N3 was set equal to either 3 or 4 and N1 = N2 was assigned values 4, 6 and 8; in E3 and E4, N1 = N2 = 4 and N3 was given values 3, 4, 6 and 8; and in E5, N1 = N2 = N3 were assigned values 4, 6 and 8. Thus, a total of 13 combinations of N1, N2 and N3 were evaluated for each value of a/h, and values of u, v, w, σxx, σyy, σxy and σxz at typical points were compared with their analytical values. For a/h = 10, N1 = N2 = 8, N3 = 3 (total degrees of freedom (DOFs) = 9,477) provided an error of less than 0.1% in the value of each of these seven variables. Keeping N1 = N2 = 8 and increasing N3 = 4, 5 marginally decreased the error, suggesting that there is no benefit derived from the increased computation cost. For a/h = 4, all 13 combinations of values of N1, N2 and N3 resulted in an error of 7.25% in w and the error in σxz decreased from 37.7% for N1=N2=N3 = 4 to 0.03% for N1 = N2 = N3 = 8 (DOFs = 24,057).
1.3.2. Simply supported sandwich plate
We now study deformations of a sandwich plate with the top and bottom facesheets made of an orthotropic Aragonite crystal and the core made of a softer material. Material properties of the facesheets and the core are the same as those in Srinivas and Rao (1970). The top surface of the plate is loaded by sinusoidal tractions given by equation [1.12] and the bottom surface is traction-free. Since the load on the top surface given by equation [1.12] is different from that in Srinivas and Rao (1970), we first solved the problem analytically using their methodology, then verified our software by ensuring that values of displacements and stresses at several points agreed with those reported in Srinivas and Rao (1970). The thickness of each facesheet equals 0.1h and that of the core 0.8h, where h equals the sandwich plate thickness.
Defining β = Ex1/Ex2, where Ex1 and Ex2 are, respectively, elastic moduli of the facesheets and the core in the x direction, we compare the results from the least-squares method and the analytical solution for β = 1, 10, 100, 1000, 10,000, and aspect ratio a/h = 10, 5 and 2 in Tables 1.2–1.4. Quantities compared are the centroidal deflection, stresses σxx, σyy and σxz, normalized as wEx2/q0, σxx/q0, σyy/q0 and σxz/q0. Note that z = 0.4h+ and 0.4h-, respectively, represent interfaces between the top facesheet and the core. For the three aspect ratios and for all values of β except β = 10,000, the least-squares methodology gives results that differ from their analytical values by less than 0.4%. Even for β = 10,000, the maximum error in all quantities is less than 1.3%, except for that in σxz. Thus, we should not use the least-squares method for β = 10,000. We have not investigated if increasing values of N1, N2 and N3 in equation [1.9] will reduce this error.
For a/h = 2 and β = 1000, the error in the 15 variables listed in Table 1.4 was less than 1% for N1 = N2 = 6 and N3 = 4 (DOFs = 7,497) and less than 0.6% for N1 = N2 = 8 and N3 = 4 (DOFs = 12,393). Thus, the present method provides accurate values of the variables at critical points for a rather modest computational cost.
1.3.3. Laminate with arbitrary boundary conditions
We now present results for a [0/90/0] laminate simply supported on edges y = 0, b, with the other two edges either clamped or traction-free, and loaded on the top surface by the surface tractions listed in equation [1.12]. For the plate with two edges free, the present results have been computed with N1 = N2 = 11 and N3 = 5 in equation [1.9] for a total of 14,157 DOFs. These values could not be increased due to memory limitation of the Dell laptop used for the computational work. We are now converting the code from MATLAB to C++, which will enable us to use larger values of N1, N2 and N3. In Table 1.5, the numerical results for different quantities for the laminates with a/h = 5 and 10 are compared with their values reported by Vel and Batra (1999), who used the Eshelby–Stroh formalism. The maximum difference of 4.2% in the two sets of values of the six quantities suggests that the least-squares method provides a reasonably accurate solution for this problem.
Table 1.2. Normalized results for the sandwich plate of the aspect ratio a/h = 10. Values in the upper (lower) row are from the analytical (least-squares) solution
| β→ | 1 | 10 | 100 | 1000 | 10,000 | |
|---|---|---|---|---|---|---|
|
|
(0.5, 0.5, -0.5h) | -436.824 (-436.820) | -102.422(-102.422) | -32.9858 (-32.9858) | -21.6946 (-21.6942) | -10.5531(-10.5452) |
|
|
(0.5, 0.5, z) | |||||
| z = 0.5h | -24.6675(-24.6687) | -45.1109(-45.1110) | -52.1977(-52.1980) | -155.576(-155.574) | -694.701(-694.429) | |
| z = 0.4h+ | -19.4585(-19.4537) | -32.7715(-32.7713) | -11.4111(-11.4113) | 116.821 (116.819) | 676.521(676.846) | |
| z = 0.4h- | -19.4585(-19.4537) | -3.29448(-3.29446) | -0.13298(-0.13300) | 0.09864(0.09864) | 0.05351(0.05349) | |
| z = -0.4h+ | 19.3977(19.3929) | 3.26259(3.26259) | 0.11506(0.11507) | -0.11458(-0.11457) | -0.06898(-0.06911) | |
| z = -0.4h- | 19.3977(19.3929) | 32.6356(32.6355) | 11.6322(11.6324) | -112.483(-112.471) | -631.675(-632.023) | |
| z = -0.5h |
24.5962(24.5973)
|