On the Use of the Continuum Mechanics Method for Describing Interactions in Discrete Systems with Rotational Degrees of Freedom

Abstract

Elastic interactions in a system of two body-points possessing both translational and rotational degrees of freedom are studied for the most general case of motion in 3D space. The continuum mechanics method is used as a theoretical foundation for describing the interactions. A definition of strain measures for the discrete system is given by analogy with that in continuum mechanics. Constitutive equations for force and moment vectors are derived based on the energy balance equation. Several new interaction potentials are suggested.

Appendices

Appendix A: Tensor Methods in the Kinematics of Spinorial Motion

There exist various approaches to describe spinorial (rotational) motion. Further, we employ an approach based on tensor calculus. An advantage of tensor methods is that they are compatible with continuum mechanics methods. Employing tensor methods for discrete systems, we can, firstly, transform discrete systems into their continuous versions with comparative ease, and secondly, efficiently use methods that are developed in continuum mechanics.

Definition

Tensor $\mathbf{P}$ is called the rotation tensor if it is a proper orthogonal tensor, i.e., it satisfies the equations

$$ \mathbf{P} \boldsymbol{\cdot } \mathbf{P}^{T} = \mathbf{P}^{T} \boldsymbol{\cdot } \mathbf{P} = \mathbf{E}, \qquad \mathrm{det}\, \mathbf{P} = 1, $$

(165)

where $\mathbf{E}$ is the unit tensor. In accordance with Euler’s theorem, the rotation tensor can be represented as

$$ \mathbf{P}(\theta \mathbf{m}) = \bigl[1 - \cos \theta (t)\bigr] \, \mathbf{m}(t) \mathbf{m}(t) + \cos \theta (t)\, \mathbf{E} + \sin \theta (t)\, \mathbf{m}(t) \times \mathbf{E}, $$

(166)

where $\theta (t)$ is the rotation angle, $\mathbf{m}(t)$ is the unit vector directed along the axis of rotation. Euler’s representation (166) is convenient in the case of rotation about a fixed axis. If $\mathbf{m} \neq {\mathrm{const}}$, then it is more convenient to introduce the rotation vector $\boldsymbol{\theta }(t) = \theta (t) \, \mathbf{m}(t)$ and to use Zhilin’s representation for the rotation tensor [42, 46]:

$$ \mathbf{P}({\boldsymbol{\theta }}) = \mathbf{E} + \frac{\sin \theta }{ \theta }\,\mathbf{R}(t) + \frac{1 - \cos \theta }{\theta ^{2}}\, \mathbf{R}^{2}, \qquad \mathbf{R} = \boldsymbol{\theta }(t) \times \mathbf{E}, \quad \theta = | \boldsymbol{\theta }|. $$

(167)

The antisymmetric tensor $\mathbf{R}$ is called the logarithmic rotation tensor. This name is due to the fact that tensor $\mathbf{R}( \boldsymbol{\theta })$ is related to the rotation tensor $\mathbf{P}( \boldsymbol{\theta })$ as $\mathbf{P}(\boldsymbol{\theta }) = \exp \mathbf{R}(\boldsymbol{\theta })$.

Definition

The angular velocity vector is defined by the following formulas which are equivalent to each other:

$$ \dot{\mathbf{P}}(t) = \boldsymbol{\omega }(t) \times \mathbf{P}(t), \qquad \boldsymbol{\omega }(t) = - \frac{1}{2} \bigl( \dot{ \mathbf{P}}(t) \boldsymbol{\cdot } \mathbf{P}^{T}(t) \bigr) _{\times }, $$

(168)

where $(\mathbf{a} \mathbf{b})_{\times } \equiv \mathbf{a} \times \mathbf{b}$. The first formula in Eq. (168) represents one of the formulations of Poisson’s equation. The second formula in Eq. (168) expresses the fact that the angular velocity vector is the accompanying vector of the spin tensor. The derivative of the rotation vector $\boldsymbol{\theta }$ and the angular velocity vector $\boldsymbol{\omega }$ are related as

$$ \dot{\boldsymbol{\theta }} = \mathbf{Z}(\boldsymbol{\theta }) \cdot \boldsymbol{\omega }, $$

(169)

where $\mathbf{Z}(\boldsymbol{\theta })$ is the Zhilin tensor. It can be expressed in terms of the rotation tensor by the formula given in [42, 46]:

$$ \mathbf{Z}(\boldsymbol{\theta }) = \mathbf{E} - \frac{1}{2}\, \mathbf{R} + \frac{1 - g(\theta )}{\theta ^{2}}\,\mathbf{R}^{2}, \qquad g(\theta ) = \frac{\theta \, \sin \theta }{2(1 - \cos \theta )} , \quad \theta = |\boldsymbol{\theta }|. $$

(170)

In [46] it is proved that the Zhilin tensor possesses the following properties:

$$ \mathbf{Z}^{T}(\boldsymbol{\theta }) = \mathbf{P}( \boldsymbol{\theta }) \cdot \mathbf{Z}(\boldsymbol{\theta }) = \mathbf{Z}( \boldsymbol{\theta }) \cdot \mathbf{P}( \boldsymbol{\theta }), \qquad \boldsymbol{ \theta } \cdot \mathbf{Z}( \boldsymbol{\theta }) = \mathbf{Z}(\boldsymbol{\theta }) \cdot \boldsymbol{\theta } = \boldsymbol{\theta }. $$

(171)

In [46] it is also proved that the determinant of $\mathbf{Z}(\boldsymbol{\theta })$ is not equal to zero (with the exception of the singular points $\theta = 2 \pi k$, where $k$ is integer), and the inverse tensor is

$$ \mathbf{Z}^{-1}(\boldsymbol{\theta }) = \mathbf{E} + \frac{1 - \cos \theta }{\theta ^{2}}\,\mathbf{R} + \frac{\theta - \sin \theta }{\theta ^{3}}\,\mathbf{R}^{2}. $$

(172)

