Three surface kinematic hardening model

Three surface kinematic hardening model by Stallebrass [7] and Stallebrass and Taylor [8] with small strain elasticity by Viggiani and Atkinson [9].

Required parameters:

• Parameter 1 - number of the constitutive model (in this case 2).
• Parameter 2 - bulk modulus of water $K_w$ for undrained analysis using the penalty approach. In drained or consolidation analysis $K_w$ should be set to 0.
• Parameters 3-12 - parameters of the model ($A$, $n$, $m$ (stiffness parameters), $M$, $\kappa^*$, $\lambda^*$, $T$, $S$, $\psi$, $N$)

Therefore, parameters should be specified as follows: 2, $K_w$, $A$, $n$, $m$, $M$, $\kappa^*$, $\lambda^*$, $T$, $S$, $\psi$, $N$

Required state variables:

18 state variables required:

• State v. 1 - void ratio $e$.
• State v. 2 - centre of the bounding surface $p_0$ (for postprocessing only).
• State v. 3-8 - centre of the history surface $\boldsymbol\sigma_a$ in Voigt notation ( $\sigma_{a11}$, $\sigma_{a22}$, $\sigma_{a33}$, $\sigma_{a12}$, $\sigma_{a13}$, $\sigma_{a23}$).
• State v. 9-14 - centre of the yield surface $\boldsymbol\sigma_b$ in Voigt notation.
• State v. 15 - value of the yield function $f$ (for postprocessing only).
• State v. 16 - Excess pore pressure $u$ for undrained analysis (compression positive sign convention).
• State v. 17 - Suggested size of the first time substep (for calculation controll).
• State v. 18 - Number of evaluation of the constitutive model in one global time step (for postprocessing only).

If positions of kinematic surfaces not initialised (values of state v. 3 and 9 are 0), they are initialised such that they are centered about the initial stress state. State v. 2 ($p_0$) is meant for post-processing only, it is always calculated from void ratio $e$.

Typical parameters:

Table 4: Typical parameters of the 3-SKH model

	$A$	$n$	$m$	$M$	$\kappa^\ast$	$\lambda^\ast$	$T$	$S$	$\psi$	$N$
Speswhite kaolin [8]	1964	0.65	0.2	0.89	0.005	0.073	0.25	0.08	2.5	1.1

Example input file and initialisation file for the FE program Abaqus$^{TM}$.
Example input file for the FE program Tochnog Professional.