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Module #4Module #4
Fundamentals of strainThe strain deviator
Mohr’s circle for strain
READING LISTDIETER: Ch. 2, Pages 38-46
Pages 11-12 in HosfordCh. 6 in Nye
HOMEWORKFrom Dieter
2-7
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StrainStrain• When a solid is subjected to a load, parts of the solid are
displaced from their original positions.• Think of it like this; the atoms making up the solid are displaced
from their original positions.
• This displacement of points or particles under an applied stressis termed strain.
OO´
Load
Load
AA´
BB´ C
C´
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Rotation
x
y
v
-v
Pure Shear
x
y
v
u
+
TotalDisplacement
= x
y
x
y
Translation
+
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Where to beginWhere to begin• Consider a point A in a solid located at position x,y,z.
• Apply force to the body and point A (x,y,z) is displaced to A′ (x+u,y+v,z+w).
A (x,y,z)
x
y
z
A′ (x+u,y+v,z+w)
u
v
w
uA
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Displacement of pointsDisplacement of points• Displacement vector:
uA = f(u,v,w)where u, v, and w are units of translation along the x, y, zaxes.
• Solids are composed of many particles.
• If uA is constant for all particles, no deformation occurs (only translation).
• If uA varies from particle to particle, i.e., ui = f(xi), the solid deforms.
Displacement = Translation + Rotation + Shear
uAA′ (x+u,y+v,z+w)
x
y
z
A (x,y,z)
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11--D Linear StrainD Linear Strain
• Points A and B are displaced from their original positions
• The amount of displacement is a function of x. Point B moves farther than Point A.• Let distance A A = u.
• Thus, distance B B = u+(Δu/Δx)dx = u+(u/x)dx.
A B
A´ B´F
dx
0 1 2 3 4 5
x
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• Strain is defined by the following relationship:
• Integrating yields the displacement.
• uo ≈ rigid body translation, which we can subtract, yielding:xxu e x
11--D Linear Strain D Linear Strain –– contcont’’d.d.
xx
udx dx dxL A B AB u uxeL AB dx x x
o xxu u e x
A B
A´ B´F
dx
x
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Generalization to 3Generalization to 3--DD• Displacement is related to the initial coordinates of the
point.
• Normal / Linear Strains:
xy xz
yx yz
xx
yy
zzzx zy
i ij j
u x e y e z
e x y e z
e x e y z
u e
e
e
e
xv
w
, ,
, ,
xx yy zz
xx yy zz
u v we e ex y zu v we e ex y z
A (x,y,z)A′ (x+u,y+v,z+w)
x
y
z
uA
If we orient the system such that the load is applied parallel to the x-axis. The variables u, v, and w are displacements parallel to the x, y, and z axes.
u
v
w
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Shear Strains in 2Shear Strains in 2--D and 3D and 3--DD• Consider a square or cubic element that is distorted by shear.
Incremental displacement in x-direction = u.
Incremental displacement in y-direction = v.
Incremental displacement in z-direction = w.
x
y
A
D
B
C
C´
D´
B´
u
v
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• Displacement of AD increases with distance along the y-axis resulting in an angular distortion of y-axis.
2D2D
shear distortion of they-axis in the x-direc
or
n
tio
xyu uDD
h DA y ye
shear distortion of thex-axis in the y-direc
or
n
tio
yxv vBB
h AB x xe
x
y
A
D
B
C
C´
D´
B´
u
v
• An analogous event occurs along the x-axis.
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Strain in 3Strain in 3--DD• The displacement strain is defined by nine strain components:
– exx, exy, exz, eyy, eyx, eyz, ezz, ezx, ezy
– The strains on the negative faces are equal to satisfy the requirements for equilibrium.
• Notation is similar to stress; subscripts reversed:
– eij: i = direction of displacementj = plane on which strain acts
• Convention– (+)ive when both i & j are (+)ive– (+)ive when both i & j are (-)ive– (-)ive when both i & j are opposite
• Tension: eij = positive• Compression: eij = negative
x
y
z
exx
eyx
ezx
ezz
eyzexz
eyy
exy
ezy
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3D Displacement Strain Matrix
xx xy xz
ij yx yy yz
zx zy zz
u u u u u ux y z x y ze e e
e e e ex y z x y z
e e e
x y z x
v v v v v v
w w w w w wy z
• The displacement strain matrix.
• Can produce pure shear strain and rigid-body rotation.
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(2)Rotation
x
y
-v
u
exy = -eyx
(1)Pure Shear
w/o Rotation
x
y
v
u
exy = eyx
(3)Simple Shear
x
yu
exy = u/y eyx = 0
• We need to break the displacement matrix into strain and rotational components.
