Saturday, June 20, 2020

The stress tensor

It's a tricky bit of business--something I (shamefully) never fully understood in graduate school. That always bothered me.

Continuum mechanics is wonderful. There, I said it, and now we can move on.

Ok, so the stress tensor.

It is like a vector, but not really. It's bigger than that. In reality, a scalar (such as temperature) is a zeroth order tensor. It only has a value. You can still have scalar fields (temperatures at given points), but they don't have direction. They just are, in the place where they are.

A vector on the other hand, like wind velocity, has direction and a magnitude (size or length) at some location. Therefore wind velocity can be described as a vector field. At each point you have a direction and magnitude (length) of velocity--speed is the magnitude of the velocity vector (how long it is, regardless of direction). A vector is a first order tensor.

Tensors (they can be anything second order and above), are a bit different. Some say they have two directions, and lack an intuitive sense. I prefer to think of one of the directions as the plane of a cube. As far as physical intuitive relationships go this works, because you can think of each dimension in space (x, y, and z or i, j, and k) as having a vector associated with it. Consider the cube. You can push on the cube on three different sets of opposite faces. Each face has one resultant vector applied (perhaps a force vector), and if the forces are equal and opposite, the resultant force vector on that face is zero.

This idea of tensors is commonly applied by chemical engineers with respect to the stress tensor. It describes an infinitely small fluid element in equilibrium. Think a tiny smidge (cube) of water in a pipe flow--it can be any bit of water (fluid) anywhere.
Components stress tensor cartesian

The little cube will have forces acting:
normal (perpendicular, like a bottle standing on a table, putting a bit of weight on the table) to its faces (hydrostatic stress terms),
and it may have some forces that act on the sides in parallel directions (picture placing your hand flat on top of a slinky and moving it in a parallel direction to the ground as the bottom remains in place) to the sides / faces (shear stress terms) of the cube.
Uni-axial terms (contribute to Hydrostatic stress): σ11, σ22, and σ33
Hydrostatic stress = (σ11 + σ22 + σ33) / 3 = σh
 {\displaystyle \sigma _{h}=\displaystyle \sum _{i=1}^{n}\rho _{i}gh_{i}}
Shear stress terms: σ12, σ13, σ21, σ23, σ31, and σ32

Stress terms in the full tensor, using matrix notation:
{\boldsymbol {\sigma }}=\left[{{\begin{matrix}\sigma _{{11}}&\sigma _{{12}}&\sigma _{{13}}\\\sigma _{{21}}&\sigma _{{22}}&\sigma _{{23}}\\\sigma _{{31}}&\sigma _{{32}}&\sigma _{{33}}\\\end{matrix}}}\right]\equiv \left[{{\begin{matrix}\sigma _{{xx}}&\sigma _{{xy}}&\sigma _{{xz}}\\\sigma _{{yx}}&\sigma _{{yy}}&\sigma _{{yz}}\\\sigma _{{zx}}&\sigma _{{zy}}&\sigma _{{zz}}\\\end{matrix}}}\right]\equiv \left[{{\begin{matrix}\sigma _{x}&\tau _{{xy}}&\tau _{{xz}}\\\tau _{{yx}}&\sigma _{y}&\tau _{{yz}}\\\tau _{{zx}}&\tau _{{zy}}&\sigma _{z}\\\end{matrix}}}\right]

Hydrostatic stress is isotropic, meaning that it acts the same way in all directions.
{\displaystyle \sigma _{h}\cdot I_{3}=\sigma _{h}\left[{\begin{array}{ccc}1&0&0\\0&1&0\\0&0&1\end{array}}\right]=\left[{\begin{array}{ccc}\sigma _{h}&0&0\\0&\sigma _{h}&0\\0&0&\sigma _{h}\end{array}}\right]}

Under the conservation of angular momentum the moments (with respect to an arbitrary point) from the stresses on the faces must balance, and it becomes clear that the tensor must be symmetric:
{\displaystyle {\begin{aligned}M_{O}&=\int _{S}(\mathbf {r} \times \mathbf {T} )dS+\int _{V}(\mathbf {r} \times \mathbf {F} )dV=0\\0&=\int _{S}\varepsilon _{ijk}x_{j}T_{k}^{(n)}dS+\int _{V}\varepsilon _{ijk}x_{j}F_{k}dV\\\end{aligned}}}

or in simpler terms:
{\displaystyle \sigma _{12}=\sigma _{21}} AND {\displaystyle \sigma _{23}=\sigma _{32}} AND {\displaystyle \sigma _{13}=\sigma _{31}} OR:
{\displaystyle \sigma _{ij}=\sigma _{ji}}

