A Toddler at Night as Predicted by Material Science

A few days ago, at around 8:00 pm, my 17-month-old daughter was standing in the living room, when she suddenly began to spin herself in circles in an effort to induce dizziness and fall down.  She seemed to enjoy it a lot.  After a few nice, relatively harmless tumbles, she eventually made herself so dizzy, that she fell, smacking her face directly onto the hardwood floor.  At that moment, the “witching hour” ended with a brief fit of crying, and it was time for bed.

As a parent of a toddler, I am beginning to see the importance of recognizing toddler fatigue.  The witching hour is an unofficial domain of time where an otherwise normal child goes bananas.  Imagine how a prison inmate gradually loses his mind in isolation over a period of months, and condense that experience into one hour, and you begin to depict the decline of a toddler in the evening.  For my daughter, this strange yet entertaining timeframe seems to occur between the end of dinner and bedtime.


After my daughter’s spinning/dizzy/falling routine, I began to see how her experience related to an elastic material in gradually increasing tension.  Her final outburst of tears was representative of the ultimate failure of a metal rod, when it finally cracks into two pieces.  If only I could predict it with accuracy... Well, with the help of introductory material science, I think I can.

Material science is a very complex field, and has been the most important driver for the technological progress of man.  Wood has always been useful for shelter, aluminium made the bicycle feasible (high strength to density ratio), carbon fibers have been used in various sport and aeronautic applications (higher strength to density ratio), and carbon nanotubes may one day make the Space Elevator a reality (perhaps the highest strength to density ratio possible). 

Most materials are elastic in nature; that is, they deform proportionally as increasing tensile load is applied across them.  Let us focus on metals, as these materials are representative of toddlers, particularly when they are stressed, or tired.

Metals dominate the periodic table as toddlers overwhelm their parents.  Metals conduct both heat and electricity very well, and toddlers are tiny balls of energy.  Like small children, metals come in many varieties, each with their own unique overall properties.  It is the similarities between the mechanical properties of metals and the evening behaviour of young children that I would like to focus on.

Many different metals are used for practical applications: iron, copper, bronze, aluminium, titanium, etc.  Each of these metal types composes the majority of several different alloys, like steel, which is about 99% iron, with some carbon mixed in.  The alloys are usually prepared in a high temperature state, and are cured (transformed into a solid state) at a lower temperature for a certain duration.  This process is actually quite complex; it is an art, like cooking. 

Material scientists who prepare the recipes for these alloys must consider the effects that time spent at various temperatures will have on the atoms of the metal.  In the end, the alloy will exhibit many different qualities.  Its mechanical properties are mainly defined by its “stress-strain” curve, which the remainder of this article will ponder.

If a metallic rod of a given length were gripped tightly at both of its end and gradually pulled apart, two measurements could be taken at any point in time.  We could measure the applied load (tensile force) passing through the rod.  This value, in units of Newtons [N], when divided by the area of cross-section of the material in square metres, will tell how much tensile stress is in the metallic rod at that moment in Pascals [Pa].  At the same moment in time, we can verify the corresponding elongation of the rod.  If the rod was initially 2 inches long, and after some time, was stretched to 2.001 inches, the elongation would be 0.001”, and would correspond to a strain of 0.001"/2" = 0.0005 and percent elongation of 0.05%. 

If we take many readings during what is often referred to as a “pull-test”, we can place them on a graph, and call it the stress-strain curve for the given metallic alloy that was subjected to the test.  A typical stress-strain curve for a ductile metallic alloy is shown below.

Figure 1: Typical Stress-Strain Curve for a Ductile Metallic Alloy

There are several observations we can make from the stress strain curve shown in Figure 1.  The elastic region is that between points 1 and 2.  This region is defined by its linear shape.  The constant slope in the elastic range is the Elastic Modulus, E, of the material.  Something critical happens at point 2: the material yields.  The stress of the material at this point is known as its yield strength.

After this point, the ductile nature of the material can be seen (a brittle material snaps in two shortly after it yields, and has a simpler shape).  The behaviour of the inelastic or plastic range of the material is seen between points 2 and 4.  The yield strength is like the point of no return for the material; if the stress is released before this point is reached, it returns elastically to its original length, whereas any deformation beyond the yield strength results in permanent deformation.  If the stress is released at point 3, the rod will be slightly stretched (0.2%) in its unstressed state.  The elongation of the rod is greater for a given stress increase in the plastic range than in the elastic range.  Eventually, the applied stress is too great for the rod, and it fails, breaking in two.  The stress at which this occurs is known as the ultimate strength of the material (point 4 on the graph).

For a toddler, between say, 7:00 pm and 8:00 pm, a similar graph exists: it is the graph of child chaos versus time.  Below is the rough chaos-time curve for my daughter.    

Figure 2: Typical Chaos-Time Curve for a Toddler

The shape of the chaos-time curve for a toddler in the time domain between the end of supper (7:00 pm) and bedtime (8:00 pm), known as the witching hour, is clearly similar to that of the stress-strain curve.  Note that the minimum value (unstressed state) of chaos does not correspond to zero chaos, but rather some inherent chaos level that the parent is accustomed to. 

After 7:00 pm, the child begins to get more hyper at an astonishing rate.  By 7:10, she is running laps around the living room.  At around 7:15 (point 2), she yields, at her yield chaos value, Cy.  Beyond this point, her chaos increases in a less predictable way; she gets loopy.  She invents new words, new games, as she exhibits her ductile qualities.

At around 7:30 pm (point 3), she begins to spin herself in circles.  Now would be a good time to consult the chaos-time curve (CTC) and observe that the end may be near – she could go at any time.  We are in ultimate chaos territory.  Now would be a good time for the parent to intervene, so that ultimate chaos occurs in a more controlled environment (bath time?).  A few nights ago, ultimate chaos, Cu, occurred around 8:00 pm, like clockwork, but my daughter was not in a bath, and was instead sprawled on the floor wishing this daily acid trip would stop.

Every child is different, and if you have one, you might want to consider charting his or her CTC.  You could then calculate his/her Elastic Modulus in units of Chaos per minute (C/min), and both yield and ultimate chaos values.  These are all unique to toddlers.

Fortunately, children are so ductile that they bounce back the next day.  Unlike a material, which, when extended beyond its elastic range undergoes permanent deformation, the changes our children experience are temporary.  As a parent, the best we can do is to manage this timeframe well.  We can choose an appropriate environment to put our children in during the witching hour, or maybe treat ourselves to a stiff drink and just enjoy the show. 

In the end, it is usually quite entertaining.  And, we can seek comfort in the knowledge that they will wake up tomorrow morning in their regular state of chaos, which has by now become welcome.  Maybe we can figure a way to be busy during tomorrow’s critical chaos-time zone and have our spouse or a babysitter be the one to supervise it.

We are not all material scientists, but many of us are parents of toddlers at some point in time.  We should not treat our young children as metal alloys, but we can appreciate that as the sun sets, and the moon becomes bright in the sky, our children begin to behave like metal bodies in tension.