Microstructural studies using X-ray diffraction
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Stress Optimizations

Warning

Stress optimizations are not reliable. It is not a software issue with Polydefix, but issues with the equations behind stress optimizations. They are based on elastic strain theories and have been shown to give inconsistent results. Refer to the strain fitting options section for more details and publications about those problems.

In theory, stress should be adjusted by comparing results of models like Elasto-Plastic Self Consistent calculations to the strain measured experimentally. Elastic solutions should be used with caution.

Elastic calculation of stresses

Polydefix can use the lattice strain parameters Q fitted to the experimental data in order to estimate stress level based on elastic lattice strain theories. Equations in Polydefix where extracted from the paper

A. K. Singh, C. Balasingh, H. K. Mao, R. J. Hemley and J. Shu, Analysis of lattice strains measured under non-hydrostatic pressure, J. Appl. Phys., 1998, 83, 7567-7575 , doi: 10.1063/1.367872

Isotropic stress model

If you selected the isotropic model when setting up your material properties, you were asked to enter parameters G0, G1, and G2. They are used to calculate an average shear modulus according to

G = G0 + G1 * P + G2 * P2

where P is the hydrostatic pressure calculated in pressure and unit cell optimizations. For each measured diffraction line, we calculate a corresponding differential stress t(hkl) according to

t = 6 * G * Q(hkl)

where Q(hkl) is the lattice strain parameter fitted in the lattice strains optimizations.

Anisotropic stress model

For cubic and hexagonal crystal symmetries, you can also use an anisotropic stress model based on the Reuss bound in the paper of Singh et al 1998. In this case, when setting up your material properties, you were asked to enter elastic moduli and pressure derivatives.

We use the hydrostatic pressure calculated in pressure and unit cell optimizations and, for each elastic modulus, calculate

Cij = Cij0 + Cij1 * P + Cij2 * P2

For cubic materials, elastic moduli are used to calculate an effective average shear moduli for the hkl diffraction line according to

1/[2 G(hkl)] = S11 - S12 - 3 ( S11- S12 - S44/2) Γ(hkl)

Γ(hkl) = (h2k2 + k2l2 + l2h2) / (h2+k2+l2)2

where the Sij are elastic compliances.

For hexagonal materials, elastic moduli are used to calculate an effective average shear moduli for the hkl diffraction line according to

1/[2 G(hkl)] = (2S11 - S12- S13)/2

+ l32 (-5S11 + S12 + 5S13 -  S33 + 3S44)
+ l34 (3S11 - 6S13  + 3S33 - 3S44)

l32 = 3 a2 l2 / M2

M2 = [ 4c2(h2+hk+k2) + 3a2l2 ]

where the Sij are elastic compliances and a and c are obtained from unit cell optimizations.

For each measured diffraction line, we then calculate a corresponding differential stress t(hkl) according to

t = 6 * G(hkl) * Q(hkl)

where Q(hkl) is the lattice strain parameter fitted in the lattice strains optimizations.