2. Linear formulation of the elements of MEMBRANE

For a membrane element, the deformation energy can be in the form of:

(2.1)\[ \ mathrm {\ Phi} =\ frac {1} {2}\ int\ mathrm {\ sigma}\ mathrm {:}\ mathrm {\ epsilon}\ text {ds}\]

with \(\mathrm{\sigma }\) the membrane stress and \(\mathrm{\epsilon }\) the membrane deformation.

The only difficulty is to obtain an expression of the type \(\mathrm{\epsilon }=\mathit{BU}\) where we write \(U\) the nodal values of the displacement.

To do this, you have to use a bit of differential geometry. By noting \(a\) the natural base (not orthogonal, only the third vector, normal to the surface, is normalized) of the reinforcement plane and \(g\) the contravariant metric associated with this base (see [R3.07.04] .for more details). We start from the expression of the contravariant derivative:

:math:`nabla u=frac{partial {u}^{i}}{partial {mathrm{xi }}_{j}}={u}^{i}{

}_{j}{a}_{i}otimes {a}^{j}`

with \(\left\{{\mathrm{\xi }}_{j}\right\}\) an acceptable surface setting and \({a}_{i}\mathrm{=}\frac{\mathrm{\partial }x}{\mathrm{\partial }{\xi }_{i}}\). Using the metric tensor \(g\), we then have:

:math:`nabla u=frac{partial {u}^{i}}{partial {mathrm{xi }}_{j}}={u}^{i}{

}_{j}{g}^{mathit{jk}}{a}_{i}otimes {a}_{k}`

The reinforcement direction is then defined by the normed vector \({e}_{1}\) which is completed, for the convenience of presenting it in an orthonormal basis \(\left\{{e}_{i}\right\}\). We call \(R\) the operator for passage between this base and the natural base such as:

\[\]

: label: eq-4

{a} _ {i}mathrm {=} {R} _ {i} ^ {p} {e} _ {p}

We write down in Greek the indices taking only the values in \((\mathrm{1,2})\), and we get:

\[\]

: label: eq-5

{varepsilon} _ {alphabeta}mathrm {=} {(mathrm {nabla} u)}} _ {alphabeta}mathrm {=} (frac {mathrm {beta}} (frac {mathrm {partial}} u} {mathrm {partial} u} {mathrm {partial} u} {mathrm {partial} u} {mathrm {partial} u} {mathrm {partial} u} {mathrm {partial} u} {mathrm {partial} u} {mathrm {partial} u} {mathrm {partial} u} {mathrm {partial} u} ^ {i}) {R} _ {i} ^ {alpha} {alpha} {R} _ {k} ^ {beta} {g} ^ {text {jk}} ^ {text {jk}}

By definition of \(R\): \({R}_{3}^{1}\mathrm{=}0\) and by definition of \(g\): \({g}^{\text{13}}\mathrm{=}{g}^{\text{23}}\mathrm{=}0\). So we get:

(2.2)\[ {\ varepsilon} _ {\ alpha\ beta}\ mathrm {=} {(\ mathrm {\nabla} u)} _ {\ alpha\ beta}\ mathrm {=} (\ frac {\ mathrm {\ beta}} (\ frac {\ mathrm {\ partial}} u} {\ mathrm {\ partial} u} {\ mathrm {\ partial} u} {\ mathrm {\ partial} u} {\ mathrm {\ partial} u} {\ mathrm {\ partial} u} {\ mathrm {\ partial} u} {\ mathrm {\ partial} u} {\ mathrm {\ partial} u} {\ mathrm {\ partial} u} {\ mathrm {\ partial} u} {} ^ {\ gamma}) {R} _ {\ gamma} _ {\ gamma}} ^ {\ alpha} {R} _ {\ theta} ^ {\ beta} {g} {g} ^ {\ delta\ theta} ^ {\ delta\ theta}\]

If we now note \(\stackrel{ˆ}{B}\) the derivative of the shape functions at the Gauss point under consideration, it comes from:

(2.3)\[ {\ mathrm {\ epsilon}}} _ {\ mathrm {\ alpha}\ mathrm {\ alpha}}\ mathrm {\ beta}} = {\ mathrm {\ gamma}}} ^ {\ mathrm {\ alpha}}} {R}} {R}} {R} _ {\ mathrm {\ theta}}} ^ {\ mathrm {\ beta}} {g} ^ {\ mathrm {\ alpha}}} {\ mathrm {\ alpha}} {R}} {R} _ {\ mathrm {\ theta}} ^ {\ mathrm {\ alpha}} {\ alpha}} {R}} {\ mathrm {\ alpha}} {R}} {\ mathrm {\ alpha}} {R}}\ mathrm {\ theta}} {\ widehat {B}}} _ {\ mathrm {\ delta}, n} {\ left ({a} ^ {\ mathrm {\ gamma}}}\ right)}\ right)} _ {i}}\ right)} _ {i, n}\]

With \(n\), the node index. Hence the \(B\) sought:

(2.4)\[ {B} _ {i, n} = {R} _ {\ mathrm {\ gamma}}} ^ {\ mathrm {\ alpha}} {R} _ {\ mathrm {\ theta}}} ^ {\ mathrm {\ theta}}} ^ {\ mathrm {\ beta}} {\ mathrm {\ beta}} {\ mathrm {\ beta}} {g}} {g} ^ {\ mathrm {\ beta}} {g} ^ {\ mathrm {\ beta}}} {g} ^ {\ mathrm {\ beta}} {g} ^ {\ mathrm {\ beta}} {g}} ^ {\ mathrm {\ beta}} {g}} ^ {\ mathrm {\ beta}} {g}}\ mathrm {\ delta}, n} {\ left ({a} ^ {\ mathrm {\ gamma}}\ right)} _ {i}\]

Starting with \(B\), we then have all the classical expressions of deformation:

(2.5)\[ \ varepsilon\ mathrm {=}\ mathit {BU}\]

nodal forces:

(2.6)\[ F\ mathrm {=}\ mathrm {\ int} {B} ^ {T}\ sigma\]

and the tangent matrix:

(2.7)\[ K\ mathrm {=}\ mathrm {\ int} {B} {B} {B} ^ {T}\ frac {\ mathrm {\ partial}\ sigma} {\ mathrm {\ partial}\ partial}\ varepsilon} B\]