Strain
? Displacement
? Strain
? Strain field
? Strain determination in rocks
Displacement vector and
displacement components
x
y
x
z
y
z
0
v
wu
DEFORMATION
1) Translation – Change in position by lateral motion
2) Rotation – Rigid body spin about 1 or more axes
3) Strain - Change in shape or distortion
1
2
3
Nature of strain
Rubber experiment
(a) initial state (b) homogeneous deformation (c) inhomogeneous
deformation （ after Pan Lizhou,1976）
homogeneous
and
inhomogeneous
deformation
Domain of Homogeneous (H) and
Inhomogeneous (I) Strain
Bending deformation
inhomogeneous deformation on the whole,homogeneous
deformation in small local areas
Extension,shear strain and the strain ellipse
l0
l
Stretch s = l/l0 = (1+e)
?? s i n?
?? t a n?
???? ??? s i nt a n
Two definitions of shear strain for large deformation
Definition of shear strain for small deformation
Strain in three dimensions,deformation of a cube with
Sides parallel to orthogonal axes a,b,and c
The strain ellipsoid,principal strain axes x,y,z
Pure shear and simple shear
In the process of pure shear (A),which involves coaxial or non-
rotational strain,the orientations of the principal strains X and Z
do not change during progressive deformation,
In simple shear (B),which involves rotational strain,principal strain
axes X and Z rotate in a clockwise manner during progressive
deformation.
Special types of homogeneous strain
A axially symmetric extension
(X>Y=Z).This is a prolate uniaxial
ellipsoid.
B Axially symmetric shortening
(X =Y>Z),This is an oblate uniaxial
ellipsoid.
C Plane strain (X >Y=1 >Z),This is a
triaxial ellipsoid,in which the
intermediate axis is unchanged.
Graphical representation of strain
Ellipsoids,the Flinn diagram.
A Different ellipsoids are described
Using the value k=(a-1)/(b-1)
B if the volume is not constant,the
line a=b(1+?) divides the field of
Constrictional strain from the field
of flattening strain,The diagram
Shows the effect of a 20% volume
reduction.
.p
(1,1) ?
k=tan ? = (a-1)/(b-1)
Progressive deformation
The finite stain is achieved by adding successive
Strain increments to the initial unstrained shape
Long axis a=1.62cm
Short axis b=0.62cm
Initial radius r=1cm
Solution for,principal elongation,principal stretch、
principal quadratic stretch and shear strain
Angular shear ?=45。
(a)initial state,(b), (d)homogeneous deformation,(c),(e), (f)
inhomogeneous deformation（ after Bell,1981）
homogeneous
and
inhomogeneous
deformation
Variation history of the line fragments in different directions
in co-axial progressive deformation( after J.G.Ramsay,1967)
Serial number of bubbles
Trajectories of principal strain
Strain Determination By Boudin
Ld
L1 L2 L3 L5L4
Lo=L1+L2+L3+L4+L5
Linear strain = (Ld — Lo)/Lo
Strain Determination By Stylolite
Linear strain = (Ld+ )
Strain Determination By
Asymmetric Fish-Head
Shear strain = d/h
Problems
? What is strain?
? What is linear strain?
? What is angular strain?
? How to distinguish inhomogeneous strain from
homogeneous strain?
? How to distinguish between rotational and non-
rotational strain?
? What are the differences between pure shear and
simple shear?
? Do principal strain axes always correspond to
principal stress axes?
? What do progressive deformation and coaxial
deformation mean,respectively?
? Displacement
? Strain
? Strain field
? Strain determination in rocks
Displacement vector and
displacement components
x
y
x
z
y
z
0
v
wu
DEFORMATION
1) Translation – Change in position by lateral motion
2) Rotation – Rigid body spin about 1 or more axes
3) Strain - Change in shape or distortion
1
2
3
Nature of strain
Rubber experiment
(a) initial state (b) homogeneous deformation (c) inhomogeneous
deformation （ after Pan Lizhou,1976）
homogeneous
and
inhomogeneous
deformation
Domain of Homogeneous (H) and
Inhomogeneous (I) Strain
Bending deformation
inhomogeneous deformation on the whole,homogeneous
deformation in small local areas
Extension,shear strain and the strain ellipse
l0
l
Stretch s = l/l0 = (1+e)
?? s i n?
?? t a n?
???? ??? s i nt a n
Two definitions of shear strain for large deformation
Definition of shear strain for small deformation
Strain in three dimensions,deformation of a cube with
Sides parallel to orthogonal axes a,b,and c
The strain ellipsoid,principal strain axes x,y,z
Pure shear and simple shear
In the process of pure shear (A),which involves coaxial or non-
rotational strain,the orientations of the principal strains X and Z
do not change during progressive deformation,
In simple shear (B),which involves rotational strain,principal strain
axes X and Z rotate in a clockwise manner during progressive
deformation.
Special types of homogeneous strain
A axially symmetric extension
(X>Y=Z).This is a prolate uniaxial
ellipsoid.
B Axially symmetric shortening
(X =Y>Z),This is an oblate uniaxial
ellipsoid.
C Plane strain (X >Y=1 >Z),This is a
triaxial ellipsoid,in which the
intermediate axis is unchanged.
Graphical representation of strain
Ellipsoids,the Flinn diagram.
A Different ellipsoids are described
Using the value k=(a-1)/(b-1)
B if the volume is not constant,the
line a=b(1+?) divides the field of
Constrictional strain from the field
of flattening strain,The diagram
Shows the effect of a 20% volume
reduction.
.p
(1,1) ?
k=tan ? = (a-1)/(b-1)
Progressive deformation
The finite stain is achieved by adding successive
Strain increments to the initial unstrained shape
Long axis a=1.62cm
Short axis b=0.62cm
Initial radius r=1cm
Solution for,principal elongation,principal stretch、
principal quadratic stretch and shear strain
Angular shear ?=45。
(a)initial state,(b), (d)homogeneous deformation,(c),(e), (f)
inhomogeneous deformation（ after Bell,1981）
homogeneous
and
inhomogeneous
deformation
Variation history of the line fragments in different directions
in co-axial progressive deformation( after J.G.Ramsay,1967)
Serial number of bubbles
Trajectories of principal strain
Strain Determination By Boudin
Ld
L1 L2 L3 L5L4
Lo=L1+L2+L3+L4+L5
Linear strain = (Ld — Lo)/Lo
Strain Determination By Stylolite
Linear strain = (Ld+ )
Strain Determination By
Asymmetric Fish-Head
Shear strain = d/h
Problems
? What is strain?
? What is linear strain?
? What is angular strain?
? How to distinguish inhomogeneous strain from
homogeneous strain?
? How to distinguish between rotational and non-
rotational strain?
? What are the differences between pure shear and
simple shear?
? Do principal strain axes always correspond to
principal stress axes?
? What do progressive deformation and coaxial
deformation mean,respectively?
