Download - The stress tensor
Transcript
![Page 1: The stress tensor](https://reader033.vdocuments.us/reader033/viewer/2022061423/56815774550346895dc5124a/html5/thumbnails/1.jpg)
The stress tensor
Section 2
![Page 2: The stress tensor](https://reader033.vdocuments.us/reader033/viewer/2022061423/56815774550346895dc5124a/html5/thumbnails/2.jpg)
“Undeformed” body means…
![Page 3: The stress tensor](https://reader033.vdocuments.us/reader033/viewer/2022061423/56815774550346895dc5124a/html5/thumbnails/3.jpg)
“Deformed” body means…
![Page 4: The stress tensor](https://reader033.vdocuments.us/reader033/viewer/2022061423/56815774550346895dc5124a/html5/thumbnails/4.jpg)
Internal stresses are…
![Page 5: The stress tensor](https://reader033.vdocuments.us/reader033/viewer/2022061423/56815774550346895dc5124a/html5/thumbnails/5.jpg)
![Page 6: The stress tensor](https://reader033.vdocuments.us/reader033/viewer/2022061423/56815774550346895dc5124a/html5/thumbnails/6.jpg)
![Page 7: The stress tensor](https://reader033.vdocuments.us/reader033/viewer/2022061423/56815774550346895dc5124a/html5/thumbnails/7.jpg)
kth component of area element df, which is along outward normal of portion
![Page 8: The stress tensor](https://reader033.vdocuments.us/reader033/viewer/2022061423/56815774550346895dc5124a/html5/thumbnails/8.jpg)
![Page 9: The stress tensor](https://reader033.vdocuments.us/reader033/viewer/2022061423/56815774550346895dc5124a/html5/thumbnails/9.jpg)
![Page 10: The stress tensor](https://reader033.vdocuments.us/reader033/viewer/2022061423/56815774550346895dc5124a/html5/thumbnails/10.jpg)
![Page 11: The stress tensor](https://reader033.vdocuments.us/reader033/viewer/2022061423/56815774550346895dc5124a/html5/thumbnails/11.jpg)
Moment of forces on portion (torque)
![Page 12: The stress tensor](https://reader033.vdocuments.us/reader033/viewer/2022061423/56815774550346895dc5124a/html5/thumbnails/12.jpg)
![Page 13: The stress tensor](https://reader033.vdocuments.us/reader033/viewer/2022061423/56815774550346895dc5124a/html5/thumbnails/13.jpg)
Vanishes if stress is symmetric, TBD
![Page 14: The stress tensor](https://reader033.vdocuments.us/reader033/viewer/2022061423/56815774550346895dc5124a/html5/thumbnails/14.jpg)
Then Mik is a surface integral, as required
![Page 15: The stress tensor](https://reader033.vdocuments.us/reader033/viewer/2022061423/56815774550346895dc5124a/html5/thumbnails/15.jpg)
Hydrostatic compression: Pressure of same magnitude acts on every unit area of the surface
of the body along the inward normal.
![Page 16: The stress tensor](https://reader033.vdocuments.us/reader033/viewer/2022061423/56815774550346895dc5124a/html5/thumbnails/16.jpg)
Arbitrary deformation
![Page 17: The stress tensor](https://reader033.vdocuments.us/reader033/viewer/2022061423/56815774550346895dc5124a/html5/thumbnails/17.jpg)
Equilibrium: Internal stresses balanceFi = 0 in every dV
![Page 18: The stress tensor](https://reader033.vdocuments.us/reader033/viewer/2022061423/56815774550346895dc5124a/html5/thumbnails/18.jpg)
External forces applied to body surface appear in equations of equilibrum as boundary conditions
![Page 19: The stress tensor](https://reader033.vdocuments.us/reader033/viewer/2022061423/56815774550346895dc5124a/html5/thumbnails/19.jpg)
![Page 20: The stress tensor](https://reader033.vdocuments.us/reader033/viewer/2022061423/56815774550346895dc5124a/html5/thumbnails/20.jpg)
Proof of sik symmetry
![Page 21: The stress tensor](https://reader033.vdocuments.us/reader033/viewer/2022061423/56815774550346895dc5124a/html5/thumbnails/21.jpg)
![Page 22: The stress tensor](https://reader033.vdocuments.us/reader033/viewer/2022061423/56815774550346895dc5124a/html5/thumbnails/22.jpg)
![Page 23: The stress tensor](https://reader033.vdocuments.us/reader033/viewer/2022061423/56815774550346895dc5124a/html5/thumbnails/23.jpg)
![Page 24: The stress tensor](https://reader033.vdocuments.us/reader033/viewer/2022061423/56815774550346895dc5124a/html5/thumbnails/24.jpg)
![Page 25: The stress tensor](https://reader033.vdocuments.us/reader033/viewer/2022061423/56815774550346895dc5124a/html5/thumbnails/25.jpg)
![Page 26: The stress tensor](https://reader033.vdocuments.us/reader033/viewer/2022061423/56815774550346895dc5124a/html5/thumbnails/26.jpg)