2. Benchmark solution

2.1. Calculation method used for the reference solution

We rely on the documentation [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 \(J\) then remains equal to 1. By noting \(d={q}_{1}f\) and \({d}_{0}={q}_{1}{f}_{0}\), we can express the equivalent constraint of GTN:

(2.1)\[ {T} _ {\ text {*}}} =\ frac {{T} _ {\ mathit {eq}}} {1-d}} {1-d}\]

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

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

With \(\mu\), the elastic deformation is equal to the shear modulus:

\[\]

: label: eq-4

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

The law of plastic evolution is reduced to:

\[\]

: 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:

(2.3)\[ {\ dot {E}} _ {\ mathit {eq}}} ^ {p} =\ frac {\ dot {\ mathrm {\ kappa}}}} {1-d}\]

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

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

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.

2.1.1. Case A: \({b}_{0}=0\)

In this case, porosity is simply written:

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

And the plastic deformation is obtained by integrating ():

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

2.1.2. Case B: \(B(\kappa )=0\)

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

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

And the plastic deformation is again obtained via ().

2.2. Benchmark results

The values of \(\kappa\) 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 \(\kappa =0.148\); for test case B, \(\kappa =0.180\). An increasing constraint is then imposed up to the value \({T}_{\mathit{eq}}(\kappa )\) according to ().

2.3. 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.

2.4. Bibliographical references

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