2. Mixed variational formulation of the problem

2.1. Formulation in the context of small deformations

Be a solid \(\Omega\) subject to:

• an imposed displacement field \(u\mathrm{=}{u}_{0}\) on \({\Gamma }_{u}\)

• a constraint field imposed \(t\mathrm{=}\sigma \mathrm{.}n\mathrm{=}{t}_{0}\) on \({\Gamma }_{t}\)

• a volume force field \(f\) out of \(\Omega\)

In the classical case of finite elements in motion (modeling 3D or ``D_ PLAN « or ``D_ » or « AXIS » in code_aster), when the problem derives from an energy, the problem solved is as follows:

find \(u\mathrm{\in }V\) with \(\sigma\) verifying the behavioral relationship, which minimizes potential energy:

\[\]

Pi (u)mathrm {=}frac {1} {2}underset {omega} {mathrm {int}}sigmamathrm {:}varepsilon domegamathrm {-}underset {mathrm {int}} fmathrm {.} udmathrm {.} udmathrm {.} udmathrm {.} udmathrm {-} dmathrm {.} udmathrm {m {-}underset {{Gamma} _ {t}} {mathrm {int}} tmathrm {.} udGamma

As we explained in section [Difficulties associated with the treatment of incompressibility], this formulation is not appropriate when trying to approximate the incompressible solution, that is to say the condition \(\text{div}(u)=0\) or \(\text{tr}(\varepsilon )=0\). To get around this difficulty, one solution is to treat separately the spherical part of the deformation tensor (the part that poses numerical problems) and its deviatory part. So we will have:

(2.1)\[ \ varepsilon (u, g) = {\ varepsilon} ^ {D} (u) +\ frac {g} {3} I\]

The previous problem therefore comes down to solving a problem with 2 variables, \(u\) and \(g\), under the constraint \(g=\text{tr}(\varepsilon (u))\). It can be reduced to solving a problem without constraints by introducing a Lagrange multiplier \(p\); it is written as:

find \(u\in V\), \(p\) and \(g\) solutions to the saddle point problem for the Lagrangian:

(2.2)\[ (u, p, g) =\ underset {\ omega} {\ omega} {\ omega} {\ omega} {\ omega} ^ {D} (u) +\ frac {g} {3} I) +p (\ text {div} (u) -g) {\ int} I) +p (\ text {div} (u) -g)\ right] d\ omega -\ underset {\ omega} {\ int} f\ mathrm {.} ud\ Omega -\ underset {{\ Gamma} _ {t}} {\ int}} {\ int} t\ mathrm {.} ud\ Gamma\]

This problem can be solved by writing the optimality conditions:

(2.3)\[\begin{split} \ left\ {\ begin {array} {c}\ frac {\ mathrm {\ partial}\ mathrm {\ partial}} {\ mathrm {\ partial} u}\ mathrm {=}\ underset {\ Omega}\ underset {\ omega} {\ omega} {\ omega} {\ omega} {\ partial}}} {\ mathrm {\ partial} u}\ mathrm {:}\ underset {\ omega} {\ omega} {\ omega} {\ omega} {\ omega} {\ omega} {\ int}}} ({\ sigma} ^ {D} +pi)\ mathrm {:}\ delta\ varepsilon d\ Omega\ mathrm {-}\ underset {\ omega} {\ mathrm {\ int}} f\ mathrm {.} \ delta ud\ omega\ mathrm {-}\ underset {{\ Gamma} _ {t}} {\ mathrm {\ int}} t\ mathrm {.} \ delta ud\ Gamma\ mathrm {=} {=} 0\\ frac {\\ frac {\ mathrm {\ partial}} {\ mathrm {\ partial} p}\ mathrm {=}\ mathrm {=}\ underset {\ omega}}\ underset {\ omega}} {\ underset {\ omega}} {\ underset}} {\ omega}} {\ mathrm {\ int}}} (\ text {div} (u)\ mathrm {-} g)\ mathrm {-} g)\ mathrm {=}}\ mathrm {=}}\ underset {\ omega}} {\ underset {\ omega}} {\ underset {\ omega}} {\ underset {\ omega}} {\ underset {{=} 0\\\ frac {\ mathrm {\ partial}\ mathrm {\ partial}\ mathrm {\ partial}} {\ mathrm {=}\ underset {\ Omega} {\ omega} {\ mathrm {\ partial}} {\ mathrm {\ partial}}}\ mathrm {\ partial}}} {\ partial}}} {\ mathrm {-} p)\ mathrm {-} p)\ mathrm {\ int}}} (\ frac {\ int}}} (\ frac {1} {3}}\ text {tr} (\ sigma)\ mathrm {-} p)\ mathrm {-} p)\ mathrm {-} p)\ mathrm {-} p)\ mathrm {\ Omega\ mathrm {=} 0\ end {array}\ right.\end{split}\]

• the first equation corresponds to the equilibrium equation,

• the second equation reflects the kinematic relationship linking \(g\) to \(u\),

• the third equation gives the expression for the Lagrange multiplier \(p\) ,

when the problem does not derive from an energy, we can directly use the system of equations (2.3).

