Energy-based plastic potential and yield functions for rockfills

Abstract

The plastic potential and yield functions are two of the most important components of elastoplastic constitutive models. Because the incremental plastic energy-based equation of the Cam-Clay model does not consider the particle breakage energy, the predicted results for crushable granular material, e.g., rockfills, are not convincing. It should be noted that the particle breakage energy is difficult to be quantified separately, and to avoid directly measuring it, the total input energy during shearing is taken as the equivalent of the energy-based equation in this study. Accordingly, a unified function with two parameters (critical stress ratio M_c and χ) is derived for defining the yield and the plastic potential surfaces for rockfills. When χ = 1.0, the yield locus f is equal to the plastic potential surface g, when χ > 1.0, f is below g, and f tends to become more bullet-shaped with an increasing χ, and when χ < 1.0, f is above g, and f tends to become more drop-shaped with a decreasing χ. Additionally, the experimental plastic strain increment vectors of the Pancrudo slate rockfill (Alonso 2016) are not normal to the yield locus, which demonstrates that the proposed unified function is reasonable.

Appendix

For the special case of an axisymmetric triaxial compression specimen (σ₂ = σ₃ and ε₂ = ε₃), the stress and strain invariants can be simplified to the following well-known functions:

$$\left\{\begin{array}{c}p=\left({\sigma }_{1}+2{\sigma }_{3}\right)/3\\ q\text{=}\left({\sigma }_{1}-{\sigma }_{3}\right)\end{array}\right.$$

(23)

Therefore,

$$p-q/3={\sigma }_{3}$$

(24)

At the critical state, p=p_c, q=q_c, and q_c = M_cp_c, thus

$$\left\{\begin{array}{c}{p}_{c}-{q}_{c}/3={\sigma }_{3}\\ {q}_{c}={M}_{c}{p}_{c}\end{array}\right.$$

(25)

which gives

$${q}_{c}=\frac{3{M}_{c}}{3-{M}_{c}}{\sigma }_{3}$$

(26)