Higher Dimensional Experiments
EM517
Bastawros 2/10/05
Higher Dimensional Experiment
For Contained Processes
Materials deform heterogeneously (non uniform deformation, ∇u ≠ const. ) at the microstructure length
scale (e.g. grain size, secondary phase inclusion…etc) and at the geometrical discontinuities (cracks,
holes, near interfaces …etc), while subjected to a macroscopic uniform loads. These heterogeneities
control the macroscopic response of the material. In addition, these microscopic heterogeneities can
lead to a macroscopic heterogeneous deformation in the form of localized necking or shear band
formation. For examples, the stresses and strains are singular near cracks (e.g. George Irwin, 1957)
KI
K
K
f ij (θ ) + II gij (θ ) + III hij (θ )
2πr
2πr
2πr
as r → 0; f ij (0) = gij (0) = hij (0) = 1
σ ij =
Well-defined
Stress state
r
θ
K: is the stress intensity factor
KI: opening, KII:Sliding, KIII:tearing
KIC: Fracture toughness of the material ( MPa m )
Fracture process
zone
KIC: is a material property which is experimentally calibrated.
It is affected by the crack growth rate,
which in turn is affected by the microstructure details.
Therefore, we would like to resolve the two dimensional deformation fields within the process zone
(near crack-tip, across shear bands or near interfaces)
Two Dimensional Full Field Displacement Measurements:
There are three basic requirements for any 2-D (or 3-D) full field displacement measurement technique,
regardless of their application or the associated gage length:
1. Illumination source
- Electron beam (Ao)
- Visible light (µm)
- Sound waves
- Radio freq.
(m)
2. Imaging Tool
SEM, TEM
CCD camera, or optical arrays
Satellite
3. Analysis Tools
Correlate pairs of collected data from the undeformed and deformed configuration to deduce the full
displacement field. This can be done in:
a) Real space: every material point is given by its coordinate relative to the coordinate basis.
b) Frequency space: distances between features are represented by wavelength (or frequencies) and
their number of repetitions by intensities.
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Full Field Deformation Measurements
EM517
Bastawros 2/10/05
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1. Geometrical Moiré Methods (Mechanical interference)
a) Over the reference configuration (the original specimen before deformation), we lay or imprint a
parallel line grid. The grid has a spacing D. We can represent the grid by a reciprocal vector G1 whose
magnitude is 1/D and normal to the grid line.
F2
G1 =
D
1
D
F1
G1
F3
x2
F4
x1
Reference (undeformed) configuration
Deformed configuration
b) Upon applying the load, the imprinted grid will deform with the surface. The spacing and orientation
of the parallel lines will change and follow the deformation.
c) Superpose another (interrogation) grid with parallel lines over the deformed configuration. The
interrogation grid has a spacing d and a reciprocal vector g1. An interference pattern will be formed
between the imprinted deformed grid on the surface and the overlaid grid.
d) In mechanical interference, a new interference fringe is formed
between two points on the deformed configuration when the
1
g1 =
number of lines of the deformed grid differ by an n-line from
d
g1
the interrogation grid (n=1,2,….).
e) If G1 = g1, then each interference fringe represent displacement
equals D. For df1 fringe number, the corresponding displacement
is
u1 = D f 1
∂f
∂u1
=D 1,
∂x1
∂x1
∂f
∂u1
=D 1
∂x2
∂x2
Full Field Deformation Measurements
EM517
Bastawros 2/10/05
f) For 2-D deformation we need two reference grids G1 and G2 , and two interrogating grid g1 and g2.
g) Then the deformation gradient in the other direction will be (G2 = g2):
u2 = D f 2
∂f
∂u2
=D 2,
∂x1
∂x1
∂f
∂u2
=D 2
∂x2
∂x2
h) The small strain component will be given by:
ε 11 ≡
∂u1
∂f
=D 1,
∂x1
∂x1
ε 12 ≡
∂f
∂f
∂u1 ∂u2
=D 1 + 2
+
∂x2 ∂x1
∂x2 ∂x1
ε 22 ≡
FG
H
∂u2
∂f
=D 2,
∂x2
∂x2
IJ
K
It is apparent that the measured strain is a function of the spacing of the reference grid, D.
