ASM Metals HandBook Vol. 14 - Forming and Forging
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(Eq 67)
(Eq 68)
1,2
= ln (1 + e
1,2
)
(Eq 69)
Equation 56 will result in two values of e
xx
, one of which will be inadmissible and can therefore be discarded. The values
of normal strains in the x and y directions are determined as follows:
e
x
= - 1
(Eq 70)
(Eq 71)
It should be pointed out that Eq 70 and 71 are based on the Lagrangian definition of strain and make no assumptions
concerning the extent of deformation. The true effective strain is given by:
= ln (1 + e)
(Eq 72)
where e is the effective engineering strain.
Therefore, the analysis of grids to compute strain variations on the meridian plane of the specimen involves three steps:
• Digitization of the undeformed and deformed grids
• Calculation of the effective strain using Eq 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71
,
and 72.
• Displays of the effective strain as contour plots
Development of Grid Mesh
Deformation in a body is characterized by the distribution of strain. The grid deformation method of strain measurement
involves applying a grid mesh on the surface or on an interior plane of a specimen. The grid can be defined as an array of
lines or dots that indicate points on the specimen, and it is usually rectangular, with dots or lines being repeated in two
perpendicular directions. The distance between the discrete nodes on the grid before and after the deformation is
measured, and the resulting data are analyzed. The grid method is an optical technique. Sighting of the grid is
accomplished with or without photography. The accuracy of the grid depends on the precision with which sighting can be
carried out.
Accuracy in the measurement of the grid does not lie in the precision of the measuring instrument but in the ability of the
investigator to view the points clearly. If a dot or point (obtained by the intersection of two grid lines) is much larger than
the resolution of the instrument, the resulting measurement will be inaccurate. The size of the point, therefore, plays an
important role in the measuring process. In addition, a point under consideration should be clearly visible with the least
amount of distortion before and after deformation has occurred.
A number of methods have been used in attempts to apply a suitable grid pattern to wedge-shaped specimens. These
include photochemical etching and engraving techniques. The engraving techniques are accomplished by means of a
computer numerical controlled engraver and are found to be accurate and appropriate. A typical specimen with engraved
grid lines is shown in Fig. 24.
Fig. 24 Undeformed wedge specimen with grids engraved on meridian plane.
Tool Setup (Segmented Die Design) for Plane-Strain Wedge Testing
To analyze the two-dimensional flow of metal and to verify the ALPID code for complex deformation, wedge-shaped
specimens with grids engraved on the meridian plane were compressed such that a range of strains varying from 0 to 0.75
was obtained. A one-piece channel-shaped die with a punch was designed and fabricated for a 90 Mg (200 kip) testing
machine. A specimen of aluminum alloy 2024-O was compressed using this die. The compressed specimen could not be
ejected undamaged from the die. The concept of segmented dies was therefore developed for convenient ejection of the
specimen without damage. These dies (Fig. 25) consisted of several segments assembled to provide the same die cavity as
that of a one-piece die, and they could be dismantled, after forging was complete, to eject the workpiece. The major
advantage of segmented dies is that, through the use of different inserts, a new family of parts can be forged. The major
disadvantage of these dies is that metal tends to extrude through the various joints of the die during metal deformation.
The segments must therefore be held together with a pressure greater than the flow stress of the metal during deformation
but must still be easy to dismantle after the forging process is complete.
Fig. 25 Components of segmented dies for two-dimensional wedge testing
Example of Experimental Procedure
To verify ALPID for a variety of materials and deformation spectra, wedge specimens were machined from aluminum
alloys 1000-F and 6061-T6. The specimens were annealed at 450 °C (840 °F) for 1 h. One 1100-O aluminum specimen
was compressed to a deformation of 54.5%, and three 6061-O aluminum alloy specimens were compressed to height
reductions of 14, 32, and 40% for this study.
Before the actual compression of the wedge specimen, the punch, die, and specimen surfaces were coated with a high-
pressure molybdenum disulfide-base spray lubricant to minimize friction along the sliding interfaces. The specimen was
then mounted in the die, and the die segments were fastened together with bolts and restrainers. The duration of the test
was determined by the extent of deformation--typically 250 s for a 50% reduction in height (12.7 mm, or 0.5 in., ram
displacement).
The deformed specimens were cleaned with soap and methanol to remove the thin film of lubricant and were then
photographed. Because of the compression of the specimens, the horizontal grid lines became thin, and the vertical ones
became thick. The thin lines could not be effectively resolved by the camera because of the reflection of light from the
shiny surface of the aluminum specimen. The specimen surface was therefore coated with carbon black in order to print
the deformed grid on photographic paper for digitizing. The computer files containing the deformed and undeformed
digitized data were used by a computer program to obtain the effective strains at various nodes.
