3.7: Electromechanical Stress Tensors

The objectives in this section are to illustrate how the stress tensor associated with any one of the force densities in Secs. 3.7 and 3.8 is determined, and to summarize the stress tensors for future reference.

The ith component of the Korteweg-Helmholtz force density, Eq. 3.7.16, written using Gauss' law to eliminate \(\rho_f\), is

\[ F_i = E_i \frac{\partial{D_j}}{\partial{x_j}} + \sum_{k=1}^m \frac{\partial{W}}{\partial{\alpha_k}} \frac{\partial{\alpha_k}}{\partial{x_i}} - \frac{\partial{}}{\partial{x_i}} \left[ \sum_{k=1}^m \alpha_k \frac{\partial{W}}{\partial{\alpha_k}} \right] \label{1} \]

The goal in the following manipulations is to express this equation in the form of a tensor divergence (in the form of Eq. 3.9.1). The second term can be replaced by Eq. 3.7.26. Also, because E is irrotational, \(\partial{E_i}/\partial{x_j} = \partial{E_j}/ \partial{x_i} \) and hence Equation \ref{1} becomes

\[ F_i = E_i \frac{\partial{D_j}}{\partial{x_j}} + \frac{\partial{}}{\partial{x_i}} ( W – E_k D_k) + D_j \frac{\partial{E_i}}{\partial{x_j}} - \frac{\partial{}}{\partial{x_i}} \left[ \sum_{k=1}^m \alpha_k \frac{\partial{W}}{\partial{\alpha_k}} \right]\label{2} \]

With the first and third terms combined and the Kronecker delta function \(\delta_{ij}\) introduced (see Eq. 3.9.15),

\[ F_i = \frac{\partial{}}{\partial{x_j}} [ E_i D_j + \delta_{ij} (W – E_k D_k - \sum_{k=1}^m \alpha_k \frac{\partial{W}}{\partial{\alpha_k}} )] \label{3} \]

It follows from a comparison of Eqs. \ref{2} and 3.9.1 that the required stress tensor is

\[ T_{ij} = E_i D_j - \delta_{ij} (W^{‘} + \sum_{k=1}^m \alpha_k \frac{\partial{W}}{\partial{\alpha_k}}) \label{4} \]

Where the coenergy density, \(W^{‘}\), is defined by Eq. 2.13.11.

Table 3.10.1 gives a summary of this and other stress tensors together with the associated force densities. It is essential that a consistent pair be used.

Table 3.10.1: Summary of force densities and associated stress tensors

Equation	Force density	Stress tensors
3.7.16	\[ \overrightarrow{F} = \rho_f \overrightarrow{E} + \sum_{k=1}^{m} \frac{\partial{W}}{\partial{\alpha_k}} \nabla \alpha_k - \nabla [ \sum_{k=1}^m \alpha_k \frac{\partial{W}}{\partial{\alpha_k}}] \nonumber \]	\[ T_{ij} = E_i D_j - \delta_{ij} (W^{‘} + \sum_{k=1}^m \alpha_k \frac{\partial{W}}{\partial{\alpha_k}}) \nonumber \]
3.8.13	\[ \overrightarrow{F} = \overrightarrow{J}_f \times \overrightarrow{B} + \sum_{k=1}^{m} \frac{\partial{W}}{\partial{\alpha_k}} \nabla \alpha_k - \nabla [ \sum_{k=1}^m \alpha_k \frac{\partial{W}}{\partial{\alpha_k}}] \nonumber \]	\[ T_{ij} = H_i B_j - \delta_{ij} (W^{‘} + \sum_{k=1}^m \alpha_k \frac{\partial{W}}{\partial{\alpha_k}}) \nonumber \]
Incompressible media
3.7.19	\[ \overrightarrow{F} = \rho_f \overrightarrow{E} + \sum_{k=1}^{m} \frac{\partial{W}}{\partial{\alpha_k}} \nabla \alpha_k \nonumber \]	\[ T_{ij} = E_i D_j - \delta_{ij} W^{‘} \nonumber \]
3.8.14	\[ \overrightarrow{F} = \overrightarrow{J}_f \times \overrightarrow{B} + \sum_{k=1}^{m} \frac{\partial{W}}{\partial{\alpha_k}} \nabla \alpha_k \nonumber \]	\[ T_{ij} = H_i B_j - \delta_{ij} W^{‘} \nonumber \]
Incompressible and electrically linear: \( \overrightarrow{D} = \varepsilon \overrightarrow{E}, \quad \overrightarrow{B} = \mu \overrightarrow{H} \)
3.7.22	\[ \overrightarrow{F} = \rho_f \overrightarrow{E} - \frac{1}{2} E^2 \nabla \varepsilon \nonumber \]	\[ T_{ij} = \varepsilon E_i E_j - \frac{\varepsilon}{2} \delta_{ij} E_k E_k \nonumber \]
3.8.14	\[ \overrightarrow{F} = \overrightarrow{J}_f \times \overrightarrow{B} + \frac{1}{2} H^2 \nabla \mu \nonumber \]	\[ T_{ij} = \mu H_i H_j - \frac{\mu}{2} \delta_{ij} H_k H_k \nonumber \]
Electrically linear, \(\varepsilon\) and \(\mu\) dependent on mass density \(\rho\) only
3.7.24	\[ \overrightarrow{F} = \rho_f \overrightarrow{E} - \frac{1}{2} E^2 \nabla \varepsilon + \nabla (\frac{1}{2} \rho \frac{\partial{\varepsilon}}{\partial{\rho}} E^2) \nonumber \]	\[ T_{ij} = \varepsilon E_i E_j - \frac{\varepsilon}{2} \delta_{ij} E_k E_k (1 - \frac{\rho}{\varepsilon} \frac{\partial{\varepsilon}}{\partial{\rho}} ) \nonumber \]
3.8.17	\[ \overrightarrow{F} = \overrightarrow{J}_f \times \overrightarrow{B} + \frac{1}{2} H^2 \nabla \mu + \nabla (\frac{1}{2} \rho \frac{\partial{\mu}}{\partial{\rho}} H^2) \nonumber \]	\[ T_{ij} = \mu H_i H_j - \frac{\mu}{2} \delta_{ij} H_k H_k ( 1 - \frac{\rho}{\mu} \frac{\partial{\mu}}{\partial{\rho}} ) \nonumber \]
Kelvin force density and stress tensor
3.6.5	\[ \overrightarrow{F} = \rho_f \overrightarrow{E} + \overrightarrow{P} \cdot \nabla \overrightarrow{E} \nonumber \]	\[ T_{ij} = E_i D_j - \frac{1}{2} \delta_{ij} \varepsilon_o E_k E_k \nonumber \]
3.5.12	\[ \overrightarrow{F} = \overrightarrow{J}_f \times \mu_o \overrightarrow{H} + \mu_o \overrightarrow{M} \cdot \nabla \overrightarrow{H} \nonumber \]	\[ T_{ij} = H_i B_j - \frac{1}{2} \delta_{ij} \mu_o H_k H_k \nonumber \]

The stress tensor makes it possible to compute the total force on an object by integrating over an enclosing surface \(S\) in accordance with Eq. 3.9.6. For an isolated object in free space, this force is the same regardless of the particular force density used. If the force is considered as the integral of the force density over the volume of the object, this fact is by no means obvious. But, note that in free space the stress tensors of Table 3.10.1 all agree, Because the enclosing surface \(S\) is in this free space region, the same total force will result from integrating Eq. 3.9.6 regardless of the force density associated with the stress tensor.