Benchmark solutions of the free vibration of simply supported laminated composite plates

Xinwei Wang1

1State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, No. 29 Yudao Street, Nanjing, 210016, China

1Corresponding author

Mathematical Models in Engineering, Vol. 3, Issue 2, 2017, p. 98-105. https://doi.org/10.21595/mme.2017.19257
Received 28 September 2017; accepted 10 October 2017; published 31 December 2017

Copyright © 2017 JVE International Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Abstract.

Due to the high specific strength and stiffness, laminated composite plates, especially mid-plane symmetric laminated composite plates, are frequently used as structural component in aeronautical and aerospace engineering. Since obtaining analytic solutions are difficult even for simply supported mid-plane symmetric laminated composite plates, numerical methods have to be used to obtain approximate solutions. To evaluate various numerical methods, benchmark solutions are needed. In this article, highly accurate frequencies of simply supported angle-ply mid-plane symmetric laminated composite plates with two sets of equivalent material properties are obtained by the modified differential quadrature method and presented to serve as the benchmark solutions.

Keywords: laminated composite plate, equivalent material properties; benchmark solution, free vibration; modified differential quadrature method.

1. Introduction

Due to the high specific strength and stiffness, laminated composite plates are frequently used as structural component in aeronautical and aerospace engineering. Their static, buckling and free vibration behavior is of important to the designers and thus has been received great attentions [1, 2]. Among various types of laminations, the mid-plane symmetric laminates are widely used in practice. The de-coupling of in-plane and out-of-plane deformation makes the production of a flat plate as well as analysis much simpler than the general laminations.

In free vibration analysis, the laminated composite plates are usually equivalent to anisotropic plates. Analytical solutions are rarely available even for rectangular anisotropic plates with simple supported boundary conditions. Therefore, various approximate approaches [1-4] and numerical methods [5-9] have been employed for solutions.

In literature, two equivalent ways of expressing the material properties are commonly used. Take the E-glass/epoxy (E/E) material as an example, the material properties expressed in one way, called the material system I (MS-I), are E1= 60.7 GPa, E2= 24.8 GPa, G12= 12.0 GPa and ν12= 0.23 [3, 8, 10], and the material properties expressed in the other way, called the material system II (MS-II), are E1/E2= 2.45, G12/E2= 0.48 and ν12= 0.23 [4, 6, 7]. Researchers often do not distinguish one set of material properties from the other since they are regarded equivalent. The choice mainly depends on their personal preference. In references [4, 6, 7], the results are obtained based on the MS-II, but compared with the upper bound solutions with the MS-I [3]. In reference [8], the material properties of MS-I are given, but the data are actually obtained with the ones with MS-II. Occasionally this might cause mis-understanding to the readers, although it is not difficult to tell that MS-II is actually used in their calculation by looking at the exact solutions of special orthotropic rectangular plates, since the exact solutions for the two sets of equivalent material systems are slightly different and the corresponding Ritz solutions of special orthotropic plates reported in [3] are also exact solutions [10]. The difference in solutions for the laminated composite plates with the two sets of equivalent material properties is really small and negligible from the practical point of view.

From the computational point of view, however, the small difference may be important in testing the accuracy and efficiency of new numerical methods. Very accurate benchmark solutions are required in such cases. The data reported in [3, 4] are not very accurate due to either the lower rate of convergence of the method or the extra constraints implicitly enforced in the test functions [11]. More terms in the series of the test functions are needed to obtained solutions with higher accuracy by using the Ritz method.

The primary objective of this paper is to provide highly accurate benchmark frequencies for simply supported square laminated composite plates with two sets of equivalent material properties. The modified differential quadrature method proposed by the author is used to obtain accurate solutions. The slight difference in the frequencies of the mid-plane symmetric laminates with two sets of equivalent material systems is clearly demonstrated.

2. Basic equations and solution procedures

2.1. Governing equation and expression of boundary condition

Denote the length, width and total thickness of the rectangular laminated composite plate by a, b, and h. The governing equation for the free vibration analysis of a mid-plane symmetric laminated composite plate is given by:

(1)
D - 11 4 w x 4 + 4 D - 16 4 w x 3 y + 2 D - 12 + 2 D - 66 4 w x 2 y 2 + 4 D - 26 4 w x y 3 + D - 22 4 w y 4 = ρ h ω 2 w ,

where D-ij are the effective bending and twisting stiffness [12], wx,y is the deflection, ρ and ω are the mass density and circular frequency, respectively.

