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首页> 外文期刊>Zeitschrift fur Angewandte Mathematik und Mechanik >A method of shell theory in determination of the surface from components of its two fundamental forms
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A method of shell theory in determination of the surface from components of its two fundamental forms

机译：一种从两种基本形式的成分确定表面的壳理论方法

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We introduce the tensor which maps the Cartesian plane into the tangent plane of the surface. Then by analogy to the polar decomposition theorem widely used in the non-linear theory of thin shells the tensor is represented as composition of the surface stretch and 3D rotation fields. Left and right polar decompositions are analysed. For each of them the position vector of the surface in space is established uniquely from the surface metric and curvature components in three subsequent steps: 1) the stretch field is found by pure algebra, 2) the rotation field is obtained by solving the system of linear first-order PDEs, and 3) the position vector of the surface follows by quadrature. Integrability conditions for the rotation field are shown to be alternative forms of the Gauss-Mainardi-Codazzi equations. The results are illustrated by a simple analytically solved example. The proposed method is expected to be more appealing and in some cases also more efficient than those used in classical differential geometry. We also briefly discuss the relation of our method to the one associated with integrable surfaces and soliton equations.
机译：我们引入了张量，该张量将笛卡尔平面映射到曲面的切平面。然后，通过类似于薄壳非线性理论中广泛使用的极坐标分解定理，将张量表示为表面拉伸和3D旋转场的组成。分析左和右极分解。对于它们中的每一个，在随后的三个步骤中，根据表面度量和曲率分量唯一地确定表面在空间中的位置矢量：1）通过纯代数找到拉伸场，2）通过求解以下方程式获得旋转场：线性一阶PDE，以及3）曲面的位置矢量遵循正交。旋转场的可积性条件是高斯-马纳迪-科达奇方程的替代形式。通过一个简单的解析示例说明了结果。与经典的微分几何中使用的方法相比，该方法有望更具吸引力，并且在某些情况下还更有效。我们还简要地讨论了我们的方法与与可积曲面和孤子方程相关的一种方法的关系。

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