ASM Metals HandBook Vol. 14 - Forming and Forging

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In some materials, m is insensitive to strain (Ref 4, 27). In other materials, however, m is sensitive to strain and strain rate

(Ref 28). In many materials, m increases and n decreases with an increase in temperature (Ref 29), sometimes to the

extent that superplastic properties develop.

Determining n and r Values

The time and facilities required for sample preparation and for performing the uniaxial tensile test make it difficult to use

for on-line process control. The following simplified tests for determining n and r are more suitable for this purpose. The

circle arc elongation test and the rapid-n test utilize tensile specimens with two sections that differ in width by about 5%

to determine n values. Fracture almost always occurs in the narrow section, but the final measurements are made on the

wide section, which elongates uniformly. The r value can be obtained from these tests, but for ferritic steels, the Modul-r

test is faster and easier to perform. This test actually measures the elastic modulus of the specimen and uses an

empirically determined correlation between the modulus and the r value.

The circle arc elongation test does not require measurement of the applied loads (Ref 30). It uses a rectangular

tensile specimen with a reduced width section produced by milling a pair of small circular arc notches on opposite sides.

The gage length is marked in the full-width section, and the specimen is pulled to fracture, which usually occurs in the

narrow section. The uniform elongation is measured in the full-width section. The value is slightly lower than that

obtained in the conventional tensile test and gives a slightly lower n value. However, it is suitable for production control.

The r value can be determined by the additional measurement of the change in width in the full-width section.

The rapid-n test provides rapid and fairly accurate measurements of yield and tensile strengths, elongation, and n and r

values (Ref 31). It requires relatively simple equipment and can be performed in less than 5 min, including specimen

preparation. The test is suitable for sheet metals whose properties are represented accurately by the Holloman equation, σ

= kε

. It has been successfully used on low-carbon and stainless steels and on a variety of nonferrous alloys.

The test specimen, which is punched directly from the sheet sample, has the dimensions shown in Fig. 12. Generally, 25

mm (1.0 in.) gage lengths are marked on both the wide and narrow sections, and the specimen is strained to fracture in a

manual or motorized load frame or in a tension-testing machine. The yield load is measured if there is discontinuous

yielding, and the maximum load is measured. Yield and tensile strengths are calculated from the measured loads and the

initial dimensions of the narrow section. If the yielding is continuous, the yield strength can be calculated from the tensile

strength and n value, as indicated below.

Fig. 12 Rapid-n test specimen. Source: Ref 31.

An empirically determined correction is applied to compensate for the effect of the sheared edges of the specimens. For

steel, this correction reduces the measured yield and tensile strengths by 13.6 MPa/mm (50.0 ksi/in.) of initial sample

thickness. The n value is calculated from:

(Eq 15)

where w

and w

are the initial widths of the wide and narrow sections, respectively; w

is the final width of the wide

section; and t

and t

are the initial and final thicknesses of the wide section, respectively. Equation 15 can be solved

iteratively in four steps, or by means of a simple computer program, beginning with a trial value of 0.24 for n.

For materials that do not have a discontinuous yield point, the yield strength can be calculated as:

(Eq 16)

where TS is the tensile strength and C is a constant. For low-carbon steels, C 0.02. For other materials, C must be

determined empirically. The r value can be computed from the initial and final dimensions of the wide section as

described previously.

The Modul-r test measures the elastic modulus (Young's modulus) of low-carbon steel samples by determining their

resonant frequencies by exciting them using an oscillating magnetic field (Ref 32). The elastic modulus is directly

proportional to the square of the resonant frequency, and simple empirical relationships exist between the directionally

averaged elastic modulus and the r

value and between the planar variation of the modulus and the ∆r value.

This test uses a flat 102 × 6.35 mm (4.0 × 0.25 in.) punched specimen and a specially designed, commercially available

magnetostrictive oscillator. The specimen is placed inside drive and pickup coils in the oscillator. An alternating current

passed through the drive coil produces an alternating magnetic field, which causes magnetostrictive oscillations in the

sample. The oscillations induce an alternating current in the pickup coil. This current is used to change the frequency of

the current in the drive coil to maximize the amplitude of the oscillations, that is, to obtain the resonant frequency, which

is displayed digitally.

