A Helical Cauchy-Born Rule for Special Cosserat Rod Modeling of Nano and Continuum Rods

Abstract

We present a novel scheme to derive nonlinearly elastic constitutive laws for special Cosserat rod modeling of nano and continuum rods. We first construct a 6-parameter (corresponding to the six strains in the theory of special Cosserat rods) family of helical rod configurations subjected to uniform strain along their arc-length. The uniformity in strain then enables us to deduce the constitutive laws by just solving the warping of the helical rod’s cross-section (smallest repeating cell for nanorods) but under certain constraints. The constraints are shown to be critical in the absence of which, the 6-parameter family reduces to a well known 2-parameter family of uniform helical equilibria. An explicit formula for the 6-parameter helical map is derived which maps atoms in the repeating cell of a nanorod to their images for the purpose of repeating cell energy minimization. A scheme for the passage from nano to continuum scale is also presented to derive the constitutive laws of a continuum rod via atomistic calculations of nanorods. The bending, twisting, stretching and shearing stiffnesses of diamond nanorods and carbon nanotubes are computed to demonstrate our theory. We show that our scheme is more general and accurate than existing schemes allowing us to deduce shearing stiffness and several coupling stiffnesses of a nanorod for the first time.

Appendices

Appendix A: Computing Inter-Atomic Energy of the Fundamental Domain ($E_{FD}$) and Its Derivatives

The energy of the fundamental domain depends on positions of the atoms within the FD as well as their images which lie within the cut-off radius. The locations of the image atoms are calculated using formula (27). Once we know these atomic locations, we can use a standard procedure to compute the inter-atomic energy of the fundamental domain. In order to minimize $E_{FD}$ (see (30)) and also to deduce the constitutive laws (see (32) and (33)), we need to calculate the derivatives of $E_{FD}$ with respect to atomic positions ($\mathbf {x}_{j}$) as well as strains which essentially implies computing the derivatives of atomic positions and their images (both $\mathbf {x}_{j}$ and $\mathbf{x}_{i,j}$) with respect to $\mathbf{x}_{j}$ and strains. The map (27) does not turn out to be useful for this purpose. In fact, the derivative of the map (27) with respect to the strain “$\underline{\mathrm{k}}$” (evaluated at zero strain) possesses a non-removable singularity. We, therefore, use the following formula equivalent to (27) for this purpose:

$$\begin{aligned} \mathbf{x}_{i,j}= \biggl( \int_{0}^{iL_{0}} \mathbf{R}_{s\theta} ds \biggr) \underline{\mathrm{v}} + (\mathbf{R}_{iL_{0}\theta } )\mathbf{x}_{j}. \end{aligned}$$

(45)

From (45), we note that computing the derivatives of $\mathbf{x}_{i,j}$ with respect to $\mathbf{x}_{j}$ and $\underline{\mathrm{v}}$ are straightforward. However, computing the derivatives with respect to $\underline{\mathrm{k}}$ necessitates taking derivatives of the rotation matrix (as well as its integral) with respect to $\underline{\mathrm{k}}$. Analytical formulas for these derivatives are now presented for the readers’ convenience.

Appendix B: Derivative of Atomic Positions with Respect to Strains

From (45), the derivatives of $\mathbf {x}_{i,j}$ with respect to strains follow as:

$$\begin{aligned} \begin{aligned} \frac{\partial\mathbf{x}_{i,j}}{\partial{\underline{\mathrm{v}}}}&= \int_{0}^{iL_{0}}\mathbf{R}_{s\theta} ds, \\ \frac{\partial^{2}\mathbf{x}_{i,j}}{\partial{\underline{\mathrm{v}}}^{2}}&= \mathbf{0}, \\ \frac{\partial^{2}\mathbf{x}_{i,j}}{\partial{\underline{\mathrm{v}}}\partial{\underline{\mathrm{k}}}}&= \int_{0}^{iL_{0}}\frac{\partial\mathbf{R}_{s\theta}}{\partial{\underline{\mathrm{k}}}} ds, \\ \frac{\partial\mathbf{x}_{i,j}}{\partial{\underline{\mathrm{k}}}} &= \biggl( \int_{0}^{iL_{0}}\frac{\partial\mathbf{R}_{s\theta}}{\partial {\underline{\mathrm{k}}}} ds \biggr) \underline{\mathrm{v}} + \frac{\partial\mathbf{R}_{iL_{0}\theta }}{\partial{\underline{\mathrm{k}}}}\mathbf{x}_{j}, \\ \frac{\partial^{2}\mathbf{x}_{i,j}}{\partial{\underline{\mathrm{k}}}^{2}} &= \biggl( \int_{0}^{iL_{0}}\frac{\partial^{2}\mathbf{R}_{s\theta }}{\partial{\underline{\mathrm{k}}}^{2}} ds \biggr) \underline{\mathrm{v}} + \frac{\partial^{2}\mathbf{R}_{iL_{0}\theta }}{\partial{\underline{\mathrm{k}}}^{2}}\mathbf{x}_{j}. \end{aligned} \end{aligned}$$

