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Wednesday
Sep262012

## L is for Lambda

Hooke's law says that the force F exerted by a spring depends only on its displacement x from equilibrium, and the spring constant k of the spring:

$F=-kx$.

We can think of k—and experience it—as stiffness. The spring constant is a property of the spring. In a sense, it is the spring. Rocks are like springs, in that they have some elasticity. We'd like to know the spring constant of our rocks, because it can help us predict useful things like porosity.

Hooke's law is the basis for elasticity theory, in which we express the law as

stress [force per unit area] is equal to strain [deformation] times a constant

This time the constant of proportionality is called the elastic modulus. And there isn't just one of them. Why more complicated? Well, rocks are like springs, but they are three dimensional.

In three dimensions, assuming isotropy, the shear modulus μ plays the role of the spring constant for shear waves. But for compressional waves we need λ+2μ, a quantity called the P-wave modulus. So λ is one part of the term that tells us how rocks get squished by P-waves.

These mysterious quantities λ and µ are Lamé's first and second parameters. They are intrinsic properties of all materials, including rocks. Like all elastic moduli, they have units of force per unit area, or pascals [Pa].

### So what is λ?

Matt and I have spent several hours discussing how to describe lambda. Unlike Young's modulus E, or Poisson's ratio ν, our friend λ does not have a simple physical description. Young's modulus just determines how much longer something gets when I stretch it. Poisson's ratio tells how much fatter something gets if I squeeze it. But lambda... what is lambda?

• λ is sometimes called incompressibility, a name best avoided because it's sometimes also used for the bulk modulus, K.
• If we apply stress σ1 along the 1 direction to this linearly elastic isotropic cube (right), then λ represents the 'spring constant' that scales the strain ε along the directions perpendicular to the applied stress.
• The derivation of Hooke's law in 3D requires tensors, which we're not getting into here. The point is that λ and μ help give the simplest form of the equations (right, shown for one dimension).

The significance of elastic properties is that they determine how a material is temporarily deformed by a passing seismic wave. Shear waves propagate by orthogonal displacements relative to the propagation direction—this deformation is determined by µ. In contrast, P-waves propagate by displacements parallel to the propagation direction, and this deformation is inversely proportional to M, which is 2µ + λ

Lambda rears its head in seismic petrophysics, AVO inversion, and is the first letter in the acronym of Bill Goodway's popular LMR inversion method (Goodway, 2001). Even though it is fundamental to seismic, there's no doubt that λ is not intuitively understood by most geoscientists. Have you ever tried to explain lambda to someone? What description of λ do you find useful? I'm open to suggestions.

Goodway, B., 2001, AVO and Lame' constants for rock parameterization and fluid detection: CSEG Recorder, 26, no. 6, 39-60.

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Nice post Evan!
Personally I like to think about things in terms of bulk and shear modulus. i.e. K and mu...bulk modulus is very easy to intuitively understand, it is just the rocks resistance to uniform comrpession. We can write P-wave or compressional velocity as Vp=sqrt((K+4*mu/3)/rho).
Personally I never think about llambda. maybe I should? Pretty much all rock physics theory is built up from K and mu too, such as Gassmann fluid substitution, Hertz-Minldin theory etc etc.
We can write any other isotropic elastic constant in terms of 2 others anyway, so we can write these equations in a number of ways, i just like the physically intuitive way which to me is to use bulk and shear modulus.
Matt

September 26, 2012 | MattS

@MattS: Hey Matt... "We can write any other isotropic elastic constant in terms of 2 others". I always liked the simplicity of this idea. The various types of curvature are the similarly symmetric. But recently I tried to write the moduli in terms of E and λ (I know it's a weird thing to do). You can do it, but unless I messed up, the result is really, really ugly: The (E, λ) problem. I thought that was curious.

September 27, 2012 | Matt Hall

@MattHall That confirms how hard it is to understand what Lambda represents physically haha...its all round a pain in the arse!

September 28, 2012 | MattS

I think that lambda was introduced to simplify Hooke's law for an isotropic elastic medium, rather than because of any particular physical significance. For an isotropic elastic medium, Hooke's law can be written as sigma = 2*mu*epsilon + lambda*tr(epsilon)*I, where sigma is the stress tensor, epsilon is the strain tensor, I is the identity tensor, and tr(epsilon) represents the trace of the strain tensor. Lambda and Mu were named after Lame (they represent the 1st and 3rd letters of his name).

September 28, 2012 | Colin Sayers

@MattS,
I agree with you about Bulk modulus; it seems to me to the most intuitive way to compress a grain or pore space or aggregate. And I think what I have learned is there there is no equivalent diagram to show what applying a Lambda deformation is like. There is no such thing. We either have shear, bulk, or uniaxial stretching / squishing I think. Should you use Lambda? I don't know if I am the best to answer that question. It does show up a lot in continuum mechanics and treatment of Hooke's law in tensor form. So there is a parsimony or economy to using it in those areas. Other applications, I am not so sure.

@Colin, thanks for the tidbit on the origin of where the 'L' and 'M', come from. Using google to search for this stuff is hard because I am never sure if I am missing references using the greek letters. Also, thanks for repeating your comment on the rock physics discussion group on LinkedIn. It's a vibrant arena with lots of fruitful commentary.

September 29, 2012 | Evan Bianco