Discrete convolution and FFT modified with double influence-coefficient superpositions (DCSS–FFT) for contact of nominally flat heterogeneous materials involving elastoplasticity

Abstract

The contact of nominally flat surfaces can be treated as a bilateral periodic contact problem considering the stochastic surface similarity to the asperity distribution in a representative region. This similarity treatment method can be extended to material inhomogeneities. A novel numerical model for simulating the elastoplastic contact between nominally flat surfaces of materials containing inhomogeneities or coatings is developed via extending the concept of the discrete convolution and FFT (DC–FFT) algorithm with double superpositions of influence coefficients, which is named the DCSS–FFT algorithm. Several cases are analyzed with this new algorithm to examine its convenience, efficiency, and accuracy in dealing with complicated nominally flat–flat contact problems. The effects of surface roughness and material inhomogeneity are explored, and the mechanisms of contact surface failure are discussed.

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Abbreviations

\(a\) :

Half-width of the target domain, mm

\(a_{0}\) :

Half-contact width, mm

\(a_{1} , a_{2} , a_{3}\) :

Element size along the x, y, z directions, respectively, mm

\(a_{x} ,a_{y} ,a_{z}\) :

Size of brick inhomogeneities along the x, y, z directions, respectively, mm

\(C_{ijkl} ,C_{ijkl}^{*}\) :

Coefficients of elastic properties for matrix and inhomogeneities, respectively

\(D\) :

Influence coefficients relating contact pressure and surface displacement

\(D_{ijk}^{*}\) :

Influence coefficients relating eigenstrain and surface displacement

\(D_{ij}^{E}\) :

Influence coefficients relating contact pressure and subsurface elastic stress

\(D^{sum}\) :

Summation of ICs

\(K_{ijk}^{P}\) :

Influence coefficients relating plastic strain and surface residual displacement

\(E,E^{*}\) :

Young’s modulus of matrix and inhomogeneities, respectively, MPa

\(G\) :

Green function relating contact pressure and surface displacement

\(h\) :

Total gap between contact surfaces, mm

\(h_{0}\) :

Initial gap (no deformation) between contact surfaces, mm

\(l_{x} , l_{y} , l_{z}\) :

Size of target domain along x, y, z directions, respectively, mm

\(2m + 1\) :

Number of copies chosen to do the ICs superposition

\(Nx, Ny, Nz\) :

Total element number along the x, y, and z directions, respectively

\(p\) :

Contact pressure, MPa

\(p_{0}\) :

Mean pressure on the target domain, MPa

\(\xi , \eta , \vartheta\) :

Element indices number along the x, y and z directions, respectively

\(f\) :

Yield function

\(\mu_{f}\) :

Friction coefficient

\(T_{ijkl}^{\left( 0 \right)} , T_{ijkl}^{\left( 1 \right)} , T_{ijkl}^{\left( 2 \right)} , T_{ijkl}^{\left( 3 \right)}\) :

Influence coefficients relating eigenstrain and eigenstress

\(\upsilon\) :

Poisson’s ratio

\(u_{Z}\) :

Normal elastic surface displacement, mm

\(u_{i}^{P}\) :

Residual plastic surface displacement, mm

\(u_{i}^{*}\) :

Surface perturbed displacement caused by inhomogeneities, mm

\(\alpha , \beta , \gamma\) :

Element indices number along the x, y, and z directions, respectively

\(\tau_{1}\) :

Maximum shear stress, MPa

\(\delta\) :

Rigid body motion, mm

\(\sigma_{VM}\) :

Von Mises stress, MPa

\(\sigma_{ij}^{P}\) :

Residual stress caused by plasticity, MPa

\(\sigma_{Y}\) :

Yield strength, MPa

\(\sigma_{ij}\) :

Total elastic stress, MPa

\(\sigma_{ij}^{E}\) :

Contact-induced subsurface elastic stress, MPa

\(\sigma_{ij}^{*}\) :

Disturbance stress (eigenstress), MPa

\(\varepsilon_{ij}^{E}\) :

Elastic strain due to the contact-induced elastic stress, %

\(\varepsilon_{ij}^{D}\) :

Disturbance strain caused by inhomogeneities, %

\(\varepsilon_{ij}^{P}\) :

Plastic strain, %

\(\varepsilon_{ij}^{*}\) :

Eigenstrain, %

\(\Gamma _{c}\) :

Set of the elements in the contact area

\(B,c,n\) :

Material constants for isotropic hardening law

\(\lambda\) :

Effective accumulative plastic strain,

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Acknowledgements

L. Sun and Q. Wang would like to thank the support from TimkenSteel, USA, and Center for Surface Engineering and Tribology at Northwestern University, Evanston, USA. L. Sun would like to thank the scholarship support from the China Scholarship Council (Grant No. 201606290100). L. Sun and N. Zhao would like to thank the support from the Fundamental Research Funds for the Central Universities (Grant No. 31020200506005). M. Zhang would like to thank the support from National Natural Science Foundation of China (Grant No. 52005419).

