Elasticity

Discussion

basics

Elasticity is the property of solid materials to return to their original shape and size after the forces deforming them have been removed. Recall Hooke's law — first stated formally by Robert Hooke in The True Theory of Elasticity or Springiness (1676)…

ut tensio, sic vis

which can be translated literally into…

As extension, so force.

or translated formally into…

Extension is directly proportional to force.

Most likely we'd replace the word "extension" with the symbol (∆x), "force" with the symbol (F), and "is directly proportional to" with an equals sign (=) and a constant of proportionality (k), then, to show that the springy object was trying to return to its original state, we'd add a negative sign (−). In other words, we'd write the equation…

F = − k∆x

This is Hooke's law for a spring — a simple object that's essentially one-dimensional. Hooke's law can be generalized to…

Stress is proportional to strain.

where strain refers to a change in some spatial dimension (length, angle, or volume) compared to its original value and stress refers to the cause of the change (a force applied to a surface).

The coefficient that relates a particular type of stress to the strain that results is called an elastic modulus (plural, moduli). Elastic moduli are properties of materials, not objects. There are three basic types of stress and three associated moduli.

The international standard symbols for the moduli are derived from appropriate non-English words — E for élasticité (French for elasticity), G for glissement (French for slipping), and K for kompression (German for compression). Some American textbooks have decided to break with tradition and use the first letter of each modulus in English — Y for Young's, S for shear, and B for bulk.

Stresses on solids are always described as a force divided by an area. The direction of the forces may change, but the units do not. The SI unit of stress is the newton per square meter, which is given the special name pascal in honor of Blaise Pascal (1623–1662) the French mathematician (Pascal's triangle), physicist (Pascal's principle), inventor (Pascal's calculator), and philosopher (Pascal's wager).

Strains are always unitless.

Which means that pascal is also the SI unit for all three moduli.

failure is an option

• elastic limit, yield strength
• breaking point, ultimate strength
• The strength of a material is a measure of its ability to withstand a load without breaking.
• Banerjee, et al. show that when nanoscale single-crystal diamond needles are elastically deformed, they fail at a maximum local tensile strength of ~89 to 98 GPa.
• Experimental results and ab initio calculations indicate that the elastic modulus of carbon nanotubes and graphene is approximately equal to 1 TPa.
• By contrast, the reported tensile strength of bulk cubic diamond is < 10 GPa

Young's modulus

Imagine a piece of dough. Stretch it. It gets longer and thinner. Squash it. It gets shorter and fatter. Now imagine a piece of granite. Try the same mental experiment. The change in shape must surely occur, but to the unaided eye it's imperceptible. Some materials stretch and squash quite easily. Some do not.

The quantity that describes a material's response to stresses applied normal to opposite faces is called Young's modulus in honor of the English scientist Thomas Young (1773–1829). Young was the first person to define work as the force displacement product, the first to use the word energy in its modern sense, and the first to show that light is a wave. He was not the first to quantify the resistance of materials to tension and compression, but he became the most famous early proponent of the modulus that now bears his name. Young didn't name the modulus after himself. He called it the elastic modulus. The symbol for Young's modulus is usually E from the French word élasticité (elasticity) but some prefer Y in honor of the scientist.

Young's modulus is defined for all shapes and sizes by the same rule, but for convenience sake let's imagine a rod of length ℓ₀ and cross sectional area A being stretched by a force F to a new length ℓ₀ + ∆ℓ.

Tensile stress is the outward normal force per area (σ = F/A) and tensile strain is the fractional increase in length of the rod (ε = ∆ℓ/ℓ₀). The proportionality constant that relates these two quantities together is the ratio of tensile stress to tensile strain —Young's modulus.

The same relation holds for forces in the opposite direction; that is, a strain that tries to shorten an object.

Replace the adjective tensile with compressive. The normal force per area directed inward (σ = F/A) is called the compressive stress and the fractional decrease in length (ε = ∆ℓ/ℓ₀) is called the compressive strain. This makes Young's modulus the ratio of compressive stress to compressive strain. The adjective may have changed, but the mathematical description did not.

The SI units of Young's modulus is the pascal [Pa]…

but for most materials the gigapascal is more appropriate [GPa].

1 GPa = 10⁹ Pa

Poisson's ratio

Extension and contraction are opposite types of linear strain. Extension means to get longer. Contraction means to get shorter. Whenever a material is extended or contracted by a linear stress in one direction (called the x axis), the reverse strain usually takes place in the perpendicular directions (the y and z axes). The direction of a linear stress is called the axial direction. All the directions that are perpendicular to this are called the transverse directions.

