Post on 24-Feb-2023
KARAKTERISTIK DEFORMASI
Strain dan Stress
HERI ANDREASMahasiswa Program Doktor
Prodi Geodesi dan Geomatika ITBE-mail : heri@gd.itb.ac.id
September 2007
1. Pengertian Deformasi
Deformasi adalah perubahan bentuk, dimensi dan posisi darisuatu materi baik merupakan bagian dari alam ataupun buatanmanusia dalam skala waktu dan ruang
3. Objek dari Deformasi
tektonik lempeng
pasut
atmosferik
proses hidrologi
ocean loading
proses geologi lokal
rotasi bumi
Alam
Manusiapelapukan
erosi
abrasi
subsidence
longsoran
tsunami
Fenomena lain
D. Sarsito, 2006
3. Objek dari Deformasi
S AMP No r t h
-25
-20
-15
-10
-5
0
5
10
15
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
35.000
35.040
35.080
35.120
35.160
35.200
35.240
35.280
35.320
35.360
35.400
35.440
35.480
35.520
35.560
35.600
35.640
35.680
35.720
35.760
35.800
35.840
35.880
35.920
35.960
36.000
36.040
36.080
36.120
36.160
36.200
36.240
36.280
36.320
36.360
36.400
36.440
36.480
36.520
36.560
22 /9/ 06 -124 /9/ 06 -1326 /9/ 06 -127 /9/ 06 -1329 /9/ 06 -130 /9/ 06 -132/10 / 06 -13/10 / 06 -135/10 / 06 -16/10 /06 -138/10 /06 -19/10 / 06 -1311 /10 /06 -112 /10 /06 -1313 /10 /06 -1915 /10 /06 -716 /10 /06 -1618 /10 /06 -419 /10 /06 -1621 /10 /06 -422 /10 /06 -1624 /10 /06 -425 /10 /06 -1627 /10 /06 -428 /10 /06 -1630 /10 /06 -431 /10 /06 -162/11 / 06 -43/11 / 06 -165/11 / 06 -46/11 /06 -168/11 / 06 -49/11 / 06 -1611 /11 /06 -412 /11 /06 -1614 /11 /06 -415 /11 /06 -1617 /11 /06 -418 /11 /06 -1620 /11 /06 -421 /11 /06 -1623 /11 /06 -424 /11 /06 -16
Tgl/Bln/Thn-Jam
Height (m)
4. Jenis dari Deformasi
Deformasi dapat dibagi menjadi 2 jenis yaitu Deformasi Statik danDeformasi sesaat
Deformasi statik bersifat permanen
Deformasi sesaat bersifat sementara / dinamis
5. Parameter Deformasi
Deformasi dari suatu benda/ materi dapat digambarkan secarapenuh dalam bentuk tiga dimensi apabila diketahui 6 parameterregangan (normal-shear) dan 3 parameter komponen rotasi
Parameter deformasi ini dapat dihitung apabila diketahui fungsipergeseran dari benda tersebut persatuan waktu
Normal strain Shear strain
6. Model dan pengamatan Deformasi
Secara praktis survey deformasi akan terpaut pada titik-titik yangbersifat diskrit, dengan demikian deformasi dari benda harusdidekati dengan model.
Fungsi dari deformasi dinyatakan oleh persamaan dalam bentukmatrik :
d = B c
Dimana :B, adalah matrik deformasi yang elemennya merupakan fungsi dariposisi dari titik yang diamati, serta waktu
C, vektor yang koefisiennya akan diketahui
6. Model dan pengamatan Deformasi
Survey deformasi: penentuan perubahan posisi, jarak, sudut,regangan : teknik geodetik, geofisika, dan lain-lain
7. Analisis Deformasi
Analisis Geometrik :
Bila kita hanya tertarik pada status geometrik (ukuran dandimensi) dari benda yang terdeformasi
Analisis Fisis :
Bila kita bermaksud untuk menentukan status fisis dari benda yangterdeformasi, regangan, dan hubungan antara gaya dengandeformasi yang terjadi
8. Analisis Deformasi aspek fisis
Dalam analisis fisis deformasi, hubungan antara gaya dandeformasi dapat dimodelkan dengan menggunakan metodaempiris (statistik), yaitu melalui korelasi antara pengamatandeformasi dan pengamatan gaya
Metoda lain dalam analisis fisis yaitu metoda deterministik, yangmemanfaatkan informasi dari gaya, jenis material dari benda, danhubungan fisis antara regangan (strain) dan tegangan (stress)pada benda
9. Normal strain :perubahan panjang
- Change of length proportional to length
- xx, yy, zz are normal component of strain
nb : If deformation is small, change of volume is xx + yy + zz (neglecting quadratic terms)
SEAMERGES GPS COURSE, 2005
10. Shear Strain : perubahan sudut
xy = -1/2 (1 + 2) = 1/2 (dydx + dxdy )xy = yx (obvious)
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11. Stress dalam 2 Dimensi
- Force = x surface
- no rotation =>xy =yx
- only 3 independent
….components :
…..xx ,yy ,xy
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12. Applied Forces
Normal forces on x axis xx(x). y xx(x+x). yy xx(x). xx(x+x) y dxx/dx . x (1)
Shear forces on x axis yx(y). x yx(y+y). x
x dyx/dy . y (2)
Total on x axis dxx/dx + dyx/dyx ySEAMERGES GPS COURSE, 2005
13. Forces Equilibrium
Total on x axis = dxx/dx + dyx/dyx y
Total on y axis = dyy/dy + dyx/dxy x
dyy/dy + dyx/dxdxx/dx + dyx/dyEquilibrium =>
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14. Solid elastic deformation
• Stresses are proportional to strains
• No preferred orientationsxx = (G) xx + yy + zzyy = xx + (G) yy + zzzz = xx + yy + (G) zz
• and G are Lamé parameters
The material properties are such that a principal strain component produces a stress (G) in the same direction
and stresses in mutually perpendicular directions
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Inversing stresses and strains give :
xx = 1/Exx -
/Eyy -
/Ezzyy = -
/Exx + 1/Eyy -
/Ezzzz = -
/Exx -
/Eyy + 1/Ezz
• E and are Young’s modulus and Poisson’s ratio
a principal stress component produces
a strain 1/E in the same direction and
strains
/E in mutually perpendicular directions
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14. Solid elastic deformation
15. Elastic deformation across a locked fault
What is the shape of the accumulated deformation ?
