1. ELEMENTS OF THE CRITICAL STATE MODEL By Ayam Jabbar Jihad

2. ELEMENTS OF THE CRITICAL STATE MODEL
Yield Surface
The equation for the yield surface is ellipse
given by :-

3. Critical State Parameters
Failure Line in space
The failure line in space is

4. Axisymmetric CompressionMc
We know
Therefore , or

5. Axisymmetric Extension
In an axisymmetric extension test , the radial stress is the
Major principal stress .Since in axial symmetry the radial stress is equal to the circumferential stress,we get
and

6. or

7. Failure line in space
Let us now find the equation for the critical
state line in space .We will use the plot as shown in Fig.(8.9c)The CSLis parallel to the normal consoldation line and is represented by

10. The essential critical state parameter are :
Compression index, which is obtained from an isotropic or a one-dimensional consolidation test .
Unloading /reloading index or recompression index, which is obtained from an isotropic or a one-dimensional consolidation test .
Critical state frictional constant .

11. Example 1-1 calculation of failure stresses
To use the critical state model , you also know the initial stresses , for example and and the initial void ratio
Example 1-1 calculation of failure stresses

12. Solution:-

13. Example 1-2 Determination of

14. Figure (8.2)
Strategy Make a sketch of the results in space to provide a visual aid for solve aid for solving this problem .
Solution:-