A geometrically exact membrane model for isotropic foils and fabrics

where R = polar(F ) ∈ SO(3) is the orthogonal part of the deformation gradient F with out-of plane component R(x, y).e3. The variable m ∈ IR accounts for a varying thickness, see [1, 2] for details. Basic idea: introduce an additional field of independently evolving viscoelastic rotations R ∈ SO(3). These rotations R are thought of as being physical meaningful but not exact continuum rotations R. With R3 ≡ R(x, y).e3 denoting the corresponding out-of plane component the dimensional reduction of a three-dimensional continuum solid to a geometrically exact membrane model results in a deformation gradient of the form F = (∇m|mR3), (2) where ∇m ∈ M is the deformation gradient of the midsurface with mx = (m1,x,m2,x,m3,x) T , my = (m1,y,m2,y,m3,y) T . The problem: find the deformation of the midsurface m : [0, T ]× ω 7→ IR and the independent local viscoelastic rotation R : [0, T ]× ω 7→ SO(3, IR) such that