L04 Stress invariants, isotropic and deviatoric stress components, stress path

[@GauravMishra-hf1pg]

Xcellent , superb explanation , total visualization. Thanks a lot Prof.

[@sanjb109]

Professor, u have so much understanding...better explanation....made us easy to understand

[@siriuslot4708]

Thank you so much Dr. Espinoza. I finally understand the principal, deviatoric, invarianst etc. Struggled with these concepts previously.

[@joisleenramirez620]

Thank you so much professor. Finally, I can understand. Saludos desde Panamá

[@dnicolasespinoza5258]

Saludos!

[@estebanpatinomarin416]

Thank you so much Dr. Espinoza. Could you tell me in which book or article to find the physical interpretation of the third invariant?

[@dnicolasespinoza5258]

Great question Esteban! I had a feeling for what it meant but not quite until you asked and I solved the problem! Here it goes:
The third invariant reduces to the determinant of the stress tensor, so when written in principal stresses
I_3 = sigma_1*sigma_2*sigma_3
Let's call D13 = sigma_1 - sigma_3 and D23 = sigma_2 - sigma_3, and use these D's (Mohr circle diameters) in the equation above.
You get
I_3 = D13*D23*sigma_3 + (D13+D23)*sigma_3 + sigma_3^3
Hence, I_3 gives you an idea of the difference between sigma_1 and sigma_2 with sigma_3.
For a compressional "polyaxial" state with sigma_1 != sigma_2 != sigma_3, I_3 is bound by the perfect triaxial extension case (sigma_1 = sigma_2 > sigma_3) and the perfect triaxial compression case (sigma_1 > sigma_2 = sigma_3) (See lectures on Mohr-Coulomb criterion with triaxial extension and compression), such that
Triax ext bound: (sigma_1 - sigma_3)*sigma_3^2 + sigma_3^3 < I_3 < Triax comp bound: (sigma_1 - sigma_3)^2*sigma_3 + 2(sigma_1 - sigma_3)*sigma_3^2 + sigma_3^3.
In summary, I_3 tell you how far you are in between triax compression and triax extension!
Notice the I_3 = 0 would happen for plane-stress (sigma_3 = 0) or I_3

[@luoxingling1980]

Thank you so much for your lecture!

[@cchrism.7586]

Wow!

[@Prophetic_heirs]

Thank you so much for this lecture!!

[@prakharmishra3726]

Waiting for "better than my university professor" comments