In the case where there is a one-to-one relationship between pressure and swelling, as for example for an elastoplastic material with a von Mises-type plasticity criterion, it is possible to explain swelling and therefore to remove the third equation of system (2.3). We then obtain the following system of two equations with two unknowns:

(2.4)\[\begin{split} \ left\ {\ begin {array} {}\ underset {\ omega} {\ omega} {\ int} ({\ sigma} ^ {D} +pi):\ delta\ varepsilon d\ Omega -\ underset {\ Omega} {\ omega} {\ omega} {\ int} f\ mathrm {.} \ delta ud\ Omega -\ underset {{\ Gamma} _ {\ Gamma} _ {t}} {\ int} t\ mathrm {.} \ delta ud\ Gamma =0\\\ underset {\ omega} {\ omega} {\ omega} {\ int} (u) -\ frac {p} {\ kappa})\ delta pd\ Omega =0\ end {array}\ right.\end{split}\]

Where \(\kappa\) is the compressibility module.

2.2. Formulation in large deformations

As for small deformations, it is possible to propose a variational formulation valid for large deformations. The principle is identical, but in this case we rely on the decomposition of the gradient tensor of transformation \(F\mathrm{=}I+\frac{\mathrm{\partial }u}{\mathrm{\partial }X}\) proposed by Flory [bib3] _:

\[\begin{split}F\ mathrm {=} {F} ^ {s}\ stackrel {} {s}\ stackrel {} {F} ^ {s}\ mathrm {=} {J} ^ {1\ mathrm {/}} ^ {s} ^ {s} ^ {1\ mathrm {/}} ^ {3}} I\\\\ text {and}}\\ stackrel {} {F}\ mathrm {=} {J}} ^ {3} ^ {3}} I\\\\ text {and}}\\ stackrel {} {F} {F}\ mathrm {=} {J}}} ^ {J} ^ {3} ^ {3} mathrm {-} 1\ mathrm {/} 3} F\\\ text {and}\\ J\ mathrm {=}\ text {det} (F)\end{split}\]

Again, the idea is to enrich the kinematics using a swelling variable \(g\), which is a priory independent of the movements, and linked weakly to the volume variation by a weak relationship:

\[B (J)\ underset {\ mathit {weak}} {\ mathrm {\ approx}} B\ mathrm {°} A (g)\ mathrm {=}} B (A (g))\]

Several relationships have been tested:

\[\]

left{begin {array} {c} {c} Jmathrm {=} 1+g\ {J} ^ {2}mathrm {=} 1+g\ text {ln} (J)mathrm {ln} (J)mathrm {=}mathrm {=}text {exp} (g)end {array} (j)mathrm {ln} (J)mathrm {=} (J)mathrm {=} (J)mathrm {=} (j)mathrm {ln} (J)mathrm {=} (J)mathrm {ln} (J)mathrm {=} (J)

For some simulations, small differences were observed. For « INCO_UPG » elements with SIMO_MIEHE deformation, It is finally the linear relationship that was implemented in the code: so \(B(J)\mathrm{=}J\) and \(A(g)\mathrm{=}1+g\). For the « INCO_UPG » elements with GDEF_LOG deformation, the logarithm relationship was used.

However, as this choice is not necessarily final, it is proposed to write the problem in the general case. A gradient is therefore introduced. enriched deformation:

\[\]

: label: eq-grad enriched

tilde {F}mathrm {=} {=} {(frac {A (g)} {J})}} ^ {frac {1} {3}}} F

The weak formulation of the problem is based on the search for the saddle point of Lagrangian \(ℒ\), in which the Lagrange multiplier \(p\) and a third field \(g\), independent of the other two, weakly ensuring that the relationship between \(J\) and \(g\) is true:

(2.5)\[ \ mathrm {} (u, g, p)\ mathrm {=}\ underset {{\ Omega} _ {0}} {\ mathrm {\ int}}\ psi (\ tilde {F}) d {\ Omega}) d {\ Omega} _ {0}\ mathrm {-} {W} _ {\ mathit {ext}}} (u) +\ underset {{\ Omega}) d {\ Omega} _ {0}\ mathrm {-} {W} _ {\ mathit {ext}} (u) +\ underset {{\ Omega}} _ {0}} {\ mathrm {\ int}} p\ left [B (J)\ mathrm {-} B\ mathrm {°} A (g)\ right] d {\ omega} _ {0}\]

where \({W}_{\mathrm{ext}}\) represents the potential of external forces and \(\psi (\tilde{F})\) represents the deformation energy.

