Divergence of Rank-2 Tensor: Motivation/Intuition

Question

I'm reading through a text on continuum mechanics (Intro to CM by Lai, Rubin, Krempl), and it defines the divergence of a rank-2 tensor $\mathbf{T}$ as follows:

The divergence of $\mathbf{T}$ is defined to be a vector field, denoted by div $\mathbf{T}$, such that for any vector $\mathbf{a}$

$(\textrm{div} \mathbf{T} )\cdot\mathbf{a}=\textrm{div}(\mathbf{T}^T\mathbf{a})-\textrm{tr}(\mathbf{T}^T(\nabla\mathbf{a}))$

where superscript $T$ denotes the transpose and tr() is trace. The book is generally very good with providing intuition and rational, but I'm having difficulty understanding (i) the motivation for the coordinate-free definition looking like this and (ii) physical intuition behind the concept of the divergence of a rank-2 tensor. I'm working with the Cauchy stress tensor, so an answer in that context would be most helpful :)

Answer 1

This looks like a very non-intuitive and quite a messy definition of divergence. Divergence is generally defined as the contraction of the gradient, and presented as a "flux density".

Some references

Try to have a look at these hand-written references. These notes introduces the description of space using coordinates, the concept of natural base and its dual, and then use them to write the expression of differential operators acting on tensor fields of general rank, both using general curvilinear coordinates and giving its abstract definition, related to some intuitive concepts (directional derivative for gradient, flux density for divergence).

• differential geometry, https://basics.altervista.org/test/Math/differential_geometry/main.html

  • coordinates for describing 3D space and definition of natural basis: https://basics.altervista.org/test/Math/differential_geometry/coordinates.html

• differential operators in tensor calculus (for tensors of any rank, not only rank-2): https://basics.altervista.org/test/Math/tensor_calculus/main.html

  • gradient, and the relation with the directional derivative, https://basics.altervista.org/test/Math/tensor_calculus/gradient.html
  • divergence, as the contraction of the gradient, and its relation with the flux across the boundaries of an elementary volume, https://basics.altervista.org/test/Math/tensor_calculus/divergence.html

If you're looking to more complete references (quite a hard way, I swear, and you need to have quite a "mathematical" mindset), take a look at the webpage of prof. Ray M. Bowen at the Texas A&M University, https://oaktrust.library.tamu.edu/handle/1969.1/2500, especially:

• Introduction to vectors and tensors, Vol 1: linear and multilinear algebra
• Introduction to vectors and tensors, Vol 2: vector and tensor analysis
• Introduction to continuum mechanics for engineers . These are among the best references I've ever used for Tensor Algebra and Calculus, and moreover they are free. I think I should say a huge thank you to prof. Bowen.

Answer 2

In Einstein notation $(\nabla T)_ia_i=\partial_j(T_{ij}a_i)-T_{ij}\partial_ja_i$, i.e. $(\nabla T)_i=\partial_jT_{ij}$. This is, of course, the only way the product rule allows us to define a useful vector-valued derivative of a rank-$2$ tensor. With $(\nabla a)_{ij}:=\partial_ja_i$, this reduces to Lai et al's equation.

So for (i), the motivation for the coordinate-free definition is that we characterize said vector with a scalar-valued and hence invariant identity for arbitrary vectors $\vec{a}$.

The physical intuition for (ii) is that $\nabla T$'s projection along a vector field $\vec{a}$ is its obvious value from the uniform-$\vec{a}$ case, minus a correction due to $\vec{a}$ not being in general uniform. In other words, parallel transport, which has a very instructive diagram. As the answer by @basics notes, this is an insight from differential geometry.

Of course, if you rearrange it as $\nabla(T^T\vec{a})=(\nabla T)\cdot\vec{a}+\operatorname{tr}(T^T\nabla\vec{a})$, so you get a $+$ instead of a $-$, you can understand parallel transport in terms of a product rule as usual. But the $-$ version has the advantage that $\nabla T$, which as per (i) rotates like any true vector does, is the focus of the definition.