The inverse Zhilin tensor possesses the properties similar to (171):

$$ \mathbf{Z}^{-T}(\boldsymbol{\theta }) = \mathbf{P}^{T}( \boldsymbol{\theta }) \cdot \mathbf{Z}^{-1}( \boldsymbol{\theta }) = \mathbf{Z}^{-1}(\boldsymbol{\theta }) \cdot \mathbf{P}^{T}( \boldsymbol{\theta }), \qquad \boldsymbol{\theta } \cdot \mathbf{Z} ^{-1}(\boldsymbol{\theta }) = \mathbf{Z}^{-1}( \boldsymbol{\theta }) \cdot \boldsymbol{\theta } = \boldsymbol{\theta }. $$

(173)

Now we consider the behavior of tensors $\mathbf{Z}( \boldsymbol{\theta })$ and $\mathbf{Z}^{-1}(\boldsymbol{\theta })$ nearby the point $\theta = 0$. Expanding the functions $\sin \theta$ and $\cos \theta$ into series and retaining the terms of the second order of smallness yields

$$ \mathbf{Z}(\boldsymbol{\theta }) \approx \mathbf{E} - \frac{1}{2}\, \mathbf{R} + \frac{1}{12}\,\mathbf{R}^{2}, \qquad \mathbf{Z}^{-1}( \boldsymbol{\theta }) \approx \mathbf{E} + \frac{1}{2}\,\mathbf{R} + \frac{1}{6}\,\mathbf{R}^{2}. $$

(174)

Thus, the point $\theta = 0$ is not a singular point of tensors $\mathbf{Z}(\boldsymbol{\theta })$ and $\mathbf{Z}^{-1}( \boldsymbol{\theta })$. However, the exact formulas (170), (172) contain the indeterminate form of type $0/0$. Therefore, for numerical calculations nearby the point $\theta = 0$, it is preferable to use the approximate formulas (174).

Next, we consider the behavior of tensors $\mathbf{Z}( \boldsymbol{\theta })$ and $\mathbf{Z}^{-1}(\boldsymbol{\theta })$ nearby the point $\theta = 2 \pi$. Putting $\theta = 2 \pi - \epsilon$, where $\epsilon \ll 1$, and expanding the functions $\sin \theta$ and $\cos \theta$ into series yields the following approximate expressions for tensors $\mathbf{Z}(\boldsymbol{\theta })$ and $\mathbf{Z}^{-1}(\boldsymbol{\theta })$:

$$ \mathbf{Z}(\boldsymbol{\theta }) \approx \mathbf{E} - \frac{1}{2}\, \mathbf{R} + \frac{1}{2 \pi \epsilon }\,\mathbf{R}^{2}, \qquad \mathbf{Z}^{-1}(\boldsymbol{\theta }) \approx - \frac{\epsilon }{2 \pi } \,\mathbf{E} + \frac{\epsilon ^{2}}{8 \pi ^{2}}\,\mathbf{R} + \frac{1}{ \theta ^{2}}\,\boldsymbol{ \theta } \boldsymbol{\theta }. $$

(175)

It is easy to see that the point $\theta = 2 \pi$ is the singular point of tensor $\mathbf{Z}(\boldsymbol{\theta })$, and it is not a singular point for tensor $\mathbf{Z}^{-1}(\boldsymbol{\theta })$. That is why to find a relation between the derivative of the rotation tensor $\boldsymbol{\theta }$ and the angular velocity vector $\boldsymbol{\omega }$, we should use rather the equation

$$ \boldsymbol{\omega } = \mathbf{Z}^{-1}(\boldsymbol{ \theta }) \cdot \dot{\boldsymbol{\theta }}, $$

(176)

than Eq. (169). Substituting the second formula in Eq. (175) into Eq. (176) yields

$$ \boldsymbol{\omega } \approx \frac{\dot{\theta }}{2 \pi }\, \boldsymbol{ \theta } + \frac{\epsilon }{2 \pi } \biggl(\frac{ \dot{\theta }}{2 \pi }\,\boldsymbol{\theta } - \dot{\boldsymbol{\theta }} \biggr) + \frac{\epsilon ^{2}}{8 \pi ^{2}}\, \boldsymbol{\theta } \times \dot{\boldsymbol{\theta }} = \frac{ \dot{\theta }}{2 \pi }\,\boldsymbol{\theta } + O( \epsilon ). $$

(177)

Thus, when the magnitude of the rotation vector approaches the value $2 \pi$, the direction of the angular velocity vector $\boldsymbol{\omega }$ tends to the direction of the rotation vector $\boldsymbol{\theta }$. Since $\mathbf{R} \cdot \boldsymbol{\theta } = \mathbf{0}$, the term ${ \frac{1}{2 \pi \epsilon }\,\mathbf{R}^{2} \cdot \boldsymbol{\omega }}$ in the formula $\dot{\boldsymbol{\theta }} = \mathbf{Z}( \boldsymbol{\theta }) \cdot \boldsymbol{\omega }$ is a finite quantity in spite of the presence of a small quantity in the denominator of the fraction. We note that solving the differential equations numerically it is not necessary to use the approximate formulas for tensors $\mathbf{Z}(\boldsymbol{\theta })$ and $\mathbf{Z}^{-1}( \boldsymbol{\theta })$ nearby the point $\theta = 2 \pi$.

The description of spinorial (rotational) motion that is based on using the rotation vector is quite convenient for deriving the constitutive equations in the case of an elastic interaction of particles. Nevertheless, another representation for the rotation tensor, namely, a representation in the form of a composition of an arbitrary number of the rotation tensors

$$ \mathbf{P} = \mathbf{P}_{n} \cdot \cdots \cdot \mathbf{P}_{2} \cdot \mathbf{P}_{1} $$

(178)

is used further. The so-called rule of quasipermutability takes place for any two rotation tensors. This rule is expressed by the following formulas [42, 47]:

$$ \mathbf{P}_{2}(\psi \mathbf{m}) \cdot \mathbf{P}_{1}(\varphi \mathbf{n}) = \mathbf{P}_{1}\bigl( \varphi \mathbf{n}'\bigr) \cdot \mathbf{P} _{2}(\psi \mathbf{m}), \quad \mathbf{n}' = \mathbf{P}_{2}(\psi \mathbf{m}) \cdot \mathbf{n}. $$

(179)

If the rotation tensor is represented as a composition of the rotation tensors, then the angular velocity vector is calculated in accordance with the sum rule for the angular velocities [42, 47]:

$$ \mathbf{P} = \mathbf{P}_{2} \cdot \mathbf{P}_{1} \quad \Longrightarrow \quad \boldsymbol{\omega } = \boldsymbol{\omega }_{2} + \mathbf{P} _{2} \cdot \boldsymbol{\omega }_{1}. $$

(180)

Appendix B: Balance Equations in Eulerian Mechanics

Below we give particular formulations of the balance equations applicable to bodies that are assumed to be closed (not exchanging mass with their surroundings) and isolated (not exchanging energy with their surroundings).

