2. Benchmark solution#

2.1. Calculation method used for the reference solution#

The reference solution is obtained by an analytical calculation.

The imposed displacement \({u}_{y}^{D}\) immediately provides the deformation \({\mathrm{\epsilon }}_{\mathit{yy}}=\frac{{u}_{y}^{D}}{1}\). Only this component (corresponding to the non-zero component of the stress tensor) is of interest to us here.

On the other hand, to calculate the behavior, it is necessary to extract the isotropic work hardening function \(R(p)\) from the data:

_images/Object_7.svg
(2.1)#\[ \begin{align}\begin{aligned} {\ mathrm {\ sigma}} ^ {t} =F (\ mathrm {\ epsilon}) = {\ mathrm {\ sigma}} _ {y} +\ frac {E\ mathrm {.} {E} _ {T}} {E- {E} _ {T}} p\\ R (p) = {\ mathrm {\ sigma}}} _ {y} +\ left (\ frac {E\ mathrm {.} {E} _ {T}} {E- {E} _ {T}} -\ frac {3} {2} C\ right) p\end{aligned}\end{align} \]

2.1.1. Elastic behavior#

The behavior is elastic up to \(t=1\). For \(t=1\), the \({\mathrm{\sigma }}_{\mathit{yy}}=E{\mathrm{\epsilon }}_{\mathit{yy}}=400\mathit{MPa}\) constraint, it just reaches the plasticity threshold.

2.1.2. Elastoplastic filler#

At \(t=2\), we reach the plasticity criterion (in charge or in discharge), we have:

(2.2)#\[ \begin{align}\begin{aligned} |\ mathrm {\ sigma} -\ frac {3} {2} C {\ mathrm {\ epsilon}} ^ {p} |=R (p)\\ \ textrm {and}\\ \ mathrm {\ sigma} =E (\ mathrm {\ epsilon} - {\ mathrm {\ epsilon}}} ^ {p})\end{aligned}\end{align} \]

For the load, it is therefore necessary to solve the following equation:

(2.3)#\[ {\ mathrm {\ sigma}} ^ {t} =F (\ mathrm {\ epsilon}) = {\ mathrm {\ sigma}} _ {y} +\ frac {E\ mathrm {.} {E} _ {T}} {E- {E} _ {T}} p= {\ mathrm {\ sigma}} _ {y} +\ frac {E\ mathrm {.} {E} _ {T}} {E- {E} _ {T}}}\ left (\ mathrm {\ epsilon} -\ frac {{\ mathrm {\ sigma}}} ^ {t}} {T}} {E}\ right)\]

Which brings back \({\mathrm{\sigma }}^{t}={\mathrm{\sigma }}_{y}+{E}_{T}\left(\mathrm{\epsilon }-\frac{{\mathrm{\sigma }}_{y}}{E}\right)\). We move on the traction curve to point \(A\) such that:

\[\]

: label: eq-4

{mathrm {sigma}} _ {A} ^ {t} =Eleft ({mathrm {epsilon}} _ {A} - {mathrm {epsilon}}} _ {A} ^ {p}right)

2.1.3. Elastic discharge#

At \(t=3\), the discharge is elastic up to point \(A’\):

\[\]

: label: eq-5

  • {mathrm {sigma}} ^ {Atext {“}}} -frac {3} {2} C {mathrm {epsilon}}} _ {A} ^ {p} =R ({p} _ {A})

{mathrm {sigma}} ^ {Atext {“}} =Eleft ({mathrm {epsilon}}} _ {Atext {“}}} - {mathrm {epsilon}}} - {mathrm {epsilon}}} _ {A} ^ {p}right)

2.1.4. Elastoplastic load under compression#

At \(t=4\):

(2.4)#\[ {\ mathrm {\ sigma}} ^ {c} =\ frac {3} {2} C {\ mathrm {\ epsilon}} ^ {p} -R (p)\]

2.2. Benchmark results#

\(t\)

 _images/Object_16.svg _images/Object_17.svg

\((\mathrm{MPa})\)

1

  1. 10—3

400

2

4.5d—3

500

3

0.1d—3

—380

4

—2. 10—3

—464

2.3. Uncertainty about the solution#

Analytical solution.

2.4. Bibliographical references#

  • Behavioral relationship to linear and isotropic nonlinear kinematic work hardening. Note [R5.03.16].