Strain Parameters

Steven Dutch, Natural and Applied Sciences, University of Wisconsin - Green Bay
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Elastic Deformation

Elastic material deforms under stress but returns to its original size and shape when the stress is released. There is no permanent deformation. Some elastic strain, like in a rubber band, can be large, but in rocks it is usually small enough to be considered infinitesimal.

Many elastic materials obey Hooke's Law behavior: the deformation is proportional to the stress. This is why spring balances work: twice the weight results in twice the deformation.

For materials, Hooke's Law is written as: Stress = E Strain. Alternatively, the relationship is sometimes written E = Stress/Strain. This is the reverse of the way the law is written in most physics texts. In physics, we can often apply the stress in a controlled way and we are interested in predicting the behavior of the spring, for example, how it oscillates. In materials science and geology, we often know the strain and want to know what stress produced it. The two versions are equivalent; the only difference is which side the constant is written on. The constant E is called Young's Modulus. Because strain is dimensionless, Young's Modulus has the units of pressure or stress, i.e. pascals.

Physical Meaning of Young's Modulus

If strain = 1, stress = E. Thus, Young's Modulus can be considered the stress it would take (theoretically only!) to result in 100 percent stretching or compression. In reality, most rocks fracture or flow when deformation exceeds a few percent, that is, at stresses a few percent of Young's Modulus.

The seismic P- and S-wave velocities in rocks are proportional to the square root of E.

For most crystalline rocks, E ranges from 50-150 Gpa, averaging about 100. If we take 100 Gpa as an average, and consider one bar (100,000 pa) of stress, we have: 105 = 1011 Strain, or Strain = 10-6. Thus, rocks typically deform elastically by 10-6 per bar of stress. This is a useful quantity to remember. Elastic strain in rocks is tiny - even ten kilobars typically results in only one percent deformation - if the rock doesn't fail first.

Poisson's Ratio

When a material is flattened, it tends to bulge out at right angles to the compression direction. If it's stretched, it tends to constrict. Poisson's Ratio is defined at the ratio of the transverse strain (at right angles to the stress) compared to the longitudinal strain (in the direction of the stress).

ex=strain in x-direction
ey=strain in y-direction
Poisson's Ratio = ey/ex

Note that the ratio is that of strains, not dimensions. We would not expect a thin rod to bulge or constrict as much as a thick cylinder.

For most rocks, Poisson's Ratio, usually represented by the Greek letter nu (ν), averages about 1/4 to 1/3. Materials with ratios greater than 1/2 actually increase in volume when compressed. Such materials are called dilatant. Many unconsolidated materials are dilatant. Rocks can become dilatant just before failure because microcracks increase the volume of the rock. There are a few weird synthetic foams with negative Poisson's Ratios. These materials are light froths whose bubble walls collapse inward under compression.

Shear Modulus

Poisson's Ratio describes transverse strain, so it obviously has a connection with shear. The Shear Modulus, usually abbreviated G, plays the same role in describing shear as Young's Modulus does in describing longitudinal strain. It is defined by G = shear stress/shear strain.

G can be calculated in terms of E and v:  G = E/2(1 + ν). Since v ranges from 1/4 to 1/3 for most rocks, G is about 0.4 E.

Bulk Modulus

The bulk modulus, K, is the ratio of hydrostatic stress to the resulting volume change, or K = pressure/volume change.

It's easy to show the relationship between K, E, and Poisson's ratio (ν). Consider the effects of pressure P acting on a unit cube equally along the x- y- and z-axes. The pressure along the x-axis will cause the cube to contract longitudinally by an amount P/E. However, it will also bulge to the side by an amount vP/E, in both the y- and z-directions. The net volume change just due to the component in the x-direction is (1 - 2ν)P/E. The minus sign reflects the fact that the bulging counteracts the volume decrease due to compression. Similarly, compression along the y- and z- axes produces similar volume changes. The total volume change is thus 3(1 - 2ν)P/E.

Since K = P/volume change, thus K = E/(3(1 - 2ν)). Since v ranges from 1/4 to 1/3 for most rocks, K ranges from 2/3E to E.

Physically, K can be considered the stress it would take to result in 100 per cent volume change, except that's physically impossible and elastic strain rarely exceeds a few percent anyway.

If ν = 1/2, then K becomes infinite - the material is absolutely incompressible. Obviously real solids cannot be utterly incompressible and therefore cannot have ν = 1/2.

Relations Between Elastic Parameters

There are really only two independent quantities, so if we know any two quantities E, v, G and K, we can calculate any others. The relations are shown below. Find the two known parameters and read across to find the other two.

