4. Formulation of the elements of GRILLE_MEMBRANE

For a uniaxial reinforcement sheet, the deformation energy can be in the form of:

(4.1)\[ \ Phi\ mathrm {=}\ frac {1} {2}\ mathrm {\ int} S\ sigma\ varepsilon\ text {ds}\]

with \(S\) the reinforcement cross section per unit length, \(\sigma\) the stress (scalar), and \(\varepsilon\) the deformation (scalar). We are looking to obtain an expression of the type \(\mathrm{\epsilon }=\mathit{BU}\) where we write \(U\) the nodal values of the displacement. Continuing the approach from the previous section, we demonstrate this time that:

(4.2)\[ \ varepsilon\ mathrm {=} {(\ mathrm {\nabla} u)}} _ {\ text {11}}\ mathrm {=} (\ frac {\ mathrm {\ partial} u} {\ mathrm {\ partial} u} {\ mathrm {\ partial} u} {\ mathrm {\ partial} u} {\ mathrm {\ partial} u} {\ mathrm {\ partial} u} {\ mathrm {\ partial} u} {\ mathrm {\ partial} u} {\ mathrm {\ partial} u} {\ mathrm {\ partial} u} {\ mathrm {\ partial} u} {\ mathrm {\ partial} u} {\ mathrm {\ partial} u} {R} _ {\ alpha} ^ {1} {1} {R} {R} _ {\ gamma} ^ {\ beta\ gamma} ^ {\ beta\ gamma}\]

By introducing the \(\widehat{B}\) derivative of form functions at the Gauss point under consideration, it comes:

(4.3)\[ \ mathrm {\ epsilon} = {R} _ {\ mathrm {\ alpha}}} ^ {1} {R} _ {\ mathrm {\ gamma}} ^ {1} {g} ^ {\ mathrm {\ beta} ^ {\ mathrm {\ beta}}\ mathrm {\ beta}} {\ left}} {\ left ({a} ^ {\ mathrm {\ alpha}}\ alpha}}\ right)} _ {i} {U} _ {i, n}\]

Hence the \(B\) sought. It will be noted that it has the shape of a vector, due to the scalar nature of the deformation sought:

(4.4)\[ {B} _ {i, n} = {R} _ {\ mathrm {\ alpha}}} ^ {1} {R} _ {\ mathrm {\ gamma}} ^ {1} {g} ^ {\ mathrm {\ beta}} {\ mathrm {\ beta}}\ mathrm {\ beta}} {\ mathrm {\ beta}} ^ {\ mathrm {\ beta}} ^ {\ mathrm {\ beta}} ^ {\ mathrm {\ beta}} ^ {\ mathrm {\ beta}} ^ {\ mathrm {\ beta}} ^ {\ mathrm {\ beta}} ^ {\ mathrm {\ beta}} ^ {\ mathrm {\ beta}} ^ {\ mathrm {\ beta}, ^ {a} ^ {\ mathrm {\ alpha}}\ right)} _ {i}\]

Starting with \(B\), we find all the classical expressions of deformation, nodal forces and the tangent matrix:

\[\]

: label: eq-71

mathrm {epsilon} =mathit {BU} F=int {B} ^ {T}mathrm {sigma} K=int {B} ^ {T}frac {partialmathrm {sigma}} {partialmathrm {epsilon}} B

It should be noted that it is the one-dimensional laws of behavior that are used to obtain the stress from the deformation. All the laws of behavior available in one dimension can be used. Otherwise, we can also use three-dimensional laws, thanks to the De Borst method.