3.1. Linear and non-linear static equilibrium
We will express the linear and non-linear static equilibrium equations in variational form. We place ourselves within the framework of the theory of large deformations. The Green-Lagrange \({E}^{u}\) strain tensor is equal to:
(3.1)\[ {E} ^ {u} =\ frac {1} {1} {2}\ left ({F} ^ {T} f-G\ right)\]
With \(F\) the classical displacement gradient tensor (see § 2.4) and \(g\) the metric tensor.
In order to be able to introduce all the methods of stabilization and correction of locks, we will start from a mixed variational formulation by Hu-Washizu [Bib2] and [Bib17], written on the initial configuration and in large deformations:
(3.2)\[ {\ pi} ^ {\ mathit {HW}} (u, E, S) = {\ int} _ {\ mathrm {\ Omega}} _ {0}} W (E) d {\ mathrm {\ Omega}}} _ {\ Omega}} _ {0}} _ {0}} S\ mathrm {\ Omega}}} _ {0}} S\ mathrm {\ Omega}}} _ {\ mathrm {\ Omega}}} _ {\ omega}} _ {0}} S\ mathrm {\ Omega}}} _ {\ omega}} {E} ^ {u} -E\ right) d {\ mathrm {\ Omega}} _ {0} - {\ pi} _ {\ text {ext}} (u)\]
With \(W(E)\) the energy of deformation of the structure and \({\pi }_{\text{ext}}(u)\) the work of external forces. The application of method EAS consists in reparameterizing the Green-Lagrange strain tensor such that:
(3.3)\[ E= {E} ^ {u} + {E} ^ {\ mathit {EAS}}\]
This gives us a new expression for () that transforms the dependency from \(E\) to \({E}^{\mathit{EAS}}\):
(3.4)\[ {\ stackrel {~} {\ pi}}} ^ {\ mathit {HW}} ^ {\ mathit {EAS}}, S) = {\ int} _ {{\ mathrm {\ omega}}} _ {\ omega}}} _ {0}} _ {0}}} _ {0}}} W ({E} ^ {u} ^ {u} + {E} ^ {\ mathit {EAS}}} _ {omega}}} _ {0}}} _ {0}}} W ({E} ^ {u}} + {E} ^ {\ mathit {}}}) d {\ mathrm {\ Omega}}} _ {0}}} _ {0}}} W ({E} ^ {u}} + {E} ^ {u} ^}} _ {0} + {\ int} _ {{\ mathrm {\ omega}} _ {0}} S\ mathrm {:} {E} ^ {\ mathit {EAS}} d {\ mathrm {\ omega}} _ {\ omega}} _ {0} - {\ pi} _ {\ text {ext}}} (u)\]
It is necessary to find correct approximations of these three fields, which can be complicated in practice. This three-field formulation is therefore generally transformed into a two-field formulation by imposing the following orthogonality condition:
(3.5)\[ {\ int} _ {{\ mathrm {\ Omega}}} _ {0}} _ {0}}} S\ mathrm {:} {E} ^ {\ mathit {EAS}}} d {\ mathrm {\ Omega}}} _ {0} =0\]
This eliminates the constraint. At the discrete level, this condition can lead to convergence problems because of a constraint space \({S}^{h}\) that is too restricted (see [Bib3]). This problem can be corrected by requiring at least piecewise constant functions to be included in this space at the discrete level. In our case, we will proceed differently to allow the pinch to be taken into account. There are two situations:
if there is no pinch, that is to say if there is no pressure applied to the lower or upper surface of the element, the orthogonality condition () will be imposed;
if there is a pinch and if we consider a pressure on the upper face \({P}_{u}\) and on the lower surface \({P}_{l}\), we choose the Piola-Kirchhoff stress of the second kind such as
(3.6)\[ {\ sigma} _ {33} =-\ frac {1} {1} {2} (1-\ zeta) {P} _ {l} -\ frac {1} {2} (1+\ zeta) {P} _ {u}\]
With \(\zeta\) the parametric coordinate in the thickness.
