# A phenomenological model of abradable wear in high performance turbomachinery

William Marscher

► To cite this version:

William Marscher. A phenomenological model of abradable wear in high performance turbomachinery. Wear, 1980, 59 (1), pp.191-211. 10.1016/0043-1648(80)90278-1. hal-01555289

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# A PHENOMENOLOGICAL MODEL OF ABRADABLE WEAR IN HIGH PERFORMANCE TURBOMACHINERY

WILLIAM D. MARSCHER

Creare Incorporated, Hanover, New Hampshire (U.S.A.)

## Summary

A phenomenological model is developed to explain the friction and wear characteristics of high speed sliding contact between blade tips and shrouds in high performance turbomachinery. Equal emphasis is placed on thermal and mechanical mechanisms, and a synergistic relation between the two is derived which yields quantitative predictions of rubbing forces, friction coefficient, total heat input, heat split and relative wear rates between the blade and the shroud. The focal point of the model is a convecting plastic "shear mix" layer on each rubbing surface which is deduced to form regardless of the rub mechanisms initially assumed to prevail.

## 1. Introduction

In modern high performance turbomachinery maintenance of close operating clearances between rotating and static parts is of importance in the achievement of maximum efficiency. In aircraft axial turbines clearance goals of 0.010 - 0.020 in are commonly set for sealing surfaces on large diameter rotors which must undergo substantial temperature and speed changes with corresponding growth. In addition, these engines are subjected to a variety of loadings ranging from gusts against the engine nacelle to g loads from a hard landing. Relatively lower deflections and loads are encountered in automotive turbochargers and gas turbines. However, the tighter clearances required to squeeze out every last mile per gallon from these devices makes interference between rotating and static parts no less inevitable. Even heavy industrial turbines, pumps and process machinery which have substantially larger clearances and static casings experience occasional rubs.

When such interference occurs the objective of the designer is to minimize any increase in operating clearance which results. Because the rotor
generally wears around its entire circumference while the static member
wears only locally, this objective is met if the wear which must occur is
taken on the static member rather than the rotating member as shown in
Fig. 1. The simple solution of choosing a gas path static shroud material that
has low hardness is limited by erosion and structural integrity constraints.
Although some success has been achieved using an intuitive "cut and try"
approach to achieve this mode of wear, it has become clear that an under-
standing of the governing wear phenomena is a necessity if truly effective
solutions to the problem are to be found.

<figure><img src="image_2.png" /><figcaption>Fig. 1. The turbomachinery abradability problem: minimization of the clearance penalty by absorbing wear on the static member.</figcaption></figure>

Research in wear has received increasing emphasis in recent years. Early
approaches such as Archard's wear equation [1] have been augmented by
diverse models emphasizing particular wear modes. Representative of these
are the work of Bates *et al.* [2] which focuses on asperity plowing,
Rabinowicz [3] and Buckley [4] on the role of adhesion effects, Suh [5] on
superplastic delamination wear, Molgaard [6] and Quinn [7] on oxidation-
assisted wear, and Duwell *et al.* [8] and Komanduri and Shaw [9] on diffu-
sion wear. Each of these wear modes has been observed in various combina-
tions and in varying degrees in turbomachinery blade tip/shroud rubs. The
evidence shown in Fig. 2 is typical of this experience, with gouged wear
scars, material transfer from one rubbing member to another and super-
plastic surface zones. On occasion other investigators have found heavily
oxidized surfaces and evidence of diffusive activity. However, the important
question is not whether these mechanisms exist but whether they are a major
influence in terms of establishing rub forces, friction coefficients and relative
blade/shroud wear rates.
<figure><img src="image_3.png" /><figcaption>Fig. 2. Typical turbomachinery rub. (Courtesy of R.C. Bill, NASA Lewis Research Center.)</figcaption></figure>

Wear situations do exist where each of the stated mechanisms has an opportunity to predominate. However, the unique form of loading in turbomachinery rubs strongly discourages most of them. Blade/shroud rubs, rather than occurring as the result of an imposed load, occur as the result of a steadily increasing imposed interference between the two rubbing members. This is because the effective compliances of the static shroud and rotor structures are sufficiently low to absorb any loads generated by the rub with minimal elastic deflection. The structural boundary conditions are thus based on deflection rather than force and the severity of the resulting sub-surface stresses decreases as wear proceeds for a given total interference. A similar concept was demonstrated within local blade contact areas by Dow and Burton [10] and Barber [11]. Conversely, on an average basis wear is forced to keep up with the interaction rate since surface forces will continue to increase until equilibrium is reached. Although the interaction rates typical of turbomachinery seem small at first glance ($10^{-7} - 10^{-5}$ in rev$^{-1}$), they are orders of magnitude higher than adhesion, delamination, oxidation and diffusion wear rates based on the rub velocities typical of turbomachinery ($400 - 1600$ ft s$^{-1}$) and surface stresses as high as material strength.

