1. Reference problem#

1.1. Geometry#

We consider an element as a material point.

1.2. Material properties#

1.2.1. Hujeux and Iwan models of behavior#

The isotropic elastic properties of the material are \(E=619.33\) MPa and \(\nu =0.3\).

The anelastic parameters specific to the Hujeux model (modeling A) are:

  • \(n=0.4\), \(\mathrm{\beta }=24\), \(b=0.2\), \(d=2.5\).

  • \(\mathrm{\varphi }=33°\), \(\mathrm{\psi }=33°\), \({P}_{c0}=-1\) MPa, \({P}_{\mathit{ref}}=-1\) MPa

  • \({a}_{\mathit{cyc}}=1\times10^{-4}\), \({a}_{\mathit{mon}}=8\times10^{-3}\)

  • \({c}_{\mathit{cyc}}=1\times10^{-1}\), \({c}_{\mathit{mon}}=2\times10^{-1}\)

  • \({r}_{\mathit{ela}}^{d}=5\times10^{-3}\), \({r}_{\mathit{ela}}^{i}=1\times10^{-3}\), \({r}_{\mathit{ela}}^{d,c}=5\times10^{-3}\), \({r}_{\mathit{ela}}^{i,c}=1\times10^{-3}\)

  • \({r}_{\mathit{hys}}=5\times10^{-2}\), \({r}_{\mathit{mob}}=9\times10^{-1}\)

  • \({x}_{m}=1\), \(\alpha=1\)

The anelastic parameters of the Iwan model (modeling B) are for their part \(\gamma_{\mathrm{ref}}=2\times10^{-4}\) and \(n=0.78\).

Note:

Using the CALC_ESSAI_GEOMECA tool, we are able to compare the behavior curves for the two models in terms of reducing the normalized secant shear modulus and reduced depreciation (see Fig. 1.1 and Fig. 1.2).

images/100002010000032000000258B5CDD4F1675D5855.png

Fig. 1.1 Predictions of the secant shear modulus normalized with distortion by Hujeux and Iwan models.#

images/1000020100000320000002589A53DF01E1E9C3F4.png

Fig. 1.2 Predictions of reduced damping with distortion by Hujeux and Iwan models.#

1.2.2. Behaviour model CSSM#

The parameters specific to the model CSSM [r7.01.44] are grouped together in the Section 1.3 concerning the C modeling.

1.3. Boundary conditions and loads#

As a reminder, the use of the SIMU_POINT_MAT command makes it possible to directly impose a field of deformations and/or constraints.

A zero evolution during loading is imposed for the following components of the stress and strain tensors:

  • \({\mathrm{d}\sigma}_{xx}={\mathrm{d}\sigma}_{yy}={\mathrm{d}\sigma}_{zz}=0\)

  • \({\mathrm{d}\varepsilon}_{yz}={\mathrm{d}\varepsilon}_{zx}=0\)

The evolution of Fig. 1.3 is imposed for \(\varepsilon_{xy}\) shear deformations:

_images/image_chargement.png

Fig. 1.3 Imposed loading chronology for the \(\varepsilon_{xy}\) (standardized) deformation component.#

Several calculations are carried out by varying the amplitude of deformations according to the values \(\left\{2\times10^{-5},2\times10^{-4},2\times10^{-3}\right\}\).

1.4. Initial conditions#

The initial stress state is isotropic and corresponds to a pressure equal to \(50\) kPa.