2. Benchmark solutions#
2.1. Calculation method used for reference solutions#
The whole of this demonstration can be read in more detail in the [bib1] document.
In the case of a linear isotropic viscoelastic material, the behavior over time can be described using two functions \(I(t)\) and \(K(t)\) in such a way that the deformations and the stresses can be written as:
\(\varepsilon (t)\mathrm{=}(I+K)\mathrm{\ast }\frac{d\sigma (t)}{d\tau }\mathrm{-}K\mathrm{\ast }\frac{d(\mathit{Tr}(\sigma (t)))}{d\tau }{I}_{3}\)
where \({I}_{3}\) refers to the identity matrix of rank 3
and \(\mathrm{\ast }\) the convolution product: \((f\mathrm{\ast }g)(t)\mathrm{=}{\mathrm{\int }}_{0}^{t}f(t\mathrm{-}\tau )g(\tau )d\tau\)
We find \(I(t)\mathrm{=}\frac{1}{E}+\mathit{kt}\), \(K(t)\mathrm{=}\frac{\nu }{E}+\frac{1}{2}\mathit{kt}\)
We impose the pressure \({P}_{0}\) at the moment \(t\mathrm{=}0\), the internal pressure is equal to \(p(t)\mathrm{=}H(t){P}_{0}\) or \(H(t)\mathrm{=}\mathrm{\{}\begin{array}{c}0\mathit{si}t\mathrm{-}\tau <0\\ 1\mathit{si}t\mathrm{-}\tau \mathrm{\ge }0\end{array}\) with in this case \(\tau \mathrm{=}0\)
We use the Laplace Carson transform \({f}^{\text{+}}(n)\mathrm{=}L(f(t))\mathrm{=}n{\mathrm{\int }}_{0}^{\mathrm{\infty }}f(t){e}^{\mathrm{-}\mathit{nt}}\mathit{dt}\)
From where \({p}^{+}\mathrm{=}{P}_{0}\)
The solution of the equivalent elastic problem is:
\({\sigma }^{\text{+}}\mathrm{=}(\begin{array}{ccc}\gamma (1\mathrm{-}\frac{{r}_{1}^{2}}{{r}^{2}})& 0& 0\\ 0& \gamma (1+\frac{{r}_{1}^{2}}{{r}^{2}})& 0\\ 0& 0& {\sigma }_{Z}^{\text{+}}\end{array})\) where \(\gamma \mathrm{=}\frac{{P}_{0}{r}_{0}^{2}}{{r}_{1}^{2}\mathrm{-}{r}_{0}^{2}}\)
We determine \({\sigma }_{Z}^{\text{+}}\) by the condition on \({\varepsilon }_{Z}^{\text{+}}\) given by the boundary conditions:
\({\varepsilon }_{Z}^{\text{+}}\mathrm{=}0\mathrm{=}({I}^{\text{+}}+{K}^{\text{+}}){\sigma }_{Z}^{\text{+}}\mathrm{-}{K}^{\text{+}}(2\gamma +{\sigma }_{Z}^{\text{+}})\mathrm{=}{I}^{\text{+}}{\sigma }_{Z}^{\text{+}}\mathrm{-}2{K}^{\text{+}}\gamma\)
Hence \({\sigma }_{Z}^{\text{+}}\mathrm{=}\gamma (1+\frac{(2\nu \mathrm{-}1)p}{p+\mathit{Ek}})\).
We find by the inverse Laplace transform \({\sigma }_{z}(t)\mathrm{=}\gamma (1\mathrm{-}(1\mathrm{-}2\nu ){e}^{\mathrm{-}\mathit{Eht}})\), similarly by applying the inverse Laplace transform on \({\sigma }_{r}\) and \({\sigma }_{\theta }\), we find
\({\sigma }^{\text{+}}\mathrm{=}(\begin{array}{ccc}\gamma (1\mathrm{-}\frac{{r}_{1}^{2}}{{r}^{2}})& 0& 0\\ 0& \gamma (1+\frac{{r}_{1}^{2}}{{r}^{2}})& 0\\ 0& 0& \gamma (1\mathrm{-}(1\mathrm{-}2\nu ){e}^{\mathrm{-}\mathit{Eht}})\end{array})\)
From this we deduce:
\(\dot{{\varepsilon }_{V}}\mathrm{=}(\begin{array}{ccc}\frac{3}{2}k\gamma (\frac{1\mathrm{-}2\nu }{3}{e}^{\mathrm{-}\mathit{Ekt}}\mathrm{-}\frac{{r}_{1}^{2}}{{r}^{2}})& 0& 0\\ 0& \frac{3}{2}k\gamma (\frac{1\mathrm{-}2\nu }{3}{e}^{\mathrm{-}\mathit{Ekt}}\mathrm{-}\frac{{r}_{1}^{2}}{{r}^{2}})& 0\\ 0& 0& \mathrm{-}k\gamma ((1\mathrm{-}2\nu ){e}^{\mathrm{-}\mathit{Eht}})\end{array})\)
and integrating with \({\varepsilon }_{V}(0)\mathrm{=}0\);
\(\dot{{\varepsilon }_{V}}\mathrm{=}(\begin{array}{ccc}\frac{3}{2}\gamma (\frac{1\mathrm{-}2\nu }{3}{e}^{\mathrm{-}\mathit{Ekt}}\mathrm{-}k\frac{{r}_{1}^{2}}{{r}^{2}}t)& 0& 0\\ 0& \frac{3}{2}\gamma (\frac{1\mathrm{-}2\nu }{3}{e}^{\mathrm{-}\mathit{Ekt}}\mathrm{-}k\frac{{r}_{1}^{2}}{{r}^{2}}t)& 0\\ 0& 0& \mathrm{-}\gamma \frac{(1\mathrm{-}2\nu )}{E}(1\mathrm{-}{e}^{\mathrm{-}\mathit{Eht}})\end{array})\).
The radial displacement is deduced
\(w(r,t)\mathrm{=}r\gamma \left[\frac{1}{E}\left[(1+\nu )\frac{{r}_{1}^{2}}{{r}^{2}}+\frac{1\mathrm{-}2\nu }{2}(3\mathrm{-}(1\mathrm{-}2\nu ){e}^{\mathrm{-}\mathit{Ekt}})\right]+\frac{3}{2}k\frac{{r}_{1}^{2}}{{r}^{2}}t\right]\)
2.2. Benchmark results#
Move \(\mathit{DX}\) on node \(B\) and constraints \(\mathit{SIXX}\), \(\mathrm{SIYY}\), and \(\mathrm{SIZZ}\) in \(B\)
2.3. Uncertainty about the solution#
\(\text{0\%}\): analytical solution
2.4. Bibliographical references#
Ph. Of BONNIERES: Two analytical solutions to axisymmetric problems in linear viscoelasticity and with unilateral contact, Note HI-71/8301