# Numerical Modelling in Geosciences. Lecture 6 Deformation

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1 Numerical Modelling in Geosciences Lecture 6 Deformation

2 Tensor Second-rank tensor stress ), strain ), strain rate ) Invariants quantities independent of the coordinate system): - First invariant trace:!! xx!!! xy!!! xz! ij =! yx! yy!!! yz "! zx! zy!!! zz - Second invariant magnitude: - Third invariant determinant!!!! tr! ij ) =! xx! yy! zz )! = 1! ij = 1! xx! xy! xz! yx! yy! yz! zx! zy! ) zz

3 Stress In geodynamics compressional extensional) stress are negative positive). Pressure is positive under compression. Stress is measured in Pa = N/m. The stress tensor contains the components of the tractions acting on the element surfaces. The first index indicate the direction of stress, the second the normal to the stressed surface!! xx!!! xy!!! xz! ij =! yx! yy!!! yz "! zx! zy!!! zz T z T x Pressure is equal to the mean normal stress: D :!!P =! tr! ) 3D :!!P =! tr! ) 3 =!! kk =!!! xx yy =!! kk 3 =!!!! xx yy zz 3 In absence of internal angolar momentum, the tensor is symmetric: T y! xy =! yx! xz =! zx! zy =! yz

4 Deviatoric Stress Stress can be divided into a deviatoric and an isotropic components. The deviatoric components produce flow, the isotropic components i.e., pressure) compaction/dilation.!! ij =! ij " kk ij 3 =! ij " P" ij!!!where!!!" ij = 3D :!! ij = 1!!0!!0 0!!1!!0 ;! 0!!0!!1! xx!!! xy!!! xz! xx P!!!!!!! xy!!!!!!!!!!!!!! xz! yx! yy!!! yz =! yx!!!!!!! yy P!!!!!!!!!!!! yz! zx! zy!!! zz! zx!!!!!!!!!!!! zy!!!!!!!!! zz P Normal!deviatoric!stress :!! ii =! ii P;! Shear!deviatoric!stress :!! ij =! ji =! ij =! ji Please,!show!that!tr! ij ) = 0 Second!!invariant!of!deviatoric!stress!tensor :! zz )!! II = 1! ij = 1! xx! yy!! xy! xz!! yz

5 Deviatoric Stress Stress can be divided into a deviatoric and an isotropic components. The deviatoric components produce flow, the isotropic component i.e., pressure) compaction/dilation D :!! ij = "! xx!!!! yx!! xy! yy " =! xx P!!!!!!! xy!! yx!!!!!!! yy P Please,!show!that!in!D! xx = )! yy Second!!invariant!of!deviatoric!stress!tensor :! II = 1! ij =! xx! xy

6 Tectonic pressure If a medium is at rest, then no deviatoric stresses exist and the total pressure is equal to the lithostatic pressure. P TOT = P LITH = P 0 g z! 0!z)dz When deformation is applied, the total pressure is equal to the lithostatic pressure the tectonic pressure P TOT = P LITH P TECT P TECT =!! kk 3! P LITH Exercise: Calculate the lithostatic, total and tectonic pressures, and the total and deviatoric stress tensors at 10 km depth in the crust density = 500 kg/m 3, g z =10 m/s ), when a compressional stress of 150 MPa is applied along the x direction.

7 Displacement, its gradient and velocity! Displacement :! u! = " u x u y u z Rate!of!displacement :! D! u Dt =! v =! " v x v y v z Displacement!gradient :!!! u = "u i "j = "u x "x!!"u x "y!!"u x "u y "x!!"u y "y!!"u y "u z "x!!"u z "y!!"u z Pure shear!u y!y " L L Simple shear!u x!y " L L

8 Displacement gradient, strain, rotation Displacement gradient adimensional)= Strain symmetric) Rotation anti-symmetric)!! u = "u i "j = "u x "x!!"u x "y!!"u x "u y "x!!"u y "y!!"u y "u z "x!!"u z "y!!"u z! u! = 1!! u! u! T ) 1!! u "! u! T ) = 1! ij = 1 "!u i!j!u j!i " ij = 1!u i!j "!u j!i!! u =! "!!Strain)!!Rotation) "!u i!j!u j!i 1!u i!j "!u j!i )! u! 1 = 1!!!!!! "u x "x!!!!!!!!! 1 "u y "x "u x "y "u z "x "u x "u x "y "u y "x!! 1!!!!!!!!!! "u y "y!!!!!!! 1! 1 "u z "y "u y "u x "u z "x "u y "u z "y!!!!!!!!!! "u z, ).!!!!!!!!!!0!!!!!!!!!! 1 "u x. "y / "u y!! 1 "x.. 1 "u y. "x / "u x!!!!!!!!!!!!0!!!!!!!!!! 1 "y.. 1 "u z -. "x / "u x!! 1 "u z "y / "u y!!!!!!!!!!0 "u x / "u z "x "u y / "u z "y,