The linear momentum balance equation. The rate of change of the body $\mathscr{A}$ linear momentum is equal to the force vector acting on the body from its surroundings $\mathscr{A}^{e}$:

$$ \dot{\mathbf{K}}_{1}(\mathscr{A}) = \mathbf{F}\bigl( \mathscr{A}, \mathscr{A}^{e}\bigr). $$

(181)

The angular momentum balance equation. The rate of change of the body $\mathscr{A}$ angular momentum calculated relative to the reference point $Q$ is equal to the moment vector acting on the body from its surroundings $\mathscr{A}^{e}$ calculated relative to the same reference point $Q$:

$$ \dot{\mathbf{K}}_{2}^{Q}(\mathscr{A}) = \mathbf{M}^{Q}\bigl(\mathscr{A}, \mathscr{A}^{e}\bigr). $$

(182)

If the body $\mathscr{A}$ consists of body-points $\mathscr{A}_{i}$ then its angular momentum is calculated as

$$ \mathbf{K}_{2}^{Q}(\mathscr{A}) = \sum _{i} \bigl[ ( \mathbf{r} _{i} - \mathbf{r}_{Q} ) \times \mathbf{K}_{1}(\mathscr{A}_{i}) + \mathbf{K}_{2}(\mathscr{A}_{i}) \bigr], $$

(183)

where the position vector $\mathbf{r}_{i}$ identifies the current position of the body-point with number $i$ and the position vector $\mathbf{r}_{Q}$ identifies the position of the reference point $Q$. The external moment $\mathbf{M}^{Q}(\mathscr{A},\mathscr{A}^{e})$, by definition, is

$$ \mathbf{M}^{Q}\bigl(\mathscr{A},\mathscr{A}^{e} \bigr) = \sum_{i} \bigl[ ( \mathbf{r}_{P_{i}} - \mathbf{r}_{Q} ) \times \mathbf{F}\bigl( \mathscr{A}_{i}, \mathscr{A}^{e}\bigr) + \mathbf{L}^{P_{i}}\bigl( \mathscr{A}_{i}, \mathscr{A}^{e}\bigr) \bigr]. $$

(184)

Here $( \mathbf{r}_{P_{i}} - \mathbf{r}_{Q} ) \times \mathbf{F}(\mathscr{A}_{i},\mathscr{A}^{e})$ is the moment of force, $\mathbf{L}^{P_{i}}(\mathscr{A}_{i},\mathscr{A}^{e})$ is the proper moment calculated relative to the datum point $P_{i}$, which position is identified by the position vector $\mathbf{r}_{P_{i}}$. The datum point can be moving and for each body-point it can be chosen arbitrary and independently of the choice of the datum points of other body-points forming the body $\mathscr{A}$.

The energy balance equation. The rate of change of the body $\mathscr{A}$ total energy is equal to the power of external actions $N(\mathscr{A},\mathscr{A}^{e})$:

$$ \dot{E}(\mathscr{A}) = N\bigl(\mathscr{A},\mathscr{A}^{e} \bigr). $$

(185)

The total energy of the body $\mathscr{A}$ is the sum of its kinetic and internal energies: $E(\mathscr{A}) = K(\mathscr{A}) + U(\mathscr{A})$. If the body $\mathscr{A}$ consists of body-points $\mathscr{A}_{i}$ then the power of external actions, by definition, is calculated as the bilinear form of velocities and external actions, namely

$$ N\bigl(\mathscr{A},\mathscr{A}^{e}\bigr) = \sum _{i} \bigl[\mathbf{F}\bigl( \mathscr{A}_{i}, \mathscr{A}^{e}\bigr) \boldsymbol{\cdot } \mathbf{v}_{i} + \mathbf{L}\bigl(\mathscr{A}_{i},\mathscr{A}^{e}\bigr) \boldsymbol{\cdot } \boldsymbol{\omega }_{i} \bigr]. $$

(186)

In Newtonian mechanics, the linear momentum balance equation, the angular momentum balance equation and the energy balance equation for a system of mass points follow from the second and third Newton laws. In Eulerian mechanics, the said balance equations are formulated as the laws that are true for arbitrary bodies. That is why in Eulerian mechanics, there is no necessity to accept any additional postulates like the third Newton law. In Eulerian mechanics, the third Newton law for force vectors and its analogue for moment vectors follow from the linear momentum balance equation and the angular momentum balance equation, respectively.

Appendix C: The Frame Indifference

Now we briefly discuss the question of frame indifference of the reduced energy balance equations (14), (15) and (20) with respect to any time-dependent reference frame. First of all, in addition to the inertial reference frame that is used to formulate all balance equations, we introduce a new reference frame arbitrary moving relative the given reference frame. A location of the new reference frame origin in the given reference frame is determined by the position vector $\mathbf{r}_{O}(t)$. An orientation of the new reference frame relative to the given reference frame is determined by the rotation tensor $\mathbf{Q}(t)$. The corresponding translational and angular velocities are calculated as

$$ \mathbf{v}_{O}(t) = \dot{\mathbf{r}}_{O}(t), \qquad \dot{\mathbf{Q}}(t) = \boldsymbol{\omega }_{Q}(t) \times \mathbf{Q}(t). $$

(187)

