2. Benchmark solution#
2.1. Calculation method used for the reference solution#
2.1.1. Displacement \(\mathrm{DY}\)#
The displacement \(\mathrm{DY}\), which refers to point \(A\), corresponds to the imposed displacement.
2.1.2. Constraint \(\mathrm{SIXX}\)#
Constraint \(\mathrm{SIXX}\) corresponds to the load applied.
2.1.3. Stress \(\mathit{SIYY}\) and cumulative plastic deformation \(V1\)#
Comparison with an analytical solution.
Triaxial calculation under drained conditions with a law of DRUCK_PRAGER or DRUCK_PRAG_N_A. The calculation is done for law DRUCK_PRAG_N_A (the most general case) and the solution can be extended to case DRUCK_PRAGER by taking \(\mathrm{\beta }(p)=\mathrm{\alpha }=\mathit{constante}\).
\({\mathrm{\sigma }}_{\mathit{eq}}+\mathrm{\beta }(p)\mathit{tr}(\mathrm{\sigma })-R(p)=0\)
We impose \({\mathrm{\sigma }}_{\mathit{xx}}={\mathrm{\sigma }}_{\mathit{yy}}=-2\mathit{MPa}={\mathrm{\sigma }}^{0}\)
\({\sigma }_{\mathrm{eq}}=\sqrt{\frac{3}{2}}{S}_{\mathrm{II}}\)
\(S=\left(\begin{array}{c}{\mathrm{\sigma }}_{\mathit{xx}}-\frac{1}{3}\mathit{tr}\mathrm{\sigma }\\ {\mathrm{\sigma }}_{\mathit{yy}}-\frac{1}{3}\mathit{tr}\mathrm{\sigma }\\ {\mathrm{\sigma }}_{\mathit{zz}}-\frac{1}{3}\mathit{tr}\mathrm{\sigma }\end{array}\right)\) with \(\mathit{tr}\mathrm{\sigma }={\mathrm{\sigma }}_{\mathit{xx}}+{\mathrm{\sigma }}_{\mathit{yy}}+{\mathrm{\sigma }}_{\mathit{zz}}={\mathrm{\sigma }}_{\mathit{zz}}+2{\mathrm{\sigma }}^{0}\)
\(S=\frac{1}{3}\left(\begin{array}{c}-{\mathrm{\sigma }}_{\mathit{zz}}+{\mathrm{\sigma }}^{0}\\ -{\mathrm{\sigma }}_{\mathit{zz}}+{\mathrm{\sigma }}^{0}\\ 2{\mathrm{\sigma }}_{\mathit{zz}}-2{\mathrm{\sigma }}^{0}\end{array}\right)\)
\({S}_{\mathit{II}}=S\mathrm{.}S=\frac{1}{3}\sqrt{2{({\mathrm{\sigma }}_{\mathit{zz}}-{\mathrm{\sigma }}^{0})}^{2}+{(2{\mathrm{\sigma }}_{\mathit{zz}}-2{\mathrm{\sigma }}^{0})}^{2}}\)
\({\sigma }_{\mathrm{eq}}=\sqrt{(\frac{3}{2})}{S}_{\mathrm{II}}\)
Which gives us:math: {mathrm {sigma}}} _ {mathit {eq}} =left| ({mathrm {sigma}} _ {mathit {zz}}} - {mathrm {sigma}} - {mathrm {sigma}}} - {mathit {zz}}} - {mathrm {sigma}} - {mathrm {sigma}} - {mathrm {sigma}} - {mathrm {sigma}} - {mathrm {sigma}} - {mathrm {sigma}} - {mathrm {sigma}}} - {mathrm {sigma}} - {mathrm {sigma}}} sigma}} _ {mathit {zz}}
So, in a plastic diet, we have: \({\mathrm{\sigma }}_{\mathit{zz}}=\frac{1}{\mathrm{\alpha }-1}(R(p)-(1+2\mathrm{\alpha }{\mathrm{\sigma }}^{0}))\) (1)
We also have \(\dot{{\mathrm{ϵ}}^{P}}=\mathrm{\lambda }\left(\frac{3}{2}\frac{S}{{\mathrm{\sigma }}_{\mathit{eq}}}+\mathrm{\beta }(p)I\right)=\dot{p}\left(\left(\begin{array}{c}1/2\\ 1/2\\ -1\end{array}\right)+\mathrm{\beta }(p)I\right)\)
Noting \(B(p)\) the primitive of \(\mathrm{\beta }(p)\) cancelling out at 0, he comes up with:
\({\mathrm{ϵ}}^{P}=p\left(\begin{array}{c}1/2\\ 1/2\\ -1\end{array}\right)+B(p)I\)
The stress-deformation relationship gives:
\({\mathrm{ϵ}}_{\mathit{zz}}^{e}=\frac{1}{E}{\mathrm{\sigma }}_{\mathit{zz}}-\frac{2\mathrm{\nu }}{E}{\mathrm{\sigma }}^{0}-\frac{1-2\mathrm{\nu }}{E}{\mathrm{\sigma }}^{0}\)
Using the expression for \({\mathrm{\sigma }}_{\mathit{zz}}\), we then get:
\({\mathrm{ϵ}}_{\mathit{zz}}=\frac{1}{E(a-1)}(R(p)-3a{\mathrm{\sigma }}^{0})-p+B(p)\) (2)
The expressions (1) and (2) give a parameterization of the strain-strain response as a function of the work-hardening variable \(p\).
