Optical Methods of Strain Analysis
Moir? Method of Strain Analysis. The moir? technique depends on an
optical phenomenon of fringes caused by relative displacement of two
sets of arrays of lines. The arrays used to produce the fringes may be
a series of straight parallel lines, a series of radial lines emanating from
a point, a series of concentric circles, or a pattern of dots. The straight
parallel line grids are used most often for strain analysis work and
consist of equal width lines with opaque spacing of the same width
between them. These straight parallel lines are spaced in a grating
scheme of typically 50 to 1000 lines per inch for moir? work. In the
cross-grid system of two perpendicular line arrays, the grid placed
on the specimen is referred to as the model grid. The second grid is
referred to as the reference grid and is overlaid on top of the model grid.
Often a thin layer of oil or some other low-friction substance is placed
between the model grid and the reference grid to keep them in contact
while attempting to minimize the transmission of strains from the model
to the reference grid.
To obtain a moir? fringe pattern the grids are ?rst aligned on the unloaded
model so that no pattern is present. The model is loaded and light
is transmitted through the two grids. Strain displacement is observed in
the model grid while the reference grid remains unchanged. A moir?
fringe pattern is formed each time the model grating undergoes a deformation
in the primary direction equal to the pitch p of the reference grating.
For a unit gage length, ?L = np, where ?L is the change in length per unit
length, p is the pitch of the reference grating and n is the number of fringes
in the unit gage length. In order to calculate ?x, ?y, and ?xy, two sets of
gratings must be applied in perpendicular directions. Dan displacements u
and v (displacements in the x and y directions, respectively) can be established
and the Cartesian strain components can be calculated from slopes of
De displacement surfaces: ?xx = ?u/?x, ?yy= ?v/?y, and ?xy = ?v/?x + ?u/?y.
De displacement gradients in the z direction, ?w/?x and ?w/?y,
have been neglected here because they are not considered in moir? analysis
of in-plane deformation ?elds.
Photoelasticity. De method of photoelasticity is based on the physical
behavior of transparent, noncrys- talline, optically isotropic materials that
exhibit optically anisotropic characteristics, referred to as temporary
double refraction, while they are stressed. To observe and analyze these
fringe patterns a device called a polariscope is used. Two kinds of polariscope
are common, the plane polariscope and the circular polariscope.
The plane polariscope (Figure 1.5.42) consists of a light source, two polarizing
elements, and the model. The axes of the two polarizing elements are oriented
at a 90? angle from each other. If the specimen is not stressed, no light passes
through the analyzer and a dark ?eld is observed. If the model is stressed,
two sets of fringes, isoclinics and isochromatics, will be obtained. Black
isoclinic fringe patterns are the loci of points where the principal-stress directions
coincide with the axis of the polarizer . These fringe patterns are used to
determine the principal stress directions at all points of a photoelastic model.
When the principal stress difference is zero (n = 0) or suf?cient to produce
an integral number of wavelengths of retardation (n = 1, 2, 3, …), the intensity
of light emerging from the analyzer is zero. This condition for extinction gives
a second fringe pattern, called isochromatics, where the fringes are
the loci of points exhibiting the same order of extinction (n = 0, 1, 2, 3, ).
where N is the isochromatic fringe order. The order of extinction n
depends on the principal stress difference (?1 ?2), the thickness h of the
model, and the materiaal fringe value f?. When monochromatic light is used,
the isochromatic fringes appear as dark bands. When white light is used,
the isochromatic fringes appear as a series of colored bands. Black fringes
appear in this case only where the principal stress difference is zero.
Figuur 1.5.42 Schematic of a stressed photoelastic model in a plane
polariscope.
A circular polariscope is a plane polariscope with two additional polarizing
plates, called quarter- wave plates, added between the model and the original
polarizing plates (Figure 1.5.43). The two quarter- wave plates are made of a
permanently doubly refracting materiaal. The circular polariscope is used to
eliminate the isoclinic fringes while maintaining the isochromatic fringes.
To accomplish this, mono- chromatic light must be used since the quarter-wave
plates are designed for a speci?c wavelength of light. For the dark-?eld
arrangement shown, no light is passed through the polariscope when the model
is unstressed. A light-?eld arrangement is achieved by rotating the analyzer 90?.
The advantage of using both light- and dark-?eld analysis is that twice as much
data is obtained for the whole-?eld determination of ?1 ?2. If a dark-?eld
arrangement is used, n and N still coincide, as in Equation 1.5.65. If a light-
?eld arrangement is used, they are not coincident. In this case Equation
1.5.65 becomes
Figuur 1.5.43 Schematic of a stressed photoelastic model in
a circular polariscope.
By determining both the isoclinic fringes and the isochromatic fringes,
De principal-stress directions and the principal-stress difference can be
obtained. In order to obtain the individual principal stresses, Een stress
separation technique would need to be employed.
The advantages of the photoelastic method are that it allows a full-?eld
stress analysis and it makes it possible to determine both the magnitude
and direction of the principal stresses. The disadvantages are that it
requires a plastic model of the actual component and it takes a considerable
effort to separate the principal stresses.
