2. Benchmark solution

2.1. Calculation method

We rely on the result of [bib1]. The state of plane deformations makes it possible to write the uniform displacement field in the cube very easily:

(2.1)\[\begin{split} \ {\ begin {array} {c} {u} {u} _ {1} = {a} _ {1}\ mathrm {.} {x} _ {1}\\ {u} _ {2} =w\ mathrm {.} {x} _ {2}\\ {u} _ {3} =0\ end {array}\end{split}\]

with \(w\) the vertical (negative) displacement of the upper face and \({a}_{1}\) an arbitrary constant.

The incompressibility condition makes it possible to write:

(2.2)\[ {a} _ {1}\ mathrm {=}\ frac {\ mathrm {-} w} {1+w}\]

And we find the relationship between the applied force \(F\) and the displacement \(w\) of the upper face:

(2.3)\[ F=\ mathrm {2S.} \ frac {w\ mathrm {.} (2+w)\ mathrm {.} (1+ {(1+w)} ^ {2})} {{(1+w)}} {{(1+w)}} ^ {3}}\ mathrm {.} (\ frac {\ partial\ Psi} {\ partial {J} _ {1}}} +\ frac {\ partial\ Psi} {\ partial {J}} _ {2}})\]

\(S\) is the area, \(\Psi\) is the deformation potential, and \({J}_{1}\), \({J}_{2}\) are the invariants of the Green-Lagrange tensor. The deformation potential used by ELAS_HYPER is as follows:

\[\]

: label: eq-4

PSI = {C} _ {10}mathrm {.} ({J} _ {1} -3) + {C} _ {01}mathrm {.} ({J} _ {2} -3) + {C} _ {20}mathrm {.} {({J} _ {1} -3)}} ^ {2} + {Psi} _ {text {vol}}

\({\Psi }_{\text{vol}}\) is the potential corresponding to incompressibility. It depends on the invariants \({J}_{1}\) and \({J}_{2}\) and on \({C}_{10}\), \({C}_{01}\) and \({C}_{20}\) which are the material characteristics. As in addition \(S=1\) we get:

\[\]

: label: eq-5

F=2. frac {wmathrm {.} (2+w)mathrm {.} (1+ {(1+w)} ^ {2})} {{(1+w)}} {{(1+w)}} ^ {3}}mathrm {.} left [{C} _ {10} + {C} _ {01} +2. {C} _ {20}mathrm {.} frac {{w} ^ {2}mathrm {.} {left (2+wright)} ^ {2}}} {{left (1+wright)} ^ {2}}}right]

Solving this nonlinear equation in \(w\) is simply done by dichotomy for \(w<0\).