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.

Saturday, February 15, 2020

Why Calvin?

I was 18.

My dad said something like "you should go to Calvin. I would like it if you went to Calvin, and we will help you."

Calvin College had a chemical engineering program, and at the time I told people that I wanted to study engineering at a Christian college. I told them that. I probably thought it. Probably.

I definitely went to Calvin because it offered chemical engineering, and I probably went because it was a Christian college--I grew up in a Christian community you see.

At this moment [actually a few years ago now], Calvin College is conducting Vision 2029. Vision 2029 is a college-wide program and forum aimed at asking good questions regarding the role and trajectory that Calvin will adopt.

Today I attended the panel discussion: “Will Calvin be a Comprehensive Liberal Arts College or a Liberal Arts University?”

Question that arose were "why Calvin?" and "what makes Calvin distinctive?" and "what are our non-negotiables?" with the required (and hopefully not overlooked) subtext: "what are the preceding for an 18 year old?"

I don't pretend to speak for the rest of 18 year olds in the history of 18 year olds considering Calvin, but for my part, that list contained three items:

1. chemical engineering*
2. attractive (reformed) women**
3. residence in the Christian bubble***

*Ok. Yes. Still very pleased about this one
**"The basis of optimism is sheer terror" --Oscar Wilde
***Not the desire for growth in my faith or a deep connection with Christ--more of a cultural imperative

Fortunately for me, I did attend Calvin despite a less than impressive (and retrospectively embarrassing) rationale.

There are many reasons I count myself fortunate for attending Calvin despite being an idiot 18 year old (many would say still an idiot 28 year old [I wrote this in the fall of 2017 and now return to publish it]). I value:

the paper I wrote (truly, truly) poorly analyzing Biblical themes in the 'Russian Primary Chronicle' for History of the West and the World I,

the physics seminars I attended to earn honors credit, and for which I wrote an insipid description of gravitational lenses in Introductory Physics: Mechanics and Gravity,

the immediate, thoughtful (and truly mind-blowing) criticism of the Bible, and exposure to a diverse body of scholarship and views regarding who Jesus was, as a first-semester freshman taking Biblical Literature and Theology,

attacking (engaging really) secular and Christian philosophy directly, with abandon, and without fear in Fundamental Questions in Philosophy with a man who gave us students dignity and the best example of a Christian intellectual I ever met,

an international internship which had me training, training, and planing about Europe and learning sophisticated industrial organic chemistry development in a foreign language (driving my eventual pursuit of graduate study),

and

a broad, functional knowledge of not just my own discipline of chemical engineering, but the ability to speak and work intelligently and effectively with other engineers and scientists in different fields.

As a senior in high school, I question that any of these would have thrilled me as they do now, even if I could now posit their existence to a past self. I recognize them now, after completing graduate study, and taking up teaching.

I certainly would not have relished the thought that, most importantly, Calvin prepared me well to lose my faith in graduate school. For that I am grateful--putting together the pieces of a shattered mess has given my resurrected faith a life it never had.

Vision 2029 will fail if it only addresses "why Calvin" and "what are our non-negotiables" for parents of prospective students that want them to come here. I believe that everyone will agree with me on that point, but to me it bears crying from the rooftops and not treating as a polite, obvious observation. Calvin was truly transformative for me in the most dramatic way after I left.

However it wasn't transformative in any of the ways I would have wanted it to be, or expected it to be as a prospective student.

We cannot foist perspective onto students, we can only give them the opportunities to earn it themselves--that begins with their introduction to Calvin as high school seniors. I wonder if it will take walking a line between humble self-promotion, and vulnerability rendered salient.

[2020 perspective]
The move to 'university' has been completed. Let's add ", or just calling ourselves a university." to the last sentence.
I just read a flowery and not particularly substantive vision of the university structure written for the internal Calvin community, and all it did was to confirm my bias against an ever expanding corps of college administrators projecting a disconnected vision. The questions posited in the aforementioned panel appear to be foregone conclusions looking back. I still value Calvin and the place it is, but I've been struck lately by a lot of self-congratulation and lip service paid, when simply creating relationships between departments and companies, organizations, etc. doing similar work (as an alternative to saying) pays the real bills to establish community and inclusion and brings value to students.