## Abstract

We describe a theoretical and numerical analysis of an existing model of anelasticity owing to grain boundary sliding. Two linearly elastic layers having finite thickness and identical material constants are separated by a given fixed spatially periodic interface across which the normal component of velocity is continuous, whereas the tangential component has a discontinuity determined by the shear stress *σ*^{*}_{ns} and the boundary sliding viscosity *η**. We derive asymptotic forms giving the complex rigidity for the extremes of low-frequency forcing and of high-frequency forcing. Using those forms, we create master variables allowing results for different interface shapes, and arbitrary forcing frequency, to be collapsed (very nearly) into a single curve. We then analyse numerically, with finite interface slope, three proposed factors that may weaken and broaden the theoretical prediction of a single Debye peak in the loss spectrum. They are, namely, stress concentrations at interface corners, spatial variation in grain size and spatial variation in boundary sliding viscosity *η**. Our results show that all these factors can, indeed, contribute to a moderate weakening of the loss peak. By contrast, the loss peak markedly broadens only when the boundary sliding viscosity *η** differs by an order of magnitude across adjacent interface. The shape of the loss spectrum (self-similar to a single Debye peak) is insensitive to the other two factors.

## 1. Introduction

High-temperature viscoelasticity in finely grained polycrystals is commonly attributed to diffusion and grain boundary sliding (Cooper 2002). When the time scales for diffusion and grain boundary sliding are widely separated, as in recent attenuation experiments (Cooper 2002; Jackson *et al.* 2004), the two processes can be treated separately (Morris & Jackson 2009). In the simplest case of elastically accommodated sliding, individual crystal grains behave as perfectly elastic solids, and mechanical energy is dissipated only within the thin disordered regions separating the individual grains. Those boundary regions are modelled as having a so-called boundary (or slip) viscosity, *η** (e.g. Raj & Ashby 1971). A polycrystal undergoing torsional oscillations thus dissipates mechanical energy at a local rate proportional to the product of the local boundary viscosity *η** and discontinuity in tangential velocity across the grain boundary region. That dissipation rate vanishes both at very low, and at very high frequencies, because vanishes in either extreme. As a result, the dissipation rate for the entire polycrystal is predicted to have a maximum at an angular frequency *ω** of order *μ**ℓ*/*η***d*^{*}_{m}; here, *d*^{*}_{m} is the mean grain size, ℓ* is a constant length scale measuring the thickness of the disordered boundary region separating the grains, and *μ** is the rigidity (shear modulus) of the perfectly elastic grains (modelled as isotropic).

Experiments showing this effect are described by Mosher & Raj (1974) and Ashby (1972) for finely grained pure metals. Although such a basic effect might be expected to be very commonly observed, that is not so. Instead, according to Cooper (2002, p. 268), the loss peak in the attenuation spectrum is ‘seen only rarely in metals’; and experiments usually ‘reveal a broad absorption peak of low magnitude in polycrystalline material, one barely competing to be seen over the power-law background absorption’. Loss spectra fitting such a description can also be found in the review by Schaller & Lakki (2001). Their experiments on three types of ceramics consistently show a mild and broad absorption peak. However, only in one of those cases can the peak be attributed to viscous grain boundary sliding.

To explain the small and broad peak found in experiments, in contrast to the single Debye peak predicted by theoretical models, e.g. in Ghahremani (1980), three suggestions have been proposed:

— variation in grain sizes (Pezzotti 1999),

— variation in boundary viscosity,

*η** (Cooper 2002, p. 268),— sharp corners at triple junctions (Faul

*et al.*2004, p. 58).

