In the problem of the synthesis of metamaterials, the pantographic architecture revealed remarkable potentialities. Indeed it allowed for the synthesis of second gradient 2D (nonlinear) continua: i.e. 2D shells whose deformation energy depends also on the second derivatives of displacements in the tangent directions to the reference configuration. Moreover, pantographic architecture seems to be able to produce metamaterials whose macroscopic elongations are large, albeit remaining in the elastic regime. The theoretically shown potentialities have started to become of ≫practical≪ interest thanks to a series of experiments, which were made possible by the recent 3D additive manufacturing. The actual construction of pantographic architecture has been based on the design of two arrays of beams interconnected by small cylinders, whose behavior can be modeled in different ways: if they are very short they can be regarded as clamps, while if they are short enough as elastic (or inelastic for large rotations) cylindrical hinges connecting the beams of different arrays. Otherwise, they must be modeled as elastic (or inelastic) elements allowing for relative rotations and displacements. In this paper, we focus on this particular case and we introduce, after a homogenization based on heuristic arguments, a 2D generalized continuum model whose kinematics is characterized by two placement and rotation fields (one for each array of beams) and whose deformation energy depends on relative displacements and rotations. The offset between the two beams arrays is proven to be an essential tool for defining effective invariant kinematical deformation measures. In facts, one wants to postulate a deformation energy for the introduced 2D generalized continuum which gives predictions in agreement with those given by the more refined 3D model where the pantographic architecture is described with its maximum geometric complexity and where the constituting material is assumed to be modelable as a standard 3D first gradient continuum. In the present paper, in order to arrive at the correct conjecture for the postulated energy, we consider the concept of averages of rotations in SO(3) Lie group. The used enriched kinematics is seen to be a possible alternative to the adoption of second gradient 2D models. Some rather surprising deformation processes are studied, where interesting non-symmetric post-buckling phenomena are observed in both the models used. Mentioned post-buckling has been observed experimentally.

在超材料的合成问题中，全景图结构显示了巨大的潜力。实际上，它允许合成第二个梯度2D（非线性）连续体：即2D壳，其变形能还取决于相对于参考配置的切线方向上的位移的二阶导数。此外，全景图结构似乎能够产生宏观伸长率大的超材料，尽管仍保留在弹性状态中。由于一系列的实验，理论上显示的潜力已开始变得具有“实际意义”，这是由于最近的3D增材制造技术得以实现的。全景图架构的实际构建基于两个由小圆柱体互连的梁阵列的设计，其行为可以用不同的方式建模：如果它们很短，则可以看作是夹具，而如果它们足够短，则可以看作是弹性（对于大旋转而言是非弹性的）圆柱形铰链，用于连接不同阵列的梁。否则，必须将它们建模为弹性（或非弹性）元素，以允许相对旋转和位移。在本文中，我们将重点放在这种特殊情况上，并在基于启发式参数进行均质化之后，介绍一个二维广义连续体模型，该模型的运动学特征是两个放置和旋转场（每个梁阵列一个），并且变形能取决于相对位移和旋转。事实证明，两个光束阵列之间的偏移是定义有效的不变运动学变形量度的必要工具。据实，一个人想为引入的2D广义连续体假设变形能，该变形能给出的预测与更精细的3D模型所给出的预测相一致，在更精确的3D模型中，以最大几何复杂度描述了全景建筑，并假定构成材料可建模为标准3D第一渐变连续体。在本文中，为了对假定的能量得出正确的猜想，我们考虑了SO（3）Lie组中平均旋转的概念。所使用的丰富运动学被认为是采用第二种梯度2D模型的一种可能的替代方法。研究了一些相当令人惊讶的变形过程，在这两种模型中都观察到了有趣的非对称后屈曲现象。实验中已经提到了屈曲后。