 Known: E = ν = G = K = E, ν E ν (E/2)/(1 + ν) (E/3)/(1 - 2ν)) E, G E (E/2G) - 1 G (E/3)/(3 - E/G)) E, K E (1 - E/3K)/2 E/(3 - E/3K) K G, ν 2G(1 + ν) ν G (2/3)G(1 + ν)/(1 - 2ν) G, K 12G2/(3K + 4G) (2G - 3K)/(3K + 4G) G K K, ν 3K(1 - 2ν) ν (3/2)K(1 - 2ν)/(1 + ν) K

Viscous Deformation

Viscous materials deform steadily under stress. Purely viscous materials like liquids deform under even the smallest stress. Rocks may behave like viscous materials under high temperature and pressure.

Viscosity is defined by N = (shear stress)/(shear strain rate). Shear stress has the units of force and strain rate has the units 1/time. Thus the parameter N has the units force times time or kg/(m-sec). In SI terms the units are pascal-seconds. Older literature uses the unit poise; one pascal-second equals ten poises.

Few if any physical parameters show such a tremendous range as viscosity

Material Viscosity Pa-Sec
Hydrogen gas, 15 degrees K 0.0000006
Air, 0 degrees C 0.000017
Liquid Hydrogen 0.00001
Water 0 degrees C 0.0018
Water 100 degrees C 0.0003
Heavy Machine Oil 15 C 0.66
Glycerin, 20 degrees C 1.5
Honey 20 degrees C 1.6
Basalt Lava 1000
Rhyolite Lava 1,000,000
Rock Salt 1014
Marble 1016
Granite, Quartzite 1018 - 1020
Asthenosphere 1019 - 1020
Deep Mantle 1021 - 1022
Shallow Mantle 1023 - 1024

Viscosity is very dependent on temperature. If it seems that water out of a boiling teakettle splashes more and soaks through clothing more quickly than cold water, that's no illusion. The viscosity of water at 100 C is only one-third as much as room temperature and one sixth what it is at 0 C. The viscosity of glycerine drops from 6700 at -40 C to .63 at 30 C, a factor of 10,000 in only 70 degrees.

Note that, strictly speaking, solid rocks aren't viscous. The figures given reflect their flow rates at the temperatures and pressures typically found during crustal deformation, and they are highly approximate and extremely dependent on temperature and pressure. Variations by several orders of magnitude are perfectly possible and commonly seen.

Plastic Deformation

Plastic material does not flow until a threshold stress has been exceeded. Plastic flow involves many different mechanisms at the atomic level and there are many different equations for different plastic flow mechanisms. Plastic flow therefore does not lend itself to neat physical parameters the way elastic and viscous deformation do.

Power Law Creep

One of the most common forms of plastic flow is Power-Law Creep, given by the formula:
Strain Rate = C (Stress)n exp(-Q/RT)

Let's take each part of the formula in turn:

• C is a scaling constant.
• n means that the strain rate increases much faster than stress. Typically n is about 3 but can range from a bit less than 2 to 8. Recall that with viscous deformation stress is proportional to strain rate (n=1). With power-law creep it's faster: the effective viscosity drops with stress.
• Q is the activation energy required to get crystal dislocations moving. It's typically 100-300 kilojoules per mole, sometimes up to 500.
• R is the Universal Gas Constant that turns up everywhere in physical chemistry. In SI units it equals 8.3144 joules/mole-degree Kelvin.
• T is the temperature in degrees Kelvin. As T increases, Q/RT decreases and thus exp(-Q/RT) increases, though much more slowly than exp(T). At very large T, Q/RT approaches zero and the exponential term approaches 1. This does not happen, though, at geologically realistic temperatures.
 The function Exp(-Q/RT) increases approximately linearly at first, then very slowly asymptotically approaches 1. The curves here use geologically realistic values of Q. At high temperatures the temperature dependence is weak because rocks are ductile enough to deform as fast as strain can be applied. Another way to look at it is that ambient thermal energy is nearly enough to overcome the activation energy barrier.

Power-law creep is given by Strain Rate = C (Stress)n exp(-Q/RT). If n = 1 and Q=0, then we just have Strain Rate = C (Stress), or in other words viscous flow. Q = 0 means it takes no energy to dislocate atoms; that is, the material deforms under even the slightest stress. In this case C is just 1/viscosity. However, you can't simply look up values of C and equate 1/C to viscosity because the other terms in real power-law creep can be extremely large.

Some everyday materials obey power-law creep or some similar behavior.

• Mayonnaise in a jar flows very sluggishly. You can tilt the jar on its side without it spilling. On the other hand, it offers no resistance to a knife and spreads easily on bread. A slight increase in shear stress (applied by the knife) results in a huge increase in strain rate or, effectively, a great decrease in viscosity.
• Ball-point pen ink has to be viscous enough to stay in place if the pen is put upside-down in a pocket, yet flow easily through a small orifice as the ball rotates. This is another material whose viscosity drops as shear stress is applied.

Materials such as these are commonly described as shear-thinning fluids because they become less viscous with increasing shear stress. However, it's easy to see they behave in a manner similar to power-law creep.

Viscoelastic Combines elastic and viscous behavior. Models of glacio-isostasy frequently assume a viscoelastic earth: the crust flexes elastically and the underlying mantle flows viscously.