Extension of plane rubber films

Abstract

Extension of thin plane films, uneven in initial thickness and obeying the constitutive equation of rubber‐like elasticity, was analyzed mathematically with the following results. (i) The plane extension of rubber films is governed by two simultaneous partial differential equations whose independent variables are (u,v) and dependent variables are ( f,g) where (u,v) is the position in Cartesian coordinates of a rubber particle constituting the film occupied before the extension and ( f,g) is the position of the same particle after the extension. (ii) The axisymmetrical stretching mode is governed by a single ordinary differential equation which can readily be solved analytically or numerically upon specification of film thickness known at any one radial position. (iii) Uniform thickness extension is possible only when the film is initially uniform in thickness and the mode of extension is uniform biaxial, i.e., the two principal strains in x and y directions, respectively, are independent of position (u,v). This is in marked difference from the case of the extension of flat Newtonian fluid films in which a complex mode of uniform thickness extension obeying the Cauchy–Riemann equations exists. (iv) The governing equations were solved numerically for the general case by means of the Newton iteration scheme used in a finite difference approximation.