The remaining unchallenged wear mechanism is plastic plowing, and this does indeed occur though not in the simple form in which it is generally understood. Instead the plastic flow takes place in an intensely deformed shear mix layer on the contacting surface of each rubbing part. This layer is on the order of $10 - 250$ µin thick and consists of extremely fine grains with the original bulk grain boundaries completely destroyed. The existence of high speed shear mix layers is documented in papers such as those of Westcott et al. [12], Beaton and Brooks [13], Sugita et al. [14], Bill and
Ludwig [15], Bill and Shiembob [16] and Bill and Wisander [17]. Of particular note is the conclusion of Bill and Wisander that since the wear particles in their experiments were much smaller than the observed depth of the shear mix layer the wear-controlling processes must take place entirely within this layer. Another interesting result of their experiments is the circumstantial evidence that very high temperatures were being reached throughout the shear mix layer, based on the post-test layer consisting entirely of submicron grains, the interior of which were virtually dislocation free. Other investigators have found similar results.

## 2. Definition of a process model

The experimental evidence reported in the literature clearly indicates that the shear mix layer plays a major role in high speed rubbing wear. In addition, the unique interaction rate dependence of abradable shroud wear rules out milder wear mechanisms as noted earlier. Therefore a reasonable starting point for model development would be the assumption of the exclusive dependence of friction and wear on the shear mix mechanism.

Even this sweeping assumption does not make the problem analytically tractable. Further simplifications must be made concerning the nature of the shear layer plasticity and the mode of interactions at the rubbing interface. However, if these simplifications are made on a sound physical basis they should not compromise the resulting model.

Perhaps the best way to accomplish this purpose is to define what considerations would be mandatory in a physically meaningful model:

(1) an elastic stress model including thermal effects on stresses and properties which is based on interaction rate and frictional shear force;

(2) a heat conduction model including a periodic moving hot spot on the shroud;

(3) a plastic flow criterion;

(4) a relation between the energy dissipation involved in plastic flow, heat generation in the rub zone and frictional force;

(5) a wear criterion.

Each of these criteria and models need to apply to the shroud as well as the blade. Also they should be derived in a manner which does not violate equilibrium and continuity considerations. However, they should be kept as simple as possible, leaving out sophistications such as strain hardening and dynamic effects. If experimentation shows the overall model to be lacking in a manner which could be amended through these additions, the proper phenomena can be brought in at a later date. However, if all such effects are included from the start it may be impossible at the end to separate the correct aspects of the model from the chaff.

On the numerical side the model should be formulated in such a way that the calculations are tractable, at least on a digital computer. Huge storage requirements and potentially unstable techniques should be avoided.
Also, care should be taken that the model be constructed in a building block fashion so that experimentally indicated alterations can be made in the future with a minimum of effort.

### 3. Analytical model definition

Two-dimensional finite element structural and finite difference thermal decks were used in initial solution attempts but were found to be grossly inaccurate when compared with three-dimensional solutions. However, three-dimensional finite element and finite difference solutions were found to be too expensive and unwieldy, and could not represent an entire rub cycle without becoming numerically unstable. It was decided that a closed form solution should be formulated although there was some concern that this approach would lack the flexibility of numerical methods which apply for a wide range of boundary conditions and geometries. This was avoided by using closed form solutions for point loads and combining them through numerical integration. Such an approach did away with the heavy time consumption and instability of the finite element and finite difference methods while keeping their ability to handle general geometries and boundary conditions. It also kept much of the accuracy of a closed form solution and allowed the dependence of the temperatures and stresses on the parameters to be inferred directly from the point load equations.

A unified computer program was developed which determined three-dimensional temperatures, thermal stresses and contact equilibrium stresses. In all stress calculations both normal and tangential loads $F_T$ and $F_c$ were taken into account and each surface was treated as a semi-infinite half-space. A half-space might seem to be a gross representation of actual rubbing hardware but temperatures and stresses were shown to die out quickly away from the contact zone. The point solutions upon which numerical integration was performed were developed from basic formulas found in structural treatises by such authors as Sneddon, Pearson and Timoshenko, and in thermal treatises such as that of Jaeger and Carslaw. These solutions are given in Appendix A for reference.

Each equation was linearly proportional to either $F_c$ or $F_T$, and thus the principle of superposition was able to be used to execute the numerical integration, which simplified the programming effort. Solution fields close to the input contact points were solved by resorting to smoothing functions based on equilibrium arguments. Thermal cross conduction between the blade and seal was based on the method of Rohsenow and Fenech [18].