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Formula matematis
•Symetry => all derivative with y = 0
yy = 0
•No gravity => zz = 0
•What is the displacement field U in the elastic layer ?
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•Elastic equations :
(3) +zz = 0 => xx + zz = -2G zz
and (2) => yy = xx + zz = -2 G zz
xx = (G) xx + zz
(2) yy = xx + zz
(3) zz = xx + (G) zz
xy = G xy xz = G xzyz = G yz
=> xx = - (2G + zz
and (1) => xx = [- (2G)2/ + ] zz
_________________________
Formula matematis
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• Force equilibrium along the 3 axis
(x) dxx /dx + dyx /dy + dxz /dz = 0
_________________________
(y) dxy /dx + dyy /dy + dyz /dz = 0
(z) dxz /dx + dyz /dy + dzz /dz = 0
xxx x
• Derivation of eq. 1 with x and eq. 3 give : d2xx /dx2 = 0
• equation 2 becomes : dxy /dx +dyz /dz = 0
Formula matematis
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Formula matematis
relations between
stress () and displacement vector (U)
xy = 2G xy = 2G [dUx /dy + dUy /dx] .1/2
_________________________yz = 2G yz = 2G [dUz /dy + dUy /dz] .1/2
xd/dx[dUx/dy +dUy/dx] +d/dz[dUz/dy +dUy/dz] = 0
d2Uy /dx2 +d2Uy /dz2 = 0
Using dxy /dx +dyz /dz = 0 we obtain :
x
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Formula matematis
What is Uy, function of x and z, solution of this equation ?
d2Uy /dx2+d2Uy /dz2 = 0
Guess : Uy = K arctang (x/z) works fine !
Nb. datan()/d1/(1+2 )
dUy/dx=K/z(1+x2/z2) => d2Uy/dx2
= -2Kxz/(z2+x2)
dUy/dz=-Kx/z2 (1+x2/z2) => d2Uy/dz2
= 2Kxz/(x2+z2)
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Formula matematis
Boundary condition at the base of the crust (z=0)
Uy = K arctang (x/z)
Uy = K . /2 if x > 0 = K . – /2 if x < 0
=> K = 2.V0 /
And also :
Uy = +V0 if x > 0 = –V0 if x < 0
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Formula matematis
at the surface (z=h)
Uy = K arctang (x/z)
Uy = 2.V0 /arctang (x/h)
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18. Elastic Dislocation (Okada, 1985)Surface deformation due to shear and tensile faults in a half space, BSSA vol75, n°4, 1135-1154, 1985.
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The displacement field ui(x1,x2,x3)
due to a dislocation uj (1,2,3)across a surface in an isotropicmedium is given by :
Where jk is the Kronecker delta, and are Lamé’s parameters, k isthe direction cosine of the normal to
the surface element d.
uij is the ith component of the
displacement at (x1,x2,x3) due to the
jth direction point force of magnitude
F at (1,2,3)
18. Elastic Dislocation (Okada, 1985)
(1) displacements
For strike-slip For dip-slip For tensile fault
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21. Case : Sumatra subduction zone
Triyoso, 2005
Natawijaya, 2007
Segmen Mentawai-Pagai belum sepenuhnya terpatahkan ??? !!!
22. Strain rate and rotation rate tensors
2. Compute strain rate and rotation rate tensors
1. Look at station velocity residuals
Velocity mm/yrStrain =
_______=
_____= % / yr
Distance km
Matrix tensor notation : Sij = d(Vi) / d(xj) =
d(Vx) / d(x) d(Vx) / d(y)
d(Vy) / d(x) d(Vy) / d(y)
Theory says : [S] = [E] + [W]
Symetrical Antisymetrical
Strain rate rotation rate
To asses plate deformation :
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22. Strain rate and rotation rate tensors
[E] has 2 Eigen values : 1, 21 and 2 are extension/compression along principal direction defined
by angle defined as angle between 2 direction and north
[E] = ½ ([S] + [S]T) =
E11 E12
E12 E22
[W] = ½ ([S] - [S]T) =
0 W
-W 0
1 = E11 cos2 + E22 sin2 – 2 E12 sin cos2 = E11 sin2 + E22 cos2 – 2 E12 sin cos
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22. Strain rate and rotation rate tensors
Therefore we can compute strain rate and rotation rate within any
polygon, the minimum polygon being a triangle
Minimum requirement to compute strain and rotation rates is :
3 velocities (to allow to determine 3 values 1, 2, and W)
No deformation compression rotation
Strain and rotations are unsensitive to reference frame
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23. Case : Strain & Rotation on GEODYSSEA network
Strains :
extension/compression/strike-slip
Rotations :
Anti-clockwise/clockwise
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