Verification of Equation for Determining Non-uniform Residual Stresses by FEM

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The shape of this function accurately describes the deformation around drilled hole (Figure 1). Coefficients and used in equation (1) are defined according to the recommendations of the standard ASTM E837.

Figure 2. Hole geometry and residual stresses for non-uniform stresses

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2. Modeling of Specimen with Strain Gage Rosette

For modeling was used software COSMOS / M. Shaped sample was modeled as a rectangular plate (Figure 3). Due to the symmetry was modeled only one quarter of the plate what significantly reduces computation time ^[6].

Figure 3. Finite element model

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Individual strain gages were modeled using elements (Truss2D), which allowed easy retrieval of strains. After that the elements of the strain gage model were applied to the finite elements grid model of the plate (Solid8). Modulus of elasticity of the elements is E=1.10^-6 MPa. We used strain gage rosette of type B according to the standard ASTM E837 (Figure 4) – strain gage 1 is identified with the x direction and strain gage 3 is identified with the y direction (Figure 5).

Figure 4. Strain gage rosette of type B

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Figure 5. Model of the strain gage rosette

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Dimensions of used strain gage rosette:

Min D₀= 1.52 mm, R₁ = 1.77 mm, D = 5.13 mm, Max D₀ = 2.54 mm, R₂ = 3.36 mm

We chose diameter D₀ = 2* R₀ = 1.539 mm which is in our flowing interval.

For modeling of the stress field, which is varying in different depths, was used model with simulation of bending, according to Figure 6.

Figure 6. Bending beam

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The thickness Z of the sample was selected which is in according to ASTM E837 considered as a medium sample. The entire thickness of the sample was divided into 32 equal sized sections – one section will have a thickness (Figure 7).

Figure 7. Layers of FE model

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Boundary conditions and loading method is shown in Figure 8.

Figure 8. Boundary conditions

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Stress field σ_x before drilling of the hole is shown in Figure 9.

Figure 9. Stress field

before drilling of the hole

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For exact determination of the state of strains around the hole is necessary to extend the model of strain gage rosette as shown in Figure 10.

Figure 10. Arrangement of the strain gages

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By simulation of progressive drilling and determining of strains in individual strain gages we obtain searched strains. Figure 11 shows strains after drilling of the first layer.

Figure 11. Strain after drilling of the first layer

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This curve is similar as the shape of the curve in the stress graph for the equal distribution of stresses through the thickness of the sample (Figure 12) ^[5].

Figure 12. Relieved strains of different depth increments for uniform residual stresses

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Figure 13. Relieved strains of different depth increments for non-uniform residual stresses

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