1. A-D modeling#

1.1. Objectives#

The objective of these models is to verify the behavior of options RIGI_MECA and RIGI_MECA_HYST on an element by testing the real and imaginary parts of the stiffness matrix produced.

In isotropic elasticity, the law of behavior is written as: \(\sigma =2G(ϵ-\frac{1}{3}\mathit{Tr}(ϵ))-K\mathit{Tr}(ϵ)\mathit{Id}\)

If the material has complex modules, the real part/imaginary part separation is simply written:

\({\sigma }_{r}=2\mathrm{\Re }(G)(ϵ-\frac{1}{3}\mathit{Tr}(ϵ))-\mathrm{\Re }(K)\mathit{Tr}(ϵ)\mathit{Id},{\sigma }_{i}=2\mathrm{\Im }(G)(ϵ-\frac{1}{3}\mathit{Tr}(ϵ))-\mathrm{\Im }(K)\mathit{Tr}(ϵ)\mathit{Id}\)

1.2. Geometry#

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

1.3. Material properties#

Three fictional isotope materials are declared:

  • elastic, material 1: \({G}_{1}=15\mathit{Pa},{K}_{1}=20\mathit{Pa}\)

  • elastic, material 2: \({G}_{2}=10\mathit{Pa},{K}_{2}=25\mathit{Pa}\)

  • viscoelastic, v material: \(G=15+10j\mathit{Pa},K=10+25j\mathit{Pa}\)

Thus, the matrix calculated on the material 1 must be the real part of the matrix calculated on the material v and the matrix calculated on the material 2 must be the imaginary part of the matrix calculated on the material v.

1.4. Characteristics of the models#

Modeling

Material Type

Element

Mesh

A

Isotropic

3D

1 HEXA8

B

Orthotropic

3D

1 HEXA8

C

Transverse Isotropic

3D

1 HEXA8

D

Isotropic

D_ PLAN

1 QUAD4

For models of non-isotropic materials, materials 1, 2 and v are used by taking \({G}_{L}={G}_{N}={G}_{T},{E}_{L}={E}_{N}={E}_{T},{\nu }_{LT}={\nu }_{LN}={\nu }_{NT}\)

1.5. Tested sizes#

For each model, the following stiffness matrices are constructed:

  • \({K}_{i}\), the real part of the stiffness matrix on the material i or i=1.2 (built with RIGI_MECA)

:math:`{K}_{i}text{}` the stiffness matrix on the material i or i=1.2 (built with RIGI_MECA_HYST)

  • \({K}_{v}\), the complex stiffness matrix on material v (built with RIGI_MECA_HYST)

We then check: \({K}_{v}={K}_{1}+\mathit{jK}2\) and \({K}_{1}\text{*}={K}_{1}\)