2. E and F models

2.1. Objectives

The objective of these models is to verify on an element that the modeling of a viscoelastic material whose Poisson’s ratio is constant/real and whose complex Young’s modulus is equivalent to the modeling of an elastic material with a hysteretic damping coefficient.

In isotropic elasticity, the law of behavior is written as:

\(\sigma =\frac{E}{1+\nu }(ϵ+\frac{\nu }{1-2\nu }\mathit{Tr}(ϵ)\mathit{Id})\)

So if \(E={E}_{r}+j{E}_{i}\) then

\(\sigma =(1+j\frac{{E}_{i}}{{E}_{r}})\frac{{E}_{r}}{1+\nu }(ϵ+\frac{\nu }{1-2\nu }\mathit{Tr}(ϵ)\mathit{Id})\)

This corresponds to an isotropic elastic material with Young’s modulus \({E}_{r}\), Poisson \(\nu\) and hysteretic damping \(\eta =\frac{{E}_{i}}{{E}_{r}}\)

2.2. Geometry

As a stiffness matrix construction test, the geometry corresponds to a cube for 3D elements and a square plane for 2D elements.

2.3. Material properties

The two declared materials are:

• viscoelastic, material 1: \(G=\mathrm{0,53e9}+\mathrm{9,3e6}j\mathit{Pa},\nu =\mathrm{0,3}\)

• elastic, material 2: \(E=\mathrm{1,3e9}\mathit{Pa},\nu =\mathrm{0,3,}\eta =\mathrm{1,8e-2}\)

2.4. Characteristics of the models

Modeling	Element	Mesh
E	3D	1 HEXA8
F	C_ PLAN	1 QUAD4

2.5. Tested sizes

For each model, complex stiffness matrices are constructed:

:math:`{K}_{1}text{}` of material 1

:math:`{K}_{2}text{}` of material 2

We then check \({K}_{1}\text{*}={K}_{2}\text{*}\)