PSI - Issue 33

Davide Palumbo et al. / Procedia Structural Integrity 33 (2021) 528–543

530

Q i,i r, θ

Heat flux through the surface of the body whose outward direct normal is n i

Polar coordinates

R Stress ratio s First stress invariant s� T Reference temperature T� α ε ij Strain tensor ε �� � Strain tensor rate ε � � Principal strain rate ε ΔT ε ΔTr% First stress invariant rate Temperature variation rate

Coefficient of linear thermal expansion

Absolute error in ΔT evaluation Percentage relative error in ΔT evaluation Absolute error in K I evaluation Percentage relative error in K I evaluation

ε KI

ε KI %

ε i

Principal strain First strain invariant Kronecker’s delta

ε kk δ ij

ΔT nc ΔT c µ, λ

Non-correct value of temperature variations Correct value of temperature variations

Lamé constants

ρ 0 σ ij

Density

Stress tensor σ i Principal stress σ � � σ m

Principal stress rate Mean uniaxial stress Amplitude uniaxial stress

a

υ

Poisson’s ratio

2. Theory of Thermoelastic Stress Analysis

The thermoelastic effect, Stanley (1997) describes the change of temperature and the change of the sum of normal stresses for an isotropic material in linear elastic and adiabatic conditions. In particular, temperatures and stresses are related by the thermoelastic constant that is generally assumed to be constant independently on the applied stress. For some materials, a linear dependence of the thermoelastic signal from the mean stress value is present (Belgen (1968)), the physical explanation of this effect and a review of the thermoelastic theory was presented by Wong et al. (1987). For an isotropic material without any internal heat source, the temperature variations can be described by the following equation: � � � � �� �� �� � �� � � ��� ������� � � ��� where ρ 0 is the density of the material, C ε is the specific heat under constant strain, T is the temperature, σ ij is the stress tensor, ε ij is the strain tensor (summing over i,j with i,j=1-3 ) and Q i,i is the heat flux through the surface of the body whose outward direct normal is n i . The dotted symbols represent derivatives with respect to the time Wong et al. (1987). The constitutive law is: �� � �� �� � � �� � �� � �� ����������������������������������� � � ����