Next, we denote the characteristics of motion of the considered body-points in the new reference frame as $\tilde{\mathbf{r}}_{i}$, $\tilde{\mathbf{v}}_{i}$, $\tilde{\mathbf{P}}_{i}$, $\tilde{\boldsymbol{\omega }}_{i}$ where $i=1,2$. It is well known formulas relating the characteristics of motion in two different reference frames:

$$ \textstyle\begin{array}{l} \mathbf{r}_{i} = \mathbf{r}_{O} + \mathbf{Q} \boldsymbol{\cdot } \tilde{\mathbf{r}}_{i}, \qquad \mathbf{v}_{i} = \mathbf{v}_{O} + \boldsymbol{\omega }_{Q} \times \mathbf{Q} \boldsymbol{\cdot } \tilde{\mathbf{r}}_{i} + \mathbf{Q} \boldsymbol{\cdot } \tilde{\mathbf{v}}_{i}, \qquad \tilde{\mathbf{v}}_{i} = \dot{\tilde{\mathbf{r}}}_{i}, \\ \mathbf{P}_{i} = \mathbf{Q} \boldsymbol{\cdot } \tilde{\mathbf{P}} _{i}, \qquad \boldsymbol{\omega }_{i} = \boldsymbol{\omega }_{Q} + \mathbf{Q} \boldsymbol{\cdot } \tilde{\boldsymbol{\omega }}_{i}, \qquad \dot{\tilde{\mathbf{P}}}_{i} = \tilde{\boldsymbol{\omega }}_{i} \times \tilde{\mathbf{P}}_{i}, \quad i = 1,2. \end{array} $$

(188)

From Eq. (188) it follows

$$ \textstyle\begin{array}{l} { \displaystyle \mathbf{r} = \mathbf{Q} \boldsymbol{\cdot } \tilde{\mathbf{r}}, \qquad \tilde{\mathbf{r}} = \tilde{\mathbf{r}}_{2} - \tilde{\mathbf{r}}_{1}, \qquad \boldsymbol{\omega }_{2} - \boldsymbol{\omega }_{1} = \mathbf{Q} \boldsymbol{\cdot } ( \tilde{\boldsymbol{\omega }}_{2} - \tilde{\boldsymbol{\omega }}_{1}),} \\ { \displaystyle \dot{\mathbf{r}} - \frac{1}{2} (\boldsymbol{\omega }_{1} + \boldsymbol{\omega }_{2}) \times \mathbf{r} = \mathbf{Q} \boldsymbol{\cdot } \biggl[ \dot{\tilde{\mathbf{r}}} - \frac{1}{2} ( \tilde{\boldsymbol{\omega }}_{1} + \tilde{\boldsymbol{\omega }}_{2}) \times \tilde{\mathbf{r}} \biggr].} \end{array} $$

(189)

Substituting Eq. (189) into Eq. (14), we obtain

$$ \dot{U} = \tilde{\mathbf{F}} \boldsymbol{\cdot } \biggl[ \dot{ \tilde{\mathbf{r}}} - \frac{1}{2} (\tilde{\boldsymbol{\omega }} _{1} + \tilde{\boldsymbol{\omega }}_{2}) \times \tilde{ \mathbf{r}} \biggr] + \tilde{\mathbf{M}} \boldsymbol{\cdot } ( \tilde{\boldsymbol{ \omega }}_{2} - \tilde{\boldsymbol{\omega }}_{1}), $$

(190)

where force vector $\tilde{\mathbf{F}}$ and moment vector $\tilde{\mathbf{M}}$ are related to vectors $\mathbf{F}$ and $\mathbf{M}$ as

$$ \mathbf{F} = \mathbf{Q} \boldsymbol{\cdot } \tilde{\mathbf{F}}, \qquad \mathbf{M} = \mathbf{Q} \boldsymbol{\cdot } \tilde{\mathbf{M}}. $$

(191)

It is well known that Eq. (191) expresses the frame indifference or, that is the same thing, the material objectivity principle [51]. It is easy to see that Eq. (191) has the exactly same form as Eq. (14). Thus, if the force and moment vectors satisfy the material objectivity principle then the reduced energy balance equation (14) possesses the property of frame indifference and vice versa. It means that if a definition of strain measures are based on Eq. (14) then the strain measures will satisfy the material objectivity principle and the corresponding constitutive equations will possess the property of frame indifference. It is not difficult to show that the aforesaid is also true in relation to Eqs. (15) and (20).

Appendix D: Relations Between Stiffness Characteristics in the Linear Theory

Taking into account Eqs. (33), (40), (42), it is not difficult to show that the stiffness characteristics $C_{\varepsilon }$, $C_{\varepsilon \varPhi }$, $C_{\varPhi }$, $\mathbf{C}_{\varepsilon 1}$, $\mathbf{C}_{\varepsilon 2}$, $\mathbf{C}_{\varPhi 1}$, $\mathbf{C}_{\varPhi 2}$, $\mathbf{C}_{\chi 1}$, $\mathbf{C}_{\chi \chi }$, $\mathbf{C}_{\chi 2}$ relate to the stiffness tensors $\hat{\mathbf{C}}_{1}$, $\hat{\mathbf{C}}_{2}$, $\hat{\mathbf{C}}_{3}$ by the formulas

