3. G modeling#

3.1. Reference problem description#

3.1.1. Geometry#

We consider a slender sandwich structure with a square section composed of two skins and a heart. This is the same calculation case as that presented by Lucie Rouleau in chapter 4 of her thesis [bib1]

_images/10000201000002C30000024B43968A4756A7B573.png

x=1m

3.1.2. Material properties#

The skins are composed of an isotropic elastic material whose properties are:

  • Young’s module \(E=2.1{10}^{11}\mathit{Pa}\)

  • Poisson’s ratio \(\mathrm{\nu }=0.3\)

  • Density \(\mathrm{\rho }=7800\mathit{kg}/m\mathrm{³}\)

The core is composed of a linear viscoelastic material whose shear coefficient is modelled by a fractional Zener model, i.e.: \(G(\omega )=\frac{{G}_{0}+{G}_{\mathrm{\infty }}{(j\omega )}^{\alpha }}{1+{(j\omega )}^{\alpha }}\), the compressibility coefficient is taken to be constant. Numeric values are:

  • \({G}_{0}=\mathrm{1,4e6}\mathit{Pa}\)

  • \({G}_{\mathrm{\infty }}=\mathrm{0,54e9}\mathit{Pa}\)

  • \(\tau =\mathrm{0,52e-6}s\)

  • \(\alpha =\mathrm{0,59}\)

  • Density \(\rho =1460\mathit{kg}/m\mathrm{³}\)

  • Compressibility coefficient \(K=\mathrm{2,2e9}\mathit{Pa}\)

3.1.3. Boundary conditions and loads#

This structure is excited punctually by a unitary harmonic effort at point \((1m,\mathrm{0,5}m,\mathrm{0,5}m)\), we will be interested in the displacement along \(z\) at this same point. The structure is embedded in x=0.

3.2. Benchmark solution#

3.2.1. Reference quantities and results#

We compare the modulus values of the displacement along z at given frequencies calculated by the new Code_Aster methods and the calculation code of [bib1]

3.2.2. Uncertainties about the solution#

It is considered that there is a regression if the results obtained are not identical.

3.3. Tested sizes and results#

3.3.1. Characteristics of the mesh#

The mesh contains 1750 elements of type HEXA20 and 35 elements of type QUAD8.

3.3.2. Results#

The following 4 frequencies corresponding to the first resonances and anti-resonances of the structure are tested.

Frequency

Calculated value

Reference value

45 Hz

-2.19729959e-06-5.31250496e-06j

-2.52885263e-06-5.26288448e-06j

179 Hz

1.15296519e-09-3.39972232e-09j

1.11783628e-09-3.30841894e-09j

255

1.04664929e-07-2.81759671e-07j

1.01911277e-07-2.96162398e-07j

696

1.57582867e-08-6.87887584e-08j

1.48442697e-08-7.13562822e-08j