The expressions of simply supported boundary conditions are:

(2)
w = 0 ,           M x = 0 x = 0 , a , w = 0 ,           M y = 0 y = 0 , b ,

where the expressions of bending moments Mx and My are:

(3)
M x = D - 11 2 w x 2 + 2 D - 16 2 w x y + D - 12 2 w y 2 , M y = D - 12 2 w x 2 + 2 D - 26 2 w x y + D - 22 2 w y 2 .

2.2. Modified differential quadrature method and solution procedures

For completeness considerations, the modified differential quadrature method (modified DQM) and solution procedures are briefly introduced.

Denote Nx and Nythe numbers of grid points in x and y directions, and (xi, yj) (i= 1, 2,..., Nx; j= 1, 2,..., Ny) the grid points. In the modified DQM, two additional derivative degrees of freedom at end points are introduced by using the method of modification of weighting coefficient-3 (MMWC-3) proposed by the author [9].

For simplicity and demonstration of the method, take a one-dimensional problem as an example. In the ordinary differential quadrature method, the first order derivative of the solution wx with respect to x at grid point xi is approximated as:

(4)
d w d x x = x i = j = 1 N x A i j x w j ,           i = 1,2 , . . . , N x ,

where Aijx is called the weighting coefficient, which can be explicitly computed by:

(5)
A i j x = k = 1 , k i , j N x x i - x k k = 1 , k j N x x j - x k ,           i j , k = 1 , k i N x 1 x i - x k ,             i = j ,           i , j = 1,2 , . . . , N x .

To apply multiple boundary conditions rigorously, two additional degrees of freedom (DOFs) are introduced during formulation the weighting coefficient of the second order derivatives at two end points by using the MMWC-3 [9], namely:

(6)
d 2 w d x 2 x = x i = k = 1 N x A i k x d w d x x = x k = k = 2 N x - 1 A i k x j = 1 N x A k j x w j + A i 1 x w 1 ' + A i N x w N '
          = j = 1 N x B ~ i j x w j + A i 1 x w 1 ' + A i N x w N ' = j = 1 N + 2 B ~ i j x δ j ,             i = 1 ,   N x ,

where δj=wj (j=1, 2, ..., Nx), δN+1=w1', δN+2=w'N, B~i(N+1)x=Ai1x, B~i(N+2)x=AiNx.

At all inner points, the weighting coefficients of the second order derivative are the same as the ordinary DQM, namely:

(7)
B ~ i j x = B i j x = k = 1 N x A i k x A k j x ,           i = 2,3 , . . . , N x - 1 ,         j = 1,2 , . . . , N x , B ~ i j x = 0 ,           i = 2,3 , . . . , N x - 1 ,         j = N x + 1 , N x + 2 , B i j x = k = 1 N x A i k x A k j x ,             B ~ i j x = k = 2 N x - 1 A i k x A k j x ,           i = 1 , N x ,         j = 1,2 , . . . , N x .

In the modified DQM, the weighting coefficients of the third and the fourth order derivatives, denoted by C~ijx, D~ijx are computed by:

(8)
C ~ i j x = k = 1 N x A i k x B ~ k j x ,       i = 1,2 , . . . , N x ,       j = 1,2 , . . . , N x + 2 ,
(9)
D ~ i j x = k = 1 N x B i k x B ~ k j x ,       i = 1,2 , . . . , N x ,       j = 1,2 , . . . , N x + 2 .

The weighting coefficients of the first to fourth-order derivatives with respect to y can be calculated in a similar way, simply replacing x and Nx in Eq. (5) to Eq. (9) by y and Ny. Since only square plates (a=b) are considered, thus Nx=Ny=N. In terms of the modified differential quadrature (DQ), the bending moments at corresponding boundary points can be expressed as:

(10)
M x i l = D - 11 k = 1 N + 2 B ~ i k x w ~ k l + 2 D - 16 j = 1 N k = 1 N A i j x A l k y w ~ j k + D - 12 k = 1 N + 2 B ~ l k y w ~ i k ,             i = 1 , N ,           l = 1,2 , . . . , N ,
(11)
M y i l = D - 12 k = 1 N + 2 B ~ i k x w ~ k l + 2 D - 26 j = 1 N k = 1 N A i j x A l k y w ~ j k + D - 22 k = 1 N + 2 B ~ l k y w ~ i k ,             i = 1,2 , . . . , N ,           l = 1 , N .

