The strain energy for incompressible anisotropic non-linearly elastic materials is decomposed into an isotropic part representing the mechanical response of an isotropic matrix and an anisotropic part representing the contribution to the mechanical response from the presence of fibres. It is the form of the anisotropic component that is of interest here. We note that the invariants can themselves be divided into two classes: the invariants that are homogeneous functions of degree two and those of degree four in the principal stretches. The approach adopted here is straightforward: assume that there is a linear proportional relationship between terms in the general stress–strain law that are of the same degree in the principal stretches. Setting these constants identically zero recovers many of the simplified strain energies commonly found in the literature. The proportionality constants are interpreted as being a measure of the fibre–matrix interaction and a measure of the interaction between fibres in anisotropic non-linear elasticity. An influential model of fibre dispersion is recovered as a special case. The results are illustrated using the homogeneous deformation of simple shear.

用于不可压缩的各向异性非线性弹性材料的应变能被分解为代表各向同性基体的机械响应的各向同性部分和代表存在纤维对机械响应的贡献的各向同性部分。此处关注的是各向异性成分的形式。我们注意到不变量本身可以分为两类：不变量是二阶次函数的齐次函数，而在主拉伸中是四次阶的齐次函数。这里采用的方法很简单：假定一般应力-应变定律中的项之间的线性比例关系与主拉伸中的度数相同。将这些常数设置为相同的零可恢复许多文献中常见的简化应变能。比例常数被解释为是纤维-基体相互作用的量度，也是各向异性非线性弹性中纤维之间相互作用的量度。作为一种特殊情况，可以恢复影响力较大的纤维分散模型。使用简单剪力的均匀变形来说明结果。