Heat generation caused by frictional sliding

ProductsAbaqus/StandardAbaqus/Explicit

For coupled temperature-displacement and coupled thermal-electrical-structural analyses in Abaqus/Standard the user can introduce a factor, $η$ , which defines the fraction of frictional work converted to heat. The fraction of generated heat into the first and second surface, $f_{1}$ and $f_{2}$ , respectively, can also be defined. This heat generation capability is to be used only in coupled temperature-displacement and coupled thermal-electrical-structural analyses.

The heat fraction, $η$ , determines the fraction of the energy dissipated during frictional slip that enters the contacting bodies as heat. Heat is instantaneously conducted into each of the contacting bodies depending on the values of $f_{1}$ and $f_{2}$ . The contact interface is assumed to have no heat capacity and may have properties for the exchange of heat by conduction and radiation.

Refer to Small-sliding interaction between bodies and Finite-sliding interaction between deformable bodies for explanations of the notation used for the shape functions and contact parameters involved in the small-sliding and slide line theory. Note that the shape functions for interpolation of the temperature field may be different from the interpolation functions for the displacements; for example, if the underlying elements are of second order, the displacements are interpolated using quadratic functions, whereas the temperature field is interpolated using linear shape functions. Hence, the temperature interpolator will be denoted with the symbol $M$ and the displacement interpolation will be denoted with the symbol $N$ . Only the heat transfer terms will be discussed in the following.

The heat flux densities— $q_{1}$ , going out the surface on side 1, and $q_{2}$ , going out the surface on side 2—are given by

q_{1} = q_{k} + q_{r} - f_{1} q_{g}

and

q_{2} = - q_{k} - q_{r} - f_{2} q_{g},

where $q_{g}$ is the heat flux density generated by the interface element due to frictional heat generation, $q_{k}$ is the heat flux due to conduction, and $q_{r}$ is the heat flux due to radiation.

The heat flux density generated by the interface element due to frictional heat generation is given by

q_{g} = η τ \dot{s} = η τ \frac{Δ s}{Δ t},

where $τ$ is the frictional stress, $Δ s$ is the incremental slip, and $Δ t$ is the incremental time. The frictional stress is dependent on the contact pressure, $p$ ; the friction coefficient, $μ$ ; and the temperatures on either side of the interface.

The heat flux due to conduction is assumed to be of the form

q_{k} = κ (h, p, \bar{θ}) (θ_{1} - θ_{2}) = κ (h, p, \bar{θ}) Δ θ,

where the heat transfer coefficient $κ (h, p, \bar{θ})$ is a function of the average temperature at the contact point, $\bar{θ} = \frac{1}{2} (θ_{1} + θ_{2})$ ; the overclosure, $h$ ; and the contact pressure, $p$ . $θ_{1}$ and $θ_{2}$ are the temperatures of side 1 and side 2, respectively.

The heat flux due to radiation is assumed to be of the form

q_{r} = F [{(θ_{1} - θ^{Z})}^{4} - {(θ_{2} - θ^{Z})}^{4}],

where $F$ is the gap radiation constant (derived from the emissivities of the two surfaces) and $θ^{Z}$ is the absolute zero on the temperature scale used.

Using the Galerkin method the weak form of the equations can be written as

\int_{S} δ θ_{1} q_{1} d S = \int_{S} δ θ_{1} (q_{k} + q_{r} - f_{1} q_{g}) d S, \int_{S} δ θ_{2} q_{2} d S = \int_{S} δ θ_{2} (- q_{k} - q_{r} - f_{2} q_{g}) d S .