In terms of the DQ, the governing equation at all grid points can be expressed as:

(12)
D - 11 k = 1 N + 2 D ~ i k x w - k l + 4 D - 16 j = 1 N + 2 k = 1 N C ~ i j x A l k y w - j k + 2 D - 12 + 2 D - 66 j = 1 N k = 1 N B i j x B l k y w - j k           + 4 D - 26 j = 1 N k = 1 N + 2 A i j x C ~ l k y w - j k + D - 22 k = 1 N + 2 D ~ l k y w - i k = ρ h ω 2 w i l ,           i , l = 1,2 , . . . , N ,

where superscripts x and y mean that the weighting coefficients of the corresponding derivatives are taken with respect to x and y, w-ik contains the deflection wil as well as the first-order derivative with respect to x or y along boundary points, introduced by the method of modification of weighting coefficient-3 (MMWC-3), w~kl, w~jk and w~ik are only a part of w-ik. There are N+2×N+2-4 degrees of freedom (DOFs) in total. From Eq. (7), it is clearly seen that Bijx(i=1, N) are different from B~ijx(i=1,N), and Blky(l=1, N) are different from B~lky(l=1, N).

The bending moment equation is placed at the position where the DOF of the first-order derivative with respect to x or y at corresponding boundary point is. Enforcing the simply supported boundary conditions rigorously yields following partitioned matrix equations, namely:

(13)
[ K α α ] [ K α β ] [ K β α ] [ K β β ] { w α } { w β } = Ω 2 [ I ] [ 0 ] [ 0 ] [ 0 ] { w α } { w β } ,

where Ω=ωa2ρh/D0 is called the frequency parameter, D0=E1h3/[12(1-ν12ν21)], E1, v12 and v21 are the modulus of elasticity in the fiber direction, as well as the major and minor Poisson’s ratios, respectively. The vector {wα} contains only the non-zero DOFs of the deflection at all inner grid points and its dimension is N2×N2.

After eliminating {wβ}, Eq. (13) can be rewritten in the following matrix equation:

(14)
K - w α = Ω 2 I w α ,

where [K-]=[Kαα-KαβKββ-1Kβα].

Solving Eq. (14) by a standard eigen-solver yields the frequency parameters.

To achieve the fastest rate of convergence and obtain reliable and accurate solutions, following grid points are used in the modified DQM:

(15)
x k = y k = a 1 - c o s k - 1 π N - 1 2 ,       ( k = 1,2 , , N ,         a = b ) .

The exact frequency parameters (Ω) for especially orthotropic rectangular plates can be calculated analytically by [1, 3]:

(16)
Ω e x a c t = ω a 2 ρ h D 0 = π 2 D - 11 D 0 m 4 + 2 D - 12 D 0 + 2 D - 66 D 0 a b 2 m 2 n 2 + D - 22 D 0 a b 2 n 4 ,             m , n = 1,2 , ,

where m and n are the half wave number of the vibration mode in x and y directions, respectively.

3. Results and discussion

Three materials of lamina, i.e., E-glass/epoxy (E/E), Boron/epoxy (B/E) and Graphite/epoxy (G/E), are considered. The material parameters directly taken from [3, 4] are listed in Table 1. For each material, two sets of equivalent material constants are given. Among the three materials, Graphite/epoxy exhibits the highest anisotropy, since E1/E2 is the largest.

Table 1. Material property of two sets of equivalent material constants

Materials
Material system I (MS-I) [3]
Material system II (MS-II) [4]
E 1 (GPa)
E 2 (GPa)
G 12 (GPa)
ν 12
E 1 / E 2
G 12 / E 2
ν 12
E/E
60.7
24.8
12.0
0.23
2.45
0.48
0.23
B/E
209.
19.0
6.40
0.21
11.0
0.34
0.21
G/E
138.
8.96
7.10
0.30
15.4
0.79
0.30

Denote θ the fiber orientation angle. Four angles, i.e., θ= 0°, 15°, 30° and 45°, are considered. The relative bending-twisting coupling coefficients D16/D0 and D26/D0, which reflect the degrees of anisotropy, are listed in Table 2.