Figure 3 illustrates the application of the computer model to a hypothetical test simulation and gives a typical break-up of an irregular contact patch. The grid shown is fully adjustable in both shape and size, and can be used to model many individual contacts (e.g. asperities) as well as the single contact patch illustrated. Rub forces and resulting local heat loads are distributed in a discrete manner by the structural and thermal submodels at the various grid node points.
<figure><img src="image_4.png" /><figcaption>Fig. 3. Typical rubbing interface discrete break-up.</figcaption></figure>

An important aspect of the structural submodel is the calculation of normal force at the rub interface as a function of surface topography and subsequent interaction-forced deflection of the blade and shroud. This was done interactively as shown in Fig. 4: (1) estimate the normal contact forces; (2) accurately determine the blade and seal deflection due to these forces; (3) define the degree of inconsistency between the blade and seal deflections (the blade and seal can move apart but not intersect); (4) compensate for the inconsistency in the new force estimates. This non-linear routine allows the closed form point solutions to be used as influence functions in an accurate model of irregular surface contact. (Such a model might also find use in bearing calculations.)

Application of the shear mix mechanism allowed the model to define the high speed rub energy dissipation and the subsequent frictional forces as shown in Appendix B. As verification a plasticity limit analysis problem was solved for the limit load $F_c$ required to support a shear mix layer flow in a manner consistent with the boundary conditions and the kinematic constraints. The flow field established by both approaches involves the recirculation of material from the surface to a depth where the von Mises equivalent stress $\sigma_{eq}$ is approximately equal to the bulk yield strength $\sigma_y$. Although the von Mises flow criterion normally applies to ductile materials only, during wear it applies to brittle materials as well. The high hydrostatic stresses set up during wear suppress crack growth until sluggish slip planes are finally activated, even in substances like ceramics and glass.

The conclusion that the flow field involves recirculation of material, as opposed to simple “smearing over” as in the delamination wear theory, is based circumstantially on experimental evidence. Although no motion pictures have been successfully taken at a high speed rubbing interface, experiments do yield the depth of the heavily deformed shear mix layer and the wear rate. If recirculation did not occur continuity at the shear mix/bulk material interface and at least partial continuity at the shear mix/rub interface would require the material to wear at a rate of half the
<figure><img src="image_5.png" /></figure>

<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 4. Iteration logic for interaction model: 1, approximate deflection is set equal to the geometrical interference; 2, find the force caused by approximate deflection using the $\sigma_3(F_T)$ equation in Appendix A; 3, make an accurate calculation based on the elasticity of the deflection due to the force calculated in (2) and compare this interference with the geometrical interference; 4, iterate until a sufficient degree of consistency is obtained.</figcaption></figure>

shear mix layer thickness for every specimen width worth of sliding distance (Fig. 5). Since the wear rate is invariably orders of magnitude less than this, recirculation leading to a velocity field similar to that of Figs. 6 and 7 must be assumed. A simplistic analog that helps in visualizing the process is a shallow cup filled with a viscous liquid that is turned over and rubbed across a flat surface.

Wear is predicted through a simplified ductile failure criterion. The plastic layer that develops as a result of temperatures and stresses in the surface zone is trapped within an outer wall of elastic material as shown in Fig. 6. As a rub interaction proceeds stresses in the elastic wall gradually increase until the wall finally develops several flow paths along which $\sigma_{eq} \ge \sigma_y$ without interruption. This allows the superplastic material in the shear mix layer to escape from the trapped zone. The assumption is made that this
<figure><img src="image_9.png" /><figcaption>Fig. 5. Zero recirculation wear.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 6. Shear mix recirculation.</figcaption></figure>

material is either squeezed out to the edge of the wear scar, is lost immediately as a chip or is so damaged that it is removed during the succeeding rub.

It is realized that the wear mechanism model and the use of bulk material properties for $σ_y$ for instance are rather gross simplifications. Metallurgical factors such as stacking fault energy (as pointed out by Hirth and Rigney [19]) and more detailed ductile fracture criteria (such as those of Needleman and Rice [20]) should eventually be brought into the model to increase the quantitative accuracy of the approach.
<figure><img src="image_11.png" /><figcaption>Fig. 7. Shear mix average velocity field.</figcaption></figure>

## 4. Physical implications of the shear mix model

The primary importance of the proposed model lies in its ability to predict where rub energy is dissipated. This might seem like a moot point since all dissipation must occur within a hundred microinches of the interface. However, heat conduction takes a finite time to occur and the time a blade takes to rub across a given contact spot on the shroud is of the order of 20 µs. Three-dimensional conduction calculations show that this is not enough time for the great majority of the heat dissipated in the shroud to reach the rub interface. This drastically alters the moving hot spot conduction solution from that derived by authors such as Carslaw and Jaeger [21] for instance who assume that all heat dissipation occurs at the interface. It also makes a good model of interface conductance a necessity for quantitative heat transfer calculations. Since the blade contact patches rub over much longer time periods than the shroud patches the build-up of heat in the blade shear mix layer is very sensitive to interface conduction. The static contact conductance model of Fenech and Rohsenow [18] is a good starting point for a comparable model for sliding.

