In this contribution, we propose a novel continuum mechanical concept to determine the homogenised mechanical response of random fibre networks. Their free energy is calculated as an average of the fibre free energy over the distribution of stretch, which forms the core of the new theory. The fibre-scale kinematic information contained in this distribution is intrinsic to the model and available for any state of macroscopic deformation in addition to the homogenised macroscopic response. In contrast to the various approaches that use directional averaging over the unit sphere, the presented model does not rely on a relation between fibre deformation and orientation, and therefore applies to highly non-affine networks as well as affine ones. In the present Part I of the work, the new concept is formally introduced, put into perspective with current approaches, and elaborated for the case of networks of elastic fibres with uniform orientation distribution, which generate a macroscopically isotropic hyperelastic response. The essential constitutive assumption of the theory establishes a relation between the macroscopic deformation of the network and the microscopic stretch distribution within. A phenomenological approach is used to illustrate the new concept, expressing statistical moments of the stretch distribution in terms of the macroscopic principal strain invariants. Application of the concept to 2D and 3D Voronoi networks with fibres of different properties finally exemplifies the accurate agreement of the model with discrete network simulations in terms of both the macroscopic and microscopic response. While the here presented phenomenological variant of the approach therefore represents an advance in the analytical multiscale modelling of network materials itself, the work also provides the basis for further developments, where the relation between the stretch distribution and macroscopic strain is derived from alternative principles.

在这篇文章中，我们提出了一种新的连续介质力学概念来确定随机纤维网络的均质机械响应。它们的自由能被计算为纤维自由能在拉伸分布上的平均值，这构成了新理论的核心。该分布中包含的纤维尺度运动学信息是模型固有的，除了均匀的宏观响应外，还可用于任何宏观变形状态。与在单位球面上使用方向平均的各种方法相比，所提出的模型不依赖于纤维变形和方向之间的关系，因此适用于高度非仿射网络和仿射网络。在目前的第一部分工作中，正式引入了新概念，从当前的方法来看，并针对具有均匀取向分布的弹性纤维网络的情况进行了详细说明，该网络会产生宏观各向同性的超弹性响应。该理论的基本本构假设建立了网络的宏观变形与内部的微观拉伸分布之间的关系。使用现象学方法来说明新概念，用宏观主应变不变量表示拉伸分布的统计矩。将该概念应用于具有不同特性的纤维的 2D 和 3D Voronoi 网络最终证明了模型与离散网络模拟在宏观和微观响应方面的准确一致性。