1. Reference problem#

The « wedge splitting test » was initially proposed by Brühwiler and Wittmann in order to obtain stable crack propagation and thus determine the fracture properties of concrete. Slowik and Saouma then used this test to study the effect of fluid pressure in the crack. Segura and Carol modeled this splitting test under fluid pressure. The test carried out by Slowik and Saouma is repeated here.

A device, called a « corner », makes it possible to apply a force to points \(B\) and \(B\text{'}\) (see figure) in order to separate the sample in two. The experience is controlled by CMOD (Crack Mouth Opening Displacement), i.e. the displacement imposed between \(B\) and \(B\text{'}\).

1.1. Geometry#

_images/10000000000003D400000358C308052B9DCE18FC.jpg

Figure 1.1-a : Sample geometry

Coordinates of the points (in \(\mathrm{mm}\)):

The sample has a thickness of \(100\mathrm{mm}\).

1.2. Material properties#

  • Properties of interstitial fluid (liquid water):

Density

\({1.10}^{-6}{\mathrm{kg.mm}}^{-3}\)

Viscosity

\({1.10}^{-9}\mathrm{MPa.s}\)

Water is considered to be incompressible.

  • Properties of the massif:

The concrete massif is elastic and has the following properties:

Young’s module

\(27500\mathrm{MPa}\)

Poisson’s Ratio

\(\mathrm{0,2}\)

Given the time scales under consideration and the comparatively very high permeability of the crack, concrete is assumed to be impermeable. It is therefore modelled by classic D_ PLAN elements.

  • Properties of the discontinuity:

The discontinuity is broken down into three parts:

  • The rubber membrane that ensures the tightness of the notch;

  • The notch in which constant pressure is imposed;

  • The crack path along which the crack spreads.

The membrane has a linear elastic behavior. We use the JOINT_BANDIS law of behavior with the parameter \(\gamma \mathrm{=}0\) in order to make it linear. The properties of the membrane are as follows:

Initial normal stiffness \({K}_{\text{ni}}\)

\(20{\mathrm{MPa.mm}}^{-1}\)

Coefficient \(\gamma\)

0

For the notch, we use law CZM_LIN_REG by initializing the internal variables in such a way as to have elements that are initially broken.

For the crack, law CZM_EXP_REG is used with the following parameters:

Cracking energy \({G}_{c}\)

\(\mathrm{0,150}\mathrm{MPa.mm}\)

Critical stress \({\sigma }_{c}\)

\(\mathrm{3,25}\mathrm{MPa}\)

The numerical parameters of the elements of joints with cohesive law are

PENA_ADHERENCE

\({1.10}^{-3}\)

PENA_CONTACT

\(1\)

  • Properties of the crack tip

Biot module \(N\)

\(\mathrm{0,3}{.10}^{-3}{\mathrm{MPa.mm}}^{-1}\)

Fictional opening OUV_FICT

10 \(\mathrm{mm}\)

1.3. Initial conditions#

The fluid pressure is initially \(\mathrm{0,21}\mathrm{MPa}\) in the notch and \(\mathrm{0,0}\mathrm{MPa}\) in the future crack.

1.4. Boundary conditions#

The mechanical and hydraulic boundary conditions are as follows:

  • In \(F\): movements blocked in all directions and zero pressure imposed;

  • In notch \([\mathrm{GH}]\): pressure imposed from \(\mathrm{0,21}\mathrm{MPa}\);

  • In membrane \([\mathrm{HI}]\): imposed pressure of \(\mathrm{0,0}\mathrm{MPa}\);

  • In \(B\) and \(B\text{'}\), the movements are symmetric;

  • In \(B\text{'}\), a force is imposed in the \((\begin{array}{c}\mathrm{cos}(\alpha )\\ -\mathrm{sin}(\alpha )\end{array})\) direction, whose intensity is controlled by the \(\mathrm{DX}\) component of the displacement at point \(B\text{'}\). At each moment \(t\), the horizontal component of the movement in \(B\text{'}\) must be equal to \({u}_{x}=-\frac{q}{2}t\), where \(q\) is the loading speed. We therefore use piloting with option DDL_IMPO.

_images/10000000000003A000000406E2F8A0286FCEC124.jpg

Moreover, since the mass is impermeable, the exchanges of fluid between the crack and the massif are blocked. The fluid pressures at the edges of the joint element and the hydraulic Lagrange multipliers are locked at zero.