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1 Problem Set #4
Problem 2
Just NE of Los Angeles, the San Andreas fault trends approximately
N65◦W–S65◦E. To within observational error, the displacement gradient
there is observed to be (each year):
0.15 0.24
0.00 −0.15
where x1 is East and x2 is North, and the units are 10−6 strain.
a) Write the (two dimensional) strain tensor, the rotation tensor, and the
areal dilation.
b) What are the directions of maximal principal compression and extension?
c) Is this what you would expect, if the San Andreas is a strike-slip fault?
Solution
a) The strain and rotation tensor are found directly from the displacement
gradient tensor
dui
dxj
= � du1
dx1
du2
dx1
du1
dx2
du2
dx2 �
= � 0.15
0.00
0.24
−0.15 �
× 10−6/year.
The strain tensor is
εij = 1
2 � dui
dxj
+ duj
dxi �
= 1
2 � 2du1
dx1
du2
dx1 + du1
dx2
du1
dx2 + du2
dx1
2du2
dx2 �
= � 0.15
0.12
0.12
−0.15 �
×10−6/year.
The rotation tensor is
ωij = 1
2 � dui
dxj
− duj
dxi �
= 1
2 � 0
du2
dx1 − du1
dx2
du1
dx2 −
0
du2
dx1 �
= � 0.00
−0.12
+0.12
0.00 �
×10−6/year.
δA
The area dilation, A , is the trace of the strain tensor
δA = εii = ε11 + ε22 = (0.15 − 0.15)10−6/year = 0/year
A
and the area is constant.
b) To find the principal strains and directions of the strain tensor, we
follow the same procedure that we used to find the principal stresses and
directions of the stress tensor (see the solutions for problem set 3 for details).
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