The resulting temperature field predicted by the shear mix model is qualitatively shown in Figs. 8 and 9. Note that there is an apparent temperature jump across the interface owing to the presence of submicron asperities. This is the basis of the interface contact resistance as demonstrated in the analysis of Ling and Pu [22]. In reality the temperature field is continuous across the interface, becoming highly three-dimensional near the contacting asperities.
<figure><img src="image_12.png" /><figcaption>Fig. 8. Temperature field based on the velocity field of Fig. 7.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Fig. 9. Temperature field generated by two shear mix layers of comparable strength.</figcaption></figure>

The temperature distribution within each rubbing part also offends intuition. The reason why the analysis predicts such a solution is that the majority of energy dissipation is located away from the interface. The apparent quasi-static heat sink which exists at the interface in Fig. 9 occurs because, although the blade shear mix layer asymptotically approaches steady state, the shroud layer remains transient through the rub process. Hence the shroud surface lags toward its pre-rub temperature during the
contact, drawing heat from the blade surface as well as its own energy dissipating shear mix layer. Note, however, that the plastic recirculation which occurs in the shear mix layer reduces the internal peak temperature effect toward the rear of the shroud's contact zone.

If it is accepted that the peak temperature in one or both materials can occur away from the interface some useful implications follow.

(1) Since strength is a strong function of temperature at temperatures which are generally agreed (and incidentally predicted by the model) to be prevalent in the contact zone during high speed rubs, it follows that the weakest link in the contact zone may no longer be the interface. The ease with which high speed rubbing material transfer occurs is therefore at least partially explained by the shear mix model.

(2) Blades will wear more than shrouds, other factors being equal. Since a blade contact patch rubs for much longer than a comparable shroud contact patch, it develops a significantly higher temperature and hence a lower strength in the contact zone. This thermally induced disadvantage combined with the inherent geometrical disadvantage that the blades have much less surface area than the shroud area to be abraded forces the blades to pay heavily for each microinch of shroud thickness they remove.

(3) Peak temperatures are very likely to be limited below the melting point. In the first few microseconds that the rub progresses heat will be generated in each part's shear mix layer until one (or both) of the layers approaches its dynamic softening temperature. This is the temperature where the strength at high strain rates has decreased to about a tenth of its original value and continues to drop exponentially with further increases in temperature. From this point the rubbing system seeks quasi-static equilibrium in accordance with the closed loop interaction of frictional force and temperature shown in Fig. 10. Calculations carried out using the model indicate that equilibrium is reached at temperatures which are generally 100 - 400 °F below the melting point.

<figure><img src="image_14.png" /><figcaption>Fig. 10. Flow chart demonstrating the interrelation of frictional force and shear mix layer temperature.</figcaption></figure>
(4) High speed rubs will tend to be bistable phenomena with a range of high instability in between. For example if conditions such as interface lubrication or pre-existent temperature fields artificially aid one layer to soften before the other layer when the reverse would have been normally true, the softened layer will continue to remain "soft" and dissipate the majority of the system energy even if the preconditions are removed. This helps explain why hot turbine rubs are so difficult to reproduce even qualitatively.

(5) In general the subsurface physics is predicted to be much more important than the surface physics in turbine abradable rubs. Regardless of what interface frictional mechanisms are initially assumed (e.g. adhesion, mountain climbing or asperity plowing), for reasonable values of friction coefficient, interaction rate and rub speed they all predict that surface temperatures in excess of the softening temperature will occur within microseconds of rub initiation. Thus the shear mix mechanism will invariably be excited and its energy dissipating mechanisms will predominate.

The results of using the model described differ in additional important respects from other investigators. In other works it was assumed that abrasion was carried out by plastic flow and/or cracking that was a direct result of the rub forces and as such was subject to Newton's law that action equals reaction. Both members of the rub pair were therefore constrained to approximately the same stress field regardless of physical properties and relative wear appeared to depend only on material strength as represented by the hardness *P*. Hence the famous Archard wear equation

$$W = K \frac{F_c \Delta X}{P}$$

was derived where *W* is the wear, Δ*X* is the sliding distance and *K* is an empirical constant which is assumed to depend primarily on surface geometry and lubricating conditions. Temperature was assumed to be important only in how it affected *P* and whether it allowed oxidation or surface melting and hence boundary lubrication.