The stress tensor is a fundamental piece of the Cauchy momentum equation, which relates:
the material derivative, {\displaystyle {\frac {D\mathbf {u} }{Dt}}\ [\mathrm {m/s^{2}} ]}
the divergence of the stress tensor, {\displaystyle \nabla \cdot {\boldsymbol {\sigma }}={\begin{bmatrix}{\dfrac {\partial \sigma _{xx}}{\partial x}}+{\dfrac {\partial \sigma _{yx}}{\partial y}}+{\dfrac {\partial \sigma _{zx}}{\partial z}}\\{\dfrac {\partial \sigma _{xy}}{\partial x}}+{\dfrac {\partial \sigma _{yy}}{\partial y}}+{\dfrac {\partial \sigma _{zy}}{\partial z}}\\{\dfrac {\partial \sigma _{xz}}{\partial x}}+{\dfrac {\partial \sigma _{yz}}{\partial y}}+{\dfrac {\partial \sigma _{zz}}{\partial z}}\\\end{bmatrix}}[\mathrm {Pa/m=kg/m^{2}\cdot s^{2}} ]}
density at a point, {\displaystyle \rho \ [\mathrm {kg/m^{3}} ]}
and the accelerations caused by body forces (gravitational acceleration mostly), {\displaystyle \mathbf {f} ={\begin{bmatrix}f_{x}\\f_{y}\\f_{z}\end{bmatrix}}\ [\mathrm {m/s^{2}} ]}

So all put together, a form of the Cauchy Momentum equation:
{\displaystyle {\frac {D\mathbf {u} }{Dt}}={\frac {1}{\rho }}\nabla \cdot {\boldsymbol {\sigma }}+\mathbf {f} }
[further explanations of these terms below]

This describes momentum transport in any continuum (not just fluids), and from the Cauchy Momentum equation, the Navier-Stokes equations can be derived. The N-S equations describe the motion of viscous fluids in continuum mechanics. These equations are fundamental to the field of fluid mechanics (and are of mathematical interest).

Navier-Stokes equations (incompressible form):
{\displaystyle \overbrace {\underbrace {\frac {\partial \mathbf {u} }{\partial t}} _{\begin{smallmatrix}{\text{Variation}}\end{smallmatrix}}+\underbrace {(\mathbf {u} \cdot \nabla )\mathbf {u} } _{\begin{smallmatrix}{\text{Convection}}\end{smallmatrix}}} ^{\text{Inertia (per volume)}}\overbrace {-\underbrace {\nu \,\nabla ^{2}\mathbf {u} } _{\text{Diffusion}}=\underbrace {-\nabla w} _{\begin{smallmatrix}{\text{Internal}}\\{\text{source}}\end{smallmatrix}}} ^{\text{Divergence of stress}}+\underbrace {\mathbf {g} } _{\begin{smallmatrix}{\text{External}}\\{\text{source}}\end{smallmatrix}}.}

This form has been simplified for use with incompressible fluids (like water), and for a Newtonian fluid (fluid with shear stress directly proportional to fluid velocity) like water the diffusion term describes the 'diffusion of momentum' through the fluid. And if the 'external source' describes a conservative field, the internal source (thermodynamic 'specific' work) can be combined with the external source to give hydraulic head pressure, or just the fluid pressure:
{\displaystyle {\frac {\partial \mathbf {u} }{\partial t}}+(\mathbf {u} \cdot \nabla )\mathbf {u} -\nu \,\nabla ^{2}\mathbf {u} =-\nabla h.}

What about the stress tensor? Where is it? From the Cauchy Momentum equation, the stress tensor disappears but yields a direct relationship, through fluid viscosity μ, to the rate of change of velocity in each direction, which is a useful and easily measurable / observable quantity in fluid motion:
\tau =\mu {\frac {\partial u}{\partial y}},

Wikipedia has an excellent description of this constitutive equation:


That's pretty much the point of the stress tensor. It's a very useful concept that under-girds many significant and practical calculations in myriad fields of mechanics from weather models, to fluid mechanical models, to physics engines used in video games.

Descriptions of pieces of the Cauchy Momentum equation
[Material derivative]: describes the rate of change of the velocity field (local velocity of the fluid) in time and space: {\displaystyle \partial _{t}\mathbf {u} +\mathbf {u} \cdot \nabla \mathbf {u} }

[divergence of the stress tensor]: describes how the force per unit volume (dimensions and units: 'Pressure / length' SI units are Pa / m = (N / m^2) /m = N / m^3 which gives dimensions 'Force / volume' equivalence) is related to how the stress tensor components change with each direction. This is essentially telling you how the force on a face (per unit volume) is related to how much the stress components on each face are changing in the direction they act.
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