Table 2. Relative bending-twisting coefficients of angle-ply (θ/-θ/θ) laminated plates

θ °
E-glass/epoxy (E/E)
Boron/epoxy (B/E)
Graphite/epoxy (G/E)
D 16 / D 0
D 26 / D 0
D 16 / D 0
D 26 / D 0
D 16 / D 0
D 26 / D 0
0
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
15
0.122312
0.012555
0.214391
0.012882
0.205801
0.027967
30
0.176432
0.079666
0.297578
0.096069
0.291366
0.113533
45
0.147858
0.147858
0.227273
0.227273
0.233768
0.233768

It is seen that D16/D0 is the largest whenθ= 30°. This perhaps is the reason why the convergence study is performed for θ= 30° in [3], since the higher the anisotropy, the lower the rate of convergence for various approximate and numerical methods. Although D16/D0 is the second largest when θ= 45°, however, D26/D0 is the largest. Thus, convergence studies are performed for both θ= 30°and θ= 45° in present investigations. Corresponding results are listed in Table 3 and Table 4, respectively. Mid-plane symmetric angle-ply square plates with all edges simply supported, denoted by SSSS, are investigated.

From Table 3 and Table 4, it is clearly seen that the rate of convergence of the DQM is high. The rate of convergence of the DQM for θ= 30° is higher than the one for θ= 45°. This indicates that the anisotropy of the (45°/–45°/45°) square plates is higher than the one of the (30°/–30°/30°) square plates for the same material and the anisotropy of the graphite/epoxy square plates with (45°/–45°/45°) is the highest.

To ensure the high accuracy of solutions, the frequency parameters of three-layer angle-ply (θ/-θ/θ) square plates with all edges simply supported are obtained by the modified DQM with 31×31 grid points and are presented in Tables 5-7. The DQ solutions contain results using two sets of equivalent material constants listed in Table 1 and are all below the upper bound solutions cited from [3]. Note that the Ritz data reported in [3] are exact only for the case of θ= 0°.

In Table 5, Table 6, and Table 7, the exact solutions for θ= 0° are re-computed by using Eq. (16) with the corresponding material constants, since the existing exact solutions are only accurate to two places of decimals. It is observed that the DQ data are exactly the same as the re-computed exact solutions. The exact solutions with MS-I of materials E/E and G/E are slightly higher than the corresponding ones with MS-II, and the exact solutions with MS-I of material B/E are slightly lower than the corresponding ones with MS-II. This trend remains the same in the DQ solutions for other fiber orientation angles. It seems that this trend is mainly caused by the difference of G12, since G12 in MS-I of materials E/E and G/E is also slightly larger than G12 in MS-II and G12 in MS-I of material B/E is smaller than G12 in MS-II.

Table 3. Convergence of frequency parameters for angle-ply (30°/–30°/30°) SSSS square plates (MS-I)

Material
N
Mode numbers
1
2
3
4
5
6
7
8
E/E
11
15.8619
35.8018
42.5515
61.3169
71.6273
85.6521
93.5636
108.7378
15
15.8621
35.8021
42.5519
61.3176
71.6287
85.6529
93.5625
108.7262
19
15.8621
35.8021
42.5520
61.3177
71.6288
85.6529
93.5626
108.7263
23
15.8622
35.8021
42.5520
61.3177
71.6289
85.6530
93.5626
108.7264
27
15.8622
35.8021
42.5520
61.3177
71.6289
85.6530
93.5626
108.7264
[3]
15.90
35.86
42.62
61.45
71.71
85.72
93.74
108.9
B/E
11
11.9625
22.4074
35.4364
37.4339
49.2075
55.9908
70.5661
73.0071
15
11.9648
22.4100
35.4424
37.4329
49.2104
55.9665
70.4988
72.9975
19
11.9655
22.4109
35.4444
37.4329
49.2123
55.9665
70.4998
72.9982
23
11.9658
22.4112
35.4453
37.4329
49.2132
55.9665
70.5002
72.9986
27
11.9659
22.4114
35.4457
37.4329
49.2136
55.9665
70.5005
72.9988
[3]
12.21
22.78
35.86
37.90
50.04
56.70
71.36
73.57
G/E
11
11.6857
21.5346
35.4172
35.5276
48.6468
52.6563
69.1293
71.4666
15
11.6894
21.5392
35.4255
35.5259
48.6519
52.6272
69.0619
71.4086
19
11.6906
21.5407
35.4286
35.5259
48.6552
52.6272
69.0637
71.4088
23
11.6911
21.5414
35.4299
35.5259
48.6569
52.6272
69.0645
71.4089
27
11.6914
21.5417
35.4306
35.5259
48.6576
52.6272
69.0649
71.4090
[3]
11.97
21.97
35.88
36.04
49.60
53.43
70.04
72.35

