Benchmark solution
=====================

The modeling verifies the behavior of the linear hardening law.


Calculation method
-----------------

The equations we are interested in for analytical calculation are (:math:`{I}_{1}=\text{tr}(\sigma )`: stress tensor trace, :math:`{\varepsilon }_{v}^{p}`: volume plastic deformation):


* plastic constitutive law on the volume part:


.. math::
: label: EQ-None

    {I} _ {1} =\ mathrm {3K} ({\ varepsilon} _ {v} - {\ varepsilon} _ {v} ^ {p})


* criterion area, setting the Von Mises constraint to zero (:math:`{\sigma }_{\mathrm{eq}}=0`):


.. math::
: label: EQ-None

    F (\ sigma, p) =\ alpha {I} _ {1} -R (p)


* relationship between volume plastic deformation and cumulative plastic deformation (internal plastic law variable):


.. math::
: label: EQ-None

    \ dot {{\ varepsilon} _ {v}} ^ {p}} =3\ alpha\ dot {p}


* expression of work hardening


    * linear:


.. math::
: label: EQ-None

    \ begin {array} {} R (p) = {\ sigma} _ {Y}} _ {Y} +hp\ mathrm {si} p\ le {p} _ {\ mathrm {ult}}}\\ R (p) = {\ sigma} = {\ sigma}} = {\ mathrm {ult}}} = {\ sigma} _ {Y}} ^ {\ mathrm {ult}} ^ {\ mathrm {ult}} ^ {\ mathrm {ult}} ^ {\ mathrm {ult}} ^ {\ mathrm {ult}} ^ {\ mathrm {ult}} ^ {\ mathrm {ult}} ^ {\ mathrm {ult}} ^ {\ mathrm {ult}} ^ {\ mathrm {ult}}}\ mathrm {si} p > {p} _ {\ mathrm {ult}}\ end {array}


*
    * parabolic:


.. math::
: label: EQ-None

    \ begin {array} {} R (p) = {\ sigma} _ {Y}} {(1- (1-\ sqrt {\ sigma} _ {Y} ^ {\ mathrm {ult}}}}} {{\ sigma}}}} {{\ sigma}}}} {{\ sigma} _ {\ sigma}}}} {{\ sigma}}}}} {{\ sigma}}}} {{\ sigma}}}} {{\ sigma}}}} {{\ sigma}}}} {{\ sigma}}}} {{\ sigma}}}} {{\ sigma} _ {Y}}}) {{\ sigma} _ {Y}}}) {{\ sigma}}}} {{\ sigma}}}} {{\ sigma}}}} {{\ sigma} thrm {si} p\ le {p} _ {\ mathrm {ult}} {\ mathrm {ult}}}\\ R (p) = {\ sigma} _ {Y} ^ {\ mathrm {ult}}}\ mathrm {ult}}\\ mathrm {ult}}\ end {ult}}\ end {array}}\ mathrm {ult}}\

 We observe that, as in the linear case, :math:`R(p)={\sigma }_{Y}` if :math:`p=0` and we have perfect plasticity if :math:`p>{p}_{\mathrm{ult}}`.


Deformation at the initial elastic limit
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

This deformation is obtained for :math:`{\varepsilon }_{v}^{p}=p=0`.

If we ask :math:`F(\sigma ,p)=0` (plastic evolution) we have:

