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Interfacial micromechanics study on contact modeling for bolted joints

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Abstract

Bolted joints represent the discontinuity of the assembled structures, so their contact characteristics contribute significantly to the overall static and dynamic performances of the mechanical system. For the multi-scale geometrical properties of contacting surfaces, an interfacial micromechanics modeling method is proposed to predict contact characteristics using the fractal theory. Meanwhile, the interaction effect caused by the successive tightening of multiple bolts is incorporated into the contact analysis, which characterizes the residual preload of bolts to improve the contact load model. Three contact models for the bolted joint are examined by combining the interfacial micromechanics model with the transfer matrix method for multi-body systems, the finite element method, and the virtual material method. A comparison with the experimental data of a dumbbell-shaped bolted structure is conducted to validate the contact models and estimate their practicality and accuracy. The models show their advantages and drawbacks, which depend on the complexity of the bolted structure, the requirements of computational efficiency, and the research focus.

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Data availability

The data and material generated or analyzed during this study are available from the corresponding author on reasonable request.

Code availability

The code generated or analyzed during this study is available from the corresponding author on reasonable request.

Abbreviations

\(A_{a}\) :

Nominal contact area of joint, m2

\(A_{b1}\) :

Nominal cross-sectional area of bolt, m2

\(A_{rf}\) :

Real contact area of joint, m2

\(a\) :

Contact area of asperity, m2

\(a^{\prime}\) :

Truncated area of asperity, m2

\(a^{\prime}_{c}\) :

Critical truncated area, m2

\(a^{\prime}_{l}\) :

Largest truncated area of asperity, m2

\({\varvec{B}}\) :

Strain–displacement matrix

\(b\) :

Width of interface, m

\(D\) :

Fractal dimension

\({\varvec{D}}_{e}\) :

Elastic matrix

\(d\) :

Distance from bolt hole, m

\(\overline{E}\) :

Equivalent elastic modulus, Pa

\(E_{bi}\) :

Elastic modulus of bolt #\(i\), Pa

\(E_{V}\) :

Elastic modulus of virtual material layer, Pa

\(E_{Ve} ,E_{{Vep}{\varvec{I}}} ,E_{{Vep}{\varvec{II}}}\) :

Total elastic moduli in elastic and elastic–plastic stages, Pa

\(e_{e} ,e_{{ep}{\varvec{I}}} ,e_{{ep}{\varvec{II}}}\) :

Elastic moduli of asperity in elastic and elastic–plastic stages, Pa

\(F_{i}\) :

Preloads applied to bolt #\(i\), N

\(F_{i,\;n}\) :

Residual preload of bolt #\(i\) after tightening \(n\) bolts, N

\(G\) :

Fractal roughness parameter, m

\(\overline{G}\) :

Equivalent shear modulus, Pa

\(G_{V}\) :

Shear modulus of virtual material layer, Pa

\(G_{Ve} ,G_{{Vep}{\varvec{I}}} ,G_{{Vep}{\varvec{II}}}\) :

Total shear moduli in elastic and elastic–plastic stages, Pa

\(g_{e} ,g_{{ep}{\varvec{I}}} ,g_{{ep}{\varvec{II}}}\) :

Shear moduli of asperity in elastic and elastic–plastic stages, Pa

\(H\) :

Hardness, Pa

\(H_{c}\) :

Hardness coefficient

\(h\) :

Thickness of virtual material layer, m

\(h_{1} ,h_{2}\) :

Thicknesses of two asperity layers, m

\({\varvec{J}}\) :

Jacobian matrix

\({\boldsymbol{J}}_{Body}\) :

Inertia matrix

\({\varvec{K}}_{B} ,{\varvec{K}}_{C}\) :

Structural stiffness matrixes of Model B and C

\(K_{bi}\) :

Stiffness of bolt #\(i\), N/m

\(K_{i,i}\) :

Local compression stiffness at bolt hole #\(i\), N/m

\(K_{i,j}\) :

Interaction stiffness between bolt #\(i\) and bolt #\(j\), N/m

\(K_{jx} ,\;K_{jy} ,\;K_{jz}\) :

Stiffness coefficients of the \(jth\) set of springs in three directions, N/m

\(K_{ne} ,K_{{nep}{\varvec{I}}} ,K_{{nep}{\varvec{II}}}\) :

Total normal contact stiffnesses at different stages, N/m

\({\boldsymbol{K}}_{L} ,\;{\boldsymbol{K}}_{R}\) :

Longitudinal stiffness matrix and rotary stiffness matrix

\(K_{Lx} ,K_{Ly} ,K_{Lz}\) :

Longitudinal stiffnesses in three directions, N/m

\({\varvec{K}}_{P1} ,{\varvec{K}}_{P2}\) :

Stiffness matrixes of Parts #1 and #2

\(K_{Rx} ,\;K_{Ry} ,\;K_{Rz}\) :

Rotary stiffnesses in three directions, Nm/rad

\({\varvec{K}}_{S}\) :