So we have:
\[\]
: label: eq-62
pi (u, {E} ^ {mathit {EAS}}) = {mathit {}}) = {int} _ {0}} W ({E} ^ {u} + {E} ^ {E} ^ {mathit {}}) = {mathit {}}}) = {int}} _ {mathrm {EAS}}}) d {mathrm {}}}) d {mathrm {Omega}}} _ {0} - {pi} _ {pi} _ {pi} _ {text {ext}} (u)
Or even:
(3.7)\[ \ pi (u, {E} ^ {\ mathit {EAS}}) = {\ mathit {}}) = {\ pi} _ {\ text {int}}} (u, {E} ^ {\ mathit {EAS}}}) - {\ mathit {}}}) - {\ pi} _ {\ pi} _ {\ text {int}} (u)\]
The first variation of () makes it possible to obtain the weak form:
(3.8)\[ \begin{align}\begin{aligned} \ delta\ pi (u, {E} ^ {\ mathit {EAS}}}) =\ delta {\ pi} _ {\ text {int}} (u, {E} ^ {\ mathit {EAS}}}) -\ delta {\ mathit {}}}) -\ delta {\ mathit {}}}) -\ delta {\ mathit {}}}) -\ delta {\ pi} _ {\ text {int}}} (u)\\ \ textrm {with}\\ \ delta {\ pi} _ {\ text {int}}} (u, {E}}} (u, {E}}}) (u, {E} ^ {\ mathit {EAS}}) = {\ mathrm {\ Omega}}} _ {0}}}\ left (\ delta {E}}}}\ left (\ delta {E}}}}\ left (\ delta {E}}}}\ left (\ delta {E}}}\ left (\ delta {E}}}\ left (\ delta {E}} ^ {u} +\ delta {E}} ^ {u} +\ delta {E}} ^ {u} ^ {u} ^ {\ mathit {EAS}}}\ right)\ mathrm {:} Sd {\ mathrm {:} Sd {{\ Omega}} _ {0}\end{aligned}\end{align} \]
The problem is non-linear. We will apply Newton’s method (see [R5.03.01]), with the notations in this document, \(\delta {\pi }_{\text{int}}\) are the internal forces. It is therefore necessary to express the different variations according to the unknowns of the system.
For deformation \({E}^{u}\), the variation being small, we only consider the linear part. In the parametric coordinate system of the element, we have:
(3.9)\[ \ delta {E} ^ {u} =\ widehat {B}\ delta U\]
And for deformation EAS, on the element, using an approximation with a single parameter \(\alpha\):
(3.10)\[ \ delta {E} ^ {\ mathit {EAS}}} =\ widehat {\ mathrm {\ Gamma}}\ delta\ alpha\]
So we have:
(3.11)\[ \ delta {\ pi} _ {\ text {int}}} = {\ int}} = {\ int} _ {\ mathrm {\ omega}} _ {0}} {\ left (\ widehat {B}\ delta U\ right)}} ^ {T}}}} ^ {T}} Sd {\ mathrm {\ int}} {\ left (\ widehat {\ mathrm {\ Gamma}}}\ delta\ alpha\ right)} ^ {T} Sd {\ mathrm {\ Omega}}} _ {0}\]
And the internal force:
(3.12)\[ {L} _ {\ text {int}} =\ underset {{L}} _ {\ text {int}}} ^ {u}} {\ underset {} {{\ int} _ {{\ mathrm {\ Omega}}} _ {\ omega}}} _ {\ omega}}} _ {\ omega}}} _ {\ omega}}} _ {\ omega}}} _ {\ omega}}} _ {\ omega}}} _ {\ omega}}} _ {\ omega}}} _ {\ omega}}} _ {\ omega}}} _ {\ omega}}} _ {\ omega}}} _ {\ omega}}} _ {\ omega}}} _ {\ omega}}} _ {\ omega}}} _ {\ omega}}} _ {\ omega}} {{L} _ {\ text {int}}} ^ {\ mathit {EAS}}}} {\ underset {} {{\ int} _ {\ mathrm {\ Omega}} _ {0}}} {\ widehat {\ widehat {\ mathrm {\ Gamma}}}} {\ mathrm {\ Omega}}} _ {0}}} {\ widehat {\ mathrm {\ Gamma}}} {\ widehat {\ mathrm {\ Gamma}}} {\ widehat {\ mathrm {\ Gamma}}} {\ widehat {\ mathrm {\ Gamma}}} {\ widehat {\ mathrm {\ Gamma}}\]