:math:`{I}_{1}^{\mathrm{el}}=\frac{R(p)}{\alpha }=\frac{{\sigma }_{Y}}{\alpha }`

:math:`{\varepsilon }_{v}^{\mathrm{el}}=\frac{{I}_{1}^{\mathrm{el}}}{3K}`


Ultimate deformation
~~~~~~~~~~~~~~~~~~~

The ultimate deformation :math:`{\varepsilon }_{v}^{\mathrm{ult}}` is called the one obtained for :math:`p={p}_{\mathrm{ult}}`.

The corresponding stress trace :math:`{I}_{1}^{\mathrm{ult}}` and the plastic deformation :math:`{\varepsilon }_{v}^{\mathrm{pult}}` can easily be found:

:math:`{I}_{1}^{\mathrm{ult}}=\frac{R(p)}{\alpha }=\frac{{\sigma }_{Y}^{\mathrm{ult}}}{\alpha }`

:math:`{\varepsilon }_{v}^{\mathrm{pult}}=3\alpha {p}_{\mathrm{ult}}`

:math:`{\varepsilon }_{v}^{\mathrm{ult}}=\frac{{I}_{1}^{\mathrm{ult}}}{3K}+{\varepsilon }_{v}^{\mathrm{pult}}`


Deformation between the elastic limit and the ultimate deformation
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

First, the cumulative plastic deformation is calculated.


*Combining equations (), (), () and () with :math:`F(\sigma ,p)=0` for**linear work hardening** we have:


.. math::
: label: EQ-None

    p=\ frac {3KA {\ varepsilon} _ {\ mathrm {v1}} - {\ sigma} _ {Y}} {9K {\ alpha} ^ {2} +h}


*By combining equations (), (), () and () with :math:`F(\sigma ,p)=0` for**parabolic work hardening** we arrive at the equation of degree 2:


.. math::
: label: EQ-None

    {A} _ {1} {\ stackrel {} {p}}} ^ {2}} + {B} _ {1}\ stackrel {} {p} {p} + {C} _ {1} =0
    \ begin {array} {} {A} _ {1} = {\ sigma} _ {1} _ {Y} {(1-\ gamma)} ^ {2}\\ {B} _ {1} =9K {\ alpha} ^ {2} {2} {p} = {p} _ {p} _ {p} _ {p} _ {p} _ {p} _ {p} _ {p} _ {p} _ {p} _ {p} _ {p} _ {p} _ {p} _ {p} _ {p} _ {p} _ {p} _ {p} _ {p} _ {p} _ {p} _ {p} _ {p} _ {p} _ {p} _ {p} _ {p} _ {p} _ {p} _ {p} _ {sigma} _ {Y} -3K\ alpha {\ varepsilon} _ {v\ text {1}}\\\ gamma =\ sqrt {\ frac {{\ sigma} _ {Y} ^ {\ mathrm {ult}}} {\ mathrm {ult}}}}} {{\ sigma}}} {{\ sigma} _ {Y}}}}\\\ stackrel {} {p} {p}} =\ frac {p}} =\ frac {p} {p} _ {\ mathrm {ult}}}}\ end {array}

We then use the equations () () to find the plastic deformation :math:`{\varepsilon }_{v}^{p}` and the stress trace :math:`{I}_{1}`.

If the material is unloaded elastically until zero stress is reached, a residual deformation equal to the plastic deformation is obtained; on the other hand, it is necessary to load the material under compression to obtain zero total deformation. This second branch is also elastic, because the Drücker-Prager material cannot plasticize in a state of hydrostatic compression. In the latter case, the negative stress trace is:


.. math::
: label: EQ-None

    {I} _ {1} ^ {c} =-3K {\ varepsilon} _ {v} ^ {p}


Deformation greater than ultimate deformation
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

All the quantities of interest are easily found, because the constraint trace is known a priory and equal to :math:`{I}_{1}^{\mathrm{ult}}`.

:math:`{\varepsilon }_{v}^{p}={\varepsilon }_{v}-\frac{{I}_{1}^{\mathrm{ult}}}{3K}`

:math:`p=\frac{{\varepsilon }_{v}^{p}}{3\alpha }`


Reference quantities and results
-----------------------------------

The compressibility module :math:`K` is:

:math:`K=\frac{E}{3(1-2\nu )}=2000\mathrm{MPa}`


Deformation at the elastic limit
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

For both models we can easily find:

:math:`{I}_{1}^{\mathrm{el}}=30\mathrm{MPa}`

:math:`{\varepsilon }_{v}^{\mathrm{el}}=\mathrm{0,005}`


Ultimate deformation
~~~~~~~~~~~~~~~~~~~~~

For both models we find:

:math:`{I}_{1}^{\mathrm{ult}}=50\mathrm{MPa}`

:math:`{\varepsilon }_{v}^{\mathrm{pult}}=\mathrm{0,024}`

:math:`{\varepsilon }_{v}^{\mathrm{ult}}\approx \mathrm{0,03233}`


Deformation equal to 0.018 and zero deformation discharge
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

