Permanent Deformation (PD): Phenomenological Model, Part I

Part of my graduate research study aimed to incorporate the shear stress component into various predictive models for permanent (plastic) deformation under cyclic loading in granular materials. In particular, granular materials referred, herein, are aggregate base and subgrade materials constructed under pavement or roadway subjected to repetitive vehicular loading as opposed to saturated sand and silt subjected to cyclic loading during seismic shaking events (i.e. undrained response). In that context, total stress was used instead of effective stress to simplify the problem. 

One should expect accumulation of permanent strain (or deformation) plotted as a function of load cycle (or time) as above. Note that the curve resembles a highly nonlinear relationship between plastic strain and load cycle; a phenomenological model that describes empirical relationship of each parameter. In one of the early studies, Monismith et. al. (1975) describes the power relationship to best fit the data by least sum of squared errors. Taking log at both sides, the power relationship becomes a linear relationship, shown in the figure below, where \(\epsilon_{p}\) = permanent strain; \(N\) = number of load cycle; \(A\) and \(b\) = experimentally determined coefficients.

$$\epsilon_{p} = AN^b$$

$$log(\epsilon_{p}) = log(A) + b\cdot log(N)$$

It now becomes obvious that coefficients \(A\) and \(b\), respectively, represent the "intercept" and "slope" of the data. The exponent \(b\) is a material constant; the exponent \(A\) must be a function of other factors such as stress level, previous stress history, placement conditions, etc. Furthermore, the exponent \(b\) is typically around 0.1, and \(A\) is harder to define due to other various factors. However, other studies have shown that \(A\) is strongly dependent on repeated stress state and material strength (Khedr, 1985; Garg et. al., 1997).

Using the laboratory test data obtained in my study, we will write a Python code to curve fit the data next.   

See Next: Curve Fitting with Python

Reference:

Gard, N, Tutumluer E, Thompson, MR, 1998. Structural modeling concepts for the design of airport pavements for heavy aircraft. Proceedings of the 5th International Conference on the Bearing Capacity of Roads and Airfields, Trondheim, Norway, 1998.

Khedr, S, 1985. Deformation characteristics of granular base course in flexible pavement. Transportation Research Record: Journal of the Transportation Research Board, 1043: 131-138.

Monosmith, CL, Ogawa, N,  Freeme, CR, 1975. Permanent deformation characteristic of subgrade soils due to repeated loading. Transportation Research Record: Journal of the Transportation Research Board, 537: 1-17.