I. Introduction
The creep-angle analysis was proposed by Reference Haefeli and KingeryHaefeli (1963, Reference Haefeli and Ōura1967) as a practical method of obtaining the stress in the neutral zone of inclined planar slabs composed of compressible viscous materials such as snow or clay. By neutral zone, Haefeli meant that portion of the slab that is free from edge effects, essentially, the central region of the slab. With reference to Figure 1, the neutral zone is modeled as a planar slab extending to infinity in two directions, x and z. For this simple geometry, spatial gradients exist only in the y-direction, that is, ∂/∂x and ∂/∂z are null. Furthermore, the neutral zone is assumed to be a region of plane deformation-rate, wherein the creep velocity in the z-direction is zero. The creep angle, β, in Haefeli’s original treatment, is simply the angle formed by the x-axis and the trajectory of the moving snow particles.
Assuming coincidence of principal stress and principal deformation-rate axes, Haefeli utilized geometrical constructions to determine the neutral zone stress components as
where ρ is the slab density, θ is the slab inclination to the horizontal, and H is the slab thickness. It will be the purpose of this short note to extend Haefeli’s analysis to include non-neutral regions and nonplanar slabs. A more detailed discussion of this particular problem, with application to snow slabs, is found in Reference PerlaPerla (1971).
II. The Deformation-Rate Coefficient
We choose a purely viscous constitutive law to relate the following slab variables: stress tensor t ij , deformation-rate tensor d kl , temperature T, and density ρ. The constitutive law is expressed in cartesian tensor notation as
where f ij is a tensor function. A spatial reference system is attached to the slab’s substratum at a convenient point, and d ij is computed from the creep components u i , according to
Material isotropy is assumed; Equation (2) must then reduce to (Reference Truesdell, Noll and FlüggeTruesdell and Noll, 1965)
where ϕ i are functions of T, ρ, and the scalar invariants of d kl . In Equation (4), the summation convention is observed for double subscripts, and δ ij is the Kronecker delta. The expansion of Equation (4) for the components t xy , t yy , and t xx is
where the summation is on i only. For plane deformation-rate problems, wherein d iz vanishes, Equation (5) may be manipulated algebraically to give the simple relationship
where D (x, y), a quantity we call the deformation-rate coefficient, consists of
In the neutral zone, ∂/∂x is null, and hence Equation (7) reduces to
In the special case, where u x and u y vary linearly with y, D(y) represents the tangent of the creep angle.
III. Generalization
Plane problems require simultaneous solution of Equation (6) and the equations of equilibrium which are
The set of Equations (6) and (9) are hyperbolic partial differential equations in the unknowns t xy , t yy , and t xx . Solutions can be found for arbitrary slab geometries, provided D(x, y) can be specified throughout the region of interest, and boundary conditions can be specified at the slab-atmosphere interface. Figure 2 illustrates the general problem. If D(x, y) can be specified within the region bounded by npmn, then it is possible to find the characteristics of Equations (6) and (9), namely nq and mq. The point q must be contained within npmn. If t xy and t yy can be specified along nm, then t xy , t yy and t xx can be determined uniquely within the region bounded by nqmn. This is the Cauchy problem in hyperbolic partial differential equations; numerical solutions are available (Reference PanovPanov, 1963).
The hyperbolic system, Equations (6) and (9), is linear and avoids the non-linear problems of the alternative, elliptical system which consists of Equations (4), (9), and the compatibility equations for plane deformation-rate. Moreover, solution of the elliptical system requires knowledge of the elusive phenomenological functions ϕ i , and knowledge of the boundary conditions around the entire slab region of interest. In contrast, the hyperbolic system requires in situ measurement of D(x, y). Such measurement seems feasible for a wide variety of natural slabs.
It is worthwhile to note that the above analysis could be repeated for deformations instead of deformation-rates. A deformation tensor could replace d kl in Equation (2). The analysis would carry through with displacements replacing velocities, u x and u y . In this case, the experimental task is in situ measurements of deformations instead of deformation-rates.
Acknowledgement
Consultation with Professor J. E. Fitzgerald, University of Utah, is gratefully acknowledged.
