The core of X-ray stress measurement technology is to invert macroscopic stress by accurately measuring changes in interplanar spacing. Its physical foundation is deeply rooted in the combination of Bragg's law and elastic mechanics theory.1、 The cornerstone of diffraction geometry: Bragg's law
The premise of this technology is Bragg's law: n λ=2dsin θ. Among them, λ is the known X-ray wavelength, θ is the diffraction angle, and d is the spacing between specific crystal planes (hkl). In a stress free state, the material has a specific interplanar spacing d ₀ and corresponding diffraction angle θ ₀. When there is stress in the material, the lattice undergoes elastic strain, resulting in a change in d (becoming d psi), which in turn causes a shift in the diffraction angle θ psi. By measuring the change in θ ψ, we can accurately calculate the relative change in interplanar spacing, i.e. strain:
εψ=(dψ-d0)/d0≈-cotθ0·(θψ-θ0)
2、 Deep derivation of stress-strain relationship: from lattice to macroscopic
The above measurement is the lattice strain εψ in a specific direction (the direction perpendicular to the normal direction of the sample surface). To correlate it with macroscopic stress, we need to rely on elastic mechanics.
Assumptions and Models: Typically, it is assumed that the material is a continuous, isotropic polycrystalline material and is in a plane stress state (with a shear stress of 0). At this point, according to the generalized Hooke's law, the relationship between the strain in any direction and the principal stress (∑₁₁, ∑₂₂) can be derived in the sample coordinate system.
Key formula: sin ² ψ method:
The derivation result establishes the relationship between the measured directional strain εψ and the stress tensor components. The relationship between the normal direction of a given crystal plane and the normal direction of the sample surface can be simplified as:
εψ =[(1+ν)/E]σφsin²ψ-[ν/E](σ11+σ₂₂)
Among them, E is the Young's modulus, ν is the Poisson's ratio, and σφ is the stress on the surface of the sample at an angle of φ to the rotation axis of the goniometer (σφ=σ₁₁ cos ² φ+σ₂₂ sin ² φ+τ₁₂ sin2 φ).
Stress calculation:
This formula indicates that for a fixed direction of φ, there is a linear relationship between εψ and sin ②ψ. By measuring a series of diffraction angles θ psi at different angles, calculate the corresponding ε psi, and then perform linear fitting on sin ² psi. The slope M of the fitted line is:
M=[(1+ν)/E]σφ
Therefore, we can ultimately calculate the actual stress in that direction:
σφ=[E/(1+ν)]·M
At this point, we have completed a complete depth derivation from microscopic diffraction geometry to macroscopic stress calculation, laying a solid theoretical foundation for quantitative analysis of X-ray stress measuring instruments.