The relationships used to determine r

and ∆r from the resonant frequency, f, are:

E = 4ρL

(Eq 17)

(Eq 18)

∆r = 0.031 - 0.0468 ∆E

(Eq 19)

where E is the elastic modulus in gigapascals; ρ and L are the density and length of the specimen, respectively; and E

and ∆E are defined analogously to r

and ∆r.

This test provides a more reproducible measure of r

and ∆r than the conventional tensile-testing method and is less

sensitive to differences between operators. It can be performed in 5 min, including specimen preparation. When alloy

steels or ferritic stainless steels are tested, a different correlation between the modulus and r value must be used. This

must be determined experimentally. When the test is used on coated products, the coating must be removed chemically

prior to testing.

References cited in this section

4. A.K. Ghosh, The Influence of Strain Hardening and Strain-Rate Sensitivity on Sheet Metal Forming,

Trans.

ASME, Vol 99, July 1977, p 264-274

24.

H.C. Wu and D.R. Rummler, Analysis of Misalignment in the Tension Test, Trans. ASME,

Vol 101, Jan

1979, p 68-74

25.

G.E. Dieter, Mechanical Metallurgy, 2nd ed., McGraw-Hill, 1976, p 347, 349, 681

26.

R.L. Whitely, Correlation of Deep Drawing Press Performance With Tensile Properties,

STP 390,

American Society for Testing and Materials, 1965

27.

S.J. Green, J.J. Langan, J.D. Leasia, and W.H. Yang, Material Properties, Including Strain-

Rate Effects, as

Related to Sheet Metal Forming, Met. Trans. A, Vol 2A, 1971, p 1813-1820

28.

G. Rai and N.J. Grant, On the Measurements of Superplasticity in an Al-Cu Alloy, Met. Trans A.,

Vol 6A,

1975, p 385-390

29.

W.J. McGregor Tegart, in Elements of Mechanical Metallurgy, Macmillan, 1966, p 29-38

30.

R.H. Heyer and J.R. Newby, Measurement of Strain Hardening and Plastic Strain Ratio Using the Circle-

Arc Specimen, Sheet Met. Ind., Vol 43, Dec 1966, p 910-914

31.

D.C. Ludwigson, A Rapid Test for the Assessment of Steel Sheet and Tinplate Properties, J. Test. Eval.,

Vol 7 (No. 6), 1979, p 301-309

32.

P.R. Mould and T.E. Johnson, Rapid Assessment of Drawability at Cold-Rolled Low-

Carbon Steel Sheets,

Sheet Met. Ind., Vol 50, June 1973, p 328-348

Formability Testing of Sheet Metals

Brian Taylor, General Motors Corporation

Plane-Strain Tensile Testing

In conventional uniaxial tensile testing, the sample is strained in the region of drawing; that is, the minor or width strain is

negative. The test does not provide information on the response of sheet materials in the plane-strain state, in which the

minor strain is zero. However, it can be modified to produce this strain state in part of the sample. This modification

involves the use of a very wide, short sample or the use of knife-edges to prevent transverse (width) strain in part of the

sample.

Wide Sample Methods. Increasing the width of the sample and decreasing the gage length changes the strain state

from one with a large negative minor strain component toward the plane-strain state, in which the minor strain component

is zero. In the rectangular sheet tension test, samples with length-to-width ratios of 1 to 1, 1 to 2, and 1 to 4 are used to

approach the plane-strain conditions (Ref 33). Gage lengths are constrained further by reinforcements welded onto each

side of the sample at both ends, thus making the samples three layers thick except in the gage length.

The minimum minor strain obtained with the 1 to 4 length-to-width ratio is -0.05 times the major strain, which is close to

the plane-strain condition of zero minor strain. The in-plane strains are measured by means of grid markings on the

samples, and through-thickness deformations can be observed by holographic interferometry.