A number of compression tests were performed. Figure 26 shows a plot of ram travel versus compression load for the two
materials. It can be seen that the load increases with the amount of deformation, but the rate of increase is larger for 6061
aluminum than for 1100 aluminum. The deformed grid meshes for the two materials for various amounts of deformation
are shown in Fig. 27. The ALPID code was then used to generate the distorted grid for both materials for the amount of
deformation used in the experimental tests. In Fig. 27, the flow patterns as depicted by the grid meshes are almost
identical in all cases; the difference in flow behavior is attributed to the estimated value of the die/workpiece friction used
to generate the ALPID simulations.
Fig. 26 Load versus ram travel for wedge specimens of aluminum alloys 1100-O and 6061-O
Fig. 27 Grid patterns showing metal flow in 6061-O (a) and 1100-
O (b) aluminum alloy wedge specimens at
40% deformation. At top are experimental results; at bottom are ALPID simulations.
To quantify the verification of the ALPID code, the experimental and ALPID-generated values of effective true strain
were used to plot the contour maps (Fig. 28). The ALPID-generated maps are very similar to the experimental ones.
Fig. 28 Effective true strain contour maps for 6061-O (a) and 1100-
O (b) aluminum alloy wedge specimens at
40% deformation. Experimental results are given at top; ALPID simulations are at bottom.
The reasons for the probable minor differences at some isolated locations in the experimental and ALPID-generated
values of effective strains are:
• Error introduced in digitizing the deformed and undeformed grids on an electronic tablet
• Error in aligning the crosshairs of the cursor with the node points
• Error introduced in locating the grid node point precisely
• Distortion of the grid network during the printing process
• Undeformed grids not being identical on all specimens
• Imprecise constitutive equations for the materials under consideration
•
Uncertainty in estimating the friction factor at the tool/specimen interface, which may vary from
specimen to specimen and with pressure and type of lubricant used
Development of Three-Dimensional Grid for Strain Calculation
To analyze the flow of material in deformation processing, it is necessary to determine the magnitude of strains in the
deformed body. The formulas defining the strains are derived in terms of the position of an element before and after
deformation with reference to a coordinate system. Therefore, the strains are computed by measuring the deformation of
some geometrical pattern, for example, a grid mesh or an array of circles. Choice of the pattern depends on such factors as
the type and accuracy of the information sought, the material, and the extent of specimen deformation.
Although in most plastic deformation processes the resulting strain is not homogeneous, the deformation will be assumed
to be homogeneous. The rationale behind this assumption is that deformed material can be divided into numerous smaller
regions where the deformation is homogeneous.
The Indirect Method. The development of a specimen shape to study metal flow in three dimensions is a tedious task.
Therefore, an indirect approach has been developed (Ref 73). In this method, a rectangular block 19 × 12.7 × 6.4 mm
(0.75 × 0.50 × 0.25 in.) thick with fine (0.38 mm, or 0.015 in., diam) through holes drilled and grid lines (2-2-2, 3-3-3,
etc.) engraved at 1.27 mm (0.050 in.) intervals is used, as shown in Fig. 29. The holes are filled with 0.38 mm (0.015 in.)
diam wires made of the same material. The specimen is then compressed between flat dies to a specified deformation, for
example, 40%. The distance between the horizontal grids (1-1-1, 2-2-2, etc.) is measured to determine the strain in the z
direction. The top surface (1-1-1-1) is photographed to obtain the flow pattern in the x-y plane. The specimen is machined
to the level 2-2-2, and the hole pattern is photographed again. The process is repeated to the level 5-5-5. An indirect 3-D
grid pattern is thus generated. This pattern is used to determine the true effective strain in each element.
Fig. 29 Standard rectangular block specimen used to study three-dimensional material flow
The multicolored workpiece method was developed for determining the velocity field in true 3-D deformation (Ref
74). Figure 30 shows the workpiece. After every deformation, thin slices are removed from the workpiece, and each
newly exposed surface is photographed.
Fig. 30 Multicolored model material workpiece consisting of four-color checkerboard
Software has been developed that can compute the coordinates of each nodal point in the deformed workpiece from
measurements of the above-mentioned photographs. The coordinates of these nodal points form the basis for determining
strain rates. Two different methods for the strain calculations have been developed: the parallelepipedon method and the
tetrahedron method. Figure 31 shows some results from an analysis of the upsetting of a cube with sticking friction
between workpiece and tools. These results were obtained through application of the multicolored workpiece method.
Fig. 31
Equivalent strain distribution in a cube after upsetting. (a) Distribution near the upper horizontal
tool/workpiece interface (see arrow at right). (b) Distribution near the horizontal symmetry plane (see arrow)
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H. L. Gegel and J.C. Malas, Air Force Wright Aeronautical Laboratories/Materials Laboratory; S.M. Doraivelu and V.A. Shende,
Universal Energy Systems, Knowledge Integration Center
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