## 2 Noncontact Mechanical Property Testing Methods
### Photoacoustic (PA) Techniques
The advantage of acoustic methodologies is their multimode propagation which is applicable for both internal and superficial stress-strain evaluation. However, the generation and detection of ultrasonic waves invariably relies on contact transducers. Based on the PA effect investigated in the 1980s and 1990s, it is possible to realize noncontact acoustic stress-strain characterization. [PERSON] and [PERSON] first discussed the possibility of stress distribution measurements using a PA microscope [59]. Because of the existence of photon-phonon energy conversion, the heat equation was modified as:  
\[\frac{Q}{T_{0}}+\rho C\frac{\Delta T}{T_{0}}=-\left(\alpha-\frac{1}{E^{2}} \frac{\partial E}{\partial T}\sigma\right)\Delta\sigma \tag{6}\]  
where, \(E\) is [PERSON]'s modulus, \(\alpha\) is linear expansion coefficient, \(\sigma\) and \(\Delta\sigma\) are static stress and stress change after deformation, \(Q\) is the heat flux induced by photon (laser) irradiation and \(T\) is the temperature, the subscript 0 refers to stress-free state. Equation (6) shows the coupling of stress and temperature change. Using piezoelectric devices, the amplitude of the detected PA signal is proportional to the \(\Delta\sigma\) for small signal approximation. However, Equation (6) does not seem to be comprehensive enough and thus no experimental procedure was proposed in Ref. [59]. [PERSON] et al. published a series of theoretical and experimental reports [60; 61; 62] on PA residual stress evaluation. They started from the basic form of the first law of thermodynamics and [PERSON]'s nonlinear theory and wrote the modified wave equation as:  
\[\frac{\partial P_{ij}}{\partial x_{j}}=\rho_{0}\frac{\partial^{2}u_{i}}{ \partial t^{2}} \tag{7}\]  
where, \(u_{i}\) is the displacement field and \(\rho_{0}\) is the density before deformation. Equation (7) can be regarded as an alternative form to Equation (4) by introducing the Piola-Kirchhoff stress tensor \(P_{ij}\)[63]. Repeated subscripts stand for summation and the equation is in Euler coordinates (\(x_{1}\), \(x_{2}\), \(x_{3}\)). \(P_{ij}\) is a function of strain energy due to pre-stressing (Equation (3)) and photon-phonon conversion, the latter being expressed as:  
\[W_{0}=\left(\frac{3}{2}\lambda+\mu\right)\alpha\left[(1+\beta_{0}l_{1}^{0}) \delta_{ij}+\beta_{1}\varepsilon_{ij}^{0}\right]\left(\frac{\partial u_{i}}{ \partial x_{j}}+\frac{\partial u_{j}}{\partial x_{i}}+\frac{\partial u_{i}}{ \partial x_{k}}\frac{\partial u_{j}}{\partial x_{k}}\right)\Delta T \tag{8}\]  
where, \(W_{0}\) is the free-energy function of laser radiation, \(\beta_{0}\) and \(\beta_{1}\) are thermoelastic coupling coefficients of static strain and repeated subscript summation is used. Further derivation shows that the PA signal is proportional to the vertical displacement component. In their experiments, [PERSON] et al. used several levels of force to produce indentations on a silicon nitride sample and the PA signal was plotted  
Figure 4: Stress measurement scheme by pulsed ultrasound. as a function of laser scanning distance. Figure 5 shows the result using a certain level of indentation with an obvious change of PA signal in the region of indentation-induced flaws. In their reports, [PERSON] et al. did not quantify the correlation between PA signal and the actual local residual stress [60]. Moreover, the system used does not qualify as totally noncontact because of the use of contacting piezoelectric crystals as detectors. [PERSON] et al. developed a noncontact PA stress-strain measurement method using a narrow-bandwidth immersion ultrasonic transducer [64]. The use of a tensile machine and adhesive strain gauge enabled simultaneous measurements of actual strain and PA signal. As shown in Figure 6, in the MHz range, the detected PA signal phase and amplitude show linear-to-nonlinear evolution of local strain from elastic to plastic deformation.