Here we study those proposed explanations using the Raj–Ashby bicrystal model for grain boundary sliding. Because the strains are small for our attenuation problem, we can assume the grain interface to be a given function of position. The resulting boundary value problem (b.v.p.) for the displacement field within the bicrystal is linear. Though we use that linear model to estimate the mechanical loss for sliding along specific interfaces, we do not follow Raj & Ashby (1971) in superposing the mechanical loss for different sliding surfaces to estimate the loss in polycrystal. For, even though displacements satisfy the principle of superposition in our linear problem, strain energy and dissipation do not. (This issue is, of course, entirely separate from the one discussed recently by Kim *et al.* (2009) and Wheeler (in press) in their studies of steady diffusion creep. In that problem, the strains are finite and the grain boundaries themselves evolve in time. In the resulting free b.v.p., the principle of superposition does not apply even for the displacements.)

Following the statement in §2 of the b.v.p., in §3, we state the analytical constraints on the numerical solution: these are a new elementary solution of the Raj–Ashby model for a sawtooth interface making angles ±*π*/4 with the direction of mean sliding; the local solution of Picu & Gupta (1996) describing the behaviour near a sharp corner; and asymptotes for the extremes . In that section, we also obtain master variables for the mechanical loss spectrum using the asymptotes. In §4, we demonstrate that our numerical solution agrees quantitatively with the constraints stated above and that it also approaches the perturbation solution in the limit of a vanishing slope. Using the master variables, we then show that the effect of corners is small. In §5, we study the effects on the mechanical loss spectrum owing to spatial variations in grain size *d**, and in boundary viscosity *η**. We show that the loss peak is reduced and broadened if the boundary viscosity varies with position within a system; whereas the loss peak does not broaden significantly if grain size is varied with position. Our result suggests that of the three proposed explanations, only a large variation in boundary viscosity along grain boundaries is able to produce a loss peak that is significantly wider than a single Debye peak. Concurrent with that effect, the proposed explanations are also found to be able to reduce the loss peak moderately.

Throughout this work, dimensional quantities are denoted by asterisks. A list and explanation of the notation used in this paper is also given in table 1

## 2. Boundary value problem

Figure 1 shows the geometry of the bicrystal model. The sample consists of two linear elastic grains of thickness *a*/*ξ** having elastic shear modulus *μ** and Poisson ratio *ν*. The interface is periodic with a wavelength 2*π*/*ξ**, where *ξ** is the wavenumber. Because samples in the attenuation experiments are subjected to small strain of approximately 10^{−6} (Cooper 2002), we follow the Raj & Ashby assumption that the interface position is given by a time-independent function *f**(*x**). Unit vectors in the coordinate directions are denoted by and . The unit tangent and unit normal vectors of the interface are denoted by and , respectively. At the upper and lower boundaries at *y**=±*a*/*ξ**, the displacement varies sinusoidally in time with angular frequency *ω** and amplitude , i.e. . The grains are assumed to be undergoing plane deformation and the *x* and *y* components of the displacement vector ** u*** are denoted by

*u**(

*x**,

*y**) and

*v**(

*x**,

*y**), respectively. Similarly, the Cartesian components of the stress and strain tensors are denoted by

*σ*

^{*}

_{ij}(

*x**,

*y**) and

*e*

^{*}

_{ij}(

*x**,

*y**).

On 𝒮_{I}, the constitutive equation and a normal constraint are given, respectively, as
2.1a
and
2.1b
where ℓ* and *η** are constants. Equation (2.1*a*) states that the shear stress along 𝒮_{I} is proportional to the discontinuity in tangential velocity across 𝒮_{I}. As described by Raj & Ashby (1971), the interface acts as if it were a thin liquid film having uniform viscosity *η** and constant thickness ℓ*. In equation (2.1*b*), we take the normal displacement to be continuous across the interface; this model does not include cavitation.

