PSI - Issue 39

Riccardo Cappello et al. / Procedia Structural Integrity 39 (2022) 179–193 Author name / Structural Integrity Procedia 00 (2019) 000–000

181

3

Phase of the Second Harmonic SH [°]

φ 2 ω

Angular frequency of the load fundamental harmonic [rad/s]

2. Theoretical background 2.1. Thermoelastic Effect Law and signal processing

According to the first order thermoelastic effect law [1], when an isotropic material or structure is loaded in the elastic field under adiabatic conditions, the temperature change ∆ T in each point of the tested medium is linearly related to the first stress invariant: ( ) o th xx yy T T K σ σ ∆ + = − ∆ (1) where T o is the mean body temperature and K th is the thermoelastic constant, that depends on the physical properties of the tested material. Thermoelastic Stress Analysis (TSA) is the full-field experimental, non-contact, technique that is capable of retrieving information about the stresses of the tested medium by means of Eq. (1). Since the thermoelastic effect is a reversible phenomenon, the adiabatic conditions are usually ensured applying a cyclical load, having a loading frequency (LF) sufficiently high (usually LF>1 Hz for steels), to avoid thermal exchanges. If a sinusoidal load varying with a ω L pulsation is applied, the temperature will follow the modulation imposed from the load. A filtering operation is required to extract the temperature variation associated with the thermoelastic effect from the sampled temperature signal. The Least Squares Fitting approach of the following multiparameter expression has been adopted in this work: The coefficient b is included to also consider any linear increase of temperature, while ∆ T 1 is the First Harmonic (FH) amplitude, that is correlated to the thermoelastic effect (Eq. (1)), ∆ T 2 is the temperature variation at twice the loading frequency, hence called Second Harmonic (SH) amplitude. φ 1 and φ 2 are respectively the FH phase shift and the SH phase shift. Higher order terms could also be included that could further improve the quality of the fitting. Including harmonic components at frequencies higher than the FH is generally useful to detect the effects of dissipative phenomena that may arise in the material, which may introduce changes in the temperature evolution that have a significant signature at twice the loading frequency [24], [25]. To obtain an accurate evaluation of the thermoelastic metrics, some signal sampling features should be carefully controlled, such as frame drop or spectral leakage, which are taken into account and opportunely corrected [2]. 2.2. Williams’ stress formulation The mode I Williams’ series formulation of the stress field, valid for linear elastic isotropic media, can be expressed in terms of the sum of normal stresses as [3]: 1 1 2 2 ( ) ) + + ) ... ) sin( sin(2 sin( T n t L mean L n L n T t b t T T t ω φ + T t ω φ ω φ + + ⋅ = + + + ∆ ∆ ∆ (2)

   

   

  



2 n

∞ ∑

2 / 1 −

(

)

2 n

co

s

1

A nr

xx σ σ

θ

+ = 

(3)

 − 

yy

n

 

1

n

=

Where r and θ are the coordinates in a polar coordinate system centered on the crack tip, while A n is the n th unknown coefficient of the Williams’ series to be determined. If m is the number of available experimental points from TSA and N w is the number of Williams’ series terms considered, eq. (3) can be re-written in matrix form. Combining Eqs. (1) and (3) it is possible to directly correlate ∆ T 1 and the Williams’ stress formulation, obtaining: