Kinematics

The micromorphic continuum theory developed by Eringen [3, 13] accounts for behavior of a micro-structure coupled with a macro-scale. The approach of classical nonlinear continuum mechanics is enriched by embedding micro-position vectors corresponding to micro-scale differential elements.

Figure 42 by a body undergoing deformation from the reference configuration (\(B_0\), \(\partial B_0\)) to the current configuration (\(B\), \(\partial B\)), with differential elements and their encompassing areas (\(dV\), \(dA\)) and (\(dv\), \(da\)), respectively. Micro-scale quantities are identified with a superscript alpha \((\bullet)^{\left(\alpha\right)}\), and the same capitalized notation distinguishes between the reference (uppercase) and current (lowercase) configurations.

Fig. 42 The reference and current configurations with detail of differential elements

The micro-position vectors in the reference Eq. (\(\mathbf{X}^{\left(\alpha\right)}\)) and current configurations (\(\mathbf{x}^{\left(\alpha\right)}\)), respectively, are shown in (15). Vectors \(\mathbf{X}\) and \(\mathbf{x}\) point to the centers of mass of the differential macro-elements and \(\mathbf{\Xi}\) and \(\bf{\xi}\) point to the centers of mass of the micro-elements relative to the centers of mass of the differential elements. It is assume that the magnitudes of micro-position vectors are significantly small, e.g., \(||\mathbf{\Xi}|| << 1\) and \(||\mathbf{\xi}|| << 1\).

(15)\[ \begin{align}\begin{aligned}X_i^{\left(\alpha\right)} &= X_i + \Xi_i\\x_i^{\left(\alpha\right)} &= x_i + \xi_i\end{aligned}\end{align} \]

Deformation Gradient

Equation (17) shows the smooth map between differential micro-elements in the reference and current configurations through the the micro-deformation tensor \(\mathbf{\chi}\). While the macro-position vectors (\(\mathbf{X}\) and \(\mathbf{x}\)) are mapped to their respective differential elements (\(d\mathbf{X}\) and \(d\mathbf{x}\)) through the deformation gradient (\(\mathbf{F}=\mathbf{I}+\partial \mathbf{u} / \partial \mathbf{X}\)), the micro-scale relative position vectors (\(\mathbf{\Xi}\) and \(\mathbf{\xi}\)) are assumed to map in their entirety through the linear transformation \(\mathbf{\chi}\).

(16)\[dx_i = F_{iI} dX_I\]

(17)\[\xi_i = \chi_{iI} \Xi_I\]

The micro-deformation tensor \(\mathbf{\chi}\) is related to the micro-displacement tensor \(\mathbf{\Phi}\) as \(\mathbf{\chi}= \mathbf{I} + \mathbf{\Phi}\).

The micro-position deformation gradient \(\mathbf{F}^{\left(\alpha\right)}\) must now be computed to relate the differential micro-elements positions \(d\mathbf{X}^{\left(\alpha\right)}\) and \(d\mathbf{x}^{\left(\alpha\right)}\).

(18)\[ \begin{align}\begin{aligned}F_{iI}^{\left(\alpha\right)} &= \frac{\partial}{\partial X_I^{\left(\alpha\right)}} (x_i + \xi_i)\\&= \frac{\partial x_i}{\partial X_L} \frac{\partial X_L}{\partial X_{I}^{\left(\alpha\right)}} + \frac{\partial}{\partial X_{I}^{\left(\alpha\right)}} (\chi_{iK} \Xi_{K})\\&= F_{iL} \frac{\partial X_L}{\partial X_I^{\left(\alpha\right)}} + \frac{\partial \chi_{iK}}{\partial X_L} \frac{\partial X_L}{\partial X_I^{\left(\alpha\right)}}\Xi_{K} + \chi_{iK}\frac{\partial \Xi_K}{\partial X_L}\frac{\partial X_L}{\partial X_I^{\left(\alpha\right)}}\\&= \left( F_{iL} + \frac{\partial \chi_{iK}}{\partial X_L}\Xi_K + \chi_{iK}\frac{\partial \Xi_K}{\partial X_L} \right) \frac{\partial X_L}{\partial X_I^{\left(\alpha\right)}}\end{aligned}\end{align} \]

Since \(X_L^{\left(\alpha\right)} = X_L + \Xi_L\), then \(X_L = X_L^{\left(\alpha\right)} - \Xi_L\), so

(19)\[ \frac{\partial X_L}{\partial X_I^{\left(\alpha\right)}} = \frac{\partial X_L^{\left(\alpha\right)}}{\partial X_I^{\left(\alpha\right)}} - \frac{\partial \Xi_L}{\partial X_I^{\left(\alpha\right)}} = \delta_{LI} - \frac{\partial \Xi_L}{\partial X_I^{\left(\alpha\right)}}\]

Similarly,

(20)\[\frac{\partial X_I^{\left(\alpha\right)}}{\partial X_L} = \frac{\partial X_I}{\partial X_L} + \frac{\partial \Xi_I}{\partial X_L} = \delta_{IL} + \frac{\partial \Xi_I}{\partial X_L}\]

Substitute Eq. (19) back into \(\mathbf{F}^{\left(\alpha\right)}\) in Eq. (18) and manipulate.

(21)\[ \begin{align}\begin{aligned}F_{iI}^{\left(\alpha\right)} &= \left( F_{iL} + \frac{\partial \chi_{iK}}{\partial X_L}\Xi_K + \chi_{iK}\frac{\partial \Xi_K}{\partial X_L} \right) \left( \delta_{LI} - \frac{\partial \Xi_L}{\partial X_I^{\left(\alpha\right)}} \right)\\&= F_{iL}\delta_{LI} + \frac{\partial \chi_{iK}}{\partial X_L}\Xi_K \delta_{LI} + \chi_{iK}\frac{\partial \Xi_K}{\partial X_L}\delta_{LI} - \left( F_{iL} + \frac{\partial \chi_{iK}}{\partial X_L}\Xi_K + \chi_{iK}\frac{\partial \Xi_K}{\partial X_L} \right) \frac{\partial \Xi_L}{\partial X_I^{\left(\alpha\right)}}\\&= F_{iI} + \frac{\partial \chi_{iK}}{\partial X_I}\Xi_K + \chi_{iK}\frac{\partial \Xi_K}{\partial X_I} - \left( F_{iL} + \frac{\partial \chi_{iK}}{\partial X_L}\Xi_K + \chi_{iK}\frac{\partial \Xi_K}{\partial X_L} \right) \frac{\partial \Xi_L}{\partial X_I^{\left(\alpha\right)}}\end{aligned}\end{align} \]

We may inspect the \(\frac{\partial \mathbf{\Xi}}{\partial \mathbf{X}^{\left(\alpha\right)}}\) term and substitute the expression from Eq. (20).

(22)\[ \begin{align}\begin{aligned}\frac{\partial \Xi_{L}}{\partial X_{I}^{\left(\alpha\right)}} &= \frac{\partial \Xi_{L}}{\partial X_{M}} \frac{\partial X_{M}}{\partial X_{I}^{\left(\alpha\right)}} = \frac{\partial \Xi_{L}}{\partial X_{M}} \left( \frac{\partial X_{I}^{\left(\alpha\right)}}{\partial X_{M}} \right)^{-1}\\&= \frac{\partial \Xi_{L}}{\partial X_{M}} \left( \delta_{IM} + \frac{\partial \Xi_I}{\partial X_M}\right)^{-1}\end{aligned}\end{align} \]

Substitute back into \(\mathbf{F}^{\left(\alpha\right)}\). Eq. (23) shows the textit{general} form of deformation gradient!

(23)\[ \boxed{F_{iI}^{\left(\alpha\right)} = F_{iI} + \frac{\partial \chi_{iK}}{\partial X_I}\Xi_K + \chi_{iK}\frac{\partial \Xi_K}{\partial X_I} - \left( F_{iL} + \frac{\partial \chi_{iK}}{\partial X_L}\Xi_K + \chi_{iK}\frac{\partial \Xi_K}{\partial X_L} \right) \frac{\partial \Xi_{L}}{\partial X_{M}} \left( \delta_{IM} + \frac{\partial \Xi_I}{\partial X_M}\right)^{-1}}\]

Simplification for small variation in density

The micro-position deformation gradient may be further simplified for different cases. One such case is when \(\frac{\partial \mathbf{\Xi}}{\partial \mathbf{X}} << 1\) which indicates that the distribution of \(\mathbf{\Xi}\) is nearly the same at every location within the body. Thus, the variation in the mass distribution is small between different \(dV\), however, the variation in mass itself is not necessarily small.

