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Stress concentration around a rectangular cuboid hole in a three-dimensional elastic body under tension loading

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Abstract

Stress concentration caused by holes can be investigated by numerical and analytical methods. Current analytical methods can only solve two-dimensional problems. This paper proposes an analytical study on a three-dimensional stress concentration problem that involves a rectangular cuboid hole in a three-dimensional elastic body under tension loading. Based on the finite element method and U-transformation method, the problem can be expressed as a set of uncoupled equations with cyclic periodicity. Displacements of the three-dimensional elastic body are derived in analytical form to study stress distribution in it. Numerical simulation is conducted using ABAQUS to verify the analytical solution. Stress concentration factors in cases of uniaxial, biaxial, and triaxial tensions and the effect of the side ratio of the hole on them are discussed.

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Acknowledgements

This work was performed with the partial support from the National Natural Science Foundation of China through Award (Nos. 11772100, 11372113, and 51508202) and China Scholarship Council through Award No. 201606155085.

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Correspondence to Weidong Zhu.

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Appendix: Calculation of stresses in the cuboid under biaxial and triaxial tensions

Appendix: Calculation of stresses in the cuboid under biaxial and triaxial tensions

Under biaxial and triaxial tensions, the cuboid still has Eq. (14) as its governing equation. Using the trilinear shape function to derive the stiffness matrix \({\varvec{K}}^{\mathrm{e}}\), one obtains components of the generalized stiffness matrix in Eq. (20). Generalized loading vectors are different from those in Eq. (21). In Fig. 8a, the cuboid is subjected to biaxial tensions along x and z directions with \(p_x =p_z =p\) and \(p_y =0\). Substituting them into Eqs. (5) and (17) yields

$$\begin{aligned} f_{1\left( {r,s,t} \right) }= & {} \left( {1-e^{-{i}\theta _1 }} \right) \left( {1+e^{-{i}\theta _2 }} \right) \left( {1+e^{-{i}\theta _3 }} \right) \nonumber \\&\left( {-\frac{bc\left( {\mu -1} \right) }{a}u_{1\left( {h_1 ,j_1 ,k_1 } \right) } +c\mu v_{1\left( {h_1 ,j_1 ,k_1 } \right) } +b\mu w_{1\left( {h_1 ,j_1 ,k_1 } \right) } -\frac{pbc}{D}} \right) \nonumber \\ f_{2\left( {r,s,t} \right) }= & {} \left( {1+e^{-{i}\theta _1 }} \right) \left( {1-e^{-{i}\theta _2 }} \right) \left( {1+e^{-{i}\theta _3 }} \right) \nonumber \\&\left( {c\mu u_{1\left( {h_1 ,j_1 ,k_1 } \right) } -\frac{ac\left( {\mu -1} \right) }{b}v_{1\left( {h_1 ,j_1 ,k_1 } \right) } +a\mu w_{1\left( {h_1 ,j_1 ,k_1 } \right) } } \right) \nonumber \\ f_{3\left( {r,s,t} \right) }= & {} \left( {1+e^{-{i}\theta _1 }} \right) \left( {1+e^{-{i}\theta _2 }} \right) \left( {1-e^{-{i}\theta _3 }} \right) \nonumber \\&\left( {b\mu u_{1\left( {h_1 ,j_1 ,k_1 } \right) } +a\mu v_{1\left( {h_1 ,j_1 ,k_1 } \right) } -\frac{ab\left( {\mu -1} \right) }{c}w_{1\left( {h_1 ,j_1 ,k_1 } \right) } -\frac{pab}{D}} \right) \end{aligned}$$
(A1)

In Fig. 8b, the cuboid subjected to triaxial tensions with \(p_x =p_y =p_z =p\). Substituting them into Eqs. (5) and (17) yields

$$\begin{aligned} f_{1\left( {r,s,t} \right) }= & {} \left( {1-e^{-{i}\theta _1 }} \right) \left( {1+e^{-{i}\theta _2 }} \right) \left( {1+e^{-{i}\theta _3 }} \right) \nonumber \\&\left( {-\frac{bc\left( {\mu -1} \right) }{a}u_{1\left( {h_1 ,j_1 ,k_1 } \right) } +c\mu v_{1\left( {h_1 ,j_1 ,k_1 } \right) } +b\mu w_{1\left( {h_1 ,j_1 ,k_1 } \right) } -\frac{pbc}{D}} \right) \nonumber \\ f_{2\left( {r,s,t} \right) }= & {} \left( {1+e^{-{i}\theta _1 }} \right) \left( {1-e^{-{i}\theta _2 }} \right) \left( {1+e^{-{i}\theta _3 }} \right) \nonumber \\&\left( {c\mu u_{1\left( {h_1 ,j_1 ,k_1 } \right) } -\frac{ac\left( {\mu -1} \right) }{b}v_{1\left( {h_1 ,j_1 ,k_1 } \right) } +a\mu w_{1\left( {h_1 ,j_1 ,k_1 } \right) } -\frac{pac}{D}} \right) \nonumber \\ f_{3\left( {r,s,t} \right) }= & {} \left( {1+e^{-{i}\theta _1 }} \right) \left( {1+e^{-{i}\theta _2 }} \right) \left( {1-e^{-{i}\theta _3 }} \right) \nonumber \\&\left( {b\mu u_{1\left( {h_1 ,j_1 ,k_1 } \right) } +a\mu v_{1\left( {h_1 ,j_1 ,k_1 } \right) } -\frac{ab\left( {\mu -1} \right) }{c}w_{1\left( {h_1 ,j_1 ,k_1 } \right) } -\frac{pab}{D}} \right) \end{aligned}$$
(A2)

When displacements of node 1 of the hole element are worked out, nodal displacements of other elements can be obtained. Stress distribution of the cuboid can then be analyzed using the stress matrix, as mentioned in Sect. 5.

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Yang, Y., Cheng, Y. & Zhu, W. Stress concentration around a rectangular cuboid hole in a three-dimensional elastic body under tension loading. Arch Appl Mech 88, 1229–1241 (2018). https://doi.org/10.1007/s00419-018-1369-7

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