• We can decompose the total strain matrix into symmetric and anti-symmetric components.
y
x
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Decomposition of StrainDecomposition of Strain
12
12
12
12
ij
ij ji
ji
j i
iij j
ij ji
ji
j i
e e
uu
e e
x
e
xuuxx
Symmetric
Shear
Anti-symmetric
Rotation
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Displacement strain[matrix]
Shear strain[tensor]
Rotation[tensor]
ij
1 12 2
1 1ε2 21 12 2
xx xy yx xz zx
xx xy xz
yx yy yz xy yx yy yz zy
zx zy zz
xz zx yz zy zz
e e e e e
e e e e e
e e e e e
ij
1 102 2
1 102 21 1 02 2
xy yx xz zx
xx xy xz
xy yy yz yx xy yz zy
xz yz zz
zx xz zy yz
e e e e
e e e e
e e e e
xx xy xz
yx yy yz
zx zy zz
e e ee e ee e e
=
+
=
+
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Shear StrainShear Strain• Total angular change from a right angle.
(1)Pure Shear
w/o Rotation
x
y
v
u
exy = eyx
2 ( 0)
2 (engineering shear strain)
xy yx xy ij
ij ij
xy
xz
yz
e e
u vy xw ux zw vy z
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Transformation of StrainsTransformation of Strains• Equations for strain, analogous to those for stress, can be
written by substituting for and /2 for .
• We can also define a coordinate system where there will be no shear strains. These will be principal axes
2 2 2
2 2
normal
normal2
2 2 2xx yy zz xy yz zx
xx yy zz xy yz zx
l m n lm mn nl
l m n lm mn nl
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3 2 2 2 2
2 2 2
1 33 2
2
1- - - -4
1 1- - 04 4
or- - 0
xx yy yy zz xx zz xy yz xz
xx yy zz xy yz xz xx
xx yy zz
yz yy xz zz xy
II I
• The directions in which the principal strains act are determined by substituting 1, 2, and 3, each for in:
and then solving the resulting equations simultaneously for l, m, and n(using the relationship l2 + m2 + n2 = 1).
(a) Substitute 1 for in & solve; (b) Substitute 2 for in & solve; (c) Substitute 3 for in & solve.
( - )2 0( - )2 0
( - )2 0
xx yx zx
xy yy zy
xz yz zz
l m nl m nl m n
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Equations for Principal Shearing StrainsEquations for Principal Shearing Strains
• Deformation of a solid involves a combination of volumechange and shape change.
• We can separate strain into hydrostatic (volume change) and deviatoric (shape change) components.
1 2 3
max 2 1 3
3 1 2
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xy
z
dxdy
dz
Hydrostatic ComponentHydrostatic Component
• The volume strain is:
• If we neglect the products of strains (i.e., εii×εjj), this becomes:
which is equal to the first invariant of the strain tensor
• Volume = dxdydz
• Volume of strained element = 1 1 1xx yy zz dxdydz
1 1 1
1 1 1 1
xx yy zz
xx yy zz
dxdydz dxdydzdxdydz
xx yy zz
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• The hydrostatic component of strain, i.e., the mean strain, is:
The mean strain does not induce shape change. It causes volume change. It is the hydrostatic component.
• The part that causes shape change is called the strain deviator.We get the strain deviator by subtracting the mean strain from the normal strain components.
3 3 3xx yy zz ii
mean
xx mean xy xz
ij yx yy mean yz
zx zy zz mean
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23
23
23
xx mean xy xz
ij yx yy mean yz
zx zy zz mean
xx yy zzxy xz
yy zz xxyx yz
zz xx yyzx zy
The Strain DeviatorThe Strain Deviator
3 3ij ij m ij ij ij
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MohrMohr’’s Circle for Strains Circle for Strain
Intersection with the -axis is min=2
Intersection withthe -axis ismax = 1
ε
max/2
C(average, 0)
+γ/2
2
CW
CCW
V(xx, -xy/2)
H(yy, yx/2)
(1, )(2, )
Allows us to determine the magnitude and directions of the principal strains.
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tan 2
tan 2
xynormal
xx yy
xx yyshear
xy
MAXIMUM & MINUMUM PRINCIPAL STRAINS IN 2-D STATE
2 2max 1
min 2 2 2 2xx yy xx yy xy
2 2max 3 xx yy xy
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Strain MeasurementStrain Measurement• Strain can be measured using a strain gauge.
• When an object is deformed, the wires in the strain gate are strained which changes their electrical resistance, which is proportional to strain.
• Strain gauges make only direct readings of linear strain. Shear strain must be determined indirectly.
45°
45°
60° 60°
60°
Rectangular Delta
y
x
45
x
60 120
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State of stress at a point:
State of strain at a point:
There are many different systems of notation.BE WARY!
11 12 13
21 22 23
31 32 33
11 12 13
12 22 23
13 23 33
xx xy xz
yx yy yz
zx zy zz
11 12 13 11 12 13
21 22 23 12 22 23
31 32 33 13 23 33
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Matrix NotationMatrix Notation
• We often replace the indices with matrix notation for simplicity
• This will be particularly important when we discuss higher order tensors and tensor relationships (i.e., elastic properties)
11 1 11 2 33 323 4 13 5 12 6
11 12 13 1 6 522 23 2 4
33 3
xx yy zzyz xz xy
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General forms for stress and strain in matrix notationGeneral forms for stress and strain in matrix notation
1 6 5
6 2 4
5 4 3
6 51
6 42
5 53
2 2
2 2
2 2
1 11 2 22 3 33
4 23 23
5 13 13
6 12 12
NOTE; ;
222
Special definitions