Generalized contact and improved frictional heating in the material point method

Abstract

The material point method (MPM) has proved to be an effective particle method for computational mechanics modeling of problems involving contact, but all prior applications have been limited to Coulomb friction. This paper generalizes the MPM approach for contact to handle any friction law with examples given for friction with adhesion or with a velocity-dependent coefficient of friction. Accounting for adhesion requires an extra calculation to evaluate contact area. Implementation of velocity-dependent laws usually needs numerical methods to find contacting forces. The friction process involves work which can be converted into heat. This paper provides a new method for calculating frictional heating that accounts for interfacial acceleration during the time step. The acceleration terms is small for many problems, but temporal convergence of heating effects for problems involving vibrations and high contact forces is improved by the new method. Fortunately, the new method needs few extra calculations and therefore is recommended for all simulations.

Appendix

For the sliding block problem in Fig. 2 the sliding traction for $t>t_2$ is:

$$\begin{aligned} S = {RN(t-t_2)\over t_s} \end{aligned}$$

(46)

where $t_s=t_f-t_2$ is total time for increasing tangential force. The sliding will start when this traction reaches the zero-velocity sliding force using the static coefficient of friction ($\mu _s N + S_a$), which occurs when

$$\begin{aligned} t_i = t_2 + {1\over R}\left( \mu _s + {S_a\over N}\right) t_s \end{aligned}$$

(47)

Once sliding begins, the total force in the x direction is $F_t-lb\bigr ((\mu _d + k v(t))N + S_a\bigl )$, which leads to an acceleration of:

$$\begin{aligned} a(t) = {dv(t)\over dt} = \alpha (t-t_i) + \beta - \gamma v(t) \end{aligned}$$

(48)

where

$$\begin{aligned} \alpha ={RN\over \rho h t_s},\quad \beta = {(\mu _s-\mu _d)N\over \rho h},\quad \mathrm{and}\quad \gamma = {kN\over \rho h} \end{aligned}$$

(49)

Solving the above differential equation for v(t), the sliding velocity is

$$\begin{aligned} v(t)=\left\{ \begin{array}{ll} {1\over 2}\alpha T^2 + \beta T &{}\quad \mathrm{for\ } \quad k=0 \\ \psi T + \phi \left( 1 - e^{-\gamma T}\right) &{}\quad \mathrm{for\ }\quad k\ne 0 \end{array} \right. \end{aligned}$$

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where

$$\begin{aligned} T = t - t_i,\quad \psi = {\alpha \over \gamma },\quad \mathrm{and}\quad \phi = {\beta - \psi \over \gamma } \end{aligned}$$

(51)

The total friction work is

$$\begin{aligned} Q(T) = lb \int _0^T \bigl ((\mu _d + k v(t))N + S_a\bigr )v(t) dt \end{aligned}$$

(52)

For $k=0$, the result is

$$\begin{aligned} {Q(T)\over lb} = \bigl (\mu _d N + S_a\bigr )\left[ {1\over 6}\alpha T^3 + {1\over 2}\beta T^2\right] \end{aligned}$$

(53)

For $k\ne 0$, the result is

$$\begin{aligned}&{Q(T)\over lb} = \bigl (\mu _d N + S_a\bigr )\left[ {1\over 2}\phi T^2 + \psi \left( T - {1-e^{-\gamma T}\over \gamma }\right) \right] \nonumber \\&\quad \text{+ }\, kN\Biggl [{1\over 3}\psi ^2 T^3 + \phi ^2\left( T - {3-4e^{-\gamma T}+e^{-2\gamma T}\over 2\gamma }\right) \nonumber \\&\quad \text{+ }\, \psi \phi \left( T^2-{2\over \gamma ^2}\Bigl (1-(1+T\gamma )e^{-\gamma T}\Bigl )\right) \Biggr ] \end{aligned}$$

(54)