(46)

Now, we show the derivatives of the rotation matrix as well as its integral with respect to $\underline{\mathrm{k}}$. Let $\psi= \vert\underline{\mathrm{k}}\vert$, $\underline{\mathrm{u}}=\frac{\underline{\mathrm{k}}}{\psi}$ denote the axis of rotation and $\theta= s\psi$ be the angle of rotation. Also, let $[\underline{\mathrm{u}} ] _{\times}$ and $[\mathbf {e}_{i} ]_{\times}$ denote the skew symmetric matrices whose axial vectors are $\underline{\mathrm{u}}$ and $\mathbf {e}_{i}$ respectively. Note that the definition of $\theta$ here is different from the earlier definition but simplify our notations below. Using Rodrigues’ formula for rotation, we get

$$ \mathbf{R}(s) =\exp(s\underline{\underline{\mathrm{K}}}) = \cos(\theta ) \mathbf{I}+ \sin(\theta) [\underline{\mathrm{u}} ] _{\times} +\bigl(1-\cos( \theta)\bigr)\underline{\mathrm{u}} \otimes\underline {\mathrm{u}}. $$

(47)

Now, $\underline{\mathrm{u}}$ being a constant, Rodrigues’ formula (47) can be integrated analytically which is shown below in (48). The formulas for the derivatives of (47) and (48) are also shown below. Since these formulas contain a removable singularity at $\theta= 0$, a Taylor expanded version (with singularity removed) is used in such situations.

$$\begin{aligned} \int\mathbf{R} =& s \biggl(\frac{\sin(\theta)}{\theta}\mathbf{I}+ \frac{1-\cos(\theta )}{\theta} [ \underline{\mathrm{u}} ]_{\times}+\frac{\theta-\sin(\theta)}{\theta} \underline {\mathrm{u}} \otimes\underline{\mathrm{u}} \biggr), \\ =&s \biggl(\biggl(1-\frac{\theta^{2}}{6}\biggr)\mathbf{I}+\frac{\theta }{2} \biggl(1-\frac{\theta^{2}}{12} \biggr) [ \underline{\mathrm{u}} ]_{\times} + \frac{\theta^{2}}{6}\biggl(1-\frac{\theta^{2}}{20}\biggr)\underline{\mathrm{u}}\otimes \underline{ \mathrm{u}} \biggr) \quad(\mbox{if } \theta\approx0), \\ =&s\mathbf{I}\quad(\mbox{if} \ \theta=0). \end{aligned}$$

(48)

$$\begin{aligned} \frac{\partial\mathbf{R}}{\partial k_{i}} =& s \biggl[-\sin(\theta )\mathrm{u}_{i}\mathbf{I}+ \frac{\cos(\theta)\theta- \sin(\theta)}{\theta} \mathrm{u}_{i} [\underline{\mathrm{u}} ]_{\times} + \frac{\sin(\theta)}{\theta} [\mathbf{e}_{i} ]_{\times } \\ &{}+ \frac{\sin(\theta)\theta-2(1-\cos(\theta))}{\theta} \mathrm{u}_{i} \underline{\mathrm{u}} \otimes \underline{\mathrm{u}}+ \frac{1-\cos(\theta)}{\theta} (\mathbf {e}_{i} \otimes \underline{\mathrm{u}} +\underline{\mathrm{u}}\otimes\mathbf{e}_{i} ) \biggr], \\ =& s \biggl[-\sin(\theta)\mathrm{u}_{i}\mathbf{I} -\frac{\theta^{2}}{3} \biggl(1-\frac{\theta^{2}}{10}\biggr) \mathrm{u}_{i} [\underline{\mathrm{u}} ]_{\times} + \biggl(1-\frac{\theta^{2}}{6}+\frac{\theta^{4}}{120}\biggr) [ \mathbf {e}_{i} ]_{\times}-\frac{ \theta^{3}}{12} \mathrm{u}_{i} \underline{\mathrm{u}} \otimes \underline{\mathrm{u}} \\ &{}+\frac{\theta}{2}\biggl(1-\frac{\theta^{2}}{12}\biggr) (\mathbf {e}_{i}\otimes \underline{\mathrm{u}} +\underline{\mathrm{u}}\otimes \mathbf{e}_{i} ) \biggr] \quad (\mbox{if } \theta\approx0), \\ =&s [\mathbf{e}_{i} ]_{\times} \quad(\mbox{if } \theta= 0). \end{aligned}$$

