2. Benchmark solution

2.1. Calculation method

The displacement field in spherical coordinates is written in the form:

\(U(r,\theta ,\varphi )={u}_{r}(r,\theta ,\varphi ){e}_{r}+{u}_{\theta }(r,\theta ,\varphi ){u}_{\theta }+{u}_{\varphi }(r,\theta ,\varphi ){e}_{\varphi }\)

In our case, pressure is applied only in the \({e}_{r}\) direction. So, we can write the displacement field in the form:

\(U(r)={u}_{r}(r){e}_{r}\)

The Lamé Navier equation for this problem is written as:

\(\rho f=(\lambda +2\mu )\nabla \wedge (\nabla \wedge u)+\rho b\)

We have: \(f=0\) and \(b=0\).

The displacement field solution of this equation is of the form:

\(U(r)=({C}_{1}r+{C}_{2}/{r}^{2}){e}_{r}\)

The boundary conditions make it possible to obtain the expression of the constants:

\({C}_{2}=\frac{(-{P}_{i}+{P}_{e}){R}_{e}^{3}{R}_{i}^{3}}{4\mu ({R}_{i}^{3}-{R}_{e}^{3})}\)

\({C}_{1}=\frac{1}{3\lambda +2\mu }(-{P}_{i}+\frac{4\mu {C}_{2}}{{R}_{i}^{3}})\)

The deformation tensor is written,

\({ϵ}_{\mathit{rr}}={C}_{1}-\frac{{\mathrm{2C}}_{2}}{{r}^{3}}\)

\({ϵ}_{\theta \theta }={C}_{1}+\frac{{C}_{2}}{{r}^{3}}\)

\({ϵ}_{\varphi \varphi }={C}_{1}+\frac{{C}_{2}}{{r}^{3}}\)

\({ϵ}_{\theta \varphi }={ϵ}_{\theta r}={ϵ}_{r\varphi }=0\)

and the stress tensor is written as:

\({\sigma }_{\mathit{rr}}={\mathrm{3C}}_{1}\lambda +2\mu ({C}_{1}-\frac{{\mathrm{2C}}_{2}}{{r}^{3}})\)

\({\sigma }_{\theta \theta }={\mathrm{3C}}_{1}\lambda +2\mu ({C}_{1}+\frac{{C}_{2}}{{r}^{3}})\)

\({\sigma }_{\varphi \varphi }={\mathrm{3C}}_{1}\lambda +2\mu ({C}_{1}+\frac{{C}_{2}}{{r}^{3}})\)

\({\sigma }_{\theta \varphi }={\sigma }_{\theta r}={\sigma }_{r\varphi }=0\)

The boundary conditions that make it possible to obtain the coefficients \({C}_{1},{C}_{2}\) are then:

\(\begin{array}{c}{\sigma }_{\mathit{rr}}({R}_{i})=-{P}_{i}\\ {\sigma }_{\mathit{rr}}({R}_{e})=-{P}_{e}\end{array}\)

2.2. Reference quantities and results

Displacements and contact pressure at the interface level are used to validate the approach for all of the four models presented in this test case.

For the value of the radial displacement field tested as a formula in \(r={R}_{f}\), we have:

\({U}_{r}({R}_{f})=\frac{1}{3\lambda +2\mu }(\frac{-{P}_{i}{R}_{i}^{3}+{P}_{e}{R}_{e}^{3}}{{R}_{i}^{3}-{R}_{e}^{3}}){R}_{f}+\frac{(-{P}_{i}+{P}_{e}){R}_{i}^{3}{R}_{e}^{3}}{4\mu {R}_{f}^{2}({R}_{i}^{3}-{R}_{e}^{3})}\)

For the value of the contact pressure field tested as a formula in \(r={R}_{f}\), we have:

\({\sigma }_{\mathit{rr}}({R}_{f})=(\frac{-{P}_{i}{R}_{i}^{3}+{P}_{e}{R}_{e}^{3}}{{R}_{i}^{3}-{R}_{e}^{3}})-\frac{(-{P}_{i}+{P}_{e}){R}_{i}^{3}{R}_{e}^{3}}{{R}_{f}^{3}({R}_{i}^{3}-{R}_{e}^{3})}\)

All numerical values are given in SI units.

Test	Identification	Analytical Reference Value
\({\sigma }_{\mathit{rr}}({R}_{f})\)	Min, Max LAGS_C for all interface post-processing nodes	-1.5046
\({U}_{r}({R}_{f})\)	Min, Max UR for all interface post-processing nodes	7.1133E-05

2.3. Uncertainty about the solution

There is no uncertainty about the solution, as it is analytical.

2.4. Bibliographical references

A.F. Bower, Applied Mechanics of Solids, Taylor and Francis, 2010.