1991;12:293±305. [9] Fallstrom KE. Determining material properties in anisotropic plates using Rayleigh's method. Polymer Comp 1991;12:306±14. [10] Moussu F, Nivoit M. Determination of elastic constants of orthotropic plates by a modal analysis/method of superposition. J Sound Vib 1993;165:149±63.
[11] Frederiksen PS. Single-layer plate theories applied to the ¯exural vibration of completely free thick laminates. J Sound Vib 1995;186:743±59. [12] Ayorinde EO. Elastic constants of thick orthotropic composite plates. J Comp Mater 1995;29:1025±39. [13] Frederiksen PS. Experimental procedure and results for the identi®cation of elastic constants of thick orthotropic plates. J Comp Mater 1997;31:360±82. [14] Dickinson SM. The buckling and frequency of ¯exural vibration of rectangular isotropic and orthotropic plates using Rayleigh's method. J Sound Vib 1978;61:1±8. [15] Kim CS, Dickinson SM. Improved approximate expressions for the natural frequencies of isotropic and orthotropic rectangular plates. J Sound Vib 1985;103:142±9. [16] Edwins DJ. Modal testing: theory and practice. Research Studies Press, 1986. [17] McConnell KG. Vibration testing. New York: Wiley, 1995. [18] The STAR reference manual. Structural Measurement Systems, 1994. [19] Mallick PK. Fiber-reinforced composites: materials manufacturing and design. New York: Marcel Dekker, 1993. [20] The ANSYS user's manual. Swanson Analysis Systems Inc, 1995. [21] Chang CS. The Determination of elastic constants of materials by a vibration testing and optimum design method, M.S. thesis. Department of Mechanical Engineering, National Yunlin University of Science & Technology. [22] Metals handbook. Vol. 2 ± Properties and selection: nonferrous alloys and pure metals, ninth ed. American Society for Metals, 1979.