However, the shear mix model showed the surprising result that thermal stresses close to the interface were many times higher than the mechanical stresses which resulted directly from *F*<sub>c</sub> and *F*<sub>T</sub>. This became known as the "thermal leverage" effect. Thermal energy was stockpiled close to the surface of each rub member as the rub proceeded while the mechanical stress field derived directly from *F*<sub>c</sub> and *F*<sub>T</sub> remained distributed in a much less concentrated manner. The large thermal strains which resulted were compressive and led to substantial plastic flow. One possible result of such a situation was predicted to be the occurrence of surface "mud-flat" cracks which could substantially accelerate wear. Hence wear would be dependent not only on *P* as Archard predicted, but also on α, *E*, *C*<sub>p</sub>, *K* and any other parameter which influences the ratio of thermal stress to strength. Indeed, mud-flat cracks did occur in many high speed tests in which high magnification post-test pictures were taken such as those of Sugita *et al.* [14], Bill and Ludwig [15] and Komanduri and Shaw [9].
It is interesting that if thermal effects do predominate during high speed rubs then for many material pairs a transition in relative abradability is predicted to occur at rub speeds of the order of 500 ft s⁻¹. One such pair is Invar and 650 stainless steel which have similar properties except that Invar has a hardness (and yield strength) that is 35% lower but has an α that is 10 times lower as well. At low speeds Invar would fare poorly but at high speeds it should do very well. Thus the understanding of the abrasive rub mechanism that this model gives can allow engineers to alter physical properties to take advantage of the phenomena involved. By making the blade tip highly conductive with a low α (thermal expansion coefficient), high C<sub>P</sub> (specific heat) and high softening temperature and giving it low prestress, it should preferentially wear a shroud material of opposite characteristics.

As a final note it should be pointed out that the model concept presented is not limited to abradable rubs and can be used for erosion as well as rub studies. In this regard the phenomenon of thermal leverage between the interacting pair takes on an additional significance because it allows the shroud to wear preferentially during rub without requiring a decrease in surface toughness and hence erosion resistance.

## 5. Proposed experimental work

Preliminary work with the proposed model has projected shear mix layer thicknesses, frictional forces, surface temperatures and relative blade/ shroud wear that correlate well with known turbomachinery experience. However, the theory must be quantitatively verified and the circumstantial evidence upon which certain assumptions were based must be replaced by direct experimentation. This is particularly true of the dynamic characteristics of the thin shear mix layer which for lack of basic research in this area are currently represented by bulk material plasticity data. Strain rate and size effects (e.g. the breakdown of dislocation theory in microscale interactions) are almost certainly important considerations.

An experimental program has been outlined to provide the basic research to verify the shear mix model and investigate the details of its application. We plan to undertake these tests on a level commensurate with available funding using high speed turbomachinery component test rigs. These rigs are capable of tip speeds in excess of 1600 ft s⁻¹ and can withstand rubbing forces of the order of 100 lbf with minimal shaft/bearing assembly deflection. Power inputs of several hundred horsepower are possible, and both hot and cold assemblies can be tested.

The program can be summarized as follows.

1. Tests will be performed to verify the existence of the shear mix layer in high speed rubs and to investigate its characteristics. Evidence of plastic convection, surface cracking, recrystallization, peak surface temperature (as indicated by phase transitions, chemical reactions and eutectic
formation), normal and shear rub forces, chip morphology (after Ruff [23]), and diffusion and oxidation (as a function of depth into the surface) will be obtained for both the blade and the shroud. Initially only a single blade will be allowed to interact in an attempt to keep the rub as simple as possible. Scanning and transmission electron microscopy and Auger techniques will be applied liberally after the fashion of Komanduri and Shaw [9] and Bill and Ludwig [15]. In addition, the constant interaction device of Komanduri and Shaw [9] and a variation of the biaxial dynamometer of Crisp et al. [24] will be used in all tests. A variety of interaction rates, rub speeds and current and proposed gas path materials will be tested.

(2) Specific tests will be performed using the above procedures to verify the predicted model parametric dependencies. When possible only one parameter will be varied at a time and repeatability tests will be performed. In particular, the interesting transition in relative abradability predicted for Invar (low $\alpha$) and molybdenum (high softening temperature) against harder tougher stainless steels will be investigated.

(3) Scale model tests based on the parameters predicted to be important and the principles of dimensional analysis will be performed with materials like clay to investigate shear layer mechanics. Motion pictures will be taken to document the results.

(4) The principles of the shear mix model will be applied in the development of prototype abradable shroud systems. Various material combinations and surface treatments predicted to be beneficial by the model will be investigated.

## 6. Conclusions

(1) Shear mix layer formation in high speed rubs has been experimentally and now analytically established. This is true regardless of the initially assumed interface friction model.

(2) The shear mix layer is predicted to be the controlling mechanism in high speed wear. Heat generation, heat split, friction force and relative wear rate can be defined exclusively as a function of recirculating plastic flow in the rub zone.

(3) Experimental observations such as large apparent temperature jumps, material transfer, reasonably sized friction coefficients and lack of consistency in seemingly identical rub tests are accounted for at least in part by the shear mix mechanism.

(4) Approaching the abradable shroud problem from a hot hardness standpoint has not led to sufficiently satisfactory results. The proposed model of the shear mix mechanism adds a new dimension to the rub problem which can be exploited through careful manipulation of thermally associated properties. We propose to pursue this approach.
References

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3 E. Rabinowicz, Surface energy effects in sliding phenomena, *Rep. 9889*, Surface Laboratory, Massachusetts Institute of Technology, September 12, 1966.

