# The Key Role of Laser Ultrasonics in the Context of Sustainable Production in an I 4.0 Value Chain
## 4 The Key Role of LUS in a Smart Manufacturing Production Site
### Operating Principle
#### 4.1.2 Attenuation  
The second important aspect, which can be analyzed from the ultrasonic signals and related to microstructural properties of the material, is attenuation. This manifests itself in the reduction of the amplitudes of subsequent echoes. The total attenuation can be attributed to three significant phenomena and added as:  
\[\alpha(f,T)=\alpha_{sc}(f,T)+\alpha_{IF}(f,T)+\alpha_{D}(f) \tag{4}\]  
The three contributions are made by diffraction \(\alpha_{\rm D}\), internal friction \(\alpha_{IF}\), and the contribution of grain scattering \(\alpha_{sc}\). The contribution of diffraction can be estimated qualitatively by the Fresnel parameter, while the contribution of internal friction (caused for example by magneto-mechanical damping, interstitial atoms, or dislocation motion) can be minimized by suitable algorithms, since this contribution is frequency independent to a large extent [110]. In turn, conclusions about recovery processes can be derived from the dislocation motion [111]. The contribution to be extracted is that of the grain boundary scattering, which provides information about the predominant grain size in the material. The contribution of grain scattering is primarily noticeable in metals with high elastic anisotropy and is the dominant contributor to the total damping, especially in metals of this type, such as steel, nickel, or cobalt. The grain size \(D\) is a temperature-dependent factor and damping increases as grain size increases. The contribution of the grain size to the total damping can be calculated via the power-law function  
\[\alpha_{Sc}(f,T)=C(T)D^{n-1}f^{n} \tag{5}\]  
where \(C\) corresponds to a temperature dependent material constant. The value of the exponent \(n\) depends on the ratio of the wavelength to the grain size. Below are the following three regimes:  
Rayleigh regime \((\lambda\gg D):\alpha_{Sc}=C_{c}D^{3}f^{4}\)  
Stochastic regime \((\lambda\approx D):\alpha_{Sc}=C_{S}Df^{2}\)  
Diffusion regime \((\lambda\ll D):\alpha_{Sc}=C_{D}D^{-1}\)  
It is usually assumed that this ratio lies between the Rayleigh and the Stochasitic regime, which is why the value of 3 is usually chosen for n. However, this value can also be fitted specifically for a material, as described in [112].  
To estimate the grain size, the power-law Formula (6) can be fitted into the measured attenuation curves:  
\[a(f,T)=a+bf^{n} \tag{6}\]  
Here, \(a\) represents a frequency-independent contribution that includes, for example, internal friction or external factors such as variations in laser intensity. The expression \(b\) represents the frequency dependent grain size contribution and can be assigned to the actual grain size via  
\[b=\Gamma(T)D^{n-1} \tag{7}\]  
with \(\Gamma(T)=\sqrt{\frac{1}{C(T)}}\), which contains the material and temperature dependent Information.  
Appropriate model calibration with ex-situ tests can be used to draw in-situ conclusions about grain size evolution in thermal and thermomechanical tests, based on the reference echo model or the single echo technique.  
The correlations of damping as well as velocity with microstructure have been tested and published based on numerous investigations on a variety of materials. For example, grain size or grain growth investigations are discussed in [113, 114, 115, 116, 117, 118, 119, 120]. Recrystallization effects can be found in [121, 122, 123, 124, 125, 126, 127, 128, 129], whereas texture measurements where conducted for instance in [130, 131, 132]. Further information of phase transformation and composition can be found in [133, 134, 135, 136, 137].