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Final

EARTH 135 Lecture Notes - Lecture 2: Shear Stress, Stress (Mechanics), Sign ConventionExam


Department
Earth Science
Course Code
EARTH 135
Professor
Tanimoto
Study Guide
Final

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Earth&135&Lecture&2& &
Stress&(Chapter&2&in&Turcotte&and&Schubert)&
&
(1)!Unit:!Pa!(kg/m/s2)!!!
Pressure!is!force!per!unit!area.!Its!unit!is!Pa!(Pascal).!It’s!also!kg/m/s2.!
Unit!for!stress!is!also!Pa.!It!is!a!coupled!force!per!unit!area.!
!
(2)!What&do&two&indices,&i&and&j,&in&
σ
ij
&mean?!
‘i’!specifies!the!direction!of!force!and!‘j’!specifies!the!plane!on!which!the!force!acts.!!
Normal!stress!(
σ
xx,
σ
yy,
σ
xx
)!and!shear!stress!(𝜎"#!,!𝜎#",!𝜎#$,%𝜎$#!,!𝜎"$,!𝜎$")!
!Draw!a!picture!and!explain!𝜎""!,!𝜎#",!𝜎$"!
!Draw!a!picture!and!explain!𝜎"#!,!𝜎##,!𝜎$#!
! !
(3)!Stress&is&a&coupled&force&(per&unit&area)!
&&&&&&&&Symmetry:!!
σ
ij =
σ
ji
!which!makes!the!number!of!independent!components!of!stress!to!6!
!!!!!!!!But!!𝜎"#!%𝑎𝑛𝑑%𝜎#"!are!different.!!!
!!!!!!!!!!!!!!The!symmetry!means!their!magnitudes!are!the!same.!
!
(4)!Sign&convention.!!Tension!is!positive!for!normal!stresses.!
!What!does!positive!!𝜎""!!mean?!Draw!a!picture.!
!What!does!negative!𝜎""!!mean?!Draw!a!picture.!
!What!does!positive!!𝜎"#!!mean?!Draw!a!picture.!
!What!does!negative!𝜎"#!!mean?!Draw!a!picture.!
!
!Draw!a!picture!of!𝜎"#!if!it!is!negative.!
!
(5)!Transformation&of&stress&components&due&to&rotation&of&coordinates.!
Physics!of!the!problem!does!not!change!with!a!choice!of!the!coordinates.!But!the!
values!of!normal!stress!and!shear!stress!are!different!for!different!coordinates.!Let’s!
take!a!two-dimensional!(2D)!case!and!assume!(x’,y’)!are!the!coordinates!that!are!
rotated!from!(x,y)!by!an!angle!q.!Then!the!stress!components!change!as!
!
σ
x'x'=
σ
xx cos2
θ
+
σ
yy sin2
θ
+
σ
xy sin 2
θ
σ
y'y'=
σ
xx sin2
θ
+
σ
yy cos2
θ
σ
xy sin 2
θ
σ
x'y'=1
2(
σ
xx +
σ
yy )sin 2
ϑ
+
σ
xy cos2
θ
! ! ! ! (1)
!
!
Let’s!do!the!following!exercises!and!consider!what!the!above!equations!mean.!
!
!Draw!the!x-y!axes!and!draw!a!picture!for!𝜎""!,!𝜎##,!𝜎"#.!
! Using!the!same!origin,!draw!the!x’-y’!axes!that!are!rotated!by!an!angle!q
Then!draw!a!picture!of!𝜎")")!,!𝜎#)#),!𝜎")#)!
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