A similar approach was used in testing many wide specimen designs to determine the effect of edge profile and length-to-

width ratio on strain state (Ref 34, 35, 36). The specimen geometry that yielded the highest center strain at failure with a

large region of plane strain is shown in Fig. 13. The plane-strain region, which is arbitrarily taken as the region where

| is less than 0.2, occupies about 80% of the specimen width. The outer part of the specimen deforms in a similar

manner to a standard tensile test specimen.

Special grips were developed that exert a high

clamping force at the inner contact lines. This

minimizes distortion and slip-page in these regions,

giving the test well-defined boundary conditions. The

results of both types of wide specimen tensile tests

described above correlated well with stress-strain

predictions obtained by finite-element modeling using

material properties obtained in the standard tensile test

(Ref 34, 37).

Width Constraint Method. In the width constraint

method, a rectangular sample is used that has a central

gage section reduced in width by circular notches (Ref

38). The gage section is clamped between two pairs of

opposing parallel knife-edges (stingers) aligned with

the sample axis. The knife-edges prevent transverse

(width) strain in this region. The sample is pulled to

fracture in a tension-testing machine, and the plane-

strain limit (necking) and fracture strains are

determined from thickness measurements made on the

fractured sample. This procedure is described in detail in Ref 38. The use of a spring-loaded clamp around the knife-edges

makes adjustment of the clamp during testing unnecessary.

References cited in this section

33.

M.L. Devenpeck and O. Richmond, Limiting Strain Tests for In-Plane Sheet Stretching, in

Novel

Techniques in Metal Deformation Testing, The Metallurgical Society, 1983, p 79-88

34.

R.H. Wagoner and N.M. Wang, An Experimental and Analytical Investigation of In-

Plane Deformation of

2036-T4 Aluminum, Int. J. Mech. Sci., Vol 21, 1979, p 255-264

35.

R.H. Wagoner, Measurement and Analysis of Plane-Strain Work Hardening, Met. Trans. A,

Vol 11A, Jan

1980, p 165-175

36.

R.H. Wagoner, Plane-

Strain and Tensile Hardening Behavior of Three Automotive Sheet Alloys, in

Experimental Verification of Process Models,

Symposium proceedings, Cincinnati, OH, Sept 1981,

American Society for Metals, 1983, p 236

37.

E.J. Appleby, M.L. Devenpeck, L.M. O'Hara, and O. Richmond, Finite Element Analysis and Experimental

Examination of the Rectangular-Sheet Tension Test, in

Applications of Numerical Methods to Forming

Processes,

Vol 28, Proceedings of the ASME Winter Annual Meeting, San Francisco, Applied Mechanics

Division, American Society of Mechanical Engineers, Dec 1978, p 95-105

38.

H. Sang and Y. Nishikawa, A Plane Strain Tensile Apparatus, J. Met., Feb 1983, p 30-33

Formability Testing of Sheet Metals

Brian Taylor, General Motors Corporation

Biaxial Stretch Testing

Two tests that determine the properties of sheet metals in biaxial stretching without involving surface friction effects are

the Marciniak biaxial stretching test (Ref 39) and the hydraulic bulge test (Ref 40). The Marciniak test subjects the

sample to in-plane biaxial stretching, but does not determine the stresses. In the hydraulic bulge test, the stresses can be

determined, but the sample is deformed into a dome, which involves out-of-plane stresses and strains.

Fig. 13 Plane-strain tensile test specimen. Source: Ref 36.

Marciniak Biaxial Stretching Test. A disk of the test material is stretched over a flat-bottomed punch of cylindrical

or elliptical cross section. This creates uniform in-plane biaxial strain in the center of the sample, with a strain ratio that is

determined by the ratio of the major and minor diameters of the punch. Most testing has been performed with a

cylindrical punch, which produces balanced biaxial stretching.

The center of the punch is hollowed out to eliminate friction in this area, and a spacer is placed between the sample and

the punch. The spacer is a disk of material similar to that under test--with the same diameter, but with a hole at the center.

