KARAKTERISTIK DEFORMASI Strain dan Stres

KARAKTERISTIK DEFORMASI

Strain dan Stress

HERI ANDREAS

Mahasiswa Program Doktor

Prodi Geodesi dan Geomatika ITB

E-mail :

heri@gd.itb.ac.id

1. Pengertian Deformasi

2. Penyebab Deformasi

4. Jenis dari Deformasi

Deformasi dapat dibagi menjadi 2 jenis yaitu Deformasi Statik dan

Deformasi sesaat

Deformasi statik bersifat permanen

5. Parameter Deformasi

Deformasi dari suatu benda/ materi dapat digambarkan secara

penuh dalam bentuk tiga dimensi apabila diketahui 6 parameter

regangan (normal-shear) dan 3 parameter komponen rotasi

Parameter deformasi ini dapat dihitung apabila diketahui fungsi

pergeseran dari benda tersebut persatuan waktu

6. Model dan pengamatan Deformasi

Secara praktis survey deformasi akan terpaut pada titik-titik yang

bersifat diskrit, dengan demikian deformasi dari benda harus

didekati dengan model.

Fungsi dari deformasi dinyatakan oleh persamaan dalam bentuk

matrik :

d = B c

Dimana :

B, adalah matrik deformasi yang elemennya merupakan fungsi dari

posisi dari titik yang diamati, serta waktu

6. Model dan pengamatan Deformasi

7. Analisis Deformasi

Analisis Geometrik :

Bila kita hanya tertarik pada status geometrik (ukuran dan

dimensi) dari benda yang terdeformasi

Analisis Fisis :

8. Analisis Deformasi aspek fisis

Dalam analisis fisis deformasi, hubungan antara gaya dan

deformasi dapat dimodelkan dengan menggunakan

metoda

empiris (statistik)

, yaitu melalui korelasi antara pengamatan

deformasi dan pengamatan gaya

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)

10. Shear Strain : perubahan sudut



xy = -1/₂

(



1 +



2

)

= 1/₂

(

d



y



d

x +

d



x



d

y

)



xy =



yx (obvious)

11. Stress dalam 2 Dimensi

-

Force =



x

surface

-

no rotation =>



xy

=



yx

-

only 3 independent

….

components :

…..



xx

,



yy

,



xy

12. Applied Forces

Normal forces on x axis





xx

(x)

.



y





xx

(x+



x)

.



y





y



_xx

(x)

_.





_xx

(x+



x)





 

y

d



xx

_/

d

x .



x

(1)

Shear forces on x axis





yx

(y)

.



x





yx

(y+



y)

.



x

13. Forces Equilibrium

14. Solid elastic deformation

• Stresses are proportional to strains

• No preferred orientations



xx = (





G)



xx +





yy +





zz



yy =





xx + (





G)



yy +





zz



zz =





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

Inversing stresses and strains give :



xx = 1

/

_E



xx -



/

_E



yy -



/

_E



zz



yy = -



/

_E



xx + 1

/

_E



yy -



/

_E



zz



zz = -



/

_E



xx -



/

_E



yy + 1

/

_E



zz

•

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

SEAMERGES GPS COURSE, 2005

15. Elastic deformation across a locked fault

What is the shape of the accumulated deformation ?

Formula matematis

Formula matematis

•Symetry =>

all derivative with y = 0



yy

= 0

•No gravity =>



zz

= 0

•What is the displacement field U in the elastic layer ?

•Elastic equations :

• Force equilibrium along the 3 axis

Formula matematis

relations between

stress (



) and displacement vector (U)



xy = 2

G



xy = 2

G

[

d

U

x

/

d

y +

d

U

y

/

d

x

]

.1

/

₂

Formula matematis

What is

U

_y, function of x and z, solution of this equation ?

d

2

_U

y

/

d

x

2

+

d

2

U

y

/

d

z

2

= 0

Formula matematis

Boundary condition at the base of the crust (z=0)

U

_y = K arctang (x

/

_z)

U

_y = K . /2 if x > 0 = K . – /2 if x < 0

=>

K =

2.

V₀

/

_

And also :

U

_y = +V₀ if x > 0 = –V₀ if x < 0

Formula matematis

at the surface (z=h)

U

_y = K arctang (x

/

_z)

U

_y =

2.

V₀

/

_{arctang (}x

/

_h)

16. Arctang Profiles

_U

_{y =}

2.

V₀

/

_{arctang (}x

/

_h)

17. Deeping Fault

18. Elastic Dislocation (Okada, 1985)

Surface deformation due to shear and tensile faults in a half space, BSSA vol75, n°4, 1135-1154, 1985.

SEAMERGES GPS COURSE, 2005

The displacement field u_i(x₁,x₂,x₃) due to a dislocation u_j (₁,₂,₃) across a surface  in an isotropic medium is given by :

Where _jk is the Kronecker delta,  and  are Lamé’s parameters, _k is the direction cosine of the normal to the surface element d.

u_ij _{is the i}th _{component of the}

displacement at (x₁,x₂,x₃) due to the jth _{direction point force of magnitude}

18. Elastic Dislocation (Okada, 1985)

(1) displacements

For strike-slip For dip-slip For tensile fault

18. Elastic Dislocation (Okada, 1985)

Where :

19. Case : Sagaing Fault Nyanmar

20. Case : Palu Koro Fault

20. Case : Palu Koro Fault (more complex)

21. Case : Sumatra subduction zone

Triyoso, 2005

Natawijaya, 2007

22. Strain rate and rotation rate tensors

2. Compute strain rate and rotation rate tensors 1. Look at station velocity residuals

Velocity mm/yr Strain rate rotation rate To asses plate deformation :

22. Strain rate and rotation rate tensors

[E] has 2 Eigen values :



₁,



₂



₁ and



₂ are extension/compression along principal direction defined by angle



defined as angle between



₂direction and north



[E] =

_½

(

[S] + [S]T

) =

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

23. Case : Strain & Rotation on GEODYSSEA network

Strains :

extension/compression/strike-slip

Rotations :

Anti-clockwise/clockwise

24. Case : intensity of strain in GEODYSSEA network

25. Case : Strain & Rotation in Nyanmar