Following Mosher & Raj (1974), we define the sliding time scale *t*^{*}_{η} by
2.2
Physically, *t*^{*}_{η} is the time scale on which the two sides of equation (2.1*a*) balance. We also define *ε* to be the characteristic slope of the interface; specifically, . Dimensionless variables (without asterisks) are then defined as follows:
2.3a
2.3b
2.3c
2.3d
and
2.3e
and
2.3f
The dimensionless b.v.p. is as follows:

in grain 1 and grain 2,
2.4a
on *y*=±*a*,
2.4b
and
2.4c
on *y*=*εf*(*x*),
2.4d
and
2.4e
2.4f
on *x*=2*π* and *x*=0,
2.4g
and
2.4h
We note here that in equation (2.4*a*), we formulate our b.v.p. using the plane elastostatic equation rather than the dynamical equation. This is appropriate because the elastic wavelength for our frequencies of interest is large when compared with the sample size. Also, because the grain interface is assumed to have a fixed location, problem (2.4*a*–*h*) is linear and separable in time. As a result, the solution of problem (2.4*a*–*h*) for time-periodic boundary displacement is time-periodic with the same angular frequency *ω*.

However, because equation (2.4*d*) contains a time derivative, the displacement within the sample, in general, lags the displacement imposed at the sample boundaries. As a result, the deformation and stress lag the imposed displacement. That phase lag between boundary displacement and stress is the expression of dissipation occurring at the interface. According to equation (2.4*d*), the phase lag vanishes if *ω*→0, or if . In the first case, equation (2.4*d*) requires that *σ*_{ns}=0; and, in the second, that [*u*_{s}]=0. Thus, the interface is asymptotically stress-free for *ω*→0, but is asymptotically welded for . In both extremes, the boundary displacement and stress are in phase. The dissipation then vanishes, and the sample behaves as if it were perfectly elastic. In §3*b*, that property is used to derive asymptotes to the loss spectrum.

By solving equation (2.4*a*–*h*), we are able to obtain the *x*-averaged shear stress *τ* applied at *y*=±*a*. *τ* is defined as
2.5
The sample shear modulus *G* is then defined by the equation:
2.6
where *γ*(*t*)=e^{iωt}/*a* is the sample shear strain. Because both *τ* and *γ* are proportional to e^{iωt} in equation (2.6), the modulus *G* is independent of *t*.

The mechanical loss ℒ is defined, as usual, by the equation
2.7
If the material can be modelled as a network of springs and dampers, the quantity defined by equation (2.7) is equal to the ratio of the loss per cycle to 4*π* times the mean strain energy stored within the grains (Bland 1960; O’Connell & Budiansky 1978).

We represent the interface position by a finite Fourier sine series having *N* terms
2.8
For a piecewise linear interface defined by *f*(−*x*)=−*f*(*x*) and
2.9
*N* is infinite and the Fourier coefficients *f*_{k} are given by
2.10

The specific values *α*=1/4 and *α*=1/2 correspond to two orthogonal sliding surfaces called ‘mode 1’ and ‘mode 2’ surfaces by Raj & Ashby (1971) for a two-dimensional array of regular hexagons. We refer to these cases as the type *TS* and type *S* interfaces, respectively. These interfaces are shown in figure 2. In the limit as , the derivative *f*′ is discontinuous at *x*=±*πα* and at *x*=±*π*(1−*α*). The type *S* interface then approaches a sawtooth and the type *TS* interface approaches a truncated sawtooth, i.e. a sawtooth with the sharp crests and troughs cut off. For large but finite *N*, those discontinuities in *f*′ are smoothed over an interval Δ*x* with an order *N*^{−1}.

To relate the characteristic slope *ε* to the true slope angle *φ* between the sloping side (for ) and the horizontal, we use equations (2.11*a*) and (2.11*b*) for the type *S* and the type *TS* interface, respectively.
2.11a
and
2.11b
For the type *S* and type *TS* surfaces in a regular hexagonal array, the slope angle *φ*=30° and 60°, respectively. Because there are three axes that correspond with each of the two sliding surfaces as shown in figure 2*a*, the response of that sample has a threefold rotational symmetry.

## 3. Analytical constraints on the numerical solution

Problem (2.4*a*–*h*) is solved using the standard finite element method. Grain 1 and grain 2 are discretized using linear quadrilateral elements and boundary conditions (2.4*b*–*h*) are imposed using the standard penalty method (Zienkiewicz & Taylor 2000). Here, we derive analytical constraints that our numerical results have to satisfy.