$$ \textstyle\begin{array}{l} C_{\varepsilon }= \mathbf{e}_{0} \boldsymbol{\cdot } \hat{\mathbf{C}} _{1} \boldsymbol{\cdot } \mathbf{e}_{0}, \qquad C_{\varepsilon \varPhi } = \mathbf{e}_{0} \boldsymbol{\cdot } \hat{\mathbf{C}}_{2} \boldsymbol{\cdot } \mathbf{e}_{0}, \qquad C_{\varPhi }= \mathbf{e}_{0} \boldsymbol{\cdot } \hat{\mathbf{C}}_{3} \boldsymbol{\cdot } \mathbf{e}_{0}, \\ \mathbf{C}_{\varepsilon 1} = \mathbf{e}_{0} \boldsymbol{\cdot } \hat{\mathbf{C}}_{1} \boldsymbol{\cdot } (\mathbf{E} - \mathbf{e}_{0} \mathbf{e}_{0}), \qquad \mathbf{C}_{\varepsilon 2} = \mathbf{r}_{0} \times \hat{\mathbf{C}}_{1} \boldsymbol{\cdot } \mathbf{e}_{0} - \mathbf{e}_{0} \boldsymbol{\cdot } \hat{\mathbf{C}}_{2} \boldsymbol{\cdot } (\mathbf{E} - \mathbf{e}_{0} \mathbf{e}_{0}), \\ \mathbf{C}_{\varPhi 1} = (\mathbf{E} - \mathbf{e}_{0} \mathbf{e}_{0}) \boldsymbol{\cdot } \hat{\mathbf{C}}_{3} \boldsymbol{\cdot } \mathbf{e}_{0}, \qquad \mathbf{C}_{\varPhi 2} = \mathbf{r}_{0} \times \hat{\mathbf{C}}_{2} \boldsymbol{\cdot } \mathbf{e}_{0} - (\mathbf{E} - \mathbf{e}_{0} \mathbf{e}_{0}) \boldsymbol{\cdot } \hat{\mathbf{C}} _{2} \boldsymbol{\cdot } \mathbf{e}_{0}, \\ \mathbf{C}_{\chi 1} = (\mathbf{E} - \mathbf{e}_{0} \mathbf{e}_{0}) \boldsymbol{\cdot } \hat{\mathbf{C}}_{3} \boldsymbol{\cdot } ( \mathbf{E} - \mathbf{e}_{0} \mathbf{e}_{0}), \\ \mathbf{C}_{\chi \chi } = \mathbf{r}_{0} \times \hat{\mathbf{C}}_{2} \boldsymbol{\cdot } (\mathbf{E} - \mathbf{e}_{0} \mathbf{e}_{0}) - ( \mathbf{E} - \mathbf{e}_{0} \mathbf{e}_{0}) \boldsymbol{\cdot } \hat{\mathbf{C}}_{3} \boldsymbol{\cdot } (\mathbf{E} - \mathbf{e}_{0} \mathbf{e}_{0}), \\ \mathbf{C}_{\chi 2} = - \mathbf{r}_{0} \times \hat{\mathbf{C}}_{1} \times \mathbf{r}_{0} - 2\, \mathbf{r}_{0} \times \hat{\mathbf{C}} _{2} \boldsymbol{\cdot } (\mathbf{E} - \mathbf{e}_{0} \mathbf{e}_{0}) + (\mathbf{E} - \mathbf{e}_{0} \mathbf{e}_{0}) \boldsymbol{\cdot } \hat{\mathbf{C}}_{3} \boldsymbol{\cdot } (\mathbf{E} - \mathbf{e}_{0} \mathbf{e}_{0}). \end{array} $$

(192)

Appendix E: The Moment Vector in the Case of Transversely Isotropic Potential

In the case of transversely isotropic interaction potential, the moment vector $\mathbf{M}$, which corresponds to the energetic moment vector $\mathbf{M}_{*}$ given by Eq. (81), is calculated as

$$\begin{aligned} \mathbf{M} =& \biggl[ 2 \frac{\partial U(\varPhi ^{2}, \mathbf{k} \boldsymbol{\cdot } \boldsymbol{\varPhi })}{\partial \varPhi ^{2}} + \frac{ \partial U(\varPhi ^{2}, \mathbf{k} \boldsymbol{\cdot } \boldsymbol{\varPhi })}{\partial (\mathbf{k} \boldsymbol{\cdot } \boldsymbol{\varPhi })} \frac{2 (1 - \cos \varPhi ) - \varPhi \sin \varPhi }{2 \varPhi ^{2} (1 - \cos \varPhi )} (\mathbf{k} \boldsymbol{\cdot } \boldsymbol{\varPhi }) \biggr] \! (\mathbf{P}_{1} \boldsymbol{\cdot } \boldsymbol{\varPhi }) \\ &{}+ \frac{\partial U(\varPhi ^{2}, \mathbf{k} \boldsymbol{\cdot } \boldsymbol{\varPhi })}{\partial (\mathbf{k} \boldsymbol{\cdot } \boldsymbol{\varPhi })} \biggl[ \frac{\varPhi \sin \varPhi }{2(1 - \cos \varPhi )}\, (\mathbf{P}_{1} \boldsymbol{\cdot } \mathbf{k}) + \frac{1}{2}\, (\mathbf{P}_{1} \boldsymbol{\cdot } \boldsymbol{\varPhi }) \times (\mathbf{P}_{1} \boldsymbol{\cdot } \mathbf{k}) \biggr] \\ \equiv& \biggl[ 2 \frac{\partial U(\varPhi ^{2}, \mathbf{k} \boldsymbol{\cdot } \boldsymbol{\varPhi })}{\partial \varPhi ^{2}} + \frac{ \partial U(\varPhi ^{2}, \mathbf{k} \boldsymbol{\cdot } \boldsymbol{\varPhi })}{\partial (\mathbf{k} \boldsymbol{\cdot } \boldsymbol{\varPhi })} \frac{2 (1 - \cos \varPhi ) - \varPhi \sin \varPhi }{2 \varPhi ^{2} (1 - \cos \varPhi )} (\mathbf{k} \boldsymbol{\cdot } \boldsymbol{\varPhi }) \biggr] (\mathbf{P}_{2} \boldsymbol{\cdot } \boldsymbol{\varPhi }) \\ &{} + \frac{\partial U(\varPhi ^{2}, \mathbf{k} \boldsymbol{\cdot } \boldsymbol{\varPhi })}{\partial (\mathbf{k} \boldsymbol{\cdot } \boldsymbol{\varPhi })} \biggl[ \frac{\varPhi \sin \varPhi }{2(1 - \cos \varPhi )}\, (\mathbf{P}_{2} \boldsymbol{\cdot } \mathbf{k}) - \frac{1}{2}\, (\mathbf{P}_{2} \boldsymbol{\cdot } \boldsymbol{\varPhi }) \times (\mathbf{P}_{2} \boldsymbol{\cdot } \mathbf{k}) \biggr] . \end{aligned}$$

(193)

It is easy to see that, in the case of transversely isotropic interaction potential, $|\mathbf{M}| \neq |\mathbf{M}_{*}|$ and the direction of the moment vector $\mathbf{M}$ depends on the relative rotation vector $\boldsymbol{\varPhi }$ in non-obvious way.

Appendix F: The Interaction of Dipoles

Two identical rigid dipoles of length $l$ are considered. (Here the term “dipole” means two close rigidly connected mass points.) In the reference configuration, the dipole centers are on the axis $z$, and the dipoles are oriented along the axis $z$. The interaction of the mass points belonging to the different dipoles is described by the Lennard–Jones potential.

