Benchmark solution ===================== Calculation method used for the reference solution -------------------------------------------------------- Displacement :math:`\mathrm{DY}` ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The displacement :math:`\mathrm{DY}`, which refers to point :math:`A`, corresponds to the imposed displacement. Constraint :math:`\mathrm{SIXX}` ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Constraint :math:`\mathrm{SIXX}` corresponds to the load applied. Stress :math:`\mathit{SIYY}` and cumulative plastic deformation :math:`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 :math:`\mathrm{\beta }(p)=\mathrm{\alpha }=\mathit{constante}`. .. image:: images/Shape4.gif .. _RefSchema_Shape4.gif: :math:`{\mathrm{\sigma }}_{\mathit{eq}}+\mathrm{\beta }(p)\mathit{tr}(\mathrm{\sigma })-R(p)=0` We impose :math:`{\mathrm{\sigma }}_{\mathit{xx}}={\mathrm{\sigma }}_{\mathit{yy}}=-2\mathit{MPa}={\mathrm{\sigma }}^{0}` :math:`{\sigma }_{\mathrm{eq}}=\sqrt{\frac{3}{2}}{S}_{\mathrm{II}}` :math:`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 :math:`\mathit{tr}\mathrm{\sigma }={\mathrm{\sigma }}_{\mathit{xx}}+{\mathrm{\sigma }}_{\mathit{yy}}+{\mathrm{\sigma }}_{\mathit{zz}}={\mathrm{\sigma }}_{\mathit{zz}}+2{\mathrm{\sigma }}^{0}` :math:`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)` :math:`{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}}` :math:`{\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: :math:`{\mathrm{\sigma }}_{\mathit{zz}}=\frac{1}{\mathrm{\alpha }-1}(R(p)-(1+2\mathrm{\alpha }{\mathrm{\sigma }}^{0}))` (1) We also have :math:`\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 :math:`B(p)` the primitive of :math:`\mathrm{\beta }(p)` cancelling out at 0, he comes up with: :math:`{\mathrm{ϵ}}^{P}=p\left(\begin{array}{c}1/2\\ 1/2\\ -1\end{array}\right)+B(p)I` The stress-deformation relationship gives: :math:`{\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 :math:`{\mathrm{\sigma }}_{\mathit{zz}}`, we then get: :math:`{\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 :math:`p`. Below are the expressions for :math:`B(p)` (primitive of :math:`\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: :math:`B(p)=\mathrm{\alpha }p` * DRUCK_PRAG_N_A, linear and parabolic work hardening: :math:`B(p)=\mathrm{\beta }(1-\frac{p}{2{p}_{\mathit{ultm}}})p` * DRUCK_PRAG_N_A, exponential work hardening: :math:`B(p)=\mathrm{\beta }{p}_{c}(1-\mathrm{exp}(\frac{-p}{{p}_{c}}))` Reference quantities ---------------------- * Constraint :math:`\mathit{SIZZ}` at node :math:`A` * Cumulative plastic deformation :math:`\mathrm{V1}` at node :math:`A` * Move :math:`\mathrm{DY}` to node :math:`A` Benchmark result ---------------------- For the law of behavior DRUCK_PRAGER: +-------------------------------------------------+-----+---------------------+--------------------------------+-----------------------------------+ |Grandeur |Point|:math:`\mathrm{Inst}`|Reference, linear work hardening|Reference, parabolic work hardening| +-------------------------------------------------+-----+---------------------+--------------------------------+-----------------------------------+ |:math:`\mathrm{SIXX}(N/{m}^{2})` |A |:math:`2.0` |:math:`-2.0\mathrm{E6}` |:math:`-2.0\mathrm{E6}` | +-------------------------------------------------+-----+---------------------+--------------------------------+-----------------------------------+ |:math:`\mathit{SIZZ}(N/{m}^{2})` |A |:math:`1.07` |:math:`-8.09\mathrm{E6}` |:math:`-8.09\mathrm{E6}` | + + +---------------------+--------------------------------+-----------------------------------+ | | |:math:`1.16` |:math:`-8.20\mathrm{E6}` |:math:`-8.01\mathrm{E6}` | + + +---------------------+--------------------------------+-----------------------------------+ | | |:math:`1.34` |:math:`-6.89\mathrm{E6}` |:math:`-6.63\mathrm{E6}` | + + +---------------------+--------------------------------+-----------------------------------+ | | |:math:`1.53` |:math:`-5.80\mathrm{E6}` |:math:`-5.81\mathrm{E6}` | +-------------------------------------------------+-----+---------------------+--------------------------------+-----------------------------------+ |:math:`\mathrm{V1}` |A |:math:`1.07` |:math:`0` |:math:`0` | + + +---------------------+--------------------------------+-----------------------------------+ | | |:math:`1.16` |:math:`1.99E-3` |:math:`2.04E-3` | + + +---------------------+--------------------------------+-----------------------------------+ | | |:math:`1.34` |:math:`6.35E-3` |:math:`6.42E-3` | + + +---------------------+--------------------------------+-----------------------------------+ | | |:math:`1.53` |:math:`1.09E-2` |:math:`1.09E-2` | +-------------------------------------------------+-----+---------------------+--------------------------------+-----------------------------------+ |:math:`\mathit{DY}\phantom{\rule{0.5em}{0ex}}(m)`|A |:math:`1.07` |:math:`-1.05E-3` |:math:`-1.05E-3` | + + +---------------------+--------------------------------+-----------------------------------+ | | |:math:`1.16` |:math:`-2.40E-3` |:math:`-2.40E-3` | + + +---------------------+--------------------------------+-----------------------------------+ | | |:math:`1.34` |:math:`-5.10E-3` |:math:`-5.10E-3` | + + +---------------------+--------------------------------+-----------------------------------+ | | |:math:`1.53` |:math:`-7.95E-3` |:math:`-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| +--------------------------------+-----+----------+--------------------------------+-----------------------------------+-------------------------------------+ |:math:`\mathit{SIZZ}(N/{m}^{2})`|A |9 |:math:`-8.77\mathrm{.}{10}^{6}` |:math:`-8.76\mathrm{.}{10}^{6}` |:math:`-8.77\mathrm{.}{10}^{6}` | + + +----------+--------------------------------+-----------------------------------+-------------------------------------+ | | |34 |:math:`-7.20\mathrm{.}{10}^{6}` |:math:`-6.44\mathrm{.}{10}^{6}` |:math:`-7.55\mathrm{.}{10}^{6}` | + + +----------+--------------------------------+-----------------------------------+-------------------------------------+ | | |60 |:math:`-5.86\mathrm{.}{10}^{6}` |:math:`-5.80\mathrm{.}{10}^{6}` |/ | +--------------------------------+-----+----------+--------------------------------+-----------------------------------+-------------------------------------+ |:math:`\mathrm{V1}` |A |9 |:math:`4.69\mathrm{.}{10}^{-5}` |:math:`5.11\mathrm{.}{10}^{-5}` |:math:`4.69\mathrm{.}{10}^{-5}` | + + +----------+--------------------------------+-----------------------------------+-------------------------------------+ | | |34 |:math:`5.3\mathrm{.}{10}^{-3}` |:math:`5.4\mathrm{.}{10}^{-3}` |:math:`5.3\mathrm{.}{10}^{-3}` | + + +----------+--------------------------------+-----------------------------------+-------------------------------------+ | | |60 |:math:`9.8\mathrm{.}{10}^{-3}` |:math:`9.8\mathrm{.}{10}^{-3}` |/ | +--------------------------------+-----+----------+--------------------------------+-----------------------------------+-------------------------------------+ |:math:`\mathrm{DY}(m)` |A |9 |:math:`-1.2\mathrm{.}{10}^{-3}` |:math:`-1.2\mathrm{.}{10}^{-3}` |:math:`-1.2\mathrm{.}{10}^{-3}` | + + +----------+--------------------------------+-----------------------------------+-------------------------------------+ | | |34 |:math:`-4.9\mathrm{.}{10}^{-3}` |:math:`-4.9\mathrm{.}{10}^{-3}` |:math:`-4.9\mathrm{.}{10}^{-3}` | + + +----------+--------------------------------+-----------------------------------+-------------------------------------+ | | |60 |:math:`-8.8\mathrm{.}{10}^{-3}` |:math:`-8.8\mathrm{.}{10}^{-3}` |/ | +--------------------------------+-----+----------+--------------------------------+-----------------------------------+-------------------------------------+ Uncertainty about the solution --------------------------- Analytical solution *