L08 Constitutive equations: Linear elasticity (orthohombic, VTI, isotropic)

This is a video recording of Lecture 08 of PGE 383 (Fall 2020) Advanced Geomechanics at The University of Texas at Austin delivered on 2020/9/15 by DN Espinoza ( / dnegeomechanics .
Topics: Constitutive equations, linearity and superposition simple, orthorhombic materials, vertical transverse isotropic (VTI) materials , isotropic materials, compliance matrix, stiffness matrix, Young modulus, Poisson ratio, Shear modulus
Demonstration of G = E/(2*(1+nu)): LINK TBA

[@benvanzon3234]

Great video explaining this theory! I'm studying and preparing for my graduation thesis regarding the simulation of phonons, and that required some new material's science theory!

[@danishzuhaidi]

You're the best sir, currently taking Geomechanics for my M.Sc. Drilling Engineering, and your channel is really helpful!!!

[@garshaspkeyvan5546]

Thank you very much, so through and complete, especially it was stated that normal stresses cannot generate shear strains.

[@dnicolasespinoza5258]

Thanks for watching and commenting!

[@garshaspkeyvan5546]

@@dnicolasespinoza5258 my absolute pleasure

[@sastraharapan2790]

I am taking my Mechanical Engineer ing M.Sc now. and it really helps me. thanks

[@dnicolasespinoza5258]

Glad it helps, I have had students from Mechanical Engineering and Material Science that enjoyed taking my class here at UT.

[@victoriage]

That was really good and clear. Thanks! Excited to see how you explain the shear decoupling :)

[@michaelsun7968]

Really great explanations, helped a lot with my research!

[@science_10523]

nice video. Due to strain the energy density changes , can we calculate "elastic stiffness constant" from these values?

[@dnicolasespinoza5258]

Yes, this is the so called "thermodynamic approach" for constitutive equations. Here is an example for isotropic poro-elasticity: ruclips.net/video/PmqfCbSMUuw/видео.html

[@wujohn598]

Hello Nicolas,
Very welled explained videos, thank you very much.
Can you please explain why did you put a 2 in front of the shear strain in the constitute equation ? You said you would explain the reasons for that but somehow I did not hear from that. thanks

[@dnicolasespinoza5258]

Hi John, this is because of a convenience using the definition of shear modulus G=tau/gamma in the stiffness matrix, where gamma is the "engineering strain" gamma = 2 *epsilon_ij with (i != j, shear strain). More equations here: en.wikipedia.org/wiki/Shear_modulus and ruclips.net/video/IB1bqLSUUBY/видео.html

[@wujohn598]

@@dnicolasespinoza5258 @D Nicolas Espinoza Thanks for your detailed replies. Can I ask one more question in related to shear modulus G ? Timber is anistropic matrial, but people usually assumes it to be orthotropic, which reduce the 21 variables to 9. However, "G32" IN ANISTROPIC MATRIX is also vanished from matrix, left only G12 G13 G23. What is the relationship between G23 and G32 in this case, are they equal together ? Moreover, if timber is assumed to be transversly isotropic material, and fibre is along dirction 3, what does G32 meant for timber in this case? Is it equal to G23?

[@dnicolasespinoza5258]

@@wujohn598 I'm not sure about what notation for G_ij your are using. The shear moduli of an orthotropic material are the inverse of C_44, C_55 and C_66 (16:42). These are all independent values for orthotropic materials. C_44=C_55 for transverse isotropic materials (more details and equations here: dnicolasespinoza.github.io/AdvancedGeomech/node5.html)

[@dnicolasespinoza5258]

@@wujohn598 You may want to check this video too: ruclips.net/video/PaW4sk3F5Mw/видео.html

[@wujohn598]

@@dnicolasespinoza5258 Thanks you very much

[@walterisraelmoscosozarate8768]

Greats from Ecuador, thanks For
Add subtitles

[@dnicolasespinoza5258]

Subtitles are automatic, they are usually pretty good but might be inaccurate for very specific non-common technical terms

[@walterisraelmoscosozarate8768]

Very Good Explaining

[@24papan]

Hello sir this is a very good explanation i found. But i am curious to know how c66 are related to c11 and c12. @30:30 sec

[@dnicolasespinoza5258]

Hi, this is equivalent to saying G=E/[2(1+nu)]. Look at the stiffness matrix of an isotropic solid (dnicolasespinoza.github.io/node16.html#SECTION00431000000000000000 - Eq. 3.19), C11 = E(1-nu)/(1+nu)/(1-2*nu), C12 = E*nu/(1+nu)/(1-2*nu), Hence, C66 =(C11-C12)/2 = E/[2(1+nu)]. The same applies to the isotropic plane 12 of the TVI solid, such that C66 =(C11-C12)/2 = E_h/[2(1+nu_h)], but be careful, it does apply for C44!
The demonstration of G=E/[2(1+nu)] is in my list of videos to upload soon.