9 Strain Indicates the amount of deformation and is adimensional! ij = 1 "!u i!j!u j!i 1 " = 1 " )!!!!!!!u x!x!!!!!!!!! 1!u y!x!u x!y!u z!x!u x!z "!u x!y u y!x!! 1 "!!!!!!!!!!!u y!y!!!!!!! 1 "! 1 "!u z!y!u y!z!u x!z!u z!x!u y!z!u z!y!!!!!!!!!!!u z!z ,- Volumetric!strain =.V = tr! ij ) =! kk =! 11!! 33 If!incompressible! "V = tr! ij ) =! " u! = 0 Deviatoric!strain :! ij! =! ij! 1 3! ij" kk

10 Velocity gradient, strain rate, vorticity Velocity gradient s -1 )= Strain rate symmetric) Vorticity anti-symmetric)!! v = "v i "j = "v x "x!!"v x "y!!"v x "v y "x!!"v y "y!!"v y "v z "x!!"v z "y!!"v z! v! = 1!! v! v! T ) 1!! v "! v! T ) = 1!! ij = 1 v i j v j i!" ij = 1 v i j " v j i!! v = "! " )!!Strain!rate) v i j v j i )!!Vorticity,!rotation!rate) ) 1 v i j " v j i ) )! v! 1 = 1!!!!!! "v x "x!!!!!!!!! 1 "v y "x "v x "y "v z "x "v x "v x "y "v y "x!! 1!!!!!!!!!! "v y "y!!!!!!! 1! 1 "v z "y "v y "v x "v z "x "v y "v z "y!!!!!!!!!! "v z, ).!!!!!!!!!!0!!!!!!!!!! 1 "v x. "y / "v y!! 1 "x.. 1 "v y. "x / "v x!!!!!!!!!!!!0!!!!!!!!!! 1 "y.. 1 "v z -. "x / "v x!! 1 "v z "y / "v y!!!!!!!!!!0 "v x / "v z "x "v y / "v z "y,

11 End-member flows Pure shear Simple shear Pure!Shear :!! v! 1!!!!0!!0 1!!!!0!!0 0!!0!!0 = 0!!"1!!0 =!!!" = 0!!"1!!0 0!!0!!0 0!!!0!!0 0!!!0!!0 0!!0!!0 0!!! 1 Simple!Shear :!! v! 0!!!1!!!0!!0!!0!!! 1!!0 1 = 0!!!0!!0 =!!!" =!!0!!0 " 1!!0!!0 0!!!0!!0 0!!!0!!0!!0!!!0!!0

12 Strain rate The strain rate tensor can be divided into a deviatoric and isotropic components. Dimension is s -1 ). The trace of the strain rate tensor gives the rate of volume change!!! xx!!!! xy!!!! xz!! ij =!! yx!!!! yy!!!! yz = 1 v i "!! zx!!!! zy!!!! j v j ) i zz!! ij =!.! ij 1 3!! kk " ij! 1 - =, ) 1 " )!!!!!! v x x!!!!!!!!! 1 v y x v x y v z x v x z ) v x y v y x -!! 1, ) -!!!!!!!!!! v y, y!!!!!!! 1 ) -! 1, ) tr!! ij ) =!! kk =!! 11!!!! 33 = v x x v y y v z z = / 0 v! = 1 V! If!incompressible 1! V = tr!! ij tr!.! ij ) = 0 ) = / 0! v = 0 v z y v y z Second!!invariant!of!deviatoric!strain!rate!tensor :!!.! II = 1!.! ij v x z v z x v y z v z y -!!!!!!!!!! v z, z -, -,!! xy =!! yx =!!! xy =!!! yx!! xz =!! zx =!!! xz =!!! zx!! zy =!! yz =!!! zy =!!! yz

13 Displacement vs deformation gradient!! u = "u i "j = Displacement gradient "u x "x!!"u x "y!!"u x "u y "x!!"u y "y!!"u y "u z "x!!"u z "y!!"u z F ij = I!! u =! ij "u i "j = Deformation gradient ) F yy = L L L =1 "u y "y ) "u y "y = L L Length variation over the initial length 1 "u x "x!!!"u x "y!!!!!!"u x!! "u y "x!!!!1 "u y "y!!!"u y )!! "u z "x!!!!!"u z "y!!!!!1 "u z Final length over the initial length Pure shear Simple shear

14 Time derivative of deformation gradient We can track the history of deformation for a given Lagrangian particle by computing the time derivative of F, and successively by isolating the amount of deformation from the amount of rotation. For homogeneous, steady-state flow:!f!t = "! v F F tt = F t "! v F t Left!stretch!tensor : B = F F T! Initial deformation gradient F t=0 ij =! ij = I =! 1!!0!!0 0!!1!!0 " 0!!0!!1 Eigenvalues and eigenvectors of B give, respectively, the square of the magnitude and orientation of the principal stretch axes. Time to practice

15 Homework Read chapter 4 of textbook: Gerya, T. Introduction to numerical geodynamic modelling. Cambridge University Press, 345 pp. 010) Practice with code we have built to track the deformation history

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