Deformation and Strain Flashcards
displacement vector and displacement field
P becomes P’, the straight line from P to P’ is the displacement vector
the actual path taken may not be a straight line
-figuring out the actual path is very difficult
-cant be certain where it started from and what path was taken
displacement field
for every point (particle) you have a displacement vector
-unlimited displacement vectors
-deformation is the array of displacement vectors
homogeneous vs inhomogeneous
homogeneous: all points move by the same degree in all ways (orientation, distance, direction)
straight lines remain straight
Parallel lines remain parallel
In 2D circles become ellipses, in 3D
spheres become ellipsoids
even if one point is different, it is not homogeneous (ie. if all lines remain parallel except for one, its not homogeneous)
what is deformation and its 3 components
any change in:
- shape/distortion (strain)
- position (translation)
- orientation (rotation)
of a body under stress
3 components: rotation, translation, strain
as long as a rock has one of three components, you can say it is deformed
individual components make up the final deformation
most times deformation refers to strain
homogeneous strain
-change in shape of a body (distortion)
-simply describes the final shape in terms of the initial shape (dont worry about what happens in between)
what are the two measures of strain
to quantities to describe strain:
change in length
-elongation,e (change in length divided by initial length)
-stretch,S (new length over initial length = 1+e)
change in angle
-angular sheer (angle between initial angle and final angle) ψ
-shear strain (γ = tanψ)
dilation (change in volume)
Δ=δV/V0=(V-V0)/V0
what are the values of e and S when there is a) no change in length, b) lengthening, c) shortening
no length change, e=0, S=1
lengthening: e >0, S >1
shortening: e <0 , 0< S <1
strain ellipsoid and circular sections
no change in length between the two planes
x is always the longest, z is the shortest
types of homogeneous strain
a) general strain
(rectangular slab)
b) axial symmetrical extension (cigar-shaped)
(square based rectangular prism)
S1>S2=S3
X>Y>Z
c) axial symmetrical shortening (pancake shaped)
S1=S1>S3
X>Y=Z
(square slab)
d) plane strain
-no length change in y direction
S2=1
e) simple shortening (volume loss)
- gets shorter in S3 but S1 and S2 stay the same
Flinn Diagram
K tells you the shape of the strain ellipsoid
K=∞, cigar-shaped strain ellipsoid
between is general constriction
K=1, plane strain
between is general flattening
K=0, pancake-shaped strain ellipsoid
Determination of strain
shape and orientation of strain ellipsoid
measure long axis and short axis and get ratio
elongation:
-find final and initial length to get stretch
angle:
initial angle, angle after deformation, tan of angle in the middle
Progressive def. and def. path
Finite strain, infinitesimal strain
Total strain, Incremental strain
Strain path
coaxial deformation
The principal axes of the incremental strain ellipsoids always have the same orientation as those of the total strain ellipsoid during deformation.
Consequently, the principal directions of total and incremental strain are fixed to the same lines of material particles throughout deformation.
Noncoaxial deformation: The principal axes of the incremental strain ellipsoids and those of the total strain ellipsoid rotated relative to each other during
deformation. Consequently, two or more axes of the principal directions of total strain lie along different lines of particles at different times in the deformation.
pure shear
homogeneous deformation, plane strain or a general strain, in which lines of particles that are parallel to the principal axes of the strain ellipsoid have the same orientation throughout
deformation. It is the same as coaxial
deformation.
simple shear
homogeneous deformation involving plane strain
- a single family of parallel material planes remain undistorted and parallel throughout deformation
(if you bend a deck of cards, thickness of the deck remains the same and cards remain parallel)
-involves both strain and rotation
It is a noncoaxial deformation, but not all noncoaxial deformation is simple shear.
how can you tell if elongation or extension happened first
if extension first then contraction means the layers will be broken up and then folded, so the broken pieces wont line up and may overlap
if contraction then extension, the folding will have happened and then they will have broken up, so they pieces will still be aligned