An axial extension is usually accompanied by a transverse contraction. Stretching a piece of dough makes it get thinner as well as longer. This is the way Chinese hand-pulled noodles (拉面, la mian) are made. Likewise, an axial contraction is usually accompanied by a transverse extension. Flattening a piece of dough makes it get wider and longer as well as thinner. This is the way Italian fresh pasta (pasta fresca) is made.

The ratio of transverse strain to axial strain is known as Poisson's ratio (ν) in honor of its inventor the French mathematician and physicist Siméon Poisson (1781–1840). A negative sign is needed to show that the changes are usually of the opposite type (+ extension vs. − contraction). If we keep with the tradition that x is the axial direction and y and z are the transverse directions then Poisson's ratio can be written as…

The symbol that looks unfortunately like the Latin letter v (vee) is actually the Greek letter ν (nu), which is related to the Latin letter n (en).

Typical values for Poisson's ratio range from 0.0 to 0.5. Cork is an example of a material with a low Poisson's ratio (nearly zero). When a cork is pushed into a wine bottle, it gets shorter but not thicker. (There is some axial strain, but barely any transverse strain.) Rubber on the other hand, has a high Poisson's ratio (nearly 0.5). When a rubber stopper is pushed into a chemical flask, the stopper gets shorter by some amount and wider by nearly half that amount. (The axial strain is accompanied by a large transverse strain.) Corks can be pounded into bottles with a mallet. Pounding a rubber stopper into a glass flask with a mallet is likely to end in disaster.

Surprisingly, negative Poisson's ratios are also possible. Such materials are said to be auxetic. They grow larger in the transverse direction when stretched and smaller when compressed. Most auxetic materials are polymers with a crumpled, foamy structure. Pulling the foam causes the crumples to unfold and the whole network expands in the transverse direction.

shear modulus

A force applied tangentially (or transversely or laterally) to an object is called a shear stress. The deformation that results is called a shear strain. Applying a shear stress to one face of a rectangular box slides that face in a direction parallel to the opposite face and changes the adjacent faces from rectangles to parallelograms.

The coefficient that relates shear stress (τ = F/A) to shear strain (γ = ∆x/y) is called the shear modulus, rigidity modulus, or Coulomb modulus. It is usually represented by the symbol G from the French word glissement (slipping) although some like to use S from the English word shear instead.

Fluids (liquids, gases, and plasmas) cannot resist a shear stress. They flow rather than deform. The quantity that describes how fluids flow in response to shear stresses is called viscosity and is dealt with elsewhere in this book.

Their inability to shear also means fluids are opaque to transverse waves like the secondary waves of an earthquake (also known as shear waves or s waves). The liquid outer core of the Earth was discovered by the s wave shadow it cast on seismometer networks. Types of waves are discussed elsewhere in this book.

Fluids can resist a normal stress. This means that liquids and gases are transparent to the primary waves of an earthquake (also known as pressure waves or p waves). The solid inner core of the Earth was detected in p wave signals that made it all the way from one side of the Earth through the liquid outer core to the other side. P waves are also audible. You can hear them when they transmit into the air.

The resistance of a material to a normal stress is described by the bulk modulus, which is the next topic in this section.

bulk modulus

A force applied uniformly over the surface of an object will compress it uniformly. This changes the volume of the object without changing its shape.

The stress in this case is simply described as a pressure (P = F/A). The resulting volume strain is measured by the fractional change in volume (θ = ∆V/V₀). The coefficient that relates stress to strain under uniform compression is known as the bulk modulus or compression modulus. Its traditional symbol is K from the German word kompression (compression) but some like to use B from the English word bulk, which is another word for volume.

The bulk modulus is a property of materials in any phase but it is more common to discuss the bulk modulus for solids than other materials. Gases have a bulk modulus that varies with initial pressure, which makes it more of a subject for thermodynamics — in particular the gas laws.

The reciprocal of bulk modulus is called compressibility. Its symbol is usually β (beta) but some people prefer κ (kappa). A material with a high compressibility experiences a large volume change when pressure is applied.

The SI unit of compressibility is the inverse pascal [Pa⁻¹].

scaling

• no gigantic animals
• surface area is proportional to length²
• mass and volume is proportional to length³
• BMR is proportional to mass^3/4
• tension is proportional to length (Hooke's law)
• pressure is proportional to length² (stomach, bladder stretching)

surface tension

Capillarity

• The average diameter of the capillaries is about 20 μm, although some are only 5 μm in diameter. there are about 190 km of capillaries in 1 kg of muscle, the surface area of the capillaries in 1 kg of muscle is about 12 m².