Effect of the workpiece material on the heat affected zones during grinding: a numerical simulation

Abstract

A simulation of the precision grinding of steel was performed using an implicit finite element code, namely the commercial code MARC, in order to describe the temperature fields developed during the process. The input data required a model obtained via a series of experiments, grinding several steels under the same grinding conditions in order to examine the effect of the workpiece materials on the temperature fields and the depth of the heat affected zones developed during grinding.

Appendix A

The mathematical formulations for the heat transfer analysis used by the implicit finite element code MARC are briefly outlined.

The heat transfer can be written as a differential equation

$${{\left[ C \right]}{\left\{ {\dot{T}} \right\}} + {\left[ K \right]}{\left\{ T \right\}} = {\left\{ Q \right\}}}$$

where [C] is the heat capacity matrix, [K] the conductivity and convection matrix, {T} the vector of the nodal temperatures and {Q} the vector of nodal fluxes. In the case of a steady state problem, where ${\dot{T} = {{\partial T} \over {\partial t}} = 0}$, the solution can be easily obtained by a matrix inversion as

$${{\left\{ T \right\}} = {\left[ K \right]}^{{ - 1}} {\left\{ Q \right\}}}$$

In the case of transient analysis, where Ṫ≠0, which is the case described in the present work, the nodal temperature is approximated at discrete points in time as

$${{\left\{ T \right\}}^{n} = {\left\{ T \right\}}{\left( {t_{0} + n\Delta t} \right)}}$$

In the MARC program a backward difference scheme is used to approximate the time derivative of the temperature

$${{\left\{ {\dot{T}} \right\}}^{n} \cong {{{\left\{ T \right\}}^{n} - {\left\{ T \right\}}^{{n - 1}} } \over {\Delta t}}}$$

Substituting in Eq. 7, the vector {Q} is obtained as

$${{\left( {{{{\left[ C \right]}} \over {\Delta t}} + {\left[ K \right]}} \right)}{\left\{ T \right\}}^{n} - {{{\left[ C \right]}} \over {\Delta t}}{\left\{ T \right\}}^{{n - 1}} = {\left\{ Q \right\}}}$$