The experimental arrangement is shown in Fig. 14. As the disk and spacer are stretched over the punch, the hole in the

spacer enlarges, and the central part of the test sample is deformed in uniform in-plane biaxial stretching.

The function of the spacer is to reverse the direction of the

surface friction experienced by the sample. In the absence of

the spacer, the surface friction opposes the movement of the

sample over the punch and reduces the maximum strain

level attainable. The spacer deforms more easily than the

test sample because of the hole in the center, and it exerts a

frictional force on the sample directed outward over the

punch radius.

For the material to stretch freely, the punch and die radii

must be adequate for the thickness of the material under test.

A ratio of spacer hole diameter to punch diameter of 1 to 3

has been successfully used. The strains can be measured by

using grid circles, squares, or other suitable markings. The

test has the following applications:

• Determination of the limiting strains of materials in uniform in-

plane biaxial stretching without surface

friction

• Application of a carefully controlled level of uniform in-

plane biaxial strain to samples with large areas

to be used in other tests--

for example, tests to determine the effect of different strain paths on the

limiting strain levels

• Detection of defects, such as inclusions, by straining a sample of large area to

a uniformly high level.

Defects will cause early localized fracture, usually parallel to the rolling direction

Hydraulic Bulge Test. The periphery of a sheet metal sample is clamped between circular or elliptical die rings, and

hydraulic pressure is applied on one side of the sample to deform it into a dome, as shown in Fig. 15. The edge of the

sample is prevented from slipping by a lock bead placed in the die rings. This consists of a ridge with small radii on one

ring and a matching groove on the other.

Fig. 14 Schematic of Marciniak biaxial stretching test.

Fig. 15 Schematic of hydraulic bulge test.

With circular die rings, the center of the dome has been found to be nearly spherical (Ref 40). The stress and strain states

in this region can be determined from the curvature and extension and the fluid pressure. A biaxial test extensometer has

been developed that measures the extension and curvature by means of a spherometer and an extensometer that are in

direct contact with the dome (Ref 41).

More recently, a system for controlling the strain rate in this test has been developed (Ref 42) because significantly

different test results have been obtained (Ref 43) under conditions of constant strain rate and constant fluid flow. The

system uses feedback from the extensometer signal to operate a servo-valve, which controls the flow of hydraulic oil to

the bulge. This system was used to determine the strain rate sensitivity of aluminum alloys to a much higher strain level

than is possible in the tensile test.

A computerized system is available that uses electronic vision to measure the principal strains and closed-loop feedback

to control the strain rate (Ref 44). This system monitors the relative positions of the centers of three closely spaced white

dots painted on a black background at the center of the sample. Initially, the dots form a right-angled isosceles triangle.

The principal strains are computed from the change in the spacings of the dots and changes in the angle they subtend.

This information is recorded and also used to maintain a constant strain rate by controlling the hydraulic pressure. Strains

can be computed only once per second, which limits the maximum controllable strain rate.

A camera is mounted so that it maintains a constant distance from the top of the dome and the dots, except for the effect

of the curvature of the dome, which is negligible. The stress state is determined by measuring the curvature of the dome

with a contacting spherometer and by measuring the hydraulic pressure with a strain gage pressure transducer.

For thin samples, bending stresses can be neglected, and the radial (meridional) stress, σ

, is given by:

(Eq 20)

where p is the hydraulic pressure, R is the radius of curvature, and t is the instantaneous thickness. The thickness is

calculated from the measured strains using constancy of volume. At the top of the dome, the sample is in balanced biaxial

stretching, and the circumferential stress, σ

, is equal to the radial stress.

For convenience, it is customary to express the results of hydraulic bulge tests in terms of the true thickness stress and

strain. This is done by theoretically superimposing a hydrostatic compressive stress that does not influence deformation

and that has in-plane components equal to the actual radial and circumferential tensile stresses. This converts the stress

state to a simple uniaxial thickness compressive stress, as shown in Fig. 16.

Fig. 16 Addition of compressive hydrostatic stress to biaxial tensile stress.