First of all, we consider the case of purely moment interactions. Using the procedure described in Sect. 5.4, we obtain the dipole interaction potential with an asymptotic error of order $O(l^{3})$:

$$\begin{aligned} U(r, \boldsymbol{\varPhi }) =& - D \biggl[ \biggl( \frac{r_{0}}{r} \biggr) ^{12}-2 \biggl( \frac{r_{0}}{r} \biggr) ^{6} \biggr] + \frac{3 D l ^{2}}{r^{2}} \biggl[ \biggl( \frac{r_{0}}{r} \biggr) ^{12} - \biggl( \frac{r _{0}}{r} \biggr) ^{6} \biggr] \\ &{} - \frac{3 D l^{2}}{2 r^{2}} \biggl[ 7 \biggl( \frac{r_{0}}{r} \biggr) ^{12}-4 \biggl( \frac{r_{0}}{r} \biggr) ^{6} \biggr] \biggl( 4 \frac{ \varPhi _{z}^{2}}{\varPhi ^{2}} \biggl[ 1 + \biggl( 1 - \frac{\varPhi _{z}^{2}}{ \varPhi ^{2}} \biggr) \cos \varPhi \biggr] \\ &{} + \biggl( 1 - \frac{\varPhi _{z} ^{2}}{\varPhi ^{2}} \biggr) ^{2} \bigl[3 + \cos (2 \varPhi ) \bigr] \biggr) . \end{aligned}$$

(194)

Here $D$ and $r_{0}$ are the parameters of the Lennard–Jones potential. It is easy to show that the point $\varPhi = 0$ is not a singular point of potential (194). Indeed, expanding the functions $\cos \varPhi$ and $\cos (2 \varPhi )$ into series near the point $\varPhi = 0$, after simple transformations we obtain the following approximate expression

$$\begin{aligned} U(r, \boldsymbol{\varPhi }) \approx& - D \biggl[ \biggl( \frac{r_{0}}{r} \biggr) ^{12} -2 \biggl( \frac{r_{0}}{r} \biggr) ^{6} \biggr] + \frac{3 D l^{2}}{r^{2}} \biggl[ \biggl( \frac{r_{0}}{r} \biggr) ^{12} - \biggl( \frac{r_{0}}{r} \biggr) ^{6} \biggr] \\ &{} - \frac{3 D l^{2}}{r^{2}} \biggl[ 7 \biggl( \frac{r_{0}}{r} \biggr) ^{12}-4 \biggl( \frac{r_{0}}{r} \biggr) ^{6} \biggr] \bigl( 2 - \varPhi ^{2} + \varPhi _{z}^{2} \bigr), \end{aligned}$$

(195)

which is the asymptotically dominant term of potential (194).

The extension of the obtained results to the case of the overall motion (including both the spinorial motion and the translational one) is not difficult. That is why we give only the final form of the interaction potential $U(\boldsymbol{\varepsilon }_{1}, \boldsymbol{\varPhi }_{1})$, without considering the details of their derivation:

$$\begin{aligned} U(\boldsymbol{\varepsilon }_{1}, \boldsymbol{\varPhi }_{1}) =& - D \biggl[ \biggl( \frac{r_{0}}{\varepsilon _{1}} \biggr) ^{12} -2 \biggl( \frac{r _{0}}{\varepsilon _{1}} \biggr) ^{6} \biggr] + \frac{3 D l^{2}}{ \varepsilon _{1}^{2}} \biggl[ \biggl( \frac{r_{0}}{\varepsilon _{1}} \biggr) ^{12} - \biggl( \frac{r_{0}}{ \varepsilon _{1}} \biggr) ^{6} \biggr] \\ &{} - \frac{3 D l^{2}}{\varepsilon _{1}^{4}} \biggl[ 7 \biggl( \frac{r_{0}}{ \varepsilon _{1}} \biggr) ^{12} -4 \biggl( \frac{r_{0}}{\varepsilon _{1}} \biggr) ^{6} \biggr] \bigl\{ \bigl( \mathbf{k} \cdot \tilde{\mathbf{P}}(\boldsymbol{\varPhi }_{1}) \cdot \boldsymbol{\varepsilon }_{1} \bigr) ^{2} + ( \mathbf{k} \cdot \boldsymbol{\varepsilon }_{1} ) ^{2} \bigr\} , \end{aligned}$$

(196)

where

$$ \tilde{\mathbf{P}}(\boldsymbol{\varPhi }_{1}) \cdot \boldsymbol{\varepsilon }_{1} = \cos \varPhi _{1}\, \boldsymbol{\varepsilon }_{1} + \frac{\sin \varPhi _{1}}{\varPhi _{1}}\, \boldsymbol{ \varPhi }_{1} \times \boldsymbol{\varepsilon }_{1} + \frac{1 - \cos \varPhi _{1}}{\varPhi _{1}^{2}}\,(\boldsymbol{\varPhi }_{1} \cdot \boldsymbol{ \varepsilon }_{1})\, \boldsymbol{\varPhi }_{1}, \quad \varepsilon _{1} = |\boldsymbol{\varepsilon }_{1}|, \ \varPhi _{1} = | \boldsymbol{\varPhi }_{1}|. $$

(197)

Potential (196) is a generalization of potential (194). The dipole interaction potential (196), as well as potential (194), is obtained with an asymptotic error of order $O(l^{3})$ compared to the exact interaction potential of the dipoles. Taking into account Eq. (99), it is easy to show that

$$ \bigl( \mathbf{k} \cdot \tilde{\mathbf{P}}(\boldsymbol{\varPhi }_{1}) \cdot \boldsymbol{\varepsilon }_{1} \bigr) ^{2} = ( \mathbf{k} \cdot \boldsymbol{\varepsilon }_{2} ) ^{2}, \qquad ( \mathbf{k} \cdot \boldsymbol{\varepsilon }_{1} ) ^{2} = \bigl( \mathbf{k} \cdot \tilde{\mathbf{P}}( \boldsymbol{\varPhi }_{2}) \cdot \boldsymbol{\varepsilon }_{2} \bigr) ^{2}, \quad \varepsilon _{1} = \varepsilon _{2}. $$