The application of Newton’s method will reveal four tangent matrices:
(3.13)\[ \ mathrm {\ Delta} {L} _ {\ text {int}} =\ begin {array}}} =\ begin {array} {c}\ delta {U} ^ {T}\ frac {\ partial {L} _ {\ text {int}}} {\ text {int}}} ^ {int}} _ {\ text {int}}} ^ {\ text {int}}} ^ {int}} _ {\ text {int}}} ^ {int}} ^ {int}} {\ text {int}}} ^ {int}} ^ {int}} {\ text {int}}} ^ {int}} ^ {int}} ^ {int}} {\ text {int}}} ^ {int}} ^ {int}} {\ text {int}}} ^ {int}} ^\ text {int}} ^ {\ mathit {EAS}}}} {\ partial\ alpha}} {\ partial\ alpha}\ mathrm {\ Delta}\ alpha} ^ {T}\ frac {\ partial {L} _ {\ partial {L}} _ {\ text {int}} _ {\ text {int}}} {\ partial\ alpha}} {\ partial\ alpha}} {\ partial\ alpha} ^ {T}\ frac {\ partial {L} _ {\ text {int}}} ^ {u}} {\ partial\ alpha}\ mathrm {\ Delta}\ alpha\ end {array} EAS\]
With the four matrices:
(3.14)\[ \ mathrm {\ Delta} {L} _ {\ text {int}}} =\ left (\ delta {U} ^ {T} {K} ^ {\ mathit {uu}}} +\ delta {\ alpha} ^ {T} ^ {T} {K} {K}} ^ {K} ^ {K} ^ {K} ^ {K} ^ {right)\ mathrm {\ Delta} U+\ left (\ delta {U}} ^ {T} {K} ^ {K} ^ {K} ^ {K} ^ {K} ^ {u\ alpha} +\ delta {\ alpha} ^ {T} {T} {K} ^ {\ alpha\ alpha}\ right)\ mathrm {\ Delta}\ alpha\]
For \({K}^{\mathit{uu}}\) is the « classical » matrix (found in standard isoparametric elements for example), it itself includes two parts:
Either:
\[\]
: label: eq-71
{K} ^ {mathit {uu}} = {int}} = {int} _ {mathrm {omega}} _ {0}} {widehat {B}} ^ {T}frac {partial S} {partial S} {partial S}} {partial S} {partial S}} {partial S} {partial S} {partial S} {partial S} {partial S} {partial S} {partial S} {partial S} {partial S} {partial S} {partial E}} d {partial E} d {partial E} d {partial E} d {mathrm {omega}} d {partial E} d {mathrm {omega}} d {partial E} 0}} {widehat {B}}} ^ {T}frac {partial E} {partial U} Sd {mathrm {Omega}} _ {0} = {K} _ {m}} ^ {m} ^ {m} ^ {m} ^ {m} ^ {mathit {m}} ^ {mathit {uu}} ^ {mathit {uu}} ^ {mathit {uu}} ^ {mathit {uu}}
3.2. Stability calculation
This paragraph makes it possible to expose the expression of the problem to the eigenvalues to calculate the stability of a structure (buckling). The geometric matrix is used in two cases:
in non-linear when using large deformations, the linearization process reveals a contribution of geometric origin, see ();
to do stability calculations (buckling), which corresponds to option RIGI_GEOM in code_aster, see [R7.05.01].
