1. Reference problem#
In this test [Willam et al., 1987], the specimen is subjected to a specific loading path that creates a continuous rotation of the main stress directions. This determines the ability of the model to converge despite such changes.
1.1. Geometry#
The test is based on a unit cubic finite element (1 m x 1 m).
1.2. Property of the materials#
Young’s module: \(E=32000\mathrm{MPa}\)
Poisson’s ratio: \(\mathrm{\nu }=0.2\)
Parameter of fragility of concrete under tension: \(\mathit{MT}=\mathrm{1,7}\)
Parameter of fragility of concrete under compression: \(\mathit{MC}=\mathrm{1,5}\)
Equivalent tensile stress of concrete: \({\mathrm{\sigma }}_{\mathit{ft}}=7.3\mathit{MPa}\)
Equivalent stress of concrete in compression: \({\mathrm{\sigma }}_{\mathit{fc}}=38.3\mathit{MPa}\)
Angle of the Drucker Prager criterion: \(\mathrm{\alpha }=\mathrm{0,15}\mathit{rad}\)
1.3. Boundary conditions and loads#
This is a cube subjected to a uniform non-proportional loading, consisting of movements imposed in plane \((\mathrm{Ox},\mathrm{Oy})\).
The material is first subjected to uniaxial traction in the direction \(\mathrm{xx}\) up to the peak of the stress-deformation curve, in a second stage a shear \(\mathrm{xy}\) and an orthogonal tension \(\mathrm{yy}\) are superimposed on the uniaxial loading \(\mathrm{xx}\) (which continues), this loading path results in a rotation of the main directions of the stresses resulting in the appearance of a stress of shear \({\sigma }_{\mathrm{xy}}\).
This results in imposed deformations that evolve finely by pieces as a function of time, with:
to \(t=0.01\mathrm{jour}\), \(\epsilon ={10}^{-4}(\begin{array}{cc}0.84& 0\\ 0& -0.105\end{array})\)
to \(t=0.05\mathrm{jour}\), \(\epsilon ={10}^{-4}(\begin{array}{cc}5.6& 4.76\\ 4.76& 7.035\end{array})\)
The following boundary conditions apply:
for the nodes in the plane X=0 → DX = 0
for node N1 (0, 0, 0) → DX = DY = DZ = 0
for node N5 (0, 0, 1) → DY = 0