(49)

$$\begin{aligned} \int\frac{\partial\mathbf{R}}{\partial k_{i}} =&s^{2} \biggl(\frac{\cos(\theta)\theta-\sin(\theta)}{\theta ^{2}} \mathrm{u}_{i}\mathbf{I}+ \frac{\theta \sin(\theta)-2(1-\cos(\theta))}{\theta^{2}}\mathrm{u}_{i} [ \underline {\mathrm{u}} ]_{\times} + \frac{1-\cos(\theta)}{\theta^{2}} [ \mathbf{e}_{i} ]_{\times } \\ &{}+\frac{3\sin(\theta)-2\theta-\cos(\theta)\theta}{\theta^{2}}\mathrm{u}_{i}\underline{ \mathrm{u}} \otimes \underline{\mathrm{u}} + \frac{\theta-\sin(\theta)}{\theta^{2}}(\mathbf{e}_{i}\otimes \underline {\mathrm{u}}+\underline {\mathrm{u}} \otimes\mathbf{e}_{i}) \biggr), \\ =&s^{2} \biggl(-\frac{\theta}{3}\biggl(1-\frac{\theta^{2}}{10} \biggr)\mathrm{u}_{i}\mathbf{I}-\frac{ \theta^{2}}{12} \mathrm{u}_{i} [\underline{\mathrm{u}} ]_{\times}+\frac {1}{2} \biggl(1-\frac{ \theta^{2}}{12}\biggr) [ \mathbf{e}_{i} ]_{\times}- \frac{1}{60}\theta^{3}\mathrm{u}_{i}\underline{ \mathrm{u}} \otimes\underline{\mathrm{u}} \\ &{}+\frac{\theta}{6}\biggl(1-\frac{\theta^{2}}{20}\biggr) (\mathbf{e}_{i} \otimes \underline{\mathrm{u}} +\underline{\mathrm{u}} \otimes\mathbf{e}_{i}) \biggr) \quad(\mbox{if } \theta\approx 0), \\ =& \frac{s^{2}}{2} [\mathbf{e}_{i} ]_{\times} \quad(\mbox{if } \theta= 0). \end{aligned}$$

(50)

Similarly, the second derivative of the rotation matrix and its integral can be obtained.

Appendix C: Derivative of $[ \{\hat{\mathbf{x}}_{j} \},\hat{\boldsymbol{\lambda}}, \hat{\boldsymbol{\delta}}]$ with Respect to Strains

In this section, we first derive an expression for the derivative of $[ \{\hat{\mathbf{x}}_{j} \},\hat{ \boldsymbol{\lambda}}, \hat{\boldsymbol{\delta}}]$ with respect to strains and finally show that the second term in Eq. (32)(a) vanishes, i.e.,

$$\begin{aligned} \biggl\{ \frac{\partial E_{FD}}{\partial\mathbf{x}_{j}} \biggr\} \bigg\vert _{(\underline{\mathrm{v}},\underline{\mathrm{k}})}\cdot \biggl\{ \frac{\partial \hat{\mathbf{x}}_{j}}{\partial[\underline{\mathrm{v}},\underline {\mathrm{k}}]} \biggr\} =\mathbf{0}. \end{aligned}$$

(51)