In both cases, the general form of this matrix is as follows:
(3.15)\[ {K} _ {g} ^ {\ mathit {uu}} = {\ int}} = {\ int}} _ {0}} {\ widehat {B}} ^ {T}\ frac {\ mathit {uu}}\ frac {\ partial E}}\ frac {\ partial E}} {\ frac {\ partial E}} {\ frac {\ partial E}} {\ partial E} {\ partial E} {\ partial e} {\ partial u} Sd {\ mathrm {\ omega}}} _ {0}\]
We will express the operator \(G\) such that:
(3.16)\[ G= {\ int} _ {{\ mathrm {\ omega}}} _ {\ omega}} _ {0}} {\ widehat {B}} ^ {T}\ frac {\ partial E} {\ partial e} {\ partial u} {\ partial u} d {\ mathrm {\ omega}} _ {0}\]
As the approximation of \(B\) and \(E\) used a decomposition into polynomials, we will keep the same decomposition logic for \(G\) (and the same Lobatto numerical integration scheme). Likewise, we will have the same change of reference between the parametric space of the element and the real space:
(3.17)\[ G=T\ widehat {G}\]
The decomposition chosen is therefore of the following form:
(3.18)\[ \ widehat {G} = {\ widehat {G}}} ^ {0}} +\ zeta {\ widehat {G}} ^ {\ zeta} + {\ zeta} ^ {2} {\ widehat {G}}} {\ widehat {G}} +\ eta {\ widehat {G}} +\ eta {\ widehat {G}} +\ eta {\ widehat {G}} +\ eta {\ widehat {G}} +\ eta {\ widehat {G}} +\ eta {\ widehat {G}} +\ eta {\ widehat {G}} +\ eta {\ widehat {G}} +\ eta {\ widehat {G}} +\ eta {\ widehat {G}}}} ^ {\ eta} +\ xi\ eta {\ widehat {G}}} ^ {\ xi\ eta} +\ xi\ zeta {\ widehat {G}}} ^ {\ xi\ zeta} +\ eta\ zeta {\ zeta} +\ eta\ zeta {\ zeta} +\ eta\ zeta} +\ eta\ zeta} +\ eta\ zeta {\ zeta}\]
In addition, we will also have a contribution from reduced integration and one to stabilize:
(3.19)\[ \begin{align}\begin{aligned} \ widehat {G} = {\ widehat {G}}} ^ {u} + {\ widehat {G}} ^ {\ mathit {STAB}}}\\ \ textrm {with}\\ {\ widehat {G}} ^ {u} = {\ widehat {G}}} ^ {0} +\ zeta {\ widehat {G}} ^ {\ zeta} + {\ zeta} + {\ zeta} ^ {2} {\ widehat {G}} ^ {\ zeta\ zeta}} + {\ zeta\ zeta} + {\ zeta}} + {\ zeta} ^ {zeta}} + {\ zeta} ^ {zeta}} + {\ zeta} ^ {zeta}} + {\ zeta}} ^ {\ zeta}}\\ {\ widehat {G}} ^ {\ eta} +\ xi {\ mathit {STAB}}} =\ xi {\ widehat {G}} +\ eta {\ widehat {G}} ^ {\ eta} +\ xi\ eta {\ eta} +\ xi\ eta} +\ xi\ zeta {\ widehat {G}} +\ xi\ zeta {\ widehat {G}} +\ xi\ zeta {\ widehat {G}} +\ xi\ zeta {\ widehat {G}} +\ xi\ zeta {\ widehat {G}}} ^ {\ xi\ zeta}} ^ {\ xi\ zeta}} +\ eta\ zeta {\ widehat {G}}} ^ {\ eta\ zeta}\end{aligned}\end{align} \]