This deformation value :math:`{\varepsilon }_{v\text{1}}=\mathrm{0,018}` is greater than the elastic limit :math:`{\varepsilon }_{v}^{\mathrm{el}}` and less than :math:`{\varepsilon }_{v}^{\mathrm{ult}}`. We first calculate the cumulative plastic deformation with the equations () and (), then the plastic deformation and the stress trace:


* linear work hardening:

:math:`{p}_{1}=\frac{3KA{\varepsilon }_{\mathrm{v1}}-{\sigma }_{Y}}{9K{\alpha }^{2}+h}\approx \mathrm{0,019}`

:math:`{\varepsilon }_{v\text{1}}^{p}=3\alpha {p}_{1}=\mathrm{0,0114}`

:math:`{I}_{1}^{1}=3K({\varepsilon }_{v\text{1}}-{\varepsilon }_{v\text{1}}^{p})\approx \mathrm{39,51}\mathrm{MPa}`


* parabolic work hardening:

:math:`{p}_{1}\approx 0.0192`

:math:`{\varepsilon }_{v\text{1}}^{p}=3\alpha {p}_{1}\approx 0.0115`

:math:`{I}_{1}^{1}=3K({\varepsilon }_{v\text{1}}-{\varepsilon }_{v\text{1}}^{p})\approx 38.956\mathrm{MPa}`

The zero deformation stress trace is:


* linear work hardening:

:math:`{I}_{1}^{\mathrm{1c}}=-3K{\varepsilon }_{v\text{1}}^{p}\approx -\mathrm{68,49}\mathrm{MPa}`


* parabolic work hardening:

:math:`{I}_{1}^{\mathrm{1c}}=-3K{\varepsilon }_{v\text{1}}^{p}\approx -\mathrm{69,044}\mathrm{MPa}`

Indeed, the difference between the parabolic and linear case is very small.


Loading until deformation equal to 0.045 and 0.06
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The material is recharged up to deformation values :math:`{\varepsilon }_{v\text{2}}=\mathrm{0,045}` and :math:`{\varepsilon }_{v\text{3}}=\mathrm{0,06}`, which are greater than :math:`{\varepsilon }_{v}^{\mathrm{ult}}`.

The results are the same for both models.

For :math:`{\varepsilon }_{v\text{2}}=\mathrm{0,045}`:

:math:`{\varepsilon }_{v\text{2}}^{p}={\varepsilon }_{v\text{2}}-\frac{{I}_{1}^{\mathrm{ult}}}{3K}\approx \mathrm{0,03667}`

:math:`{p}_{2}=\frac{{\varepsilon }_{v\text{2}}^{p}}{3\alpha }\approx \mathrm{0,0611}`

Following the elastic discharge (up to zero stress), we find :math:`{\varepsilon }_{v}={\varepsilon }_{v\text{2}}^{p}`, :math:`p={p}_{2}`.

For :math:`{\varepsilon }_{v\text{3}}=\mathrm{0,06}`:

:math:`{\varepsilon }_{v\text{3}}^{p}={\varepsilon }_{v\text{3}}-\frac{{I}_{1}^{\mathrm{ult}}}{3K}\approx \mathrm{0,051667}`

:math:`{p}_{3}=\frac{{\varepsilon }_{v\text{3}}^{p}}{3\alpha }\approx \mathrm{0,0861}`


Stress-strain curves
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

In Figures () and () the curve (:math:`{\varepsilon }_{v},{I}_{1}`) for linear and parabolic work hardening is represented. In red are the points tested by the test case.


.. image:: images/10000000000002A2000001EADCF6892A359ABF61.jpg
    :width: 4.7244in
    :height: 3.4335in


.. _RefImage_10000000000002A2000001EADCF6892A359ABF61.jpg:

**Figure** 2.2.5-a **: stress-strain curves for linear work hardening.**


.. image:: images/10000000000002A5000001E28E1C79F903020CD2.jpg
    :width: 5.1181in
    :height: 3.6374in


.. _RefImage_10000000000002A5000001E28E1C79F903020CD2.jpg:

**Figure** 2.2.5-b **: stress-strain curves for parabolic work hardening.**


Uncertainty about the solution
----------------------------

The solution is analytical.


Bibliographical references
---------------------------


1. Document [:external:ref:`R3.01.16 <R3.01.16>`], Integration of the elasto-plastic mechanical behavior of Drücker-Prager DRUCK_PRAGER and post-treatments. Code_Aster reference manual.