### (a) A simple shear solution for a bicrystal

For all interface shapes, the simple shear field given by
3.1
satisfies all governing equations except the slip condition (2.4*d*). For an arbitrary interface shape, equation (3.1) does not satisfy that condition because the left-hand side (l.h.s.) vanishes, but the right-hand side (r.h.s.) is non-zero in general. However, for a type *S* interface with *φ*=±45°, the interface coincides with the principal axes of stress calculated from equation (3.1). Consequently, for that special case, the shear stress on the interface vanishes. Equation (3.1) then satisfies problem (2.4*a*–*h*) exactly. Thus, for a type *S* interface with *φ*=45°, the two grains are effectively welded together and the sample rigidity equals the grain rigidity (i.e. *G*=1) for all *ω*. In particular, the mechanical loss vanishes identically for this case.

### (b) Asymptotes for high frequency and for low frequency

Provided the interpretation of ℒ given below equation (2.7) is applicable, asymptotes to the mechanical loss spectrum can be obtained by evaluating the strain energy and the dissipation directly.

For the system shown in figure 1, the external power supplied at the sample boundaries is either dissipated at the time-independent grain interface 𝒮_{I}, or stored as strain energy within the perfectly elastic grains, i.e.
3.2a
3.2b
and
3.2c
define the strain energy function *W*(*t*) and the dissipation rate . We also define the total dissipation *Υ* in one oscillation period by
3.3
In equation (3.2*b*), 𝒱 is the combined, time-independent volume of the upper and lower grains, but with the dissipative interfacial region excluded. Equation (3.2*a*–*c*) is a simplified form of a more general energy balance derived by Morris & Jackson (2009).

#### (i) High-frequency limit

In the limit as , equation (2.4*d*) requires that [*u*_{s}]→0. Because the two grains then behave as a single grain in simple shear, the stress components become *σ*_{xy}=*τ* and *σ*_{xx}=0=*σ*_{yy}. By the discussion following equation (2.4*a*–*h*), we may assume that where the amplitude is independent of *t*. Using Hooke’s law to evaluate (3.2*b*), then time averaging, we find that the time mean strain energy of the system is given asymptotically by
3.4
Noting that at the interface 𝒮_{I} the shear stress is given by , we evaluate (3.2*c*), then use equation (3.3) to show that
3.5
Using equations (3.4) and (3.5), and the interpretation of ℒ given below equation (2.7), we find that in the limit as , ℒ is given asymptotically by
3.6a
and
3.6b
The geometric factor *g*(*φ*) depends only on interface geometry. This quantity is a measure of the slip along the interface; in the limit , the amplitude of is proportional to *g*/*ω*^{2}.

Evaluating equation (3.6*b*) for the type *S* and type *TS* interfaces shown in figure 2, we find that
3.7
The geometric factor varies inversely with because for a fixed wavelength, the interface length increases with slope.

#### (ii) Low-frequency limit

In the limit as *ω*→0, equation (2.4*d*) simplifies to *σ*_{ns}=0. As discussed below (2.4*a*–*h*), the sample behaviour again becomes perfectly elastic. Correspondingly, the energy balance (3.2*a*) simplifies: the l.h.s. balances the second term on the r.h.s.; power supplied at the sample boundary now balances the rate of increase of stored strain energy. Integrating that simplified balance in time, we find that at zero frequency, the rigidity and strain energy are related by
3.8
Here *W*_{0} is calculated from equation (3.2*b*) using the solution of problem (2.4*a*–*h*) for *ω*=0 and *u*=±1 at *y*=±*a*. We have used the relation *τ*∼*G*_{0}*U*/*a*, where .

