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


Calculation method used for the reference solution
--------------------------------------------------------

We rely on the documentation [:ref:`R5.03.29 <R5.03.29>`] which describes the equations of the model; we also adopt the notations introduced there.

As the stress trace is zero, the traces of elastic, plastic, and total deformations are also zero. And the Jacobian for transformation :math:`J` then remains equal to 1. By noting :math:`d={q}_{1}f` and :math:`{d}_{0}={q}_{1}{f}_{0}`, we can express the equivalent constraint of GTN:

.. math::
 :label: eq-2

    {T} _ {\ text {*}}} =\ frac {{T} _ {\ mathit {eq}}} {1-d}} {1-d}

In the plastic regime, the consistency condition establishes the link between the stress and the work-hardening variable:

.. math::
 :label: eq-3

    {T} _ {\ mathit {eq}} = (1-d) (1-d) ({R} _ {0} + {R} _ {H}\ mathrm {\ kappa})

With :math:`\mu`, the elastic deformation is equal to the shear modulus:

.. math::
: label: eq-4

    {E} ^ {e} =\ frac {{T} _ {\ mathit {eq}}}} {2\ mathrm {\ mu}} {D}} {D}} ^ {0}

The law of plastic evolution is reduced to:

.. math::
: label: eq-5

    \ dot {{E} ^ {p}} =\ frac {3} {2}\ frac {\ dot {\ mathrm {\ kappa}}} {1-d} {D} {D} {D}} ^ {0}

In particular, the equivalent plastic deformation is equal to:

.. math::
 :label: eq-6

    {\ dot {E}} _ {\ mathit {eq}}} ^ {p} =\ frac {\ dot {\ mathrm {\ kappa}}}} {1-d}

And, finally, the evolution of porosity is written as:

.. math::
 :label: eq-7

    \ dot {f} =B (\ mathrm {\ kappa})\ dot {\ mathrm {\ kappa}} + {b} _ {0}\ frac {\ dot {\ mathrm {\ kappa}}}} {1- {kappa}}}} {1- {q} _ {1} q} _ {1} f} {\ mathrm {\ chi}}} _ {[{E} _ {\ mathit {eq}}}} {\ mathit {eq}} 0} ^ {p}; +\ mathrm {\ infty}]} ({E}}} ({E} _ {\ mathit {eq}}})\ text {;}\ dot {d} = {q} _ {1} B (\ mathrm {\ infty}]]} ({E} _ {\ infty}]]} ({E} _ {1} _ {1} {1} {1} {1}} B (\ mathrm {\ kappa})\ dot {\ kappa})\ dot {\ mathrm {\ kappa})\ dot {\ mathrm {\ kappa}}} + {q} _ {1} {b} _ {1} {1} {1} {1} {1} {1} {1} {1}}\ dot {\ mathrm {\ kappa}}} {1-d} {1-d} {\ mathrm {\ chi}}} _ {[{E} _ {\ mathit {eq} 0}} ^ {p}; +\ mathrm {\ infty}]}} {\ mathrm {\ infty}]}} ({E} _ {\ mathit {eq}} 0} ^ {p}; +\ mathrm {\ infty}]]} ({E} _ {\ mathit {eq}})

In general, the above differential equation does not allow for a simple analytical solution. This is why a distinction is made between two cases, which will correspond to versions A and B of the test case.


Case A: :math:`{b}_{0}=0`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~

In this case, porosity is simply written:

.. math::
 :label: eq-8

    f (\ mathrm {\ kappa}) = {f} _ {0} +\ underset {0} +\ underset {0} {\ o{\ kappa}} {\ int}} {\ int}} B (u)\ mathit {of}} of}\ text {of}\ text {;}} + {\ int}}}} B (u)\ mathit {du}\ du}\ text {;}\ text {;} d (\ mathrm {\ kappa}) = {d} _ {0} + {q} _ {1}\ underset {0})} {\ overset {\ mathrm {\ kappa}} {\ int}} {\ int}} B (u)\ mathit {du}

And the plastic deformation is obtained by integrating ():

.. math::
 :label: eq-9

    {E} ^ {p} (\ mathrm {\ kappa}) =\ frac {3} {2}\ underset {0} {\ overset {\ mathrm {\ kappa}} {\ kappa}} {\ int}} {\ int}}} {\ int}}} {\ int}}} {\ int}}} {\ int}}} {\ int}}}\ frac {\ mathit {du}} {\ mathit {du}} {1-d (u)} {D} ^ {0}


Case B: :math:`B(\kappa )=0`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

In this case, the differential equation () has separable variables. It is easily integrated and leads to:

.. math::
 :label: eq-10

    {\ left (1-d (\ mathrm {\ kappa})\ right)}} ^ {2}} = {\ left (1- {q} _ {0}\ right)} ^ {2}\ right)} ^ {2} -2 {q} -2 {q} -2 {q} _ {q} _ {1} {b} _ {b} _ {0} ⟩\ text {;} {\ mathrm {\ kappa}}} _ {0}} = {E} _ {\ mathit {eq} 0} ^ {p} (1- {q} _ {1} _ {1} {1} {f} _ {0})

And the plastic deformation is again obtained via ().


Benchmark results
----------------------

The values of :math:`\kappa` used to obtain the reference results in cases A and B are chosen so as to remain in the positive work-hardening range of the model (otherwise, stress control is not possible). For test case A, we choose :math:`\kappa =0.148`; for test case B, :math:`\kappa =0.180`. An increasing constraint is then imposed up to the value :math:`{T}_{\mathit{eq}}(\kappa )` according to ().


Uncertainties about the solution
----------------------------

The reference results are obtained very precisely by calculating the integrals that appear in the analytical solution by means of a trapezium method whose step is chosen very finely.


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

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