Table 1.

Notation.

symbolsdefinition
Graphicgrain boundary interface
Graphictime derivative
εinterface characteristic slope
φinterface slope angle
ξ*wavenumber of the periodic interface
Graphic, Graphicnormal, tangential unit vector along interface
r, Θradial, polar coordinates measured from a corner
x, ydimensionless Graphic, Graphic coordinates. Refer to equation (2.3a)
f(x)dimensionless interface function. Refer to equation (2.3c)
Nnumber of Fourier terms used to define the interface
g(φ)geometric factor for high-frequency asymptote. Refer to equation (3.6b)
pstress singularity exponent or eigenvalue
udimensionless displacement field. Refer to equation (2.3b)
Graphicamplitude of the dimensionless displacement field
u, vdimensionless Graphic, Graphic displacements
un, usdimensionless normal, tangential displacement along interface
[us], Graphicdimensionless slip, amplitude of dimensionless slip along interface
σij, eijdimensionless cartesian stress, strain tensors. Refer to equation (2.3d,f)
σnn, σnsdimensionless normal, shear stress along interface
η*boundary slip viscosity
ℓ*thickness of the disordered boundary region separating grains
d*, d*mgrain size, mean grain size
μ*, νshear modulus, Poisson ratio of grains
τdimensionless x-average shear stress at ya. Refer to equation (2.5)
Graphicdimensionless amplitude of x-average shear stress at ya
G, γsample shear modulus, shear strain. Refer to equation (2.6)
G0zero frequency sample shear modulus
t*ηsliding timescale. Refer to equation (2.2)
tdimensionless time. Refer to equation (2.3e)
ωdimensionless angular frequency
adimensionless thickness of the grains. Refer to figure 1
GraphicGraphic-displacement imposed at y*=±a/ξ*. Refer to figure 1
Graphicmechanical loss. Refer to equation (2.7)
Graphicmaximum mechanical loss, dimensionless maximizing frequency
Graphic, ωmaster variables for the mechanical loss curve. Refer to equation (3.11a,b)
Graphicdimensionless strain energy, time mean strain energy
W0zero frequency strain energy
Φ0zero frequency mean square slip
Υdimensionless total dissipation