Table 4. Convergence of frequency parameters for angle-ply (45°/–45°/45°) SSSS square plates (MS-I)

Material
N
Mode numbers
1
2
3
4
5
6
7
8
E/E
11
16.0871
36.8624
41.7104
61.6715
76.9472
79.8778
94.4474
108.7482
15
16.0876
36.8626
41.7116
61.6726
76.9474
79.8804
94.4454
108.7347
19
16.0877
36.8627
41.7120
61.6728
76.9474
79.8810
94.4456
108.7352
23
16.0878
36.8627
41.7121
61.6728
76.9474
79.8812
94.4456
108.7354
27
16.0878
36.8627
41.7121
61.6728
76.9474
79.8813
94.4456
108.7355
[3]
16.14
36.93
41.81
61.85
77.04
80.00
94.68
109.0
B/E
11
12.3054
24.1007
33.5834
39.5290
53.7288
58.3472
64.9806
76.8154
15
12.3196
24.1000
33.6269
39.5290
53.7162
58.3201
65.0412
76.7477
19
12.3253
24.0999
33.6443
39.5297
53.7162
58.3198
65.0666
76.7506
23
12.3281
24.0999
33.6528
39.5301
53.7162
58.3197
65.0794
76.7522
27
12.3298
24.0999
33.6576
39.5303
53.7162
58.3197
65.0868
76.7532
[3]
12.71
24.51
34.44
40.23
54.44
59.40
66.38
78.00
G/E
11
11.8647
23.2991
33.2088
37.7016
53.3682
55.2007
64.6746
75.3103
15
11.8774
23.2987
33.2480
37.7018
53.3533
55.1709
64.7192
75.2411
19
11.8824
23.2987
33.2641
37.7028
53.3533
55.1707
64.7400
75.2449
23
11.8850
23.2987
33.2720
37.7033
53.3533
55.1707
64.7507
75.2470
27
11.8865
23.2987
33.2765
37.7036
53.3533
55.1707
64.7571
75.2482
[3]
12.31
23.72
34.14
38.45
54.10
56.31
66.20
76.23

Table 5. Frequency parameters of angle-ply (θ/-θ/θ) SSSS square plates (E/E, N= 31)

θ °
Methods
Mode numbers
1
2
3
4
5
6
7
8
0
DQM (I)
15.19467
33.29959
44.41877
60.77869
64.52979
90.30141
93.66415
108.5563
Exact (I)
15.19467
33.29959
44.41877
60.77869
64.52979
90.30141
93.66415
108.5563
DQM (II)
15.17055
33.24847
44.38711
60.68220
64.45675
90.14548
93.63063
108.4588
Exact (II)
15.17055
33.24847
44.38711
60.68220
64.45675
90.14548
93.63063
108.4588
15
DQM (I)
15.4150
34.0748
43.8514
60.8068
66.6413
91.3847
91.5001
108.8889
[3]
15.43
34.09
43.87
60.85
66.67
91.40
91.56
108.9
DQM (II)
15.3959
34.0299
43.8199
60.7327
66.5601
91.3403
91.3773
108.7845
30
DQM (I)
15.8622
35.8021
42.5521
61.3177
71.6289
85.6530
93.5627
108.7265
[3]
15.90
35.86
42.62
61.45
71.71
85.72
93.74
108.9
DQM (II)
15.8534
35.7679
42.5238
61.2745
71.5463
85.5891
93.4889
108.6531
45
DQM (I)
16.0880
36.8627
41.7122
61.6729
76.9474
79.8813
94.4456
108.7356
[3]
16.14
36.93
41.81
61.85
77.04
80.00
94.68
109.0
DQM (II)
16.0842
36.8321
41.6880
61.6430
76.8622
79.8129
94.3878
108.6515