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Appendix

Appendix

ICs for surface deformation of homogeneous materials based on the zero-order approximation

When a uniform pressure, p, is applied on the surface of an elastic half-space in a rectangular area of \(2a_{1} \times 2a_{2}\), the normal surface displacement at point (x, y) can be obtained from the following expression [41].

$$ \begin{aligned} \frac{\pi E}{{1 - \upsilon^{2} }} \cdot \frac{{u_{Z} }}{p} & = \left( {x + a_{1} } \right)\left[ {\frac{{\left( {y + a_{2} } \right) + \sqrt {\left( {y + a_{2} } \right)^{2} + \left( {x + a_{1} } \right)^{2} } }}{{\left( {y - a_{2} } \right) + \sqrt {\left( {y - a_{2} } \right)^{2} + \left( {x + a_{1} } \right)^{2} } }}} \right] \\ & \quad + \,\left( {y + a_{2} } \right)\left[ {\frac{{\left( {x + a_{1} } \right) + \sqrt {\left( {y + a_{2} } \right)^{2} + \left( {x + a_{1} } \right)^{2} } }}{{\left( {x - a_{1} } \right) + \sqrt {\left( {y + a_{2} } \right)^{2} + \left( {x - a_{1} } \right)^{2} } }}} \right] \\ & \quad + \,\left( {x - a_{1} } \right)\left[ {\frac{{\left( {y - a_{2} } \right) + \sqrt {\left( {y - a_{2} } \right)^{2} + \left( {x - a_{1} } \right)^{2} } }}{{\left( {y + a_{2} } \right) + \sqrt {\left( {y + a_{2} } \right)^{2} + \left( {x - a_{1} } \right)^{2} } }}} \right] \\ & \quad + \,\left( {y - a_{2} } \right)\left[ {\frac{{\left( {x - a_{1} } \right) + \sqrt {\left( {y - a_{2} } \right)^{2} + \left( {x - a_{1} } \right)^{2} } }}{{\left( {x + a_{1} } \right) + \sqrt {\left( {y - a_{2} } \right)^{2} + \left( {x + a_{1} } \right)^{2} } }}} \right] \\ \end{aligned} $$
(27)

The general relationships between pressure and vertical surface deformation are defined as the form of discrete convolution:

$$ u_{z[\alpha ,\beta ]} = \sum\limits_{\xi = 0}^{Nx - 1} {\sum\limits_{\eta = 0}^{Ny - 1} {D_{[\alpha - \xi ,\beta - \eta ]} p_{[\xi ,\eta ]} } } $$
(28)

where \(Nx\). and \(Ny\) are the total numbers of elements in the target domain along the \(x\) and \(y\) direction, and \(\alpha\), \(\beta\), \(\xi ,\) and \(\eta\) are the element indices. According to Eq. (27), the ICs can be expressed as:

$$ \begin{aligned} D_{{\left[ {\alpha - \xi ,\beta - \eta } \right]}} & = \frac{{1 - \upsilon^{2} }}{\pi E}\left\{ {\left( {x_{\alpha } - x_{\xi } + a_{1} } \right)\left[ {\frac{{\left( {y_{\beta } - y_{\eta } + a_{2} } \right) + \sqrt {\left( {y_{\beta } - y_{\eta } + a_{2} } \right)^{2} + \left( {x_{\alpha } - x_{\xi } + a_{1} } \right)^{2} } }}{{\left( {y_{\beta } - y_{\eta } - a_{2} } \right) + \sqrt {\left( {y_{\beta } - y_{\eta } - a_{2} } \right)^{2} + \left( {x_{\alpha } - x_{\xi } + a_{1} } \right)^{2} } }}} \right]} \right. \\ & \quad + \,\left( {y_{\beta } - y_{\eta } + a_{2} } \right)\left[ {\frac{{\left( {x_{\alpha } - x_{\xi } + a_{1} } \right) + \sqrt {\left( {y_{\beta } - y_{\eta } + a_{2} } \right)^{2} + \left( {x_{\alpha } - x_{\xi } + a_{1} } \right)^{2} } }}{{\left( {x_{\alpha } - x_{\xi } - a_{1} } \right) + \sqrt {\left( {y_{\beta } - y_{\eta } + a_{2} } \right)^{2} + \left( {x_{\alpha } - x_{\xi } - a_{1} } \right)^{2} } }}} \right] \\ & \quad + \,\left( {x_{\alpha } - x_{\xi } - a_{1} } \right)\left[ {\frac{{\left( {y_{\beta } - y_{\eta } - a_{2} } \right) + \sqrt {\left( {y_{\beta } - y_{\eta } - a_{2} } \right)^{2} + \left( {x_{\alpha } - x_{\xi } - a_{1} } \right)^{2} } }}{{\left( {y_{\beta } - y_{\eta } + a_{2} } \right) + \sqrt {\left( {y_{\beta } - y_{\eta } + a_{2} } \right)^{2} + \left( {x_{\alpha } - x_{\xi } - a_{1} } \right)^{2} } }}} \right] \\ & \quad \left. { + \,\left( {y_{\beta } - y_{\eta } - a_{2} } \right)\left[ {\frac{{\left( {x_{\alpha } - x_{\xi } - a_{1} } \right) + \sqrt {\left( {y_{\beta } - y_{\eta } - a_{2} } \right)^{2} + \left( {x_{\alpha } - x_{\xi } - a_{1} } \right)^{2} } }}{{\left( {x_{\alpha } - x_{\xi } + a_{1} } \right) + \sqrt {\left( {y_{\beta } - y_{\eta } - a_{2} } \right)^{2} + \left( {x_{\alpha } - x_{\xi } + a_{1} } \right)^{2} } }}} \right]} \right\} \\ \end{aligned} $$
(29)

