Heat generation caused by frictional sliding

This section defines the heat transfer terms for coupled temperature-displacement and coupled thermal-electrical-structural analyses in Abaqus/Standard.

See Also
In Other Guides
Fully Coupled Thermal-Stress Analysis
Fully Coupled Thermal-Electrical-Structural Analysis
Thermal Contact Properties

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, f1 and f2, 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 f1 and f2. 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—q1, going out the surface on side 1, and q2, going out the surface on side 2—are given by

q1=qk+qr-f1qg

and

q2=-qk-qr-f2qg,

where qg is the heat flux density generated by the interface element due to frictional heat generation, qk is the heat flux due to conduction, and qr is the heat flux due to radiation.

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

qg=ητs˙=ητΔsΔt,

where τ is the frictional stress, Δs is the incremental slip, and Δt 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

qk=κ(h,p,θ¯)(θ1-θ2)=κ(h,p,θ¯)Δθ,

where the heat transfer coefficient κ(h,p,θ¯) is a function of the average temperature at the contact point, θ¯=12(θ1+θ2); the overclosure, h; and the contact pressure, p. θ1 and θ2 are the temperatures of side 1 and side 2, respectively.

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

qr=F[(θ1-θZ)4-(θ2-θZ)4],

where F is the gap radiation constant (derived from the emissivities of the two surfaces) and θZ is the absolute zero on the temperature scale used.

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

Sδθ1q1dS=Sδθ1(qk+qr-f1qg)dS,        Sδθ2q2dS=Sδθ2(-qk-qr-f2qg)dS.

The contribution to the variational statement of thermal equilibrium is

δΠ=S(δθ1q1+δθ2q2)dS=S[δΔθ(qk+qr)-δθ^qg]dS,

where θ^=f1θ1+f2θ2. The contribution to the Jacobian matrix for the Newton solution is

(1)dδΠ=S[dδΔθ(qk+qr)+δΔθ(dqk+dqr)-dδθ^qg-δθ^dqg]dS.

At a contact point the temperatures can be interpolated with

θ1(s)=M1N(s)θN,        θ2(s)=M2N(s)θN,

where θN is the temperature at the Nth 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:

Δθ(s)=ΔMN(s)θN,        θ^(s)=M^N(s)θN,        θ¯(s)=M¯N(s)θN,

where M^N(s)=f1M1N(s)+f2M2N(s) and M¯N(s)=12(M1N(s)+M2N(s)). Substituting the above expressions into Equation 1, we obtain

dδΠ=δθNS[(qk+qr)dΔMNdsds+ΔMN(qrθ1dθ1+qrθ2dθ2+qkΔθdΔθ+qkθ¯dθ¯+qkhdh+qkpdp)-qgdM^Ndsds-M^N(qgτdτ+qgsds)]dS.

After rearranging and expanding terms, we obtain

dδΠ=δθNS[((qk+qr)dΔMNds-qgdM^Nds)ds+ΔMN(qrθ1M1M+qrθ2M2M+qkΔθΔMM+qkθ¯M¯M)dθM+ΔMN(qkhdh+qkpdp)+ΔMN(qrθ1dM1KdsθK+qrθ2dM2KdsθK+qkΔθdΔMKdsθK+qkθ¯dM¯KdsθK)ds-M^N(qgτ(τpdp+τθ1dθ1+τθ2dθ2)+qgsds)]dS.

Expanding the terms involving frictional heat generation yields

dδΠ=δθNS[((qk+qr)dΔMNds-qgdM^Nds)ds+ΔMN(qrθ1M1M+qrθ2M2M+qkΔθΔMM+qkθ¯M¯M)dθM+ΔMN(qkhdh+qkpdp)+ΔMN(qrθ1dM1KdsθK+qrθ2dM2KdsθK+qkΔθdΔMKdsθK+qkθ¯dM¯KdsθK)ds-M^Nqgτ(τθ1M1M+τθ2M2M)dθM-M^Nqgτ(τθ1dM1KdsθK+τθ2dM2KdsθK)ds-M^N(qgττpdp+qgsds)]dS.

The derivatives of qr, qk, and qg, are as follows:

qrθ1=4F(θ1-θZ)3,        qrθ2=-4F(θ2-θZ)3,
qkΔθ=κ(h,p,θ¯),        qkθ¯=κ(h,p,θ¯)θ¯Δθ,
qkh=κ(h,p,θ¯)hΔθ,        qkp=κ(h,p,θ¯)pΔθ,
qgττp=η|Δs|Δtμ,        qgs=ητΔt.

For contact pairs and slide line elements, each integration point is associated with a unique secondary node. If we associate M 1 N with the secondary 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 main surface there will be multiple nonzero entries in M 2 N , which are a function of the position on the main surface at which contact occurs.

For GAPUNIT and DGAP elements each contact (integration point) is directly associated with a node pair. Hence, M1N and M2N each have one nonzero entry that is equal to one, and all terms involving derivatives of M1N and M2N 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.