Next, using successive approximations, and assuming that the displacement vector varies sinusoidally in time, so that , we find that the first correction to the shear stress at the interface is given by . Here, is calculated from the solution of the b.v.p. posed immediately below equation (3.8). Evaluating equation (3.2*c*), we find that the dissipation rate is given by . Integrating that relation over one period, we find that
3.9a
and
3.9b
is the mean square slip at *ω*=0. Similarly, the mean strain energy is given by , where *W*_{0} is obtained using to evaluate equation (3.2*b*). Using the interpretation of ℒ given below equation (2.7), we obtain
3.10
Although *Φ*_{0} and *W*_{0} need to be computed numerically to obtain the low-frequency asymptote, this result, nevertheless, enables us to verify that our numerical solution is self-consistent.

### (c) Master variables for the mechanical loss curve

Using the above asymptotes, we introduce master variables allowing numerical results for different interface geometries to be represented on a single curve. According to equations (3.6*a*,*b*) and (3.10), in the extremes of high and of low frequency, the mechanical loss depends on interface geometry solely through the parameters *Φ*_{0}/*W*_{0} and *g*(*φ*)/*a*. Defining new variables *ω*′, ℒ′ by
3.11a
and
3.11b
then choosing the scales *ω*_{m} and ℒ_{m} so that the asymptotes (3.6*a*,*b*) and (3.10) become, respectively, ℒ′∼1/*ω*′ and ℒ′∼*ω*′, we find that
3.12a
and
3.12b
Provided no additional processes enter at intermediate frequencies, values of ℒ computed for different interface geometries should define a single curve when ℒ/ℒ_{m} is graphed against *ω*/*ω*_{m}. This prediction is tested in §4.

We may note that the master variables are particularly useful because as *N* is increased above about 10, the interface length rapidly approaches that of the limiting forms shown in figure 2, so the geometric factor *g*(*φ*) can be calculated using the results for a piecewise linear interface. By contrast, both *W*_{0} and *Φ*_{0} prove to converge slowly as *N* is increased, making it useful to be able to present results for differing *N* in terms of a single curve.

### (d) Local solution of Picu and Gupta

For interfaces having sharp corners, the stress obtained by solving problem (2.4*a*–*h*) proves to be singular at corners. The asymptotic behaviour near the corner must be compatible with a local analysis given by Picu & Gupta (1996). Specifically, because the displacements are finite, boundary condition (2.1a) requires the shear stress on the interface to remain finite as the distance *r* from the corner vanishes. Within the grains, however, the stress becomes infinite as *r*→0. Compared with that infinity, the interfacial stress appears to vanish, and so the effective interfacial boundary condition is *σ*_{ns}=0 on the interface near a corner. The local problem applying near the corner is defined by the system (2.4*a*–*h*) but with equation (2.4*d*) replaced by the simplified condition *σ*_{ns}=0. Picu and Gupta show that the problem admits a separable solution in which the stress , where *r*,*Θ* are local polar coordinates centred on the corner. The eigenvalue *p* must satisfy the condition −1<*p*<0; the first inequality follows because the strain energy of the whole sample must be finite, and the second inequality follows because in the discussion above, the stress at *r*=0 is assumed to be infinite. The Picu–Gupta solution shows that, owing to the condition of finite strain energy, the displacement must be continuous at the corner, i.e. as the corner *O* is approached along any path, the difference |** u**−

*u*_{O}|→0. In his numerical solution for a polygonal array, Ghahremani (1980) imposed, without discussion, the equivalent condition of vanishing relative displacement of the grains at corners. In §4, we demonstrate that our numerical solution of problem (2.4

*a*–

*h*) is consistent with that local analysis.

## 4. Discussion of numerical results

Figure 3*a* shows the computed values of ℒ graphed using the master variables defined by equation (3.11*a*,*b*). Results are shown for slope angles including those appropriate for a regular hexagonal array. The solid curve shows the prediction of the small slope analysis, as given by eqn (4.19) in Lee (2010). A similar master plot is also possible for |*G*|. Because fig. 6 of Ghahremani (1980) shows that an array of regular hexagons behaves as a standard solid, we assume and then verify that the same is true of the Raj–Ashby system. For a standard solid, however, depends only on , where *G*_{0} is the rigidity at zero frequency, and we have identified the time scale for the loss curve with *ω*^{−1}_{m}. Substituting for *G*_{0} and *ω*_{m} from equations (3.8) and (3.12*a*,*b*), respectively, we find that if the sample behaves as a standard solid, values of |*G*| computed for different *N* and *φ* will define a curve when is graphed against Figure 3*b* confirms that prediction. The Raj–Ashby system therefore behaves as a standard solid. This result also justifies our having used the physical interpretation of ℒ given below equation (2.7).

