A numerical method to investigate the effect of thermal and plastic behaviors on the evolution of sliding wear

Abstract

Thermo-mechanical wear widely exists in various mechanical components. This paper improves the elastic finite element wear method to solve sliding wear problems with friction heat and elastoplastic behaviors. Especially, the thermo-mechanical wear method is illustrated in details, and the wear problems with full coupled thermal and plastic behaviors are numerically simulated for the first time based on the integration of power hardening law with Archard wear model. The local contact parameters are applied to solve the change of surface geometry and wear profiles based on the proposed wear model. After validation in ball-on-flat experiments, the comparing analysis indicate that friction heat and plasticity have great influence on wear evolution, the proposed method is more accurate than the elastic simulation method. The wear profiles, contact pressure, and temperature can be predicted effectively to help the designer further understand the wear process.

Appendix

In the thermo-mechanical analysis, the stain of material is subject to the stress state. In general, the stress state at a material point can be determined by 9 stress components which in matrix notation is given as:

$$\left[ {\begin{array}{*{20}c} {\sigma_{xx} } & {\tau_{xy} } & {\tau_{xz} } \\ {\tau_{yx} } & {\sigma_{yy} } & {\tau_{yz} } \\ {\tau_{zx} } & {\tau_{zy} } & {\sigma_{zz} } \\ \end{array} } \right]$$

(22)

In the equation above, the diagonal members of this matrix σ_xx are normal stresses in the x direction. The symbol σ_xx denotes a normal stress associated with a plane whose normal is in the x direction. Shear stresses are given by τ. The symbol τ_xy denotes that the shear stresses applied on the yoz plane whose normal is the x direction (first subscript), the second subscript indicates that the direction of the applied force is in the y direction. For convenient, we consider the total stress as the sum of the average stress σ_m and the stress deviations:

$$\left[ {\begin{array}{*{20}c} {\sigma_{xx} } & {\tau_{xy} } & {\tau_{xz} } \\ {\tau_{yx} } & {\sigma_{yy} } & {\tau_{yz} } \\ {\tau_{zx} } & {\tau_{zy} } & {\sigma_{zz} } \\ \end{array} } \right]{ = }\left[ {\begin{array}{*{20}c} {\sigma_{m} } & 0 & 0 \\ 0 & {\sigma_{m} } & 0 \\ 0 & 0 & {\sigma_{m} } \\ \end{array} } \right] + \left[ {\begin{array}{*{20}c} {\sigma_{xx} - \sigma_{m} } & {\tau_{xy} } & {\tau_{xz} } \\ {\tau_{yx} } & {\sigma_{yy} - \sigma_{m} } & {\tau_{yz} } \\ {\tau_{zx} } & {\tau_{zy} } & {\sigma_{zz} - \sigma_{m} } \\ \end{array} } \right]$$

(23)

where the average stress σ_m is defined as:

$$\sigma_{m} = \frac{1}{3}\left( {\sigma_{xx} + \sigma_{yy} + \sigma_{zz} } \right)$$

(24)

In the Eq. (23), the first term on the right is the spherical stress tensor, the second term presents the deviatoric stress tensor, S donates the second-order deviator stress which is defined as:

$${\varvec{S}} = \left[ {\begin{array}{*{20}c} {s_{11} } & {s_{12} } & {s_{13} } \\ {s_{21} } & {s_{22} } & {s_{23} } \\ {s_{31} } & {s_{32} } & {s_{33} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\sigma_{xx} - \sigma_{m} } & {\tau_{xy} } & {\tau_{xz} } \\ {\tau_{yx} } & {\sigma_{yy} - \sigma_{m} } & {\tau_{yz} } \\ {\tau_{zx} } & {\tau_{zy} } & {\sigma_{zz} - \sigma_{m} } \\ \end{array} } \right]$$

(25)

Substituting Eqs. (24, 25) into the Eq. (23), the deviatoric stress S_ij can be obtained as follows:

$$S_{ij} = \sigma_{ij} - \sigma_{m} \delta_{ij} /3$$

Here, δ_ij is the Kronecker delta:

$$\delta_{ij} = \left\{ \begin{gathered} 1{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\text{if}}{\kern 1pt} {\kern 1pt} j = i \hfill \\ 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\text{otherwise}} \hfill \\ \end{gathered} \right.$$

After determination of the stress state of surface material by numerical analysis, the von Mises yield criterion given by deviatoric stress tensor is utilized to solve the plastic stain. Here, the plastic yield rule can be expressed as:

$$F = \sigma_{vM} - \sigma_{Y} \left( {\overline{\varepsilon }^{pl} ,T} \right) = \sqrt {\frac{3}{2}{\varvec{S}}:{\varvec{S}}} - \sigma_{Y} \left( {\overline{\varepsilon }^{pl} ,T} \right)$$

(26)

Here, ${\sigma }_{vM}$ is the equivalent von Mises stress, σ_Y presents the flow yield strength of material. The symbol: is the product of second-order tensors.

In the thermo-mechanical contact problem, the strain hardening behavior with temperature has great influence on the material yield strength, σ_Y, and the isotropic Johnson and Cook power hardening law is applied to describe influences of plastic stain and temperature on the yield strength:

$$\sigma_{Y} \left( {\overline{\varepsilon }^{pl} ,T} \right){ = }\left[ {A + B\left( {\overline{\varepsilon }^{pl} } \right)^{n} } \right]\left[ {1 - \left( {\frac{{T - T_{0} }}{{T_{m} - T_{0} }}} \right)^{m} } \right]$$

(27)

where ${\overline{\varepsilon }}^{pl}$ donates the equivalent plastic strain. ${T}_{0}$ is the room temperature, and ${T}_{m}$ is usually the melting temperature of material. $T$ presents the local contact temperature.

For isotropic Mises plasticity, the equivalent plastic strain $\overline{\varepsilon }^{pl}$ is the function of equivalent plastic strain rate tensor ${\dot{{{\varepsilon}}}}^{pl}$ [25]:

$$\overline{\varepsilon }^{pl} = \int_{0}^{\Delta t} {\sqrt {\frac{2}{3}\varvec{\dot{\user2{\varepsilon }}}^{pl} :\varvec{\dot{\user2{\varepsilon }} }^{pl}}} dt$$

(28)

The kinematic hardening models assume associated plastic flow rate, $\dot{\user2{\varepsilon }}^{pl}$ can be written as:

$$\dot{\user2{\varepsilon }}^{pl} { = }\dot{\overline{\varepsilon }}^{pl} \frac{{3{\varvec{S}}}}{{2\sigma_{vM} }}$$

(29)