(198)

Substituting Eq. (198) into Eq. (196) yields the dipole interaction potential $U(\boldsymbol{\varepsilon }_{2}, \boldsymbol{\varPhi }_{2})$. It is obvious that potential $U(\boldsymbol{\varepsilon }_{2}, \boldsymbol{\varPhi }_{2})$ has the same structure as potential (196), and it can be obtained from potential (196) by replacing $\boldsymbol{\varepsilon }_{1}$ by $\boldsymbol{\varepsilon }_{2}$ and $\boldsymbol{\varPhi }_{1}$ by $\boldsymbol{\varPhi }_{2}$. Moreover, it is obvious that $\varepsilon _{2}$ and $\tilde{\mathbf{P}}(\boldsymbol{\varPhi }_{2}) \cdot \boldsymbol{\varepsilon }_{2}$ are expressed in terms of vectors $\boldsymbol{\varepsilon }_{2}$, $\boldsymbol{\varPhi }_{2}$ by the formulas similar to Eq. (197).

Appendix G: The Interaction of Tetrahedrons

Two identical rectilinear tetrahedrons are considered. The size of the tetrahedron is specified by the distance $a$ from the tetrahedron center to its vertex. The tetrahedrons are considered to be rigid and inertialess. There are mass points in the vertices of the tetrahedrons. In the reference configuration, the tetrahedron centers are on the axis $z$, and the tetrahedrons are uniformly oriented in space. In the reference configuration, the orientation of the tetrahedrons is specified by unit vectors $\mathbf{e}_{1}$, $\mathbf{e}_{2}$, $\mathbf{e}_{3}$, $\mathbf{e}_{4}$, which point from the tetrahedron center to its vertices. These vectors are expressed in terms of basis vector of a rectangular coordinate system as follows:

$$ \textstyle\begin{array}{l} { \displaystyle \mathbf{e}_{1} = \frac{2 \sqrt{2}}{3}\, \mathbf{i} - \frac{1}{3}\, \mathbf{k}, \qquad \mathbf{e}_{2} = - \frac{\sqrt{2}}{3}\, \mathbf{i} + \frac{\sqrt{2}}{\sqrt{3}}\, \mathbf{j} - \frac{1}{3} \, \mathbf{k},} \\ { \displaystyle \mathbf{e}_{3} = - \frac{\sqrt{2}}{3}\, \mathbf{i} - \frac{ \sqrt{2}}{\sqrt{3}}\, \mathbf{j} - \frac{1}{3}\, \mathbf{k}, \qquad \mathbf{e}_{4} = \mathbf{k}.} \end{array} $$

(199)

The relative rotation vector $\boldsymbol{\varPhi }$ can be represented by means of vectors $\mathbf{e}_{1}$, $\mathbf{e}_{2}$, $\mathbf{e}_{3}$, $\mathbf{e}_{4}$ as

$$ \textstyle\begin{array}{l} \displaystyle \boldsymbol{\varPhi } = \frac{3}{4} \bigl( \varPhi ^{(1)} \mathbf{e}_{1} + \varPhi ^{(2)} \mathbf{e}_{2} + \varPhi ^{(3)} \mathbf{e}_{3} + \varPhi _{z} \mathbf{k} \bigr) , \qquad \varPhi _{z} \equiv \varPhi ^{(4)}, \\ \displaystyle \varPhi ^{(4)} = - \bigl(\varPhi ^{(1)} + \varPhi ^{(2)} + \varPhi ^{(3)}\bigr), \qquad \varPhi ^{(i)} = \mathbf{e}_{i} \boldsymbol{\cdot } \boldsymbol{\varPhi }, \quad i = 1, 2,3,4. \end{array} $$

(200)

We assume that the interaction of the mass points belonging to the different tetrahedrons is described by the Lennard–Jones potential.

First of all, we consider purely moment interactions. In this case, using the procedure described in Sect. 5.4, we obtain the tetrahedron interaction potential with an asymptotic error of order $O(a^{4})$:

$$\begin{aligned} U(r, \boldsymbol{\varPhi }) =& - D \biggl[ \biggl( \frac{r_{0}}{r} \biggr) ^{12} -2 \biggl( \frac{r_{0}}{r} \biggr) ^{6} \biggr] \\ &{} - \frac{4 D a^{2}}{r^{2}} \biggl[ 11 \biggl( \frac{r_{0}}{r} \biggr) ^{12} -5 \biggl( \frac{r_{0}}{r} \biggr) ^{6} \biggr] - \frac{32 D a^{3}}{r^{3}} \biggl[ 14 \biggl( \frac{r_{0}}{r} \biggr) ^{12} -5 \biggl( \frac{r_{0}}{r} \biggr) ^{6} \biggr] \\ &{} \times \biggl\{ \biggl( 1 - \frac{\varPhi _{z}^{2}}{\varPhi ^{2}} \biggr) \biggl[1 + \frac{5}{3}\, \frac{\varPhi _{z}^{4}}{\varPhi ^{4}} + \frac{10}{9} \biggl( 1 - \frac{\varPhi _{z}^{2}}{\varPhi ^{2}} \biggr) \biggl( 1 + 2 \frac{\varPhi _{z}^{2}}{\varPhi ^{2}} \biggr) \cos \varPhi \\ &{}+ \frac{5}{9} \biggl( 1 - \frac{\varPhi _{z}^{2}}{\varPhi ^{2}} \biggr) ^{\!2} \cos (2 \varPhi ) \biggr] - \frac{\varPhi _{z} (\varPhi _{z} + 3 \varPhi _{1}) (\varPhi _{z} + 3 \varPhi _{2}) (\varPhi _{z} + 3 \varPhi _{3})}{12\, \varPhi ^{4}} \\ &{}\times \biggl[1 - \frac{\varPhi _{z}^{2}}{\varPhi ^{2}} + \frac{4}{3}\,\frac{\varPhi _{z}^{2}}{\varPhi ^{2}} \cos \varPhi - \biggl( 1 + \frac{1}{3}\, \frac{\varPhi _{z}^{2}}{\varPhi ^{2}} \biggr) \cos (2 \varPhi ) \biggr] \\ &{} + \frac{(\varPhi _{1} - \varPhi _{2}) (\varPhi _{2} - \varPhi _{3}) (\varPhi _{3} - \varPhi _{1})}{2 \sqrt{3}\, \varPhi ^{3}} \biggl[ \biggl( 1 - 3\, \frac{\varPhi _{z}^{2}}{\varPhi ^{2}} \biggr) \sin \varPhi \\ &{}+ \frac{1}{2} \biggl( 1 + 3\, \frac{ \varPhi _{z}^{2}}{\varPhi ^{2}} \biggr) \sin (2 \varPhi ) \biggr] \biggr\} \sin ^{2} \biggl( \frac{\varPhi }{2} \biggr) . \end{aligned}$$