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# Appendix A

## Three-dimensional solutions for stress, strain, displacement and heat conduction

The displacements for a point normal force $F_T^*$ are

$$u_1 = \frac{F_T}{2\pi E} (1 + \nu) \left\{ \frac{x_3 x_1}{R^3} - (1 - 2\nu) \frac{x_1}{R(R + x_3)} \right\}$$

$$u_2 = \frac{F_T}{2\pi E} (1 + \nu) \left\{ \frac{x_3 x_2}{R^3} - (1 - 2\nu) \frac{x_2}{R(R + x_3)} \right\}$$

$$u_3 = \frac{F_T}{2\pi E} (1 + \nu) \left\{ \frac{x_3^2}{R^3} + 2(1 - \nu) \frac{1}{R} \right\}$$

The strains are

$$\epsilon_1 = \frac{\partial u_1}{\partial x_1} = \frac{F_T}{2\pi E} (1 + \nu) \left[ \frac{x_3}{R^3} - \frac{3x_3 x_1^2}{R^5} - (1 - 2\nu) \left\{ \frac{1}{R(R + x_3)} - \frac{x_1^2}{R^2(R + x_3)^2} - \frac{x_1^2}{R^3(R + x_3)} \right\} \right]$$

$$\epsilon_2 = \frac{\partial u_2}{\partial x_2} = \frac{F_T}{2\pi E} (1 + \nu) \left[ \frac{x_3}{R^3} - \frac{3x_3 x_2^2}{R^5} - (1 - 2\nu) \left\{ \frac{1}{R(R + x_3)} - \frac{x_2^2}{R^2(R + x_3)^2} - \frac{x_2^2}{R^3(R + x_3)} \right\} \right]$$

$$\epsilon_3 = \frac{\partial u_3}{\partial x_3} = \frac{F_T}{2\pi E} (1 + \nu) \left( \frac{2\nu x_3}{R^3} - \frac{3x_3^3}{R^5} \right)$$

The stresses are

$$\sigma_1 = - \frac{F_T}{2\pi} (1 - 2\nu) \left\{ \frac{3}{1 - 2\nu} \frac{x_3 x_1^2}{R^5} - \frac{x_3}{R^3} + \frac{1}{R(R + x_3)} - \frac{(2R + x_3)x_1^2}{R^3(R + x_3)^2} \right\}$$

$$\sigma_2 = - \frac{F_T}{2\pi} (1 - 2\nu) \left\{ \frac{3}{1 - 2\nu} \frac{x_3 x_2^2}{R^5} - \frac{x_3}{R^3} + \frac{1}{R(R + x_3)} - \frac{(2R + x_3)x_2^2}{R^3(R + x_3)^2} \right\}$$

$$\sigma_3 = - \frac{F_T}{2\pi} \frac{3x_3^3}{R^5}$$

$$\sigma_{12} = \frac{F_T}{2\pi} \left\{ - \frac{3x_1 x_2 x_3}{R^5} + (1 - 2\nu) \frac{x_1 x_2}{R^3(R + x_3)^2} (2R + x_3) \right\}$$

$$\sigma_{13} = - \frac{3F_T}{2\pi} \frac{x_1 x_3^2}{R^5}$$
$$ \sigma_{23} = - \frac{3F_T}{2\pi} \frac{x_2 x_3^2}{R^5} $$

The displacements for a point shear force $F_c$ are

$$ u_1 = \frac{F_c}{2\pi E} (1 + \nu) \left[ \frac{1}{R} + \frac{x_1^2}{R^3} + \frac{1 - 2\nu}{(R + x_3)^2 R} \{ R(R + x_3) - x_1^2 \} \right] $$

$$ u_2 = \frac{F_c}{2\pi E} (1 + \nu) \left\{ \frac{x_1 x_2}{R^3} - \frac{1 - 2\nu}{(R + x_3)^2 R} x_1 x_2 \right\} $$

$$ u_3 = \frac{F_c}{2\pi E} (1 + \nu) \left\{ \frac{x_1 x_3}{R^3} + \frac{(1 - 2\nu)x_1}{(R + x_3)R} \right\} $$

The strains are

$$ \epsilon_1 = \frac{\partial u_1}{\partial x_1} = \frac{F_c}{2\pi E} (1 + \nu) \left[ \frac{x_1}{R^3} - \frac{3x_1^3}{R^5} - (1 - 2\nu) \left\{ \frac{3x_1}{(R + x_3)^2 R} - \frac{(3R + x_3)x_1^3}{(R + x_3)^3 R^3} \right\} \right] $$

$$ \epsilon_2 = \frac{F_c}{2\pi E} (1 + \nu) \left[ \frac{x_1}{R^3} - \frac{3x_1 x_2^2}{R^5} - (1 - 2\nu) \left\{ \frac{x_1}{(R + x_3)^2 R} - \frac{x_1 x_2^2 (3R + x_3)}{(R + x_3)^3 R^3} \right\} \right] $$

$$ \epsilon_3 = \frac{F_c}{2\pi E} (1 + \nu) \left( 2\nu \frac{x_1}{R^3} - \frac{3x_1 x_3^2}{R^5} \right) $$