The contribution to the variational statement of thermal equilibrium is

δ Π = \int_{S} (δ θ_{1} q_{1} + δ θ_{2} q_{2}) d S = \int_{S} [δ Δ θ (q_{k} + q_{r}) - δ \hat{θ} q_{g}] d S,

where $\hat{θ} = f_{1} θ_{1} + f_{2} θ_{2}$ . The contribution to the Jacobian matrix for the Newton solution is

\begin{matrix} (1) & d δ Π = \int_{S} [d δ Δ θ (q_{k} + q_{r}) + δ Δ θ (d q_{k} + d q_{r}) - d δ \hat{θ} q_{g} - δ \hat{θ} d q_{g}] d S . \end{matrix}

At a contact point the temperatures can be interpolated with

θ_{1} (s) = M_{1}^{N} (s) θ^{N}, θ_{2} (s) = M_{2}^{N} (s) θ^{N},

where $θ^{N}$ is the temperature at the $N$ th node associated with the interface element. Note that the summation convention will be used for all superscripts. Therefore, the temperature variables can be written as follows:

Δ θ (s) = Δ M^{N} (s) θ^{N}, \hat{θ} (s) = {\hat{M}}^{N} (s) θ^{N}, \bar{θ} (s) = {\bar{M}}^{N} (s) θ^{N},

where ${\hat{M}}^{N} (s) = f_{1} M_{1}^{N} (s) + f_{2} M_{2}^{N} (s)$ and ${\bar{M}}^{N} (s) = \frac{1}{2} (M_{1}^{N} (s) + M_{2}^{N} (s))$ . Substituting the above expressions into Equation 1, we obtain

\begin{matrix} d δ Π = δ θ^{N} \int_{S} [ & (q_{k} + q_{r}) \frac{d Δ M^{N}}{d s} d s + Δ M^{N} (\frac{\partial q_{r}}{\partial θ_{1}} d θ_{1} + \frac{\partial q_{r}}{\partial θ_{2}} d θ_{2} \\ + \frac{\partial q_{k}}{\partial Δ θ} d Δ θ + \frac{\partial q_{k}}{\partial \bar{θ}} d \bar{θ} + \frac{\partial q_{k}}{\partial h} d h + \frac{\partial q_{k}}{\partial p} d p) \\ - q_{g} \frac{d {\hat{M}}^{N}}{d s} d s - {\hat{M}}^{N} (\frac{\partial q_{g}}{\partial τ} d τ + \frac{\partial q_{g}}{\partial s} d s)] d S . \end{matrix}

After rearranging and expanding terms, we obtain

\begin{matrix} d δ Π = δ θ^{N} \int_{S} [ & ((q_{k} + q_{r}) \frac{d Δ M^{N}}{d s} - q_{g} \frac{d {\hat{M}}^{N}}{d s}) d s \\ + Δ M^{N} (\frac{\partial q_{r}}{\partial θ_{1}} M_{1}^{M} + \frac{\partial q_{r}}{\partial θ_{2}} M_{2}^{M} + \frac{\partial q_{k}}{\partial Δ θ} Δ M^{M} + \frac{\partial q_{k}}{\partial \bar{θ}} {\bar{M}}^{M}) d θ^{M} \\ + Δ M^{N} (\frac{\partial q_{k}}{\partial h} d h + \frac{\partial q_{k}}{\partial p} d p) \\ + Δ M^{N} (\frac{\partial q_{r}}{\partial θ_{1}} \frac{d M_{1}^{K}}{d s} θ^{K} + \frac{\partial q_{r}}{\partial θ_{2}} \frac{d M_{2}^{K}}{d s} θ^{K} + \frac{\partial q_{k}}{\partial Δ θ} \frac{d Δ M^{K}}{d s} θ^{K} + \frac{\partial q_{k}}{\partial \bar{θ}} \frac{d {\bar{M}}^{K}}{d s} θ^{K}) d s \\ - {\hat{M}}^{N} (\frac{\partial q_{g}}{\partial τ} (\frac{\partial τ}{\partial p} d p + \frac{\partial τ}{\partial θ_{1}} d θ_{1} + \frac{\partial τ}{\partial θ_{2}} d θ_{2}) + \frac{\partial q_{g}}{\partial s} d s)] d S . \end{matrix}

