## 2 Noncontact Mechanical Property Testing Methods
### Photoacoustic (PA) Techniques  
Regarding optical NDT techniques, the term \"photo\" is used here to refer to the phenomenon of photonic-to-acoustic (specifically ultrasonic) energy conversion. The photo-thermo-elastic or simply photoacoustic (PA) effect was discovered by [PERSON] in 1880 [41], followed by a number of theoretical and experimental research reports on the physical principle of ultrasound  
Figure 3: Scheme of residual stress detection by X-ray diffraction. generation in solids by transient optical power absorption [42; 43; 44]. From the acoustic point of view, the existence of stress in a sample can affect the propagation of elastic waves because of elastic inhomogeneity or anisotropy. Such a phenomenon was first discussed theoretically by seismologist [PERSON] [45] and experimentally validated by [PERSON] and [PERSON] [46] and [PERSON] and [PERSON] [47]. They concluded that the existence of static pre-stress can change the acoustic wave velocity inside a medium, a phenomenon described as the acoustoelastic effect [48; 49; 50]. In the case of irreversible residual stress, finite and irrecoverable deformation and strain should be considered with the result that the stress-strain relation becomes nonlinear [51]. According to [PERSON]'s finite deformation of elastic materials, the stress-strain relation should be determined with the help of the free-energy function \(W_{s}\) which is defined as [46; 52]:  
\[W_{s}=\frac{1}{2}(\lambda+2\mu)I_{1}^{2}-2\mu I_{2}+\frac{1}{3}(l+2m)I_{1}^{3} -2 mI_{1}I_{2}+nI_{3} \tag{3}\]  
Here, \(\lambda\) and \(\mu\) are the Lame constants with respect to infinitesimal deformation, \(l\), \(m\) and \(n\) are [PERSON]'s constants determined by the type of sample material, \(I_{1}\), \(I_{2}\) and \(I_{3}\) are the strain invariants of first-, second- and third-order. Conservation of energy requires Hooke's Law to be expressed as:  
\[\rho\delta W_{s}=\sigma_{ij}\frac{\partial\delta u_{i}}{\partial x_{j}} \tag{4}\]  
where \(\delta W\) and \(\delta u_{i}\) denote the finite increment in free-energy function and displacement field, respectively. \(\rho\) is the density after deformation. Combining Equations (3) and (4) results in an acoustoelastic equation which connects static loading with the elastic wave velocity under hydrostatic pressure \(P\):  
\[ \begin{array}{l}\rho_{0}v_{c}^{2}=\lambda+2\mu-\frac{P}{3\lambda+2\mu}(6l+4 m+7\lambda+10\mu)\\ \rho_{0}v_{s}^{2}=\mu-\frac{P}{3\lambda+2\mu}(3m-0.5n+3\lambda+6\mu)\end{array}  \tag{5}\]  
The subscripts \(c\) and \(s\) imply compressive (longitudinal) and shear wave respectively. Subscript 0 denotes the undeformed state. The level of stress can thus be determined by measuring the wave velocity based on Equation (5) [53]. A typical ultrasonic stress measurement setup is illustrated by Figure 4. The sample is fixed on a tensile machine and subjected to uniaxial loading. Ultrasonic transducers are used to generate and receive an acoustic pulse that propagates along the sample and measure the wave velocity from the two ends. [PERSON] and [PERSON] extended this time-of-flight velocity measurement to metallic cable strands and determined their stress-strain state [54]. [PERSON] et al. focused on stressed soft solids and explored shear modulus measurements [55]. Besides bulk waves, surface acoustic waves were also used to evaluate the existence and level of surface residual stress. A nonlinear acoustoelastic theory with respect to Rayleigh wave propagation was introduced by [PERSON] and [PERSON] [56]. They concluded that the propagation of a Rayleigh surface wave in finitely deformed solids satisfies the wave equation similar to the linear-elastic material, only the elastic modulus tensor loses some of the symmetric conditions as the regular stress-strain relation. More recently, [PERSON] et al. used an interdigitated transducer and a laser interferometer to test superficial residual stress experimentally [57; 58]. The transducer was designated for surface wave generation using a contacting approach.