When using the ICs-superposition with the DCSS–FFT algorithm,

$$ D_{{\left[ {\alpha ,\beta } \right]}}^{sum} = \sum\limits_{t = - n}^{n} {\sum\limits_{r = - m}^{m} {D_{{\left[ {\alpha + tNx,\beta + rNy} \right]}}^{{\left( {0,0} \right)}} } } $$
(30)

Then the once forever ICs-superposition becomes:

$$ \begin{aligned} D_{{\left[ {\alpha - \xi ,\beta - \eta } \right]}}^{sum} & = \frac{{1 - \upsilon^{2} }}{\pi E}\sum\limits_{t = - n}^{n} {\sum\limits_{r = - m}^{m} {\left\{ {\left( {x_{\alpha + tNx} - x_{\xi } + a_{1} } \right)\left[ {\frac{{\left( {y_{\beta + rNy} - y_{\eta } + a_{2} } \right) + \sqrt {\left( {y_{\beta + rNy} - y_{\eta } + a_{2} } \right)^{2} + \left( {x_{\alpha + tNx} - x_{\xi } + a_{1} } \right)^{2} } }}{{\left( {y_{\beta + rNy} - y_{\eta } - a_{2} } \right) + \sqrt {\left( {y_{\beta + rNy} - y_{\eta } - a_{2} } \right)^{2} + \left( {x_{\alpha + tNx} - x_{\xi } + a_{1} } \right)^{2} } }}} \right]} \right.} } \\ & \quad + \,\left( {y_{\beta + rNy} - y_{\eta } + a_{2} } \right)\left[ {\frac{{\left( {x_{\alpha + tNx} - x_{\xi } + a_{1} } \right) + \sqrt {\left( {y_{\beta + rNy} - y_{\eta } + a_{2} } \right)^{2} + \left( {x_{\alpha + tNx} - x_{\xi } + a_{1} } \right)^{2} } }}{{\left( {x_{\alpha + tNx} - x_{\xi } - a_{1} } \right) + \sqrt {\left( {y_{\beta + rNy} - y_{\eta } + a_{2} } \right)^{2} + \left( {x_{\alpha + tNx} - x_{\xi } - a_{1} } \right)^{2} } }}} \right] \\ & \quad + \,\left( {x_{\alpha + tNx} - x_{\xi } - a_{1} } \right)\left[ {\frac{{\left( {y_{\beta + rNy} - y_{\eta } - a_{2} } \right) + \sqrt {\left( {y_{\beta + rNy} - y_{\eta } - a_{2} } \right)^{2} + \left( {x_{\alpha + tNx} - x_{\xi } - a_{1} } \right)^{2} } }}{{\left( {y_{\beta + rNy} - y_{\eta } + a_{2} } \right) + \sqrt {\left( {y_{\beta + rNy} - y_{\eta } + a_{2} } \right)^{2} + \left( {x_{\alpha + tNx} - x_{\xi } - a_{1} } \right)^{2} } }}} \right] \\ & \quad \left. { + \,\left( {y_{\beta + rNy} - y_{\eta } - a_{2} } \right)\left[ {\frac{{\left( {x_{\alpha + tNx} - x_{\xi } - a_{1} } \right) + \sqrt {\left( {y_{\beta + rNy} - y_{\eta } - a_{2} } \right)^{2} + \left( {x_{\alpha + tNx} - x_{\xi } - a_{1} } \right)^{2} } }}{{\left( {x_{\alpha + tNx} - x_{\xi } + a_{1} } \right) + \sqrt {\left( {y_{\beta + rNy} - y_{\eta } - a_{2} } \right)^{2} + \left( {x_{\alpha + tNx} - x_{\xi } + a_{1} } \right)^{2} } }}} \right]} \right\} \\ \end{aligned} $$
(31)

This only need to be calculated once for all.

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Sun, L., Wang, Q.J., Zhao, N. et al. Discrete convolution and FFT modified with double influence-coefficient superpositions (DCSS–FFT) for contact of nominally flat heterogeneous materials involving elastoplasticity. Comput Mech (2021). https://doi.org/10.1007/s00466-021-01980-z

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Keywords

  • Nominally flat surfaces contact
  • Inhomogeneities
  • Periodic convolution
  • Fourier transform
  • ICs superposition