Expanding the terms involving frictional heat generation yields

\begin{matrix} d δ Π = δ θ^{N} \int_{S} & [((q_{k} + q_{r}) \frac{d Δ M^{N}}{d s} - q_{g} \frac{d {\hat{M}}^{N}}{d s}) d s \\ + Δ M^{N} (\frac{\partial q_{r}}{\partial θ_{1}} M_{1}^{M} + \frac{\partial q_{r}}{\partial θ_{2}} M_{2}^{M} + \frac{\partial q_{k}}{\partial Δ θ} Δ M^{M} + \frac{\partial q_{k}}{\partial \bar{θ}} {\bar{M}}^{M}) d θ^{M} \\ + Δ M^{N} (\frac{\partial q_{k}}{\partial h} d h + \frac{\partial q_{k}}{\partial p} d p) \\ + Δ M^{N} (\frac{\partial q_{r}}{\partial θ_{1}} \frac{d M_{1}^{K}}{d s} θ^{K} + \frac{\partial q_{r}}{\partial θ_{2}} \frac{d M_{2}^{K}}{d s} θ^{K} + \frac{\partial q_{k}}{\partial Δ θ} \frac{d Δ M^{K}}{d s} θ^{K} + \frac{\partial q_{k}}{\partial \bar{θ}} \frac{d {\bar{M}}^{K}}{d s} θ^{K}) d s \\ - {\hat{M}}^{N} \frac{\partial q_{g}}{\partial τ} (\frac{\partial τ}{\partial θ_{1}} M_{1}^{M} + \frac{\partial τ}{\partial θ_{2}} M_{2}^{M}) d θ^{M} \\ - {\hat{M}}^{N} \frac{\partial q_{g}}{\partial τ} (\frac{\partial τ}{\partial θ_{1}} \frac{d M_{1}^{K}}{d s} θ^{K} + \frac{\partial τ}{\partial θ_{2}} \frac{d M_{2}^{K}}{d s} θ^{K}) d s \\ - {\hat{M}}^{N} (\frac{\partial q_{g}}{\partial τ} \frac{\partial τ}{\partial p} d p + \frac{\partial q_{g}}{\partial s} d s)] d S . \end{matrix}

The derivatives of $q_{r}$ , $q_{k}$ , and $q_{g}$ , are as follows:

\frac{\partial q_{r}}{\partial θ_{1}} = 4 F {(θ_{1} - θ^{Z})}^{3}, \frac{\partial q_{r}}{\partial θ_{2}} = - 4 F {(θ_{2} - θ^{Z})}^{3},

\frac{\partial q_{k}}{\partial Δ θ} = κ (h, p, \bar{θ}), \frac{\partial q_{k}}{\partial \bar{θ}} = \frac{\partial κ (h, p, \bar{θ})}{\partial \bar{θ}} Δ θ,

\frac{\partial q_{k}}{\partial h} = \frac{\partial κ (h, p, \bar{θ})}{\partial h} Δ θ, \frac{\partial q_{k}}{\partial p} = \frac{\partial κ (h, p, \bar{θ})}{\partial p} Δ θ,

\frac{\partial q_{g}}{\partial τ} \frac{\partial τ}{\partial p} = η \frac{| Δ s |}{Δ t} μ, \frac{\partial q_{g}}{\partial s} = η \frac{τ}{Δ t} .

For contact pairs and slide line elements, each integration point is associated with a unique slave node. If we associate $M_{1}^{N}$ with the slave surface, then $M_{1}^{N}$ will again have only a single nonzero entry equal to one and the derivatives of $M_{1}^{N}$ with respect to $s$ vanish. In contrast, on the master surface there will be multiple nonzero entries in $M_{2}^{N}$ , which are a function of the position on the master surface at which contact occurs.

For GAPUNIT and DGAP elements each contact (integration point) is directly associated with a node pair. Hence, $M_{1}^{N}$ and $M_{2}^{N}$ each have one nonzero entry that is equal to one, and all terms involving derivatives of $M_{1}^{N}$ and $M_{2}^{N}$ with respect to $s$ vanish.

The variations of overclosure, $h$ , and slip, $s$ , can be written as linear functions of the variations of displacement. These expressions, which determine the form of the $β$ matrix for contact elements, have been derived in Small-sliding interaction between bodies and Finite-sliding interaction between deformable bodies.