Biological tissues have been shown to behave isotropically at lower strain values, while at higher strains the fibers embedded in the tissue straighten and tend to take up the load. Thus, the anisotropy induced at higher loads can be mathematically modeled by incorporating the strains experienced by the fibers. From histological studies on soft tissues it is evident that for a wide range of tissues the fibers have an oblique mean orientation about the physiological loading directions. Thus, we require a mathematical framework of tensors defined in nonorthogonal basis to capture the direction-dependent response of fibers under high induced loads. In this work, we propose a novel approach to determine the fiber strains with the aid of the contravariant tensors defined in an oblique coordinate system. To determine the fiber strains, we introduce a fourth-order contravariant fiber orientation transformation tensor. The approach helps us successfully in determining the fiber strains, for a family of symmetrically and asymmetrically oriented fibers, with the aid of a single anisotropic invariant. The proposed model was fitted with the experimental data from literature to determine the corresponding material parameters.

中文翻译：

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使用逆变张量来模拟各向异性软组织

生物组织在较低的应变值下表现出各向同性，而在较高的应变下，嵌入组织中的纤维伸直并倾向于承受载荷。因此，在较高负载下引起的各向异性可以通过结合纤维所经历的应变来进行数学建模。从对软组织的组织学研究可以明显看出，对于大范围的组织，纤维具有关于生理负荷方向的倾斜平均取向。因此，我们需要一个在非正交基础上定义的张量数学框架，以捕捉纤维在高诱导载荷下的方向相关响应。在这项工作中，我们提出了一种利用斜坐标系中定义的逆变张量来确定纤维应变的新方法。为了确定纤维应变，我们引入了一个四阶逆变纤维取向变换张量。该方法帮助我们在单个各向异性不变量的帮助下成功地确定了对称和非对称取向纤维族的纤维应变。所提出的模型与文献中的实验数据进行拟合，以确定相应的材料参数。