Typically, D= 50-1000 line/inch or 2-40 line/mm. Therefore, the minimum detectable strain is in
the range of 2% ( ε = 1x 1 40 )
This range can be increased by using digital imaging with extended gray scale. For an effective
gray scale of 100 level, minimum strain will be 0.02%.
i) Usually, |G1|=|G2|,or D1=D2 = D. However g1 ≠ g 2 , d1 ≠ d 2 ≠ D
For any two points on the surface,
The No. of interrogation grid
The No. of specimen grid
= gi . d x
= Gi .d X
f i = gi . d x − G i . d X,
(i = 1, 2)
The no. of fringes
x = X+ u
and
By differentiating both eqn., we get the full finite strain tensor.
dX
u
dx
X
x
o
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Full Field Deformation Measurements
EM517
Bastawros 2/10/05
-Using the vector representation of Moiré, number of fringes is given by:
fi = g i .d x − G i .d X,
(i = 1, 2) ;
dX
d x = d X+ d u
or
G1
fi = (g i − G i ).d X + g i .d u,
(i = 1, 2)
x2
x2
g1
interrogation
G2
g2
θ1
θ2
x1
G1
specimen
dX = dX 1i + dX 2 j ,
x1
du = du1i + du2 j
-Reference (specimen grids)
G1 =
- interrogation grids
g1 =
1
i,
D
1
1
cosθ 1 i + sin θ 1 j ,
d1
d1
G2 =
g2 =
1
j,
D
1
1
cosθ 2 i + sin θ 2 j ,
d2
d2
-Then, the fringe order difference is
⎛1
⎛1
⎞
⎛1
⎞
⎛1
⎞
1⎞
f1 = ⎜ cos θ1 − ⎟ dX 1 + ⎜ sin θ1 ⎟ dX 2 + ⎜ cos θ1 ⎟ du1 + ⎜ sin θ1 ⎟ du2
D⎠
⎝ d1
⎝ d1
⎠
⎝ d1
⎠
⎝ d1
⎠
⎛ 1
⎞
⎛ 1
⎛ 1
⎞
⎛ 1
⎞
1⎞
f 2 = ⎜ cos θ 2 ⎟ dX 1 + ⎜ sin θ 2 − ⎟ dX 2 + ⎜ cos θ 2 ⎟ du1 + ⎜ sin θ 2 ⎟ du2
D⎠
⎝ d2
⎠
⎝ d2
⎝ d2
⎠
⎝ d2
⎠
-The displacement derivatives will be given by:
⎞ du ⎛ 1
⎞ du
df1 ⎛ 1
1⎞ ⎛1
= ⎜ cos θ1 − ⎟ + ⎜ cos θ1 ⎟ 1 + ⎜ sin θ1 ⎟ 2 ,
dX 1 ⎝ d1
D ⎠ ⎝ d1
⎠ dX 1 ⎝ d1
⎠ dX 1
⎞
df1 ⎛ 1
= ⎜ sin θ1 ⎟
dX 2 ⎝ d1
⎠
⎛1
⎞ du ⎛ 1
⎞ du
+ ⎜ cos θ1 ⎟ 1 + ⎜ sin θ1 ⎟ 2 ,
⎝ d1
⎠ dX 2 ⎝ d1
⎠ dX 2
⎞
df 2 ⎛ 1
= ⎜ cos θ 2 ⎟
dX 1 ⎝ d 2
⎠
⎛ 1
⎞ du ⎛ 1
⎞ du
+ ⎜ cos θ 2 ⎟ 1 + ⎜ sin θ 2 ⎟ 2 ,
⎝ d2
⎠ dX 1 ⎝ d 2
⎠ dX 1
⎞ du ⎛ 1
⎞ du
df 2 ⎛ 1
1⎞ ⎛ 1
= ⎜ sin θ 2 − ⎟ + ⎜ cos θ 2 ⎟ 1 + ⎜ sin θ 2 ⎟ 2
dX 2 ⎝ d 2
D ⎠ ⎝ d2
⎠ dX 2 ⎝ d 2
⎠ dX 2
Reference: Shield, T.W., Kim, K.-S., 1991, “ Diffraction theory of optical interference moiré and
a device for production of variable virtual reference gratings. A moiré microscope,”
Experimental Mechanics, v 31, n 2, Jun, p 126-134
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