Table 6. Frequency parameters of angle-ply (θ/-θ/θ) SSSS square plates (B/E, N= 31)

θ °
Methods
Mode numbers
1
2
3
4
5
6
7
8
0
DQM (I)
11.03935
17.36394
30.90502
40.37093
44.15742
51.12759
53.26851
69.45577
Exact (I)
11.03935
17.36394
30.90502
40.37093
44.15742
51.12759
53.26851
69.45577
DQM (II)
11.04440
17.37677
30.92123
40.37645
44.17759
51.14502
53.30614
69.50708
Exact (II)
11.04440
17.37677
30.92123
40.37645
44.17759
51.14502
53.30614
69.50708
15
DQM (I)
11.3047
19.0789
33.1642
38.7790
45.2024
51.9267
59.1244
72.3957
[3]
11.37
19.21
33.32
38.86
45.46
52.14
59.48
72.77
DQM (II)
11.3089
19.0890
33.1790
38.7854
45.2210
51.9463
59.1566
72.4253
30
DQM (I)
11.9660
22.4115
35.4460
37.4329
49.2139
55.9665
70.5006
72.9989
[3]
12.21
22.78
35.86
37.90
50.04
56.70
71.36
73.57
DQM (II)
11.9678
22.4180
35.4527
37.4446
49.2295
55.9816
70.5315
73.0123
45
DQM (I)
12.3308
24.0999
33.6606
39.5305
53.7162
58.3197
65.0916
76.7538
[3]
12.71
24.51
34.44
40.23
54.44
59.40
66.38
78.00
DQM (II)
12.3315
24.1065
33.6659
39.5399
53.7361
58.3325
65.1028
76.7794

Table 7. Frequency parameters of angle-ply (θ/-θ/θ) SSSS square plates (G/E, N= 31)

θ °
Methods
Mode
1
2
3
4
5
6
7
8
0
DQM (I)
11.28972
17.13178
28.69169
40.74023
45.15887
45.78291
54.08234
68.14209
Exact (I)
11.28972
17.13178
28.69169
40.74023
45.15887
45.78291
54.08234
68.14209
DQM (II)
11.28718
17.12536
28.68364
40.73740
45.14874
45.77484
54.06362
68.13483
Exact (II)
11.28718
17.12536
28.68364
40.73740
45.14874
45.77484
54.06362
68.13483
15
DQM (I)
11.3927
18.5447
31.0178
39.0659
45.6444
47.9654
58.2869
67.4789
[3]
11.46
18.69
31.20
39.15
45.91
48.19
58.70
67.84
DQM (II)
11.3906
18.5398
31.0109
39.0628
45.6354
47.9567
58.2711
67.4670
30
DQM (I)
11.6916
21.5420
35.4311
35.5259
48.6582
52.6273
69.0651
71.4091
[3]
11.97
21.97
35.88
36.04
49.60
53.43
70.04
72.35
DQM (II)
11.6908
21.5389
35.4278
35.5204
48.6507
52.6203
69.0502
71.4019
45
DQM (I)
11.8875
23.2987
33.2794
37.7038
53.3533
55.1707
64.7612
75.2490
[3]
12.31
23.72
34.14
38.45
54.10
56.31
66.20
76.63
DQM (II)
11.8873
23.2956
33.2768
37.6996
53.3432
55.1652
64.7556
75.2369

4. Conclusions

The free vibration of mid-plane symmetric angle-ply laminated composite square plates with all edges simply supported is successfully solved by using the modified differential quadrature method (modified DQM). Three material systems are considered. The rate of convergence of the modified DQM is investigated. The results are tabulated for references.

Based on the results reported herein, one may conclude that the DQ data are highly accurate and can be served as the benchmark solutions. The difference in solutions of the mid-plane symmetric angle-ply laminated composite plates with two sets of equivalent material constants is clearly seen and thus care should be taken when highly accurate results are needed for comparisons in testing newly developed numerical methods. However, the difference is small and negligible from the practical point of view.

Acknowledgements

The project is partially supported by the Priority Academic Program Development of Jiangsu Higher Education Institutions.

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