2. Reference problem#
2.1. 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 3]. Edges \(\mathit{N0N1}\), \(\mathit{N0N2}\), and \(\mathit{N0N3}\) 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:
\({\varepsilon }_{\mathit{xx}}\mathrm{=}\mathit{DX}(\mathit{N1})\)
\({\varepsilon }_{\mathrm{yy}}=\mathrm{DY}(\mathrm{N2})\)
\({\varepsilon }_{\mathrm{zz}}=\mathrm{DZ}(\mathrm{N3})\)
\({\varepsilon }_{\mathrm{xy}}=\mathrm{DX}(\mathrm{N2})=\mathrm{DY}(\mathrm{N1})\)
\({\varepsilon }_{\mathrm{xz}}=\mathrm{DX}(\mathrm{N3})=\mathrm{DZ}(\mathrm{N1})\)
\({\varepsilon }_{\mathrm{yz}}=\mathrm{DY}(\mathrm{N3})=\mathrm{DZ}(\mathrm{N2})\)
With imposed deformation, it is therefore sufficient to impose the displacements at the appropriate nodes.
Blocks: \(\mathit{N0}\): \(\mathit{DX}\mathrm{=}\mathit{DY}\mathrm{=}\mathit{DZ}\mathrm{=}0\) Linear relationships: \(\mathit{DY}(\mathit{N1})\mathrm{=}\mathit{DX}(\mathit{N2})\) \(\mathit{DZ}(\mathit{N1})\mathrm{=}\mathit{DX}(\mathit{N3})\) \(\mathit{DZ}(\mathit{N2})\mathrm{=}\mathit{DY}(\mathit{N3})\) |
Loads: Phase 1: Traction in imposed displacement \(\mathit{DX}\mathrm{=}{F}^{\mathit{trac}}\) imposed on \(\mathit{N1}\) Phase 2: Traction/Shear in imposed displacement \(\mathit{DX}\mathrm{=}\mathit{DY}\mathrm{=}0.5\mathrm{\ast }{F}^{\mathit{cisa}}\) imposed on \(\mathit{N1}\) \(\mathit{DY}\mathrm{=}0.75\mathrm{\ast }{F}^{\mathit{cisa}}\) imposed on \(\mathit{N2}\) Where \({F}^{\mathit{trac}}\) and \({F}^{\mathit{cisa}}\) are increasing affine functions of time |
Figure 2.1-a Geometry, Boundary Conditions, and Willam Test Loads
2.2. Material properties#
The material characteristics are identical for the 5 tests that are presented.
The elastic characteristics of the material are as follows:
The following set of parameters is used for the law of behavior:
\(\mathit{ALPHA}\) |
|
\(\mathit{ECROB}\) (\(\mathit{MJ}\mathrm{/}{m}^{3}\)) |
() |
\(\mathit{ECROD}\) (\(\mathit{MJ}\mathrm{/}{m}^{3}\)) |
\({K}_{1}\) (\(\mathit{Mpa}\)) |
\({K}_{2}\) |
0.87 |
2.634e-4 |
0 |
0.06 |
10.5 |
6.e-4 |