2. Benchmark solution

2.1. Calculation method

The normal at the interface is noted \(n\) and the tangent vector is noted \(\tau\):

\(n=(\begin{array}{}\mathrm{cos}\theta \\ \mathrm{sin}\theta \end{array})\), \(\tau =(\begin{array}{}-\mathrm{sin}\theta \\ \mathrm{cos}\theta \end{array})\) eq 2.1-1

with \(\theta =\mathrm{arctan}\frac{y-\mathrm{11,77}}{x-5}\) eq 2.1-2

In this modeling, we consider the hypothesis of plane stresses (although here, the Poisson’s ratio being zero, there are no differences between stresses and plane deformations).

The augmented Lagrangian method is used for the treatment of touch/friction.

The interface has a slope that varies from one end of the plate to the other. However, in the places where the inclination is greatest, near the lateral edges, there is a risk of sliding. To avoid this, we increase the adhesion via the Coulomb coefficient of friction: we take \(\mu =2\).

The value of the contact pressure on the interface is a function of normal \(n\):

\(\lambda =n\mathrm{.}\sigma \mathrm{.}n={n}_{y}{\sigma }_{\mathrm{yy}}{n}_{y}\) eq .2.1-3

• where \({n}_{y}\) is the next component \(y\) of \(n\)

• where \({\sigma }_{\mathrm{yy}}\) is the following constraint \(y\) in the plane of normal \({e}_{y}\) in the structure without an interface: \({\sigma }_{\mathrm{yy}}=E\frac{{u}_{y}}{{L}_{y}}\)

The friction semi-multiplier \(\Lambda\) is defined by:

\({r}_{\tau }=\lambda \mu \Lambda\) eq 2.1.1-4

With the tangential force density being written as follows:

\({r}_{\tau }=(\tau \mathrm{.}\sigma \mathrm{.}n)\tau\) eq 2.1-5

From where:

\(\Lambda =(\frac{1}{\mu }\frac{\tau \mathrm{.}\sigma \mathrm{.}n}{n\mathrm{.}\sigma \mathrm{.}n})\tau =(\frac{1}{\mu }\frac{{\tau }_{y}}{{n}_{y}})\tau\) eq 2.1-6

With the numeric values previously introduced:

\(\lambda (x,y)=\frac{-1}{10}{\mathrm{sin}}^{2}(\mathrm{arctan}\frac{\mathrm{11,77}-y}{5-x})\mathrm{Pa}\) eq 2.1-7

\(\Lambda (x,y)=\Lambda \mathrm{.}\tau =\frac{x-5}{2(y-\mathrm{11,77})}\) eq 2.1-8

2.2. Reference quantities and results

Minimum contact pressure LAGS_C on the interface (at points A and C): -0.1

Maximum contact pressure LAGS_C on the interface (at point B): -0.06110

Minimum tangential force density LAGS_F1 on the interface (at point A): 0.39894

Maximum tangential force density LAGS_F1 on the interface (at point B): 0

2.3. Uncertainties in the solution

None (analytical solution).