# Changes in a defined power-law parameter when calculating effective viscosity with PowerLawPlaneStrain viscoelastic materials

Dear Pylith team,

I have some questions. I am sorry if this is a basic question. I would like to ask why there are changes in a defined power-law parameter when calculating effective viscosity with PowerLawPlaneStrain viscoelastic materials in a 2D model. Laboratory experiments suggest a power-law rheology: η=Cσ^(1-n) or ε ̇=σ^n/2C, where σ is differential stress, n is the power-law exponent, C is the power-law parameter associated with temperature and rock composition (C=e^((Q/RT) )/2A), and ε ̇ is strain rate. If Q, R, T, and A are constants, C is a constant, is this correct?

I calculate the effective viscosity in postseismic periods as follows: (1) I have preseismic strain rate ε ̇_pre, viscosity η_pre, and power-law-exponent n. I determine the preseismic differential stress by σ_pre=2η_pre ε ̇_pre. (2) I solve for C_pre by using C_pre=η_pre σ_pre^(n-1). Then, I calculate A_T by A_T=√3^(n+1)/4C. Using an assumed reference-strain-rate e ̇ (10-16 s-1), I calculate reference-stress S_0 by S_0=(e ̇/A_T )^(1/n). By now, I have specified the three parameters for the PowerLawPlaneStrain viscoelastic material. (3) I obtain stress and total_strain after an earthquake from the Pylith modeling results. Now, the problem comes.

I use the inferred stress and total_strain to calculate C_post by C_post=σ_post^n/2ε ̇_post, where σ_post=|σ_xx-σ_yy | and ε ̇_post= ε/dt. But I find C_post is unequal to C_pre. I am very confused about this, why does C change? Isn’t it a constant?

I attach my cfg files and my question docx file. powerlaw.cfg (5.7 KB)
pylithapp.cfg (10.1 KB)

YageZhu

Dear YageZhu,
I am sorry for the slow response. I think that some of your issues involve comparison of the simplified form of the flow law you are using with the generalized form used in PyLith (e.g., equation 5.73 in the manual). Some issues to consider:

1. Only deviatoric stresses are used in computing the viscous strain rate.
2. You need to make sure you are using the viscous strain rates and not the total strain rate.
3. When you compute \sigma^n, it appears you are neglecting the shear stress.

If you look at equation 5.73, you can see that if you divide the viscous strain rate by the stress terms in the numerator, you should get the A_T term (equation 5.71). Let me know if this helps.

Cheers,
Charles

Dear Charles,