1. Reference problem

1.1. Geometry and boundary conditions

The element used is a tetrahedron with a Gauss point. There is therefore no problem of homogeneity of the fields in the element.

The blocking conditions and the linear relationships between the nodes that must be applied are summarized on [Figure 1.1-a]. Edges \(\mathrm{N0N1}\), \(\mathrm{N0N2}\), and \(\mathrm{N0N3}\) are 1 in length.

Taking into account the geometry of the element, the blocking conditions and the linear relationships, the deformation is directly related to the movements of the nodes:

\({\varepsilon }_{\mathit{xx}}\mathrm{=}\mathit{DX}(\mathit{N1})\)

\({\varepsilon }_{\mathit{yy}}\mathrm{=}\mathit{DY}(\mathit{N2})\)

\({\varepsilon }_{\mathrm{zz}}=\mathrm{DZ}(\mathrm{N3})\)

\({\varepsilon }_{\mathit{xy}}\mathrm{=}\mathit{DX}(\mathit{N2})\mathrm{=}\mathit{DY}(\mathit{N1})\)

\({\varepsilon }_{\mathit{xz}}\mathrm{=}\mathit{DX}(\mathit{N3})\mathrm{=}\mathit{DZ}(\mathit{N1})\)

\({\varepsilon }_{\mathit{yz}}\mathrm{=}\mathit{DY}(\mathit{N3})\mathrm{=}\mathit{DZ}(\mathit{N2})\)

If we work with imposed deformation, it is therefore sufficient to impose the movements at the appropriate nodes.

If you want to work with imposed force, as is the case for modeling E, you have to impose the following loads (see [Figure 1.1-a] for the definition of faces \(\mathrm{F1}\), \(\mathrm{F2}\), \(\mathrm{F3}\) and \(\mathrm{F4}\)):

\({\sigma }_{\mathit{xx}}>0\): \(\mathrm{FX}\) on \(\mathrm{F1}\) and \(-1/\sqrt{3}\mathrm{FX}\) on \(\mathrm{F4}\), \(\mathrm{FX}<0\) (traction according to \(x\))

\({\sigma }_{\mathit{xx}}<0\): \(\mathrm{FX}\) out of \(\mathrm{F1}\) and \(-1/\sqrt{3}\mathrm{FX}\) out of \(\mathrm{F4}\), \(\mathrm{FX}<0\) (compression according to \(x\))

\({\sigma }_{\mathit{yy}}>0\): \(\mathrm{FY}\) on \(\mathrm{F2}\) and \(-1/\sqrt{3}\mathrm{FY}\) on \(\mathrm{F4}\), \(\mathrm{FY}<0\) (traction according to \(y\))

\({\sigma }_{\mathit{yy}}<0\): \(\mathrm{FY}\) out of \(\mathrm{F2}\) and \(-1/\sqrt{3}\mathrm{FY}\) out of \(\mathrm{F4}\), \(\mathrm{FY}<0\) (compression according to \(y\))

\({\sigma }_{\mathit{zz}}>0\): \(\mathrm{FZ}\) on \(\mathrm{F3}\) and \(-1/\sqrt{3}\mathrm{FZ}\) on \(\mathrm{F4}\), \(\mathrm{FZ}<0\) (traction according to \(z\))

\({\sigma }_{\mathit{zz}}<0\): \(\mathrm{FZ}\) out of \(\mathrm{F3}\) and \(-1/\sqrt{3}\mathrm{FZ}\) out of \(\mathrm{F4}\), \(\mathrm{FZ}<0\) (compression according to \(z\))

	Blocks: \(\mathrm{N0}:\mathrm{DX}=\mathrm{DY}=\mathrm{DZ}=0\) Linear relationships: \(\mathrm{DY}(\mathrm{N1})=\mathrm{DX}(\mathrm{N2})\) \(\mathrm{DZ}(\mathrm{N1})=\mathrm{DX}(\mathrm{N3})\) \(\mathrm{DZ}(\mathrm{N2})=\mathrm{DY}(\mathrm{N3})\)	Traction/compression on imposed displacement: According to \(x\) \(\mathrm{DX}\) imposed on \(\mathrm{N1}\) According to \(y\) \(\mathrm{DY}\) imposed on \(\mathrm{N2}\) According to \(z\) \(\mathrm{DZ}\) imposed on \(\mathrm{N3}\)
	Definition of faces: \(\mathrm{F1}=\mathrm{N0}\mathrm{N2}\mathrm{N3}\) \(\mathrm{F2}=\mathrm{N0}\mathrm{N1}\mathrm{N3}\) \(\mathrm{F3}=\mathrm{N0}\mathrm{N1}\mathrm{N2}\) \(\mathrm{F4}=\mathrm{N1}\mathrm{N2}\mathrm{N3}\)	Traction/compression in imposed force: According to \(x\): \(\mathrm{FX}\) out of \(\mathrm{F1}\) and \(-1/\sqrt{3}\mathrm{FX}\) out of \(\mathrm{F4}\) According to \(y\): \(\mathrm{FY}\) on \(\mathrm{F2}\) and BELOVED out \(-1/\sqrt{3}\mathrm{FY}\) \(\mathrm{F4}\) \(x\): \(\mathrm{FZ}\) on \(\mathrm{F3}\) and \(-1/\sqrt{3}\mathrm{FZ}\) on \(\mathrm{F4}\)

Figure 1.1-a: Geometry and uniaxial test boundary conditions

1.2. Material properties

The material characteristics are identical for the 5 tests that are presented.

The elastic characteristics of the material are as follows:

\(E\mathrm{=}32000\mathit{Mpa}\); \(\nu \mathrm{=}0.2\)

The tensile and compressive failure stresses are:

\({\sigma }_{\mathit{rupture}}^{\mathit{traction}}\mathrm{=}3.2\mathit{MPa}\); \({\sigma }_{\mathit{rupture}}^{\mathit{compression}}\mathrm{=}\mathrm{-}31.8\mathit{MPa}\)

The following set of parameters is used for the law of behavior:

\(\mathit{ALPHA}\)	\(\mathit{K0}(\mathit{Mpa})\)	\(\mathit{ECROB}(\mathit{MJ}\mathrm{/}{m}^{3})\)		\(\mathit{ECROD}(\mathit{MJ}\mathrm{/}{m}^{3})\)	\({K}_{1}(\mathit{Mpa})\)	\({K}_{2}\)
0.87	3.10-4	1.10-3	1.10-3	6.10-2	10.5	6.10-4

Note:

There are several sets of parameters that provide the same failure stresses. The parameters have been identified so that the rupture envelope of the biaxial tests does not swell (see doc. [R7.01.09]).

Model responses for uniaxial tests are shown below.

Figure 1 . 2-a: Law answer ENDO_ORTH_BETON in simple pull

Figure 1 *.2-b: Law response**ENDO_ORTH_BETON**in simple compression*

The internal variables, which are numbered in Aster, have the following meaning:

\(\mathit{V1}\mathrm{=}{D}_{\mathit{xx}};\mathit{V2}\mathrm{=}{D}_{\mathit{yy}};\mathit{V3}\mathrm{=}{D}_{\mathit{zz}};\mathit{V4}\mathrm{=}{D}_{\mathit{xy}};\mathit{V5}\mathrm{=}{D}_{\mathit{xz}};\mathit{V6}\mathrm{=}{D}_{\mathit{yz}};\mathit{V7}\mathrm{=}d;\)

Where \(D\) is the tensor representing orthotropic tensile damage, and \(d\) is isotropic compression damage (cf. doc. [R7.01.09]).