Below are the expressions for \(B(p)\) (primitive of \(\mathrm{\beta }(p)\) cancelling out at 0), according to the type of work hardening and the law of behavior:
DRUCK_PRAGER, for all work hardening:
\(B(p)=\mathrm{\alpha }p\)
DRUCK_PRAG_N_A, linear and parabolic work hardening:
\(B(p)=\mathrm{\beta }(1-\frac{p}{2{p}_{\mathit{ultm}}})p\)
DRUCK_PRAG_N_A, exponential work hardening:
\(B(p)=\mathrm{\beta }{p}_{c}(1-\mathrm{exp}(\frac{-p}{{p}_{c}}))\)
2.2. Reference quantities#
Constraint \(\mathit{SIZZ}\) at node \(A\)
Cumulative plastic deformation \(\mathrm{V1}\) at node \(A\)
Move \(\mathrm{DY}\) to node \(A\)
2.3. Benchmark result#
For the law of behavior DRUCK_PRAGER:
Grandeur |
Point |
\(\mathrm{Inst}\) |
Reference, linear work hardening |
Reference, parabolic work hardening |
\(\mathrm{SIXX}(N/{m}^{2})\) |
A |
\(2.0\) |
\(-2.0\mathrm{E6}\) |
\(-2.0\mathrm{E6}\) |
\(\mathit{SIZZ}(N/{m}^{2})\) |
A |
\(1.07\) |
\(-8.09\mathrm{E6}\) |
\(-8.09\mathrm{E6}\) |
\(1.16\) |
\(-8.20\mathrm{E6}\) |
\(-8.01\mathrm{E6}\) |
||
\(1.34\) |
\(-6.89\mathrm{E6}\) |
\(-6.63\mathrm{E6}\) |
||
\(1.53\) |
\(-5.80\mathrm{E6}\) |
\(-5.81\mathrm{E6}\) |
||
\(\mathrm{V1}\) |
A |
\(1.07\) |
\(0\) |
\(0\) |
\(1.16\) |
\(1.99E-3\) |
\(2.04E-3\) |
||
\(1.34\) |
\(6.35E-3\) |
\(6.42E-3\) |
||
\(1.53\) |
\(1.09E-2\) |
\(1.09E-2\) |
||
\(\mathit{DY}\phantom{\rule{0.5em}{0ex}}(m)\) |
A |
\(1.07\) |
\(-1.05E-3\) |
\(-1.05E-3\) |
\(1.16\) |
\(-2.40E-3\) |
\(-2.40E-3\) |
||
\(1.34\) |
\(-5.10E-3\) |
\(-5.10E-3\) |
||
\(1.53\) |
\(-7.95E-3\) |
\(-7.95E-3\) |
For the law of behavior DRUCK_PRAG_N_A:
Grandeur |
Point |
NUME_ORDRE |
Reference, linear work hardening |
Reference, parabolic work hardening |
Reference, exponential work hardening |
\(\mathit{SIZZ}(N/{m}^{2})\) |
A |
9 |
\(-8.77\mathrm{.}{10}^{6}\) |
\(-8.76\mathrm{.}{10}^{6}\) |
\(-8.77\mathrm{.}{10}^{6}\) |
34 |
\(-7.20\mathrm{.}{10}^{6}\) |
\(-6.44\mathrm{.}{10}^{6}\) |
\(-7.55\mathrm{.}{10}^{6}\) |
||
60 |
\(-5.86\mathrm{.}{10}^{6}\) |
\(-5.80\mathrm{.}{10}^{6}\) |
/ |
||
\(\mathrm{V1}\) |
A |
9 |
\(4.69\mathrm{.}{10}^{-5}\) |
\(5.11\mathrm{.}{10}^{-5}\) |
\(4.69\mathrm{.}{10}^{-5}\) |
34 |
\(5.3\mathrm{.}{10}^{-3}\) |
\(5.4\mathrm{.}{10}^{-3}\) |
\(5.3\mathrm{.}{10}^{-3}\) |
||
60 |
\(9.8\mathrm{.}{10}^{-3}\) |
\(9.8\mathrm{.}{10}^{-3}\) |
/ |
||
\(\mathrm{DY}(m)\) |
A |
9 |
\(-1.2\mathrm{.}{10}^{-3}\) |
\(-1.2\mathrm{.}{10}^{-3}\) |
\(-1.2\mathrm{.}{10}^{-3}\) |
34 |
\(-4.9\mathrm{.}{10}^{-3}\) |
\(-4.9\mathrm{.}{10}^{-3}\) |
\(-4.9\mathrm{.}{10}^{-3}\) |
||
60 |
\(-8.8\mathrm{.}{10}^{-3}\) |
\(-8.8\mathrm{.}{10}^{-3}\) |
/ |
2.4. Uncertainty about the solution#
Analytical solution