The true thickness strain, ε

, can be obtained from the radial strain, ε

, and circumferential strain, ε

, by using the

constancy of volume condition:

= - ε

- ε

(Eq 21)

This enables the results to be represented by a true compressive stress-strain curve in the thickness direction. The

hydraulic bulge test has the following applications:

•

Intrinsic material characterization in biaxial stretching, which is a very common strain state in

production stampings

•

Testing to much higher strain levels than those achievable in tensile testing (in some cases, by as much

as a factor of ten), particularly for heavily cold-worked materials

• Checking the validity of plasticity theories that attempt to predic

t the yielding behavior of metals in all

stress states from properties measured in uniaxial and plane-strain tensile testing

References cited in this section

39.

Z. Marciniak and K. Kuczynski, Limit Strains in the Processes of Stretch-Forming Sheet Metal,

Int. J.

Mech. Sci., Vol 9, 1967, p 609-620

40.

J.L. Duncan, J. Kolodziejski, and G. Glover, Bulge Testing as an Aid to Formability Assessment, in

Sheet

Metal Forming and Energy Conservation, Proceedings of the 9th Biennial Congress of the International

Deep Drawing Research Group, Ann Arbor, MI, American Society for Metals, 1976, p 131-150

41.

W. Johnson and J.L. Duncan, The Use of the Biaxial Test Extensometer, Sheet Met. Ind.,

Vol 42, April

1965, p 271-275

42.

J.E. Bird, Hydraulic Bulge Testing at Controlled Strain Rate, in

Novel Techniques in Metal Deformation

Testing, The Metallurgical Society, 1983, p 403-416

43.

A.J. Ranta-Eskola, Use of the Hydraulic Bulge Test in Biaxial Tensile Testing, Int. J. Mech. Sci.,

Vol 21,

1979, p 457-465

44.

D.N. Ha

rvey, Electronic Vision as Input to Calculated Variable Control of the Hydraulic Bulge Test, in

Proceedings of the International Symposium on Automotive Technology and Automation

(Wolfsburg, West

Germany), Sept 1982

Formability Testing of Sheet Metals

Brian Taylor, General Motors Corporation

Shear Testing

Two tests have been developed to determine the properties of sheet metals subjected to planar shear deformation: the

Marciniak in-plane sheet torsion test (Ref 45) and the Miyauchi shear test (Ref 46).

Marciniak In-Plane Sheet Torsion Test. A flat 50 mm (1.97 in.) square sample is effectively divided into three

zones: an inner circular zone, which is clamped; a ring-shaped middle zone, which surrounds the inner zone and is free to

deform; and an outer ring-shaped zone, which is clamped. The inner zone is rotated in its plane relative to the outer zone,

which deforms the middle zone in shear. The sample is deformed to fracture, and the angular rotations at two radii in the

middle zone are measured by means of calibrated drums that rotate with the sheet. This is shown schematically in Fig. 17.

Fig. 17 Deformed Marciniak in-plane torsion test specimen.

For materials that follow the power law shear strain-hardening relationship:

τ= Cγ

(Eq 22)

where τ is the shear stress, C is a constant, and γ is the shear strain; it can be shown that (Ref 45):

(Eq 23)

where α

and α

are the angular displacements at radii R

and R

Shear fracture occurs where the shear stress is greatest--at the inner radius, R

, of the deforming ring. The shear fracture

strain, γ

, is given by:

(Eq 24)

This test enables the forming properties of sheet metals to be determined at much higher strain levels than is possible in

the uniaxial tensile test and in a different strain state, that is, in shear.

The Miyauchi shear test determines the properties of sheet metals in planar shear deformation by means of a

modified tensile technique (Ref 46). The test uses flat, rectangular specimens whose ends are divided into three equal

sections by parallel longitudinal slits, as shown in Fig. 18(a).

Fig. 18 Miyauchi shear test specimen. (a) Undeformed. (b) Deformed. Source: Ref 46.

