The consequences of introducing the fourth order orientation tensor as an independent variable in addition to the second order one are investigated. In the first part consequences of the Second Law of Thermodynamics are exploited. The cases with the second order alignment tensor in the state space on one hand and with the second and fourth order alignment tensors on the other hand are analogous. In the latter case differential equations for the second and fourth order tensors result from linear force-flux relations with a coupling arising due to coupling terms in the free energy. In the second part the differential equations for the second order orientation tensor or the second and fourth order orientation tensors, respectively are given explicitly in the special case of a rotation symmetric orientation distribution. The Folgar-Tucker equation with a quadratic closure relation leads to a Riccati equation for the second order parameter. In comparison the Folgar-Tucker equation and the differential equation for the fourth order parameter are considered. The fourth order parameter is eliminated later. The resulting equation for the second order parameter is a Duffing equation with a behavior of solutions completely different from the solutions of the Riccati equation.

中文翻译：

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四阶取向张量对二阶取向张量的影响

研究了除二阶方向张量之外引入四阶方向张量作为自变量的后果。在第一部分中，利用了热力学第二定律的结果。一方面具有状态空间中的二阶对准张量的情况和另一方面具有二阶和四阶对准张量的情况是类似的。在后一种情况下，二阶和四阶张量的微分方程由线性力-通量关系产生，并且由于自由能中的耦合项而产生耦合。在第二部分中，在旋转对称取向分布的特殊情况下，分别明确给出了二阶取向张量或二阶和四阶取向张量的微分方程。具有二次闭合关系的 Folgar-Tucker 方程得出二阶参数的 Riccati 方程。相比之下，考虑了 Folgar-Tucker 方程和四阶参数的微分方程。第四阶参数稍后被消除。所得到的二阶参数方程是杜芬方程，其解的行为与 Riccati 方程的解完全不同。