16.2.1 Grains Model

The grains (or standard) model, proposed by Allaart [2], gives the hydrostatic pressure and the effective deviatoric stress as nonlinear functions of the isotropic and deviatoric strain invariants:

$$\begin{cases} p &= \tfrac{1}{2}\sqrt[n]{ K_{1} \varepsilon _... ...3ex] q &= 3 G_{1} p^{1-n} \varepsilon _{\mathrm{s}} \end{cases}$$ (16.10)

with the factor $\beta$ given by

$$\beta {\frac{{K_{1} \left( 1-n \right)}}{{6 G_{1} }}}$$ (16.11)

The reference values of the compression and shear moduli, K₁ and G₁ respectively, as well as the value of the power n are determined by experiments. Given the current strain state, the stress invariants p and q are given by (16.10) and the moduli K and G are determined. The (secant) stiffness matrix is then easily calculated with (16.9).