@f[
-
\text{div}\,
- (C \nabla \mathbf{u})
+ ({\mathbf C} \nabla \mathbf{u})
=
\mathbf f,
@f]
where $\mathbf u$ is the vector-valued displacement at each point,
-$\mathbf f$ the force, and $C$ is a rank-4 tensor (i.e., it has four
+$\mathbf f$ the force, and ${\mathbf C}$ is a rank-4 tensor (i.e., it has four
indices) that encodes the stress-strain relationship -- in essence,
it represents the
<a href="https://en.wikipedia.org/wiki/Hooke%27s_law">"spring constant"</a> in
-Hookes law that relates the displacement to the forces. $C$ will, in many
+Hookes law that relates the displacement to the forces. ${\mathbf C}$ will, in many
cases, depend on $\mathbf x$ if the body whose deformation we want to
simulate is composed of different materials.
@f[
-
\text{div}\,
- (C \varepsilon(\mathbf u))
+ ({\mathbf C} \varepsilon(\mathbf u))
=
\mathbf f,
@f]
which you can think of as the more natural generalization of the Laplace
equation to vector-valued problems. (The form shown first is equivalent to
-this form because the tensor $C$ has certain symmetries, namely that
-$C_{ijkl}=C_{ijlk}$, and consequently $C \varepsilon(\mathbf u)_{kl}
-= C \nabla\mathbf u$.)
+this form because the tensor ${\mathbf C}$ has certain symmetries, namely that
+$C_{ijkl}=C_{ijlk}$, and consequently ${\mathbf C} \varepsilon(\mathbf u)_{kl}
+= {\mathbf C} \nabla\mathbf u$.)
One can of course alternatively write these equations in component form:
@f[