Owing to the self-similarity demonstrated in figure 3, at arbitrary frequency, the values of *N* and *φ* affect *G* purely through their influence on the elastic solution for *ω*=0. In particular, the mechanical loss ℒ is controlled by the quantity *W*_{0}/*Φ*_{0}. Figure 4 shows *W*_{0}/*Φ*_{0} as a function of *φ* with *N* as a parameter. For the *N*-values included there, the numerical solution approaches the perturbation solution when *φ*<10°. In figure 4*a*, the vertical line corresponds to the simple shear solution described in §3*a*; for that solution *Φ*_{0}=0, so that *W*_{0}/*Φ*_{0} is infinite. The dotted curve shows the solution obtained for a piecewise linear interface, but using the Ghahremani (pinned corner) boundary condition, namely that [** u**]=0 at a vertex. That solution, computed for a large value of

*N*, is independent of the parameter

*N*itself. Apart from the curve for

*N*=1, the behaviour is independent of

*N*for sufficiently large

*φ*; and as

*N*is increased (so that the corners are made tighter), the dependence on

*N*is confined to a range of

*φ*of decreasing size. We infer that in the limit as (fixed

*φ*≠0),

*W*

_{0}/

*Φ*

_{0}approaches the solution obtained when the corners are pinned.

The existence of that limiting state strongly constrains the effect of rounding corners on the loss and frequency scales defined by equation (3.12*a*,*b*). To emphasize this conclusion, in table 2, we give a numerical example. Comparing lines 1 and 2 in the table, we see that for the *S*-type interface with *φ*=30°, increasing *N* from 10 to causes the maximum value of ℒ to decrease by only about 30 per cent; according to figure 4*a*, the limiting case shown in line 2 is attained for *N*≥100. In line 3, we show only the limiting case because *W*_{0}/*Φ*_{0} is essentially independent of *N* at *φ*=60°. We note that these values of maximum loss are roughly half the corresponding value reported by Ghahremani (1980) for elastically accommodated sliding in an array of regular hexagons. In that array, sliding occurs simultaneously on two orthogonal surfaces, and a larger loss is to be expected.

Figure 5 shows the zero frequency rigidity *G*_{0}=*W*_{0}*a*/2*π* as a function of *φ*. We note that *G*_{0}≤1, in agreement with the physical constraint that the sample cannot have a rigidity exceeding that of the individual grains. Further, *G*_{0} is a non-monotonic function of *φ*. For the *S*-type interface, that non-monotonicity is a consequence of the simple shear solution for *φ*=45°, as discussed in §3*a*. Figure 5*b* shows that even for the *TS* interface, *G*_{0} has a maximum at around 50°.

Figure 6 shows the distribution of slip [*u*_{s}] along the interface for the parameter values given in the caption. The slip decreases monotonically as *N* increases and the corners become sharper. That is consistent with the requirement that the total strain energy of the sample is finite. That condition requires that the stress tensor varies with distance *r* from a sharp corner in such a way that as , and the local solution of Picu & Gupta (1996) then requires the slip to vanish at the corner. This result is evident from the figure. By increasing *N*, the slip distribution approaches the limiting case when the corners are numerically pinned. We also note that the weak-dependence of slip [*u*_{s}] on *N* is consistent with the result shown in figure 4*a*: that the ratio *W*_{0}/*Φ*_{0} becomes almost independent of *N* when *φ*=30°.