If \(\frac{\partial \mathbf{\Xi}}{\partial \mathbf{X}} << 1\), then

\[\left( \delta_{IM} + \frac{\partial \Xi_I}{\partial X_M}\right)^{-1} \approx \left( \delta_{IM} - \frac{\partial \Xi_I}{\partial X_M}\right)\]

which we may substitute into Eq. (22)

(24)\[ \begin{align}\begin{aligned}\frac{\partial \Xi_{L}}{\partial X_{I}^{\left(\alpha\right)}} &= \frac{\partial \Xi_{L}}{\partial X_{M}} \left( \delta_{IM} + \frac{\partial \Xi_I}{\partial X_M}\right)^{-1} \approx \frac{\partial \Xi_{L}}{\partial X_{M}} \left( \delta_{IM} - \frac{\partial \Xi_I}{\partial X_M}\right)\\&\approx \frac{\partial \Xi_{L}}{\partial X_{M}} \delta_{IM} + \cancelto{0}{\frac{\partial \Xi_{L}}{\partial X_{M}}\frac{\partial \Xi_{I}}{\partial X_{M}}}\\&\approx \frac{\partial \Xi_{L}}{\partial X_{I}}\end{aligned}\end{align} \]

One may insert the results of Eq. (24) into Eq. (21) to provide

(25)\[F_{iI}^{\left(\alpha\right)} = F_{iI} + \frac{\partial \chi_{iK}}{\partial X_I}\Xi_K + \chi_{iK}\frac{\partial \Xi_K}{\partial X_I} - \left( F_{iL} + \frac{\partial \chi_{iK}}{\partial X_L}\Xi_K + \chi_{iK}\frac{\partial \Xi_K}{\partial X_L} \right) \frac{\partial \Xi_L}{\partial X_I}\]

This expression may be expanded and a single term may be canceled.

\[ F_{iI}^{\left(\alpha\right)} = F_{iI} + \frac{\partial \chi_{iK}}{\partial X_I}\Xi_K + \chi_{iK}\frac{\partial \Xi_K}{\partial X_I} - F_{iL}\frac{\partial \Xi_L}{\partial X_I} - \frac{\partial \chi_{iK}}{\partial X_L}\Xi_K\frac{\partial \Xi_L}{\partial X_I} - \cancelto{0}{\chi_{iK}\frac{\partial \Xi_K}{\partial X_L}\frac{\partial \Xi_L}{\partial X_I}}\]

The dummy indices on the third term may be switched from \(K\) to \(L\).

\[ F_{iI}^{\left(\alpha\right)} = F_{iI} + \frac{\partial \chi_{iK}}{\partial X_I}\Xi_K + \chi_{iL}\frac{\partial \Xi_L}{\partial X_I} - F_{iL}\frac{\partial \Xi_L}{\partial X_I} - \frac{\partial \chi_{iK}}{\partial X_L}\Xi_K\frac{\partial \Xi_L}{\partial X_I}\]

Finally, recollect terms to give:

(26)\[ F_{iI}^{\left(\alpha\right)} = F_{iI} + \frac{\partial \chi_{iK}}{\partial X_I}\Xi_K - \left( F_{iL} + \frac{\partial \chi_{iK}}{\partial X_L}\Xi_K - \chi_{iL} \right) \frac{\partial \Xi_L}{\partial X_I}\]

Materials with a large variation in density can include functionally graded materials, additively manufactured lattice structures, granular materials with particles spreading out, and some foams (with spatially varying density). Micromorphic descriptions of these materials need to use the micro-position deformation gradient from Eq. (23).

Deformation Measures

We now investigate several deformation measures. For now, we will assume only elastic deformations and will introduce more specific kinematics for an elastoplastic model using a multiplicative decomposition of the deformation gradient and micro deformation tensor. A full treatment of deformation measures will be added in the future, but for now we may start with the following definitions.

(27)\[ \begin{align}\begin{aligned}\mathcal{C}_{IJ} &= F_{iI} F_{iJ}\\\Psi_{IJ} &= F_{iI} \xi_{iJ}\\\Gamma_{IJK} &= F_{iI} \xi_{iJ,K}\end{aligned}\end{align} \]

These measures may be used to define the Green-Lagrange strain (Eq. (28)) and the Micro strain (Eq. (29)) which will be used for an elastic constitutive model along with the Micro-deformation gradient \(\Gamma_{IJK}\).

(28)\[E_{IJ} = \frac{1}{2} \left( \mathcal{C}_{IJ} - \delta_{IJ} \right)\]

(29)\[\mathcal{E}_{IJ} = \Psi_{IJ} - \delta_{IJ}\]