MATLAB coding for generation of various yield/ failure surfaces used in Soil-plasticity

MATLAB coding for generation of various yield/ failure surfaces used in Soil-plasticity

Introduction: In this blog we will give (i) formulation of yield criteria in some plasticity models (ii) a MATLAB code that uses the formulation to generate the 3D yield surface in principal stress-space (iii) the MATLAB generated figures for some selected values of the parameters involved in the respective models.
Formulation: The yield/ failure criteria are expressed in terms of stress invariants ( I₁, I₂, I₃ )or deviatoric stress invariants ( J₁, J₂, J₃) or combinations of them. The plotting is to be done in principal stress space (σ₁- σ₂-σ₃). For the plotting of yield surface, the Haigh- westergaard coordinates (ξ, ρ, θ) are best means.Therefore, these principal stresses are to be expressed in terms of Haigh- westergaard coordinates (ξ, ρ, θ) using the transformation shown below




For Matsuoka-Nakai model, we need to do some extra formulations to obtain the yield surface in terms of I₁, J₂ and J₃, as follows,


Using this J₂ can be determined and on applying Haigh-Westergaard transformation, we get coordinates in principal stress space.

Original Cam Clay Model

 

Modified Cam-Clay model

Algorithm : For generating 3D yield surfaces, the following steps depict the sequence of implementation of plotting 

1. Taking the input parameters
2. Discretizing the axial coordinate [hydrostatic (p) pressures] and Lode Angles (th) , and forming grid of these values in Haigh-westergaard space
3. For each set of (p,th) values, the value of the radial coordinate (r ) is calculated based on yield criteria. This step will be different for each yield criteria
4. Applying the Haigh-westergaard transformation to get the coordinates in principal stress space (σ₁, σ₂, σ₃).

MATLAB code for these steps is shown in tables below, for Tresca, Matsuako-Nakai, Drucker- Prager and HiSS yielding criteria.


The plots resulting from execution of above MATLAB codes are shown below.