The mechanics of biological entities, from single molecules to the whole organ, has been extensively analyzed during the last decades. At the smaller scales, statistical mechanics has fostered successful physical models of proteins and molecules, which have been later incorporated within constitutive models of rubber-like materials and biological tissues. At the macroscopic scale, the additive decomposition of energy functions i.e., a parallel arrangement of the tissue constituent, has been recurrently used to account for the internal heterogeneity of soft biological materials. However, it has not yet been possible to unite the mechanics at the tissue level with the actual response of the tissue components. Here, we exemplify our approach using cardiovascular tissue where the mechanical response at the tissue scale is in the range of kPa whereas the elastic modulus of collagen, the main component of the vascular tissue, is in the range of MPa GPa. In this work we develop a novel theoretical framework based on a complementary strain energy function that builds-up on a full network model. The complementary strain energy function introduces naturally an additive decomposition of the deformation gradient for the tissue constituents, i.e an arrangement in series of the constituents. We demonstrate that the macroscopic response of the tissue can be reproduced by just introducing the underlying mechanical and structural features of the micro-constituents, improving in a fundamental manner previous attempts in the mechanical characterization of soft biological tissues. The proposed theoretical framework unveils a new direction in the mechanical modeling of soft tissues and biological networks.

在过去的几十年中，从单个分子到整个器官的生物实体的力学已经得到了广泛的分析。在较小的规模上，统计力学已经建立了成功的蛋白质和分子物理模型，这些模型后来被并入了橡胶样材料和生物组织的本构模型。在宏观尺度上，能量功能的加性分解，即组织成分的平行排列，已被反复用于解释软生物材料的内部异质性。但是，尚不可能将组织层面的力学与组织组件的实际响应结合起来。这里，我们以心血管组织为例来说明我们的方法，其中组织尺度的机械响应在kPa范围内，而血管组织的主要成分胶原的弹性模量在MPa GPa范围内。在这项工作中，我们开发了一个基于互补应变能函数的新颖理论框架，该函数建立在完整的网络模型上。互补应变能函数自然会引入组织成分的变形梯度的加性分解，即一系列成分的排列。我们证明，只需介绍微成分的基本机械和结构特征，即可再现组织的宏观响应，从根本上改进了软生物组织机械特性的先前尝试。拟议的理论框架揭示了软组织和生物网络的机械建模的新方向。