We first observe that (30) holds at all strains. Differentiating the three equations in (30) with respect to strains, we then obtain

$$ \frac{\partial^{2}E_{\mathit{cons}}}{\partial[ \{\mathbf{x}_{j} \} ,\boldsymbol{\lambda}, \boldsymbol{\delta}]\partial[\underline{\mathrm{v}},\underline{\mathrm{k}}]} + \frac{\partial^{2} E_{\mathit{cons}}}{\partial[ \{\mathbf{x}_{j} \},\boldsymbol{\lambda}, \boldsymbol{\delta}]^{2}}\frac{\partial[ \{\hat{\mathbf {x}}_{j} \},\hat{ \boldsymbol{\lambda}}, \hat{\boldsymbol{\delta}}]}{\partial[\underline{\mathrm{v}},\underline{\mathrm{k}}]}= \mathbf{0}. $$

(52)

Let us define a stiffness matrix $\mathbf{K}^{3m\times3m}$ and the linearized constraint matrix $\mathbf{C}^{3m\times6}$ such that its block-components are as shown below:

$$\begin{aligned} \mathbf{K}_{ij}^{3\times3} = \frac{\partial^{2} E_{FD}}{\partial \mathbf{x}_{i} \partial\mathbf{x}_{j}}, \quad \mathbf{C}_{i}^{3\times6} = \biggl[\frac{\partial^{2} E_{\mathit{cons}}}{\partial\mathbf{x}_{i}\partial\boldsymbol{\lambda}} \frac{\partial^{2} E_{\mathit{cons}}}{\partial\mathbf{x}_{i}\partial \boldsymbol{\delta}} \biggr]= \bigl[ \mathbf{I}^{3\times3} \quad \underline{\underline{x_{i}}}^{3\times3} \bigr] . \end{aligned}$$

(53)

Here $\mathbf{I}$ is an identity matrix whereas

$$\underline {\underline{x_{i}}}= \left[\textstyle\begin{array}{c@{\quad}c@{\quad}c} 0 & x_{i,3} & x_{i,2}\\ x_{i,3} & 0 & x_{i,1}\\ x_{i,2} & x_{i,1} & 0 \end{array}\displaystyle \right] $$

is a matrix formed by three components of the position vector of $i$th atom. We can then write (52) in the following matrix form:

$$ \begin{bmatrix} \{\mathbf{K}_{ij}+\delta_{ij}\underline{\underline{\delta }} \}^{3m\times3m} & \mathbf{C}^{3m\times6}\\ \mathbf{C}^{T} & \mathbf{0}^{6\times6} \end{bmatrix} \begin{bmatrix} \bigl\{ \frac{\partial\hat{\mathbf{x}}_{i}}{\partial [\underline{\mathrm{v}},\underline{\mathrm{k}}]} \bigr\} ^{3m\times6}\\ \frac{\partial[\hat{\boldsymbol{\lambda}}, \hat{\boldsymbol {\delta}}]}{\partial [\underline{\mathrm{v}},\underline{\mathrm{k}}]}^{6\times6} \end{bmatrix} = - \begin{bmatrix} \bigl\{ \frac{\partial^{2} E_{FD}}{\partial \mathbf{x}_{i}\partial[\underline{\mathrm{v}},\underline{\mathrm{k}}]} \bigr\} ^{3m\times6} \\ \mathbf{0}^{6\times6} \end{bmatrix} . $$

(54)

Equation (54) can be inverted to obtain the derivatives of $[ \{\hat{\mathbf{x}}_{j} \},\hat{ \boldsymbol{\lambda}},\hat{\boldsymbol{\delta}}]$ with respect to strains. Now, upon block multiplication in (54), we see that $\mathbf{C}^{T} \bigl\{\frac{\partial\hat{\mathbf {x}}_{i}}{\partial [\underline{\mathrm{v}},\underline{\mathrm{k}}]} \bigr\} = \mathbf{0}$. This further implies that $\bigl \{\frac{\partial\hat{\mathbf {x}}_{i}}{\partial [\underline{\mathrm{v}},\underline{\mathrm{k}}]} \bigr\}$ is perpendicular to columns of $\mathbf{C}$. Using Q-R factorization of $\mathbf{C}$, we then obtain

$$\begin{aligned} \mathbf{C}= \bigl[ \mathbf{Q}1^{3m\times6} \quad \mathbf{Q}2^{3m\times(3m-6)} \bigr] \bigl[\mathbf{R}1^{6\times6} \quad \mathbf{0}^{6\times(3m-6)} \bigr] ^{T} =\mathbf{Q}1\ \mathbf{R}1\Rightarrow \biggl\{ \frac{\partial\hat {\mathbf{x}}_{i}}{\partial [\underline{\mathrm{v}},\underline{\mathrm{k}}]} \biggr\} = \mathbf{Q2} \ \mathbf{z}^{3m-6}. \end{aligned}$$