Stiffness matrix of all spring elements

\({\varvec{K}}_{Sj}^{e}\) :

Stiffness matrix of the \(jth\) group spring elements

\(K_{te} ,K_{{tep}{\varvec{I}}} ,K_{{tep}{\varvec{II}}}\) :

Total tangential contact stiffnesses at different stages, N/m

\({\varvec{K}}_{V}\) :

Stiffness matrix of virtual material layer

\({\varvec{K}}_{Vi}^{e}\) :

Stiffness matrix of element \(i\)

\(k_{ne} ,k_{{nep}{\varvec{I}}} ,k_{{nep}{\varvec{II}}}\) :

Normal contact stiffnesses of asperities at different stages, N/m

\(k_{t}\) :

Tangential contact stiffness of asperity, N/m

\(L_{b1}\) :

Effective bolt length, m

\(l\) :

Length of interface, m

\({\tilde{\boldsymbol{l}}}_{IC} ,{\tilde{\boldsymbol{l}}}_{IO}\) :

Coordinates cross-product matrixes of mass center \(C\) and output end \(O\)

\({\varvec{M}}_{B} ,{\varvec{M}}_{C}\) :

Structural mass matrixes of Model B and C

\({\varvec{M}}_{P1} ,{\varvec{M}}_{P2}\) :

Mass matrixes of Parts #1 and #2

\({\varvec{M}}_{V}\) :

Mass matrix of virtual material layer

\({\varvec{M}}_{Vi}^{e}\) :

Mass matrix of element \(i\)

\({\varvec{N}}\) :

Displacement interpolation matrix

\(P_{e} ,P_{{ep}{\varvec{I}}} ,P_{{ep}{\varvec{II}}} ,P_{p}\) :

Total normal contact loads at different stages, N

\(p_{e} ,p_{{ep}{\varvec{I}}} ,p_{{ep}{\varvec{II}}} ,p_{p}\) :

Normal contact loads of asperities at different stages, N

\(p_{re} ,p_{repi}\) :

Real normal contact pressures of asperities, Pa

\(R\) :

Equivalent radius of asperity, m

\(r\) :

Contact radius of asperity, m

\(r^{\prime}\) :

Truncated radius of asperity, m

\(T\) :

Tangential load, N

\(T_{e} ,T_{{ep}{\varvec{I}}} ,T_{{ep}{\varvec{II}}}\) :

Total tangential contact loads, N

\(t\) :

Tangential contact load of asperity, N

\({\boldsymbol{U}}_{{Part}\_1} ,{\boldsymbol{U}}_{{Part}\_2}\) :

Transfer matrixes for Parts #1 and #2

\({\mathbf{U}}_{Hinge}\) :

Transfer matrix for hinge

\({\varvec{u}}\) :

Nodal displacement matrix

\(v\) :

Poisson ratio of the softer material

\(\overline{v}\) :

Equivalent Poisson ratio

\(v_{V}\) :

Poisson ratio of virtual material layer

\({\boldsymbol{Z}}_{I} ,{\boldsymbol{Z}}_{O}\) :

State vectors of input end \(I\) and output end \(O\)

\(\Delta F_{1,2}\) :

Preload variation of bolt #1 caused by tightening bolt #2, N

\(\Delta u_{1,2}\) :

Deformation variation of bolt hole #1 caused by tightening bolt #2, m

\(\Delta u_{b1,2}\) :

Deformation variation of bolt #1 caused by tightening bolt #2, m

\(\delta\) :

Asperity interference, m

\(\delta_{c}\) :

Critical interference, m

\(\delta_{t}\) :

Tangential interference, m

\(\gamma\) :

Spatial frequency spectrum parameter

\(\mu\) :

Friction coefficient

\(\rho_{V}\) :

Density of virtual material layer, kg/m3

\(\rho_{1} ,\;\rho_{2}\) :

Densities of Parts #1 and #2, kg/m3

\(\psi\) :

Domain extension factor

*:

Dimensionless form

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Acknowledgements

This work was supported by Science Challenge Project (No. JDZZ2016006-0102). Author Yu Chang has received support from China Scholarship Council (No. 202106840001).

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Authors and Affiliations

  1. School of Science, Nanjing University of Science and Technology, Nanjing, 210094, People’s Republic of China

    Yu Chang & Jianguo Ding

  2. School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore, 639798, Republic of Singapore

    Yu Chang & Hui Fan

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Contributions

YC and HF were involved in the conceptualization and methodology; YC contributed to the formal analysis and investigation; YC assisted in the writing—original draft preparation; HF and JD were involved in writing—review and editing; JD and YC acquired the funding; JD contributed to resources; JD and HF were involved in the supervision.

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Correspondence to Jianguo Ding.

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Chang, Y., Ding, J. & Fan, H. Interfacial micromechanics study on contact modeling for bolted joints. Acta Mech 234, 3377–3396 (2023). https://doi.org/10.1007/s00707-023-03562-x

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