| grain boundary interface |

| time derivative |

*ε* | interface characteristic slope |

*φ* | interface slope angle |

*ξ** | wavenumber of the periodic interface |

, | normal, tangential unit vector along interface |

*r*, *Θ* | radial, polar coordinates measured from a corner |

*x*, *y* | dimensionless , coordinates. Refer to equation (2.3*a*) |

*f*(*x*) | dimensionless interface function. Refer to equation (2.3*c*) |

*N* | number of Fourier terms used to define the interface |

*g*(*φ*) | geometric factor for high-frequency asymptote. Refer to equation (3.6*b*) |

*p* | stress singularity exponent or eigenvalue |

*u* | dimensionless displacement field. Refer to equation (2.3*b*) |

| amplitude of the dimensionless displacement field |

*u*, *v* | dimensionless , displacements |

*u*_{n}, *u*_{s} | dimensionless normal, tangential displacement along interface |

[*u*_{s}], | dimensionless slip, amplitude of dimensionless slip along interface |

*σ*_{ij}, *e*_{ij} | dimensionless cartesian stress, strain tensors. Refer to equation (2.3*d*,*f*) |

*σ*_{nn}, *σ*_{ns} | dimensionless normal, shear stress along interface |

*η** | boundary slip viscosity |

ℓ* | thickness of the disordered boundary region separating grains |

*d**, *d*^{*}_{m} | grain size, mean grain size |

*μ**, *ν* | shear modulus, Poisson ratio of grains |

*τ* | dimensionless *x*-average shear stress at *y*=±*a*. Refer to equation (2.5) |

| dimensionless amplitude of *x*-average shear stress at *y*=±*a* |

*G*, *γ* | sample shear modulus, shear strain. Refer to equation (2.6) |

*G*_{0} | zero frequency sample shear modulus |

*t*^{*}_{η} | sliding timescale. Refer to equation (2.2) |

*t* | dimensionless time. Refer to equation (2.3*e*) |

*ω* | dimensionless angular frequency |

*a* | dimensionless thickness of the grains. Refer to figure 1 |

| -displacement imposed at *y**=±*a*/*ξ**. Refer to figure 1 |

| mechanical loss. Refer to equation (2.7) |

| maximum mechanical loss, dimensionless maximizing frequency |

, *ω*′ | master variables for the mechanical loss curve. Refer to equation (3.11*a*,*b*) |

| dimensionless strain energy, time mean strain energy |

*W*_{0} | zero frequency strain energy |

*Φ*_{0} | zero frequency mean square slip |

*Υ* | dimensionless total dissipation |