Ateshian Gerard A
Department of Mechanical Engineering, Columbia University, 500 West 120th Street, MC4703, 220 S.W. Mudd, New York, NY 10027, USA.
J Biomech Eng. 2007 Apr;129(2):240-9. doi: 10.1115/1.2486179.
Fibrous tissues are characterized by a much higher stiffness in tension than compression. This study uses microstructural modeling to analyze the material symmetry of fibrous tissues undergoing tension and compression, to better understand how material symmetry relates to the distribution of tensed and buckled fibers. The analysis is also used to determine whether the behavior predicted from a microstructural model can be identically described by phenomenological continuum models. The analysis confirms that in the case when all the fibers are in tension in the current configuration, the material symmetry of a fibrous tissue in the corresponding reference configuration is dictated by the symmetry of its fiber angular distribution in that configuration. However, if the strain field exhibits a mix of tensile and compressive principal normal strains, the fibrous tissue is represented by a material body which consists only of those fibers which are in tension; the material symmetry of this body may be deduced from the superposition of the planes of symmetry of the strain and the planes of symmetry of the angular fiber distribution. Thus the material symmetry is dictated by the symmetry of the angular distribution of only those fibers which are in tension. Examples are provided for various fiber angular distribution symmetries. In particular, it is found that a fibrous tissue with isotropic fiber angular distribution exhibits orthotropic symmetry when subjected to a mix of tensile and compressive principal normal strains, with the planes of symmetry normal to the principal directions of the strain. This anisotropy occurs even under infinitesimal strains and is distinct from the anisotropy induced from the finite rotation of fibers. It is also noted that fibrous materials are not stable under all strain states due to the inability of fibers to sustain compression along their axis; this instability can be overcome by the incorporation of a ground matrix. It is concluded that the material response predicted using a microstructural model of the fibers cannot be described exactly by phenomenological continuum models. These results are also applicable to nonbiological fiber-composite materials.
纤维组织的特点是在拉伸时的刚度远高于压缩时。本研究使用微观结构建模来分析纤维组织在拉伸和压缩时的材料对称性，以便更好地理解材料对称性与拉伸和屈曲纤维分布之间的关系。该分析还用于确定微观结构模型预测的行为是否可以用唯象连续介质模型完全描述。分析证实，在当前构型中所有纤维都处于拉伸状态的情况下，相应参考构型中纤维组织的材料对称性由该构型中其纤维角分布的对称性决定。然而，如果应变场呈现拉伸和压缩主法向应变的混合，纤维组织由仅由处于拉伸状态的那些纤维组成的材料体表示；该物体的材料对称性可以从应变对称面和角纤维分布对称面的叠加中推导出来。因此，材料对称性仅由处于拉伸状态的那些纤维的角分布对称性决定。针对各种纤维角分布对称性提供了示例。特别地，发现具有各向同性纤维角分布的纤维组织在受到拉伸和压缩主法向应变的混合作用时呈现正交各向异性对称性，对称面垂直于应变的主方向。这种各向异性即使在无限小应变下也会出现，并且与纤维有限旋转引起的各向异性不同。还指出，由于纤维无法沿其轴线承受压缩，纤维材料在所有应变状态下都不稳定；这种不稳定性可以通过加入基体来克服。得出的结论是，使用纤维微观结构模型预测的材料响应不能用唯象连续介质模型精确描述。这些结果也适用于非生物纤维复合材料。