The stresses are

$$ \sigma_1 = - \frac{F_c x_1}{2\pi} \left[ \frac{3x_1^2}{R^5} + (1 - 2\nu) \left\{ \frac{3}{(R + x_3)^2 R} - \frac{1}{R^3} - \frac{(3R + x_3)x_1^2}{(R + x_3)^3 R^3} \right\} \right] $$

$$ \sigma_2 = - \frac{F_c x_1}{2\pi} \left[ \frac{3x_2^2}{R^5} + (1 - 2\nu) \left\{ \frac{1}{(R + x_3)^2 R} - \frac{1}{R^3} - \frac{(3R + x_3)x_2^2}{(R + x_3)^3 R^3} \right\} \right] $$

$$ \sigma_3 = - \frac{F_c x_1}{2\pi} \frac{3x_3^2}{R^5} $$

$$ \sigma_{12} = \frac{F_c x_2}{2\pi} \left[ - \frac{3x_1^2 + x_1 x_2}{2R^5} (1 - 2\nu) \left\{ \frac{1}{(R + x_3)^2 R} - \frac{x_1^2 (3R + x_3)}{(R + x_3)^3 R^3} \right\} \right] $$

$$ \sigma_{13} = \frac{F_c}{2\pi} \left( - \frac{2x_1^2 x_3}{R^5} \right) $$

$$ \sigma_{23} = \frac{F_c x_1 x_2}{2\pi} \left\{ - \frac{x_3}{R^5} + (1 - 2\nu) \frac{2R + x_3}{(R + x_3)^2 R^3} \right\} $$

The temperatures for a point heat load are as follows:
point solution

$$
\begin{aligned}
T &= \frac{Q}{8\rho C_p (\pi \alpha_t t)^{3/2}} \exp\left(-\frac{1}{4\alpha_t} \frac{R^2}{t}\right) \\
  &= \frac{AQ \exp(-BR^2/t)}{t^{3/2}}
\end{aligned}
$$

superposition

$$
T_{klm}|_t = \sum_{\xi=0}^{\xi=t} \frac{\sum_{ij} AQ_{\xi, i, j} \exp\{-BR_{klmij}^2/(t-\xi)\}}{(t-\xi)^{3/2}}
$$

Nomenclature for Appendix A

E Young's modulus

$$
R (x_1^2 + x_2^2 + x_3^2)^{1/2}
$$

$x$ coordinate value in the 1, 2 or 3 direction relative to the 1-2-3 coordinate system
with its center at the point of application of $F_T$

$\nu$ Poisson's ratio

1 direction of sliding ($F_c$ is in this direction)

2 direction orthogonal to the 3-1 plane based on the right hand rule

3 direction into the part normal to the interface ($F_T$ is in this direction)

**Appendix B**

*Derivation of plasticity model energy equation*

The energy dissipation rate $\dot{E}$ is first determined in each grid block
volume:

$$
\dot{E}_{ijk} = \int \tau^{\mathrm{P}} \dot{\gamma}^{\mathrm{P}} \Delta \mathrm{vol} \approx \frac{\sigma_y}{2} \left. \frac{\mathrm{d}V}{\mathrm{d}y} \right|_{ijk} \Delta \mathrm{vol}_{ijk} \quad (\mathrm{B}1)
$$

where $\tau^P$ is the plastic shear stress, $\dot{\gamma}^P$ the plastic shear strain rate, $\Delta vol$ the grid block volume, $dV/dy$ the velocity gradient in the direction perpendicular to the interface, $T$ the average temperature of the grid block, $ijk$ the three-dimensional index numbers of the grid and $\sigma_{yijk} = \sigma_y(T_{ijk})$ is the yield stress.

$dV/dy$ is now defined. It is assumed that

$$
\dot{\gamma}^{\mathrm{P}} = \frac{\mathrm{d}V}{\mathrm{d}y} = C \frac{\sigma_{\mathrm{eq}} - \sigma_y}{\sigma_y}
$$

where $\sigma_{eq}$ is the von Mises equivalent stress. This is justified by the follow-
ing procedure. From the Bingham plastic equation

$$
\mu_{\text{eff}} \frac{\mathrm{d}V}{\mathrm{d}y} + \tau_y = \tau_{\text{eq}}
$$
where $\mu_{\text{eff}}$ is the effective viscosity based on a first-order plasticity flow criterion as suggested by Kennedy [B1] and $\tau_{\text{eq}}$ is the equivalent shear stress. Therefore

$$ \frac{dV}{dy} = \frac{\tau_{\text{eq}} - \tau_y}{\mu_{\text{eff}}} $$

If it is assumed that $\mu_{\text{eff}} \approx c'\tau_y$

$$ \frac{dV}{dy} = \frac{\tau_{\text{eq}} - \tau_y}{c'\tau_y} $$

Since

$$ \frac{\tau_{\text{eq}} - \tau_y}{\tau_y} \propto \frac{\sigma_{\text{eq}} - \sigma_y}{\sigma_y} $$

$$ \frac{dV}{dy} = C \frac{\sigma_{\text{eq}} - \sigma_y}{\sigma_y} \qquad (\text{B2}) $$