This problem can be solved as in small deformations by writing the optimum conditions. The Lagrangian variation is written as:

\[\]

: label: eq-lagr2

delta =underset {{Omega} _ {0}} {0}} {int}left [P:deltatilde {F} +p (frac {partial B (J)} {partial J}} Jfrac {delta J} {0}} {delta J} {0}} {delta J} {delta J} {J} -frac {partial B°A (g)} {partial g}delta g) +delta p (B (J) -B°A (g))right] d {Omega} _ {0} -delta {W} _ {mathrm {ext}}} (u)

with \(P\) the first Piola-Kirchhoff stress tensor:

By injecting the variation of the enriched transformation and the expression of the Kirchhoff constraint \(\tau \mathrm{=}P\tilde{{F}^{T}}\) [1] _, the following form is obtained for the Lagrangian variation:

(2.6)\[\begin{split} \ begin {array} {ccc}\ delta\ mathrm {} &\ mathrm {=} &\ underset {{\ omega} _ {0}} {\ mathrm {\ int}} ({\ tau}}} ({\ tau} ^ {d}} +p\ frac {\ mathrm {\ partial}} B (J)}} {\ mathrm {\ partial}}} {\ mathrm {\ int}}} ({\ tau} ^ {d}} +p\ frac {\ mathrm {\ partial}} J} JI)\ mathrm {:}\ delta Ld {\ omega} _ {0}\\ & +&\ underset {{\ omega} _ {0}} {\ mathrm {\ int}} (\ frac {\ text {tr} (\ tau)} {\ tau)} {3}\ frac {\ & +&\ underset {\ omega} _ {0}} {\ mathrm {\ int}} (\ frac {\ text {tr} (\ tau)} {\ tau)} {3}\ frac {\ mathrm {\ partial} g} {A (g)}\ mathrm {-} p\ frac {\ mathrm {\ partial} B\ mathrm {\ partial} A (g)} {\ mathrm {\ partial} g})\ delta gd {\ omega} {\ omega} _ {0}} +\ underset {{\ omega} _ {0}} {\ mathrm {\ int}})\ delta gd {\ omega} _ {0}} +\ underset {\ omega} _ {0}} +\ underset {\ omega} _ {0}} {\ mathrm {\ int}} (B (J)\ mathrm {\ int}} (B (J)\ mathrm {0}} -} B\ mathrm {°} A (g))\ delta pd {\ omega} _ {0}\\ &\ mathrm {-} &\ delta {W} _ {\ mathit {ext}} _ {\ mathit {ext}}} (u)\ end { array}\end{split}\]

where we introduced the Eulerian displacement gradient (\(x\) represents the position vector at the end of the increment): \(\delta L=\frac{\partial \delta u}{\partial x}=\delta F\mathrm{.}{F}^{-1}\)

In summary, the system to be solved is as follows:

(2.7)\[\begin{split} \ begin {array} {c}\ frac {\ mathrm {\ partial}\ mathrm {\ partial}\ mathrm {\ partial} u}\ mathrm {=}\ underset {{\ Omega} _ {0}} _ {0}}} {\ mathrm {\ partial}}\ mathrm {\ int}}} {\ mathrm {\ int}}} {\ mathrm {\ int}} {\ mathrm {\ int}} {:} ({\ tau} ^ {d}}\ underset {\ omega} _ {\ omega} _ {0}} _ {0}}} {\ mathrm {\ int}}}\ delta L\ mathrm {\ int}}\ delta L\ mathrm {:} ({\ tau} ^ {mathrm {\ partial} B (J)} {\ mathrm {\ partial}} J} JI) d {\ omega} _ {0}\ mathrm {-}\ delta {W} _ {\ mathit {ext}}}\ mathrm {ext}}}\ mathrm {\ ext}}\ mathrm {\ partial}}\ mathrm {\ partial}} {\ mathrm {\ partial}} {\ mathrm {\ partial}} {\ mathrm {\ partial}} {\ mathrm {\ partial}} {\ mathrm {\ partial}} {\ mathrm {\ partial}}}\ mathrm {=}\ underset {{\ omega} _ {0}} {\ mathrm {\ int}}\ delta g (\ frac {\ text {tr} (\ tau)} {3}\ frac {\ mathrm {\ omega} _ {0}} {\ mathrm {\ omega} _ {0}} {\ mathrm {\ omega} _ {0}} {0}} {\ mathrm {\ omega} _ {0}} {0}} {\ mathrm {\ omega} _ {0}} {0}} {\ mathrm {\ omega} _ {0}} {0}} {\ mathrm {\ omega} _ {0}} {0}} {\ mathrm {\ omega} _ {0}} {0}} {-} p\ frac {\ mathrm {\ partial} B\ mathrm {°} A (g)} {\ mathrm {\ partial} g}) d {\ omega} _ {0}\ mathrm { =} 0\\\ frac {\ mathrm {\ partial}\ mathrm {\ partial}\ mathrm {\ partial} p}\ mathrm {=}\ underset {{\ omega} _ {0} _ {0}}} {\ mathrm {\ partial}}} {\ mathrm {\ partial}}}\ mathrm {\ partial} p}\ mathrm {-}\ underset {{\ omega} _ {0}}} {0}}} {\ mathrm {\ partial}}} {\ mathrm {\ int}}}\ delta p (J)\ mathrm {-} B\ mathrm {°} _ {0}}} {\ mathrm {0}}}} {\ mathrm {\ int}}}\ delta {\ Omega} _ {0}\ mathrm {=} 0\ end {array}\end{split}\]