Figure 7 shows, as a function of distance *r* from the origin shown in the inset, the computed shear stress *σ*_{rΘ} on the interface at *Θ*=0°. The origin *O* sets a corner length scale at which the stresses are smoothed out. From the figure, we find that *σ*_{rΘ} scales with *r*^{−0.95} when *ε*=0.001. This result is close to the 1/*r* scaling predicted by the perturbation solution, and explains the logarithmic scaling for *G*_{0} in the perturbation solution given in Lee (2010). Consistent with the local solution by Picu & Gupta (1996), our numerical solution also predicts that the stress singularity weakens as *φ* increases. When *φ*=32.5°, our eigenvalue *p*=−0.53. That result is close to the solution given in fig. 5 of Picu & Gupta (1996) where we estimate, using our slope angle definition, their eigenvalue *p*=−0.55 at *φ*=32.5°. The weakened singularity with *p*>−1 ensures that the strain energy *W* remains finite.

## 5. Effects of non-uniform grain size and viscosity

The idealization of a constant grain size and a constant boundary viscosity is usually not found in real polycrystalline solids. Even in experiments on synthetic samples (e.g. Faul *et al.* 2004), grain size can vary by about a factor of four. According to the theory given in Ashby (1972), the boundary viscosity *η** may also vary by about an order of magnitude. This non-uniformity of grain size and viscosity is often invoked to explain the broad absorption peak (instead of a single Debye peak) commonly found in the experiment (e.g Schaller & Lakki 2001). We test that suggestion using the bicrystal model.

Figure 8 shows the type *S* interfaces used to assess these effects. In figure 8*a*, there are two adjacent grains of equal size *d** with viscosities *η*^{*}_{1} and *η*^{*}_{2} along their interface. Figure 8*b* shows five adjacent grains of two different sizes *d*^{*}_{1} and *d*^{*}_{2}. In that figure, the total area occupied by the ‘small’ grains is equal to that of the ‘big’ grain, and the viscosities along their interface are *η*^{*}_{2} and *η*^{*}_{1}, respectively. We note the total interfacial length available for sliding is the same in both configurations. Four systems are considered here:

configuration 1 with

*η*^{*}_{1}=*η*^{*}_{2}=*η** (uniform grain size and viscosity);configuration 1 with

*η*^{*}_{1}=*η** and*η*^{*}_{2}=10*η** (non-uniform viscosity);configuration 2 with

*d*^{*}_{1}=*d**,*d*^{*}_{2}=*d*^{*}_{1}/4 and*η*^{*}_{1}=*η*^{*}_{2}=*η** (non-uniform grain size);configuration 2 with

*d*^{*}_{1}=*d**,*d*^{*}_{2}=*d*^{*}_{1}/4,*η*^{*}_{1}=10*η** and*η*^{*}_{2}=*η** (non-uniform viscosity and grain size).

In figure 9, we graphed the mechanical loss spectra obtained from these systems. There are two main features in the figure. Firstly, the loss peak weakens as well as broadens when boundary viscosity *η** varies with position, i.e. in systems (ii) and (iv). Secondly, comparing systems (i) to (iii) and (ii) to (iv), we find that, when grain size varies spatially, the loss peak only weakens without any significant change to its shape.

The two effects (broadening and weakening) are caused by a difference in the sliding frequency *t**_{η}^{−1} across adjacent interfaces. Because sliding frequency *t**_{η}^{−1} depends on grain size and the boundary viscosity as defined in equation (2.2), sliding in systems (ii)–(iv) occurs at two different time scales and the peak broadens as a result. In the figure, the broadening effect caused by a spatial variation in boundary viscosity in system (ii) is more significant than that caused by non-uniform grain size in system (iii) because the difference in time scale is larger in the former.

Besides broadening the peak, the presence of multiple sliding frequencies within a system also cause the peak to weaken because the amount of sliding along an interface is constrained by the slip along the adjacent interface. Consequently, the response is controlled by the interface having a smaller sliding frequency. This result is evident in the figure where the peak is consistently located close to the smaller of the two sliding frequencies in systems (ii)–(iv).

