23. Simple Soil Models

This chapter describes the background theory for the Hardin-Drnevich and Ramberg-Osgood soil models. Both models are elastic models with a nonlinear shear stress-shear strain relationship. The contents of this chapter are based on Jenning [53], Konder [56], and Hardin & Drnevich [37].

In a three-dimensional isotropic linear-elastic material model, the stress-strain relationship can be characterised as:

$$\left\{\vphantom{ \negthickspace \begin{array}{c} \sigma _{xx} \\... ...\\ [1.5ex] \tau_{yx} \\ [1.5ex] \tau_{xz} \end{array} \negthickspace }\right. \begin{array}{c} \sigma _{xx} \\ [1.5ex] \sigma _{yy} \\ [1.5ex... ...} \\ [1.5ex] \tau_{xy} \\ [1.5ex] \tau_{yx} \\ [1.5ex] \tau_{xz} \end{array} \left.\vphantom{ \negthickspace \begin{array}{c} \sigma _{xx} \\ ... ...\ [1.5ex] \tau_{yx} \\ [1.5ex] \tau_{xz} \end{array} \negthickspace }\right\} \left[\vphantom{ \negthickspace \begin{array}{cccccc} c_1 & c_2 &... ...c_3 & 0 \\ [1.5ex] 0 & 0 & 0 & 0 & 0 & c_3 \end{array} \negthickspace }\right. \begin{array}{cccccc} c_1 & c_2 & c_2 & 0 & 0 & 0 \\ [1.5ex] c_2... ...[1.5ex] 0 & 0 & 0 & 0 & c_3 & 0 \\ [1.5ex] 0 & 0 & 0 & 0 & 0 & c_3 \end{array} \left.\vphantom{ \negthickspace \begin{array}{cccccc} c_1 & c_2 &... ...c_3 & 0 \\ [1.5ex] 0 & 0 & 0 & 0 & 0 & c_3 \end{array} \negthickspace }\right] \left\{\vphantom{ \negthickspace \begin{array}{c} \varepsilon _{x... ...[1.5ex] \gamma_{yz} \\ [1.5ex] \gamma_{xz} \end{array} \negthickspace }\right. \begin{array}{c} \varepsilon _{xx} \\ [1.5ex] \varepsilon _{yy} ... ...[1.5ex] \gamma_{xy} \\ [1.5ex] \gamma_{yz} \\ [1.5ex] \gamma_{xz} \end{array} \left.\vphantom{ \negthickspace \begin{array}{c} \varepsilon _{xx... ...1.5ex] \gamma_{yz} \\ [1.5ex] \gamma_{xz} \end{array} \negthickspace }\right\}$$ (23.1)

where the parameters c₁ , c₂ , and c₃ are constant. In the Hardin-Drnevich and Ramberg-Osgood models, the parameter c₃ is dependent on the shear strain $\gamma$ as described in §23.1 and §23.2. The parameters c₁ and c₂ are dependent on the Poisson's ratio $\nu$ , and on either the initial Young's modulus E or the current shear state. In a bulk modulus formulation this reads, respectively

$${\frac{{E}}{{3\left(1-2\nu\right)}}}$$ (23.2)

or

$${\frac{{2\left(1+\nu\right)\frac{\partial \tau}{\partial \gamma}}}{{3\left(1-2\nu\right)}}}$$ (23.3)

The models should behave according to the so-called extended Masing rules:

• For the initial loading, the stress-strain relationship is prescribed by a skeleton curve (or backbone curve).
• When reloading or unloading from the initial loading occurs, the stress-strain relationship forms a loop, which is obtained by scaling the skeleton curve by a factor two.
• If the previous maximum shear strain is exceeded, the stress-strain relationship again follows the skeleton curve.
• If the hysteresis loop intersects a previous loading or unloading curve, the stress-strain relationship follows that previous curve.