Integral method

According to [32], Figure 3.9 shows a schematic of the area where hole drilling is applied.

Residual stresses are assumed to be uniform through the depth, and shear stresses at the hole surfaces are neglected. The cross-sectional view, as presented in Figure 3.9a), illustrates the loading applied externally to the hole surface, which is needed to replace the initially existing stress before the hole is drilled; thus, the stress distribution and deformation are maintained.

The main assumption of the method is the fully linear elasticity of the material during the drilling of the hole. By considering the material to be linearly elastic, the principle of superposition can be applied to the model, as shown in Figure 3.9. In the “summand” case in Figure 3.9b), the loading is applied to the hole surface with the opposite sign but equal value to the force in Figure 3(a), whereas the boundaries are unstressed. The result of the superposition shown in Figure 3.9c) expresses the residual stress distribution after the hole has been drilled. In conclusion, the loading case in Figure 3.9b) represents the residual stress redistribution, which in turn causes strain relaxations at the top surface. These relaxations are measured by strain gages and then used for the calculation of residual stresses. It is worth noting that the measured strain relaxations are caused only by residual stresses initially existing at the hole surface [32] and are assumed not to be affected by the drilling process.

(a) (b) (c)

Figure 3.9 Superposition of loadings for hole drilling simulation; a) original stress state, b) stress redistribution, c) final stress state. Depicted and adopted from [32].

All destructive methods for residual stress measurements have a main feature, which involves the elimination of the material in the place where residual stresses are to be determined and the measurement of deformations in the area surrounding the removal. The different locations of stresses varying over the depth and measured deformations at the top surface make the inverse procedure computationally challenging. ESPI provides a large amount of data about displacements from certain points within the optical image, thereby substantially enhancing the measurement accuracy and reliability. However, the computational challenge is to use all the data properly and efficiently to save time and computational resources.

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For the calculation of the stress profiles [144], the measured deformations depend on the contributions of the various stresses originally existing in the removed layers of material. The integral relationship between the deformation at the top surface point and the depth-dependent stress is as follows:

𝑑𝑑(ℎ) =� 𝐺𝐺(𝐻𝐻,ℎ)𝜎𝜎(𝐻𝐻)𝑑𝑑𝐻𝐻

ℎ

0

(3.2) Here, 𝐺𝐺(𝐻𝐻,ℎ) is the main function providing the information about deformation caused by a unit stress at depth 𝐻𝐻 within a hole of depth ℎ, and 𝑑𝑑(ℎ) is the measured deformation parallel to the specimen surface, as a projection on the sensitivity vector, while the out-of-plane displacement component is neglected. 𝜎𝜎(𝐻𝐻) is the in-plane equivalent uniform stress. In case of equal biaxial stress profiles induced by LSP, the equivalent uniform stress equals each of the biaxial components. The stress profile enclosed within the integral can be determined by an inverse approach. A widely accepted way to solve this inverse problem is by the expansion of the stress as a series [144]:

𝜎𝜎(𝐻𝐻) =� 𝑐𝑐_𝑚𝑚𝑢𝑢_𝑚𝑚(𝐻𝐻)

𝑚𝑚

𝑗𝑗=1

(3.3) Here, 𝑢𝑢_𝑗𝑗(𝐻𝐻) are the pulse functions and 𝑐𝑐_𝑗𝑗 are the numerical coefficients to be calculated. Both Equations (3.2) and (3.3) compose the final matrix equation:

𝐺𝐺 ∙ 𝑐𝑐⃗=𝑑𝑑⃗ (3.4)

where

𝐺𝐺𝑚𝑚𝑗𝑗 =� 𝐺𝐺(𝐻𝐻,^ℎ𝑗𝑗 ℎ𝑚𝑚)𝑢𝑢_𝑚𝑚(𝐻𝐻)𝑑𝑑𝐻𝐻

0

(3.5) The pulse function 𝐺𝐺_{𝑚𝑚𝑗𝑗} is illustrated schematically in Figure 3.10, where coefficient 𝐺𝐺₃₂ reflects the deformation caused by a unit stress within Step 2 of a hole that is three increments deep.

More details about the pulse function 𝐺𝐺_{𝑚𝑚𝑗𝑗} can be found in the work of Schajer [21]. For a given geometry and loading situation, the corresponding relaxation matrix is obtained through FE simulations. Every component in the displacement vector 𝑑𝑑⃗ is the projection of surface displacement on the sensitivity vector, which in this study is defined to be lying on the specimen surface and passing through the center of the hole. Given that the surface deformation 𝑑𝑑 can be measured, the inverse solution can be implemented by solving the system of Equation (3.6) or Matrix Notation (3.7):

3.2 Numerical methods dependence of the material relaxations on material properties,𝑛𝑛 is total number of hole depth increments, and 𝜎𝜎⃗ is the equivalent uniform stress. The shapes of 𝐺𝐺,𝜎𝜎⃗,𝑑𝑑⃗ are as follows:

𝐺𝐺 = �𝐺𝐺11 ⋯ 0

The size of the lower triangular matrix 𝐺𝐺 depends on the level of discretization. The more elements are contained in the matrix, the more accurate is the solution provided by Integral method. In this study, the relaxation matrix is 20 × 20. This allows the determination of the stress profiles with an uncertainty less than 2% in the elastic case. This source of error can be neglected compared to the effect of plastic deformation.

Figure 3.10 Physical interpretation of matrix coefficients 𝐺𝐺_{𝑚𝑚𝑗𝑗}, according to [21].

One of the applicable computational methods for solving matrix equations is the least-squares fit.

Because of the full-field measurement, the number of unknowns greatly exceeds the number of stress quantities to be determined. A least-squares solution provides the “best fit” to the measured data in this overdetermined calculation. The data corresponding to the resulting solution differs from the measured data, wherein the difference is called “misfit.” Typically, the misfit has a random character and is associated with the measurement noise.

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