The paper presents a coordinate-free analysis of deformation measures for shells modeled as 2D surfaces. These measures are represented by second-order tensors. As is well-known, two types are needed in general: the surface strain measure (deformations in tangential directions), and the bending strain measure (warping). Our approach first determines the 3D strain tensor E of a shear deformation of a 3D shell-like body and then linearizes E in two smallness parameters: the displacement and the distance of a point from the middle surface. The linearized expression is an affine function of the signed distance from the middle surface: the absolute term is the surface strain measure and the coefficient of the linear term is the bending strain measure. The main result of the paper determines these two tensors explicitly for general shear deformations and for the subcase of Kirchhoff-Love deformations. The derived surface strain measures are the classical ones: Naghdi’s surface strain measure generally and its well-known particular case for the Kirchhoff-Love deformations. With the bending strain measures comes a surprise: they are different from the traditional ones. For shear deformations our analysis provides a new tensor ${\rho }^D$, which is different from the widely used Naghdi’s bending strain tensor ${\rho }^N$. In the particular case of Kirchhoff–Love deformations, the tensor ${\rho }^D$ reduces to a tensor ${\rho }^{\mathrm{AL}}$ introduced earlier by Anicic and Léger (Formulation bidimensionnelle exacte du modéle de coque 3D de Kirchhoff–Love. C R Acad Sci Paris I 1999; 329: 741–746). Again, ${\rho }^{\mathrm{AL}}$ is different from Koiter’s bending strain tensor ${\rho }^K$ (frequently used in this context).

AMS 2010 classification: 74B99

本文介绍了建模为2D曲面的壳体的变形量度的无坐标分析。这些度量由二阶张量表示。众所周知，通常需要两种类型：表面应变测量（切向变形）和弯曲应变测量（翘曲）。我们的方法首先确定3D壳状物体的剪切变形的3D应变张量E，然后线性化E有两个小参数：点的位移和到中间表面的距离。线性化表达式是到中间表面的有符号距离的仿射函数：绝对项是表面应变量度，线性项的系数是弯曲应变量度。本文的主要结果明确确定了这两个张量用于一般剪切变形和Kirchhoff-Love变形的子情况。派生的表面应变度量是经典度量：Naghdi的表面应变度量通常及其对于Kirchhoff-Love变形的众所周知的特殊情况。弯曲应变措施出人意料：与传统措施不同。对于剪切变形，我们的分析提供了一个新的张量${\rho }^d$不同于广泛使用的Naghdi的弯曲应变张量 ${\rho }^{ñ}$。在Kirchhoff–Love变形的特殊情况下，张量${\rho }^d$ 减少到张量 ${\rho }^{铝}$由Anicic和Léger早些时候介绍（3D的Kirchhoff–Love公式化的双向精确度。CRAcad Sci Paris I 1999； 329：741–746）。再次，${\rho }^{铝}$ 与Koiter的弯曲应变张量不同 ${\rho }^{ķ}$ （在此上下文中经常使用）。

AMS 2010分类：74B99