## 2 Noncontact Mechanical Property Testing Methods
### Photothermal (PT) Techniques  
The discussion of PA signal dependence on the stress-strain state is based on the coupling between elastic and thermophysical properties such as density and bulk modulus, the change of which has perceptible impact on the acoustic signal. However, according to the thermodynamic theory proposed by [PERSON] and [PERSON] [71] and [PERSON] [72], a change of stress-strain energy can affect another two groups of parameters apart from elasticity, namely, thermophysical properties [69] and the thermoelastic coupling coefficient [73; 74]. Therefore, it is possible to determine the mechanical performance of a material remotely by examining its thermal property changes by means of, e.g., thermal infrared detectors. The earliest implementation of this relationship is found in a patented instrumentation system named Stress Pattern Analysis by measurement of Thermal Emission, or SPATE [75]. A sample under test is fixed on a repetitive loading machine which provides a constant strain rate. The temperature associated with the strain energy change is measured with an infrared detector. The measurement is based on the relation between temperature and stress level described in Equation (6) and as a result a quantitative measurement of stress can materialize [76]. SPATE is a passive infrared thermographic (IT) system that does not require pumping external energy: the thermal radiation is the result of stress work in the sample. In 1988, [PERSON] et al. reported that IT method can also be used for characterizing static loading in addition to cyclic loading state such as in a SPATE system [77]. They exerted a static bending load on an aluminum plate and captured thermographic images with an infrared camera. By analyzing the images before and after static deformation, they validated the possibility of detecting residual stress in metallic materials [77]. Similar tests were carried out for thermographical residual stress analysis in titanium alloys in NASA by [PERSON] and [PERSON] [78]. [PERSON] et al. tested a series of steel and aluminum alloys with a commercial tensile machine and an IT setup. The thermoelastic signal even showed a qualitative stress distribution around holes on plates [79]. For hyper-elastic materials such as polymers, the use of passive IT can also diagnose residual stress and fatigues in them [80]. More recently, the development of NDT in reinforced composites drew the attention of a number of researchers. When the laminated structure was subjected to high levels of tensile, bending and torsional loading, the formation of internal delamination and micro cracks occurred along with energy release. By observing thermal infrared emission with a camera, IT techniques enables inspection of a whole piece of material and analysis of the inelastic stress-strain performance [81; 82]. Generally, the heat equation in the presence of static stress can be derived from the first law of thermodynamics, which has the following form [70; 83; 84; 85]:  
\[\rho C\frac{\partial T}{\partial t}-\left(k_{il}J_{ii}^{2}\frac{\partial^{2}T}{ \partial x_{i}^{2}}\delta_{ij}+(1-\delta_{ij})(k_{ij}+k_{jl})J_{il}J_{jj}\frac {\partial^{2}T}{\partial x_{i}\partial x_{j}}\right)=g \tag{9}\]  
where, \(J_{ij}\) is a Jacobian indicating the coordinate transformation due to loading (repeated subscripts denote summation), \(g\) is the source term and \(k_{ij}\) is the thermal conductivity tensor. It can be shown that the thermal conductivity tensor has a direct connection with the static stress-strain state [45; 60; 71; 83]. As a result, instead of the SPATE mode which is suitable for evaluating cyclic loading, the static stress-strain performance can be determined based on thermal anisotropy analysis [85; 86] with a proper heat source, usually a laser, which brings photothermal (PT) techniques into this consideration. Compared with PA techniques, PT detection decouples the stress-strain dependence of many physical properties by focusing only on thermal fields.