90 Riccardo Rossi,Vitaliani Renato,and Eugenio Onate makes use exclusively of orthogonal bases simplifying the calculations and al- lowing to express all the terms directly in Voigt Notation,which eases the implementation. Einstein's sum on repeated indices is assumed unless specified otherwise .xI=,is the position vector of the I-th node in the cartesian space(3D space) .={describes the position of a point in the local system of coordi- nates Capital letter are used for addressing to the reference configuration Nr(g)is the value of the shape function centered on node I on the point of local coordinatesξ The use of the standard iso-parametric approach allows to express the position of any point as x()=NI()xI. In the usual assumptions of the continuum mechanics it is always possible to define the transformation between the local system of coordinates and the cartesian system as 长+，P-长，P-0x5=gG ∂灰 (1) {长，7+dr-长，nr-05l=g, (2) in which we introduced the symbols gE= aN,x灯 (3) DE gn= aN,dx灯 8n (4) the vectors ge and gn of the 3D space,can be considered linearly independent (otherwise compenetration or self contact would manifest)follows immedi- ately that they can be used in the construction of a base of the 3D space.In particular an orthogonal base can be defined as v1二ge gE (5) n= g:X gn →V2=nXV1 (6) lge×gml V3=g X gn (7) Vectors vI and v2 describe the local tangent plane to the membrane while the third base vector is always orthogonal.Follows the possibility of defining a local transformation rule that links the local coordinates and the coordinates90 Riccardo Rossi, Vitaliani Renato, and Eugenio Onate makes use exclusively of orthogonal bases simplifying the calculations and al￾lowing to express all the terms directly in Voigt Notation, which eases the implementation. Einstein’s sum on repeated indices is assumed unless specified otherwise • xI = {xI , yI , zI } T is the position vector of the I–th node in the cartesian space (3D space) • ξ = {ξ, η} T describes the position of a point in the local system of coordi￾nates • Capital letter are used for addressing to the reference configuration • NI (ξ) is the value of the shape function centered on node I on the point of local coordinates ξ The use of the standard iso-parametric approach allows to express the position of any point as x(ξ) = NI (ξ)xI. In the usual assumptions of the continuum mechanics it is always possible to define the transformation between the local system of coordinates and the cartesian system as {ξ + dξ, η} T − {ξ, η} T → ∂x(ξ,η) ∂ξ dξ = gξdξ (1) {ξ, η + dη}T − {ξ, η} T → ∂x(ξ, η) ∂η dη = gηdη (2) in which we introduced the symbols gξ = ∂NI (ξ, η) ∂ξ xI (3) gη = ∂NI (ξ, η) ∂η xI (4) the vectors gξ and gη of the 3D space, can be considered linearly independent (otherwise compenetration or self contact would manifest) follows immedi￾ately that they can be used in the construction of a base of the 3D space. In particular an orthogonal base can be defined as v1 = gξ gξ (5) n = gξ × gη gξ × gη → v2 = n × v1 (6) v3 = gξ × gη (7) Vectors v1 and v2 describe the local tangent plane to the membrane while the third base vector is always orthogonal. Follows the possibility of defining a local transformation rule that links the local coordinates ξ and the coordinates