Note: The Kirchhoff constraint, derived from the law of behavior, is written as so \(\tau \mathrm{=}{\tau }^{D}+p\frac{\mathrm{\partial }B(J)}{\mathrm{\partial }J}JI\)

As far as obtaining the tangent matrix is concerned, of course, it requires a little more calculation than in small deformations, and has the particularity of not being symmetric in the general case. It is found in the code in the following form:

\[\begin{split}K\ mathrm {=}\ left [\ begin {array} {ccc} {ccc} {K} _ {\ text {UU}} & {\ text {UG}}} & {K} _ {\ text {UP}}} {\ text {UP}}}}\\ {UP}}}}\\ text {UP}}}}\\ text {UP}}}}\\ {UP}}}\\ text {UP}}}}\\ {UP}}}\\ {P}}}\\ {K}}\\ {K} _ {\ text {PU}}} & {K}} _ {\ text {PG}}} & {K} _ {\ text {PP}}}\ end {array}\ right]\end{split}\]

The calculations are not detailed here. The reader can refer to reading [bib8].

Note: *This formulation makes it possible to regulate, at a lower cost, ductile damage models where the damage variable is directly linked to the change in volume. * *In fact, to control the location of damage and deformation, the idea is to penalize the strong damage gradients. As in this 3-field formulation, local swelling is treated as a nodal variable, its gradient is easily accessible. numerically (subject to at least linear interpolation) . *

In the spirit of formulations with a second displacement gradient ([bib9] _, [bib10] _), it is enriched by a quadratic gradient term swelling. The Lagrangian variation is then written:

(2.8)\[ \ begin {array} {ccc}\ delta & =&\ underset {{\ omega} _ {\ omega} _ {0}} {\ int}\ left ({\ tau} ^ {d} +p\ frac {\ partial B (J)}} {\ partial B (J)}} {\ partial B (J)} =& =&\ underset {{\ omega} _ {0}} {0}} {\ int}\ left (\ tau)} {3}\ frac {\ partial A (g)/\ partial g} {A (g)} {A (g)} -p\ frac {\ partial B\ mathrm {°}} {\ partial g}\ right)\ delta gd {\ Omega} {\ Omega} _ {0} _ {0}} {\ int}\ right)\ delta gd {\ omega} _ {\ right)\)\ delta gd {\ omega} _ {0}\ left [\ left (B (J) -B\ mathrm {°} A (g)\ right)\ delta p+c\nabla g\ cdot\nabla\ delta g\ right] d {\ Omega} _ {0} -\ delta {W}} -\ delta {W}} _ {\ delta {W}} _ {W} _ {W} _ {W} _ {W} _ _ {W} _ _ {W} _ _ {W} _ _ {W} _ _ {W} _ _ {W} _ {W} _ {W} _ {W} _ {W} _ {W} _ {W} _\]

\(c\) is a parameter to be determined and homogeneous to a force. In a sense, this parameter introduces an internal coupling length between the material points. The term added here is isotropic: it is considered that the internal length to be introduced is identical in all directions. For application to the ductile damage of steels, this hypothesis seems entirely acceptable. This formulation can be used for the Rousselier model, [R5.03.07], provided the definition of the keyword ``C_ CARA « under the operand » NON_LOCAL « of » DEFI_MATERIAU « (see ssnap122a test)

Obtaining the formulation with two fields in large deformations follows the same principle as With respect to obtaining the tangent matrix, it of course requires a little more calculation than in small deformations, and has the particularity of not being symmetric in the general case. It is found in the code in the following form:

[1]

Il s’agit là d’un choix et l’on aurait tout aussi bien pu transporter \(P\) par la déformation compatible \(F\) instead of the enriched deformation \(\tilde{F}\). But the latter has the advantage of maintaining the symmetry of the formulation.

in the elastic case. In addition, it is better suited to the architecture of a calculation code in which integration of the laws of behavior is well differentiated from the calculation of terms specific to finite elements (the routines of behavior). do not need to know the existence of the two deformation measures) but only the enriched deformation).