Axial velocity distribution imparted on thin media transported by layered rolls (Q1960827)

The authors investigate the axial velocity distribution along a thin medium that is transported between flexible rubber-coated rolls. The rubber coatings on each roll are modeled as imcompressible hyperelastic material using a neo-Hookean model of the form \(\tau=\frac{E_L}3\overline F\cdot \overline F^T-p'\overline I\), where \(\tau\) is the Cauchy stress, and \(\overline F\) is the deformation gradient. The boundary value problem is of the form \(\operatorname {div} \tau = \overline 0\) on \(\Omega\), \(\overline v = \frac d{dt}\overline \xi(\overline X(t))=\nabla \overline \xi(\overline X(t))\cdot \frac d{dt}\overline X(t)=\overline F(\overline X(t))\cdot \overline V(t)\) on \(\Omega\), \(\overline x=\overline X\) on \(\Gamma_c\), \(t= \tau\cdot\overline n=\overline 0\) on \(\gamma_f\), \(\overline g=-g\overline n\) on \(\Gamma_n\), \(|\Delta \overline v|= 0\), if \(f<\mu g\), \(|\Delta \overline v|> 0\) if \(f=\mu g\) on \(\gamma_n.\) A fully three-dimensional finite elements analysis would require significant computational effort. Here the authors present an analysis which requires less computational effort by taking into account the advantage of the geometry, i.e. the rolls are long relative to cross-sectional dimensions, and the skew angle is small.