Abstract
The polar decomposition theorem for a two dimensional continuum (a membrane) is used to produce a set of equations that describe the evolution of the geometrical quantities of a moving surface, i.e., the metric, the unit normal, the shape operator, the second fundamental form, the mean and the Gauss curvature. A link to the kinematical quantities of the continuum is also given. The version of the polar decomposition theorem for membranes we use was proved by Chi-Sing Man and H. Cohen (J. Elast. 16:97–104, 1986). Both the geometric and the kinematical framework are coordinate-free, in an attempt to contribute to a coordinate-free description for the kinematics of membranes in analogy to the kinematics of three dimensional continuum bodies as it emerges from the classical works of Noll (Arch. Rat. Mech. Anal. 2:197–226, 1958), Truesdell and Noll (The Non-linear Field Theories of Mechanics, 3rd edn., Springer, Berlin, 2004), Truesdell (A First Course in Rational Continuum Mechanics, vol. 1, Academic Press, San Diego, 1977).
Published Version
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