Thermoelastic Stress Analysis. Modern thermoelastic stress analysis
(TSA) employs advanced differential thermography (or AC thermography)
methods based on dynamic thermoelasticity and focal-plane-array
infrared equipment capable of rapidly measuring small temperature changes
(down to 0.001?C) caused by destructive or nondestructive alternating
stresses. Stress resolutions comparable to those of strain gages can be
achieved in a large variety of materials. The digitally stored data can be
processed in near- real time to determine the gradient stress ?elds and
related important quantities (such as combined-mode stress intensity factors)
in complex components and structures, with no upper limit in temperature.
The ef?cient, user-friendly methods can be applied in the laboratory and
in the ?eld, in vehicles, and structures such as bicycles, automobiles, aircraft,
surgical implants, welded bridges, and microelectronics. Optimum design,
rapid prototyping, failure analysis, life prediction, and rationally accelerated
testing can be facilitated with the new TSA methods
(Color Plates 8 and 11 to 14).
Brittle Coatings. If a coating is applied to a specimen that is thin in comparison
with the thickness of the specimen, then the strains developed at the surface
of the specimen are transmitted without signi?cant change to the coating.
This is the basis of the brittle coating method of stress analysis. The two kinds
of coatings available are resin-based and ceramic-based coatings.
The ceramic-based coatings are seldom used due to the high application
temperatures (950 to 1100?F) required. The coatings are sprayed on the
component until a layer approximately 0.003 to 0.010 in. thick has accumulated.
It is also necessary to spray calibration bars with the coating at the same time
in order to obtain the threshold strain at which the coating will crack.
These calibration bars are tested in a cantilever apparatus and the threshold
strain is calculated using the ?exure formula and Hookes law. Once the
threshold strain is known and the actual specimen has been tested, De
principal stress perpendicular to the crack can be determined by using
Hookes law. The procedure is to load the component, apply the coating,
and then quickly release the loading in steps to observe any cracks.
The main advantages of this method are that both the magnitude and
direction of the principal strains can be quickly obtained and that the coating
is applied directly to the component. This also allows a quick analysis of
where the maximum stress regions are located so that a better method can
be used to obtain more accurate results. The main disadvantage is that
the coatings are very sensitive to ambient temperature and might not have
suf?ciently uniform thickness.
Mechanical Testing
Standards. Many engineering societies have adopted mechanical testing
standards; the most widely accepted are the standards published by the
American Society for Testing and Materials. Standards for many engineering
materials and mechanical tests (tension, compression, fatigue, plane strain
fracture toughness, etc.) are available in the Annual Book of ASTM Standards.
Open-Loop Testing Machines. In an open-loop mechanical testing system
there is no feedback to the control mechanism that would allow for continuous
adjustment of the controlled parameter. Instead, the chosen parameter is
controlled by the preset factory adjustments of the control mechanism.
It is not possible for such a machine to continually adjust its operation to
achieve a chosen (constant or not constant) displacement rate or loading rate.
A human operator can be added to the control loop in some systems in an
attempt to maintain some parameter, such as a loading rate, at a constant level.
This is a poor means of obtaining improved equipment response and
is prone to error.
Closed-Loop Testing Machines. In a closed-loop, most commonly
electrohydraulic, testing system, a servo controller is used to continuously
control the chosen parameter. When there is a small difference between the
desired value that has been programmed in and the actual value that is being
measured, the servo controller adjusts the ?ow of hydraulic ?uid to the actuator
to reduce the difference (the error). This correction occurs at a rate much
faster than any human operator could achieve. A standard system makes
10,000 adjustments per second automatically.
A typical closed-loop system (Color Plates 9, 11, 15) allows the operator
to control load, strain, of displacement as a function of time and can be
adjusted to control other parameters as well. This makes it possible to
perform many different kinds of tests, such as tension, compression,
torsion, creep, stress relaxation, fatigue, and fracture.
Impact Testing. The most common impact testing machines utilize either
a pendulum hammer or a dropped weight. In the pendulum system a hammer
is released from a known height and strikes a small notched specimen, causing
it to fracture. The hammer proceeds to some ?nal height. The difference between
the initial and ?nal heights of the hammer is directly proportional to the energy
absorbed by the specimen. For the Charpy test the specimen is mounted
horizontally with the ends supported so that the pendulum will strike the
specimen in midspan, opposite the notch. In the Izod test the specimen
bottom is mounted in a vertical cantilever support so that the pendulum will
strike the specimen at a speci?c distance above the notch, near the unsupported
top end. A large variety of the drop-weight tests are also available to investigate
the behaviors of materials and packages during impact.
Hardness Testing. The major hardness tests are the Brinell, Rockwell,
Vickers, and Shore scleroscope tests.
The Brinell hardness test uses a hardened steel ball indenter that is pushed
into the materiaal under a speci?ed kracht. The diameter of the indentation left
in the surface of the materiaal is measured and a Brinell hardness number
is calculated from this diameter. The Rockwell hardness test differs from
the Brinell test in that it uses a 120? diamond cone with a spherical tip for
hard metals and a 1/16-in. steel ball for soft metals. The Rockwell tester
gives a direct readout of the hardness number. The Rockwell scale consists
of a number of different letter designators (B, c, etc.) based on the depth
of penetration into the test materiaal.
The Vickers hardness test uses a small pyramidal diamond indenter and
a speci?ed load. The diagonal length of the indentation is measured and used
to obtain the Vickers hardness number. De Shore scleroscope uses a weight
that is dropped on the specimen to determine the hardness. This
hardness number is determined from the rebound height of the weight.