Here, we note that according to the definition of the mechanical loss given in equation (2.7), the peak magnitude is independent of both boundary viscosity and grain size if these two quantities are uniform in a sample. When these two quantities change uniformly within a system, the loss spectrum simply shifts along the frequency axis and remains self-similar to a Debye peak. Thus, the weakening of the loss peak found here is caused purely by a spatial variation in grain size and in boundary viscosity; and the amount of weakening depends on the degree of variation, as well as the distribution of these two quantities within a system.

In table 3, we give, for different slope angles *φ*, the numerical values of the maximum mechanical loss ℒ in systems (i)–(iv). Except at *φ*=45° when the systems are under simple shear as discussed in §3*a*, we find that, in system (iii), where grain size is non-uniform, the loss peak decreases by about 50 per cent when compared with system (i) with uniform grain size and boundary viscosity. That decrease is insensitive to slope angle *φ*. By contrast, the weakening effect caused by a non-uniform viscosity interface is stronger when the slope angle *φ* is large. In system (ii) with slope angle *φ*=60°, the loss peak decreases by about 25 per cent when viscosity varies spatially by an order of magnitude; whereas at *φ*=5°, we find that the peak only decreases slightly by about 8 per cent. Combining these two effects in system (iv), the loss peak can decrease up to about 60 per cent.

Here, we note that our analysis may underestimate the effect of grain size summarized in table 3 if slip occurs concurrently along multiple planes in polycrystals. For that case, the overall grain size effect may then be enhanced because grain size varies along multiple directions. As a result, variation in grain size may then cause the loss peak to be even broader and weaker than shown in figure 9.

## 6. Conclusion

We have made the first numerical solution for finite slopes of the Raj–Ashby model of elastically accommodated grain boundary sliding along a prescribed spatially periodic interface. For small interface slopes, our solution approaches the limit given by the perturbation solution of Morris & Jackson (2009); it also agrees quantitatively with other analytical checks detailed above. To explain the difference between experimental results and current theoretical predictions on the loss peak, we use the validated numerical method to examine the sensitivity of the mechanical loss spectrum to (i) variation in boundary viscosity, (ii) rounding of corners, and (iii) variation in grain size, all on a piecewise linear interface. The effects owing to these factors are summarized here:

— When boundary viscosity

*η** varies by an order of magnitude across adjacent interface, the loss peak becomes significantly broader and its magnitude is reduced. According to table 3, the loss peak decreases up to about 25 per cent and the amount of reduction depends on the interface slope angle*φ*.— When corners are made sharper, the loss peak remains self-similar to a single Debye peak and weakens. That effect becomes insensitive to the rounding of corners once the corner radius is less than one-tenth of the wavelength of the spatially periodic interface (i.e.

*N*>10). For a type*S*interface with slope angle*φ*=30°, the loss peak decreases by about 30 per cent when an interface with a corner radius of about one-tenth of the wavelength is replaced with one that has infinitely sharp corners.— When the grain size is non-uniform, the loss peak weakens. A fourfold variation in the grain size roughly halves the peak height which remains nearly self-similar to a single Debye peak.

For a finely grained sample, grain boundary diffusion will always produce a background in the measured loss spectrum. Notwithstanding that, it has been shown in Morris & Jackson (2009) and Lee *et al.* (2009) that the effects of corner stress concentrations (in the form given in §3*d*) persists even in the presence of diffusion. Whether that effect in combination with the effects owing to heterogeneities in boundary viscosity and grain size can allow the appearance of a measurable sliding peak will depend on the magnitude of the background at the sliding frequency *t**_{η}^{−1}. That matter will be addressed in a later paper.

## Acknowledgements

The authors would like to thank Prof. Ian Jackson and the reviewers for their valuable comments which have helped us improve the presentation. L. C. Lee was supported in part by a Committee on Research Faculty Research Grant to S. J. S. Morris from the University of California.

## Footnotes

- Received November 23, 2009.
- Accepted February 22, 2010.

- © 2010 The Royal Society