(55)

Upon pre-multiplying the first set of block equations in (54) by $\mathbf{Q2}^{T}$ and substituting (55), we then obtain

$$\begin{aligned} \bigl(\mathbf{Q2}^{T} \{\mathbf{K}_{ij}+ \delta_{ij}\underline {\underline{\delta}} \}\mathbf{Q2} \bigr) \mathbf{z} =& -\mathbf{Q2}^{T} \biggl\{ \frac{\partial^{2} E_{FD}}{\partial \mathbf{x}_{i}\partial[\underline{\mathrm{v}},\underline{\mathrm{k}}]} \biggr\} \\ \Rightarrow \biggl\{ \frac{\partial\hat{\mathbf{x}}_{i}}{\partial [\underline{\mathrm{v}},\underline{\mathrm{k}}]} \biggr\} =& -\mathbf{Q2} \bigl( \mathbf{Q2}^{T} \{\mathbf{K}_{ij}+\delta _{ij} \underline{\underline{\delta}} \}\mathbf{Q2} \bigr)^{-1} \mathbf{Q2}^{T} \biggl\{ \frac{\partial^{2} E_{FD}}{\partial \mathbf{x}_{i}\partial[\underline{\mathrm{v}},\underline{\mathrm{k}}]} \biggr\} . \end{aligned}$$

(56)

Now using (30) and (53), we get:

$$\begin{aligned} \biggl\{ \frac{\partial E_{\mathit{cons}}}{\partial{\mathbf{x}_{i}}} \biggr\} = \biggl\{ \frac{\partial E_{FD}}{\partial{\mathbf{x}_{i}}} \biggr\} + \mathbf{C} \begin{bmatrix} \boldsymbol{\lambda}\\ \boldsymbol{\delta} \end{bmatrix} \Rightarrow \mathbf{Q2}^{T} \biggl\{ \frac{\partial E_{\mathit{cons}}}{\partial {\mathbf{x}_{i}}} \biggr\} = \mathbf{Q2}^{T} \biggl\{ \frac{\partial E_{FD}}{\partial{\mathbf {x}_{i}}} \biggr\} . \end{aligned}$$

(57)

Finally,

$$\begin{aligned} \biggl\{ \frac{\partial E_{FD}}{\partial\mathbf{x}_{i}} \biggr\} \cdot \biggl\{ \frac{\partial\hat{\mathbf{x}}_{i}}{\partial [\underline{\mathrm{v}},\underline{\mathrm{k}}]} \biggr\} =& \biggl\{ \frac{\partial E_{FD}}{\partial{\mathbf{x}_{i}}} \biggr\} \cdot \mathbf{Q2} \bigl(\mathbf{Q2}^{T} \{\mathbf{K}_{ij}+ \delta_{ij}\underline {\underline{\delta}} \}\mathbf{Q2} \bigr)^{-1}\mathbf {Q2}^{T} \biggl\{ \frac{\partial^{2} E_{FD}}{\partial \mathbf{x}_{i}\partial[\underline{\mathrm{v}},\underline{\mathrm{k}}]} \biggr\} \\ =&\mathbf{Q2}^{T} \biggl\{ \frac{\partial E_{\mathit{cons}}}{\partial{\mathbf {x}_{i}}} \biggr\} \cdot \bigl( \mathbf{Q2}^{T} \{\mathbf{K}_{ij}+\delta_{ij} \underline {\underline{\delta}} \}\mathbf{Q2} \bigr)^{-1}\mathbf {Q2}^{T} \biggl\{ \frac{\partial^{2} E_{FD}}{\partial \mathbf{x}_{i}\partial[\underline{\mathrm{v}},\underline{\mathrm{k}}]} \biggr\} \\ =&\mathbf{0}\quad(\mbox{using (57) and (30)(a)}). \end{aligned}$$

(58)