$C$ is now determined. $C$ will be a function of the material properties and configuration. If the bulk velocity of body $i$ (relative to the interface) is $V_i$, the shear mix layer thickness is $\delta_i$ and $C$ for that body is denoted by $C_i$

$$ \int_{0}^{\delta_i} \left. \frac{dV}{dy} \right|_{i} dy = V_i $$

which leads to

$$ \int_{0}^{\delta_i} \left\{ \frac{C_i(\sigma_{\text{eq}} - \sigma_y)}{\sigma_y} \right\} \Big|_{i} dy = V_i $$

Therefore

$$ C_i = \frac{V_i}{\int_{0}^{\delta_i} \left( \frac{\sigma_{\text{eq}} - \sigma_y}{\sigma_y} \right)_i dy} \qquad (\text{B3}) $$

$dV/dy|_i$ is determined in terms of $V_i$. Equation (B3) is substituted into eqn. (B2):

$$ \left. \frac{dV}{dy} \right|_i = V_i \frac{\{(\sigma_{\text{eq}} - \sigma_y)/\sigma_y\}_i}{\int_{0}^{\delta_i} \{(\sigma_{\text{eq}} - \sigma_y)/\sigma_y\}_i dy} \qquad (\text{B4}) $$

$V_i$ is now determined. Figure B1 shows how plastic flow is assumed to occur during rubs. In order to support recirculation the sum of $V_s$ and $V_B$ must be greater than the absolute velocity $V$ of the moving body. Simplified
<figure><img src="image_15.png" /><figcaption>Fig. B1.</figcaption></figure>

flow analysis indicates that the sum $V_s + V_B$ remains roughly 50% higher than $V$ regardless of the details of the plastic flow state. Therefore

$$V \approx \frac{V_B + V_s}{1.5} \qquad (B5)$$

Geometrically we can write, based on the velocity profiles in the assumed picture,

$$V_s \approx K \left. \frac{dV_s}{dy} \right|_{surf} \delta_s$$

$$V_B \approx K \left. \frac{dV_B}{dy} \right|_{surf} \delta_B$$

where $K$ is a proportionality constant. Substituting for $dV_s/dy$ and $dV_B/dy$ and dividing gives

$$\begin{aligned} \frac{V_s}{V_B} &= \frac{(\sigma_{eqs} - \sigma_{ys})/\sigma_{ys}|_{surf} \delta_s}{(\sigma_{eqB} - \sigma_{yB})/\sigma_{yB}|_{surf} \delta_B} \\ V_s &= \frac{(\sigma_{eqs} - \sigma_{ys})/\sigma_{ys}|_{surf} \delta_s}{(\sigma_{eqB} - \sigma_{yB})/\sigma_{yB}|_{surf} \delta_B} V_B \end{aligned} \qquad (B6)$$

Substituting eqn. (B6) in eqn. (B5) and solving for $V_B$ gives

$$V_B \approx 1.5V - \left\{ \frac{(\sigma_{eqs} - \sigma_{ys})/\sigma_{ys}|_{surf}}{(\sigma_{eqB} - \sigma_{yB})/\sigma_{yB}|_{surf}} \right\} \frac{\delta_s}{\delta_B} \qquad (B7)$$

Similarly

$$V_s \approx 1.5 V - V_B \qquad (B8)$$

$\dot{E}$ is now defined in terms of $V_B$ and $V_s$. Equations (B7) and (B8) are substituted in eqn. (B4) and these results are substituted in turn in eqn. (B1):

$$\dot{E}_{Bijk} = \frac{\Delta \text{vol}}{2} V_B \frac{(\sigma_{eq} - \sigma_y)_B |_{ijk}}{\int_0^{\delta_B} \left( \frac{\sigma_{eq} - \sigma_y}{\sigma_y} \right)_B dy} \qquad (B9)$$
$$ \dot{E}_{sijk} = \frac{\Delta \text{vol}}{2} V_s \frac{(\sigma_{eq} - \sigma_y)_s|_{ijk}}{\int_0^{\delta_s} \left( \frac{\sigma_{eq} - \sigma_y}{\sigma_y} \right)_s dy} \quad (B10) $$

Finally we define $F_c$. Since $\sum_{ijk} (\dot{E}_B + \dot{E}_s) = F_c V$

$$ F_c = \frac{\dot{E}_B + \dot{E}_s}{V} \quad (B11) $$

## References to Appendix B

B1 F. E. Kennedy, Discussion, Two temperature gradients model for friction failure, *J. Lubr. Technol.*, 100 (1978) 484 - 485.