The specimen is clamped in a fixture that prevents out-of-plane deformation. The inner and outer sections are then pulled

in opposite directions in a tensile machine. This produces a shear stress in the regions between the inner and outer

sections and deforms the specimen, as shown in Fig. 18(b). The deformation in these regions is uniform, except at the

ends.

The shear strain, γ, is the tangent of angle θ, which is the change in direction of lines scribed across the specimen as they

pass through the shear zone, as shown in Fig. 18(b). The strain can also be determined from the displacement of the inner

section once a relationship has been established between the displacement and θ, which must be done for each type of

sheet metal tested. Shear stress-strain curves are given in Ref 46 for three different steels. These curves show differences

in the strain dependence of the work-hardening coefficient from that in the tensile test.

References cited in this section

45.

Z. Marciniak, Aspects of Material Formability, Metalworking Research Group, McMast

er University, 1973,

p 84-91

46.

K. Miyauchi, Stress-Strain Relationship in Simple Shear of In-

Plane Deformation for Various Steel Sheets,

in Efficiency in Sheet Metal Forming,

Proceedings of the 13th Biennial Congress, Melbourne, Australia,

International Deep Drawing Research Group, Feb 1984, p 360-371

Formability Testing of Sheet Metals

Brian Taylor, General Motors Corporation

Hardness Testing

Hardness, or the resistance to indentation by a concentrated load applied by a suitable indenter, has been used in many

stamping plants as a measure of formability. Generally, formability decreases with increasing hardness, but the fine-scale

correlation between these properties has not been reliable. The test can be used effectively to monitor changes in a

particular grade of material caused by changes in processing that may affect formability.

For steels, hardness measurements correlate well with yield strength values (Ref 47). Therefore, hardness testing is useful

in quality control to ensure that the material in use is the specified grade and has the required strength level.

The Rockwell hardness test (ASTM E 18), which is described in detail in the article "Rockwell Hardness Testing" in

Mechanical Testing, Volume 8 of ASM Handbook, formerly 9th Edition Metals Handbook, is typically used to determine

the hardness of such materials. The load and indenter must be selected for the gage and hardness range of the material

according to the test specification to ensure that the indentation is the appropriate size. If the indentation is too deep in a

sheet sample, the reading will be artificially high because of the influence of the supporting anvil. This becomes a more

serious consideration with the use of thinner gage sheet metal in many industries. For thicker and harder materials, the

Rockwell B scale is commonly used, and for thinner or softer materials, the Rockwell 30T superficial scale is used.

Hardness readings are influenced by the degree of flatness and surface conditions. In addition, the presence of cold-

worked surface layers can cause unrepresentatively high hardness readings that suggest a lower formability level than

actually exists.

Reference cited in this section

47.

R.A. George, S. Dinda, and A.S. Kasper, Estimating Yield Strength From Hardness Data, Met. Prog.,

May

1976, p 30-35

Formability Testing of Sheet Metals

Brian Taylor, General Motors Corporation

Simulative Tests

For many forming operations, tests that simulate the operation are more useful and relevant than fundamental intrinsic

property measurement tests. These tests subject the work material to deformation that closely approximates the

production operation, including the effects of factors not present in the intrinsic tests, such as bending and unbending and

friction between the work materials and die surfaces. Because these additional factors are present, simulative tests tend to

be less reproducible than intrinsic tests and must be performed under carefully controlled conditions to minimize

variability in the results.

Simulative tests can be classified on the basis of the predominant forming operation involved: bending, stretching,

drawing, and stretch-drawing. In addition, tests have been developed to measure wrinkling and the springback that occurs

after bending or another forming operation.

Bending Tests

Two types of bending tests relate to sheet metal forming: simple bending and stretch-bending tests. Simple bending tests

are useful in predicting how the sheet metal will perform when bent without tension, as in a hemming operation. Stretch-

bending tests relate to the response to combined bending and stretching, as when sheet metal is pulled over a punch or die

radius.

Simple bending tests can be performed in various ways (ASTM E 290). The simplest method for thin sheet material

is to clamp a specimen and a bending die in a vise, as shown in Fig. 19, and to bend the specimen over the die manually

or with a nonmetallic mallet.