(201)

Expanding the trigonometric functions in Eq. (201) into series near the point $\varPhi = 0$, after simple transformations we obtain the approximate expression

$$\begin{aligned} U(r, \boldsymbol{\varPhi }) \approx& - D \biggl[ \biggl( \frac{r_{0}}{r} \biggr) ^{12} -2 \biggl( \frac{r_{0}}{r} \biggr) ^{6} \biggr] - \frac{4 D a^{2}}{r^{2}} \biggl[ 11 \biggl( \frac{r_{0}}{r} \biggr) ^{12} -5 \biggl( \frac{r_{0}}{r} \biggr) ^{6} \biggr] \\ &{} - \frac{32 D a^{3}}{r^{3}} \biggl[ 14 \biggl( \frac{r_{0}}{r} \biggr) ^{12} -5 \biggl( \frac{r_{0}}{r} \biggr) ^{6} \biggr] \times \biggl\{ \bigl( \varPhi ^{2} - \varPhi _{z}^{2} \bigr) \biggl( \frac{2}{3} + \frac{15 \varPhi _{z}^{2} - 17 \varPhi ^{2}}{36} \biggr) \\ &{} - \frac{\varPhi _{z} (\varPhi _{z} + 3 \varPhi _{1}) (\varPhi _{z} + 3 \varPhi _{2}) (\varPhi _{z} + 3 \varPhi _{3})}{24} + \frac{(\varPhi _{1} - \varPhi _{2}) (\varPhi _{2} - \varPhi _{3}) (\varPhi _{3} - \varPhi _{1})}{4 \sqrt{3}} \biggr\} , \end{aligned}$$

(202)

from which it is seen that the point $\varPhi = 0$ is not a singular point of potential (201).

Now we extend the obtained results to the case of the overall motion (including both the spinorial motion and the translational one). In this case, the interaction potential $U(\boldsymbol{\varepsilon }_{1}, \boldsymbol{\varPhi }_{1})$ has the form

$$\begin{aligned} U(\boldsymbol{\varepsilon }_{1}, \boldsymbol{\varPhi }_{1}) =& - D \biggl[ \biggl( \frac{r_{0}}{\varepsilon _{1}} \biggr) ^{12} -2 \biggl( \frac{r _{0}}{\varepsilon _{1}} \biggr) ^{6} \biggr] - \frac{4 D a^{2}}{ \varepsilon _{1}^{2}} \biggl[ 11 \biggl( \frac{r_{0}}{\varepsilon _{1}} \biggr) ^{12} -5 \biggl( \frac{r_{0}}{ \varepsilon _{1}} \biggr) ^{6} \biggr] \\ &{} - \frac{8 D a^{3}}{\varepsilon _{1}^{6}} \biggl[ 14 \biggl( \frac{r _{0}}{\varepsilon _{1}} \biggr) ^{12} -5 \biggl( \frac{r_{0}}{ \varepsilon _{1}} \biggr) ^{6} \biggr] \Biggl\{ \sum_{k=1}^{4} \bigl( \mathbf{e}_{k} \cdot \tilde{\mathbf{P}}(\boldsymbol{\varPhi }_{1}) \cdot \boldsymbol{\varepsilon }_{1} \bigr) ^{3} - \sum_{k=1}^{4} ( \mathbf{e}_{k} \cdot \boldsymbol{\varepsilon }_{1} ) ^{3} \Biggr\} , \end{aligned}$$

(203)

where the unit vectors $\mathbf{e}_{k}$ are determined by Eq. (199), and quantities $\varepsilon _{1}$, $\tilde{\mathbf{P}}(\boldsymbol{\varPhi }_{1}) \cdot \boldsymbol{\varepsilon }_{1}$ are calculated by Eq. (197). Potential (203) is a generalization of potential (201). The interaction potential (203), as well as potential (201), is obtained with an asymptotic error of order $O(a^{4})$ compared to the exact interaction potential of the tetrahedrons. Taking into account Eq. (99), it is not difficult to show that

$$ \bigl( \mathbf{e}_{k} \cdot \tilde{\mathbf{P}}( \boldsymbol{\varPhi } _{1}) \cdot \boldsymbol{\varepsilon }_{1} \bigr) ^{3} = - ( \mathbf{e}_{k} \cdot \boldsymbol{\varepsilon }_{2} ) ^{3}, \qquad (\mathbf{e}_{k} \cdot \boldsymbol{\varepsilon }_{1} ) ^{3} = - \bigl( \mathbf{e}_{k} \cdot \tilde{\mathbf{P}}( \boldsymbol{\varPhi }_{2}) \cdot \boldsymbol{\varepsilon }_{2} \bigr) ^{3} , \quad \varepsilon _{1} = \varepsilon _{2}. $$

(204)

Transforming potential (203) in view of Eq. (204) yields the interaction potential $U( \boldsymbol{\varepsilon }_{2}, \boldsymbol{\varPhi }_{2})$, which has the same structure as potential (203) and can be obtained from it by replacing $\boldsymbol{\varepsilon }_{1}$ by $\boldsymbol{\varepsilon }_{2}$ and $\boldsymbol{\varPhi }_{1}$ by $\boldsymbol{\varPhi }_{2}$.

We do not give the coordinate representations of potentials (196) and (203) because of their extent and complexity and also due to the fact that the invariant representations of the interaction potentials are more convenient to calculate the interaction forces and the interaction moments.