We write the components of this tensor in vector form:
(3.20)\[ \ widehat {G} =\ left\ {{\ widehat {G}}} _ {\ mathrm {\ xi}\ mathrm {\ xi}}, {\ widehat {G}}} _ {\ mathrm {\ eta}}\ mathrm {\ eta}}\ mathrm {\ eta}}, {\ widehat {G}} _ {\ mathrm {\ zeta}} _ {\ mathrm {\ zeta}} _ {\ mathrm {\ zeta}} _ {\ mathrm {\ zeta}} _ {\ mathrm {\ zeta}}\ zeta}}, {\ widehat {G}}} _ {\ mathrm {\ xi}\ mathrm {\ eta}}}, {\ widehat {G}} _ {\ mathrm {\ eta}}\ mathrm {\ eta}}\ mathrm {\ eta}}\ mathrm {\ eta}}\ mathrm {\ eta}}\ right\}\]
For each pair of \(i\times j\) nodes, we have explicit expressions for these components. For the membrane:
(3.21)\[ \begin{align}\begin{aligned}\begin{split} {\ widehat {G}} _ {m,\ mathit {ij}}} ^ {ij}}} ^ {0} =\ left [\ begin {array} {c} {a} _ {a} _ {1} _ {1} _ {1} ^ {1} ^ {j} _ {1} ^ {a} _ {1} ^ {j} _ {1} ^ {j} _ {1} ^ {a} _ {1} ^ {i} {a} _ {2} ^ {j} + {a} _ {1} _ {1} ^ {a} _ {2} ^ {i}\\ 0\\ 0\\ 0\ end {array}\ right]\end{split}\\\begin{split} {\ widehat {G}} _ {m,\ mathit {ij}}} ^ {\ mathrm {\ zeta}} =\ frac {1} {8}\ left [\ begin {array} {c} {h} {h} _ {h} _ {3} {h} {h} {h} {j} {a} {h} {j} {a} {h} {h} {h} {h} {h} {h} {h} {h} {h} {h} {h} {h} {h} {h} {h} {h} {h} {h} {h} {h} {h} {h} {h} {h} {h} {h} {h} {h} {h} {h} {h} {h} {h} {i}\\ {h} _ {2} ^ {i} ^ {i} {a} _ {2} ^ {j} + {h} _ {2} ^ {a} _ {2} ^ {i}\\ 0\\ {h}\\ {h} _ {h} _ {2} _ {2} ^ {i} {a} _ {2} _ {3} ^ {i} {i} {a} _ {2} ^ {3} ^ {i} {i} {a} _ {2} ^ {3} ^ {i} {i} {a} _ {2} ^ {3} ^ {i} {i} {a} _ {2} ^ {3} ^ {i}} {i} {a} _ {2}} ^ {j} + {h} _ {3} ^ {3} ^ {3} ^ {j} ^ {j}} ^ {3} ^ {i} _ {1} ^ {i}\\\ 0\\ 0\\ 0\\ 0\\ 0\ end {array}\\ 0\ 0\ end {array}\\ right] `:math: `{\ widehat {G}} _ {m,\ mathit {ij} _ {1} ^ {i}}\\ i}\\ i}\\ 0\\ 0\ 0\ 0\ end {array}\ right] `:math:` {\ widehat {G}} _ {m,\ mathit {ij}} _ {m,\ mathit {ij}} _ {1} ^ {i}}\\ i}\\ m {\ zeta}\ mathrm {\ zeta}} =\ frac {1} {8}}\ left [\ begin {array} {c} {h} _ {3} ^ {i} {h} _ {3} {h} _ {3}} _ {3} ^ {h}} _ {3} {h} _ {2} ^ {j} _ {2} ^ {j} _ {2}} ^ {i} {h} _ {3} ^ {j} + {h} _ {3} ^ {i} {h} _ {2} ^ {j}\\ 0\\ 0\\ 0\ end {array}\ right]\end{split}\end{aligned}\end{align} \]