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| For a medium suffered large deformation, knots may be produced internally. Such a kind of knotted medium will reduce the elasticity of the medium. In this research, intrinsic knot production conditions are formulated by the deformation tensor in 3-space. Hence, the traditional continuity concept is replaced with path-connected continuity concept. For a spring produced by deformation, the curvatures of spring are formulated by the intrinsic local rotation angular which is determined by the asymmetrical displacement gradient components. The corresponding deformation energy functionals are formulated, also. They form the basic theoretic formulation of knots for finite deformation mechanics in Chen rational mechanics frame. Finally, the deformation energy of knotted medium is discussed with the knot representation polynomials. These results is expected to be valuable for many practical mechanic engineering problems, such as fatigue-crack, damage/risk evaluation, and elastoplastic instability, especially for understanding and formulating the non-linear dynamics of finite deformation. |
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| Keywords:knot polynomial, curvature, local rotation, gauge invariant deformation, rational mechanics |
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