## Uniform Elongation

During a tensile test, the elongating sample leads to a reduction in the cross-sectional width and thickness. The shape of the engineering stress-strain curve showing a peak at the load maximum (Figure 1) results from the balance of the work hardening which occurs as metals deform and the reduction in cross-sectional width and thickness which occurs as the sample dogbone is pulled in tension. In the upward sloping region at the beginning of the curve, the effects of work hardening dominate over the cross-sectional reduction. Starting at the load maximum (ultimate tensile strength), the reduction in cross-sectional area of the test sample overpowers the work hardening and the slope of the engineering stress-strain curve decreases. Also beginning at the load maximum, a diffuse neck forms usually in the middle of the sample.

Figure 1: Engineering stress-strain curve from which mechanical properties are derived.

The elongation at which the load maximum occurs is known as Uniform Elongation. In a tensile test, uniform elongation is the percentage the gauge length elongated at peak load relative to the initial gauge length. For example, if the gauge length at peak load measures 61 mm and the initial gauge length was 50mm, uniform elongation is (61-50)/50 = 22%.

Schematics of tensile bar shapes are shown within Figure 1. Note the gauge region highlighted in blue. Up though uniform elongation, the cross-section has a rectangular shape. Necking begins at uniform elongation, and the cross section is no longer rectangular.

Theory and experiments have shown that uniform elongation expressed in true strain units is numerically equivalent to the instantaneous n-value.

## Deformation Prior to Uniform Elongation is Not Uniformly Distributed

Conventional wisdom for decades held that there is a uniform distribution of strains within the gauge region of a tensile bar prior to strains reaching uniform elongation. Traditional extensometers calibrated for 50-mm or 80-mm gauge lengths determine elongation from deformation measured relative to this initial length. This approach averages results over these spans.

The advent of Digital Image Correlation (DIC) and advanced processing techniques allowed for a closer look. A studyS-113 released in 2021 clearly showed that each of the 201 data points monitored within a 50 mm gauge length (virtual gauge length of 0.5-mm) experiences a unique strain evolution, with differences starting before uniform elongation.

Figure 2: Strain evolution of the 201 points on the DP980 tensile-test specimen exhibits divergence beginning before uniform elongation—counter to conventional thinking.S-113

## R-Value, The Plastic Anisotropy Ratio

The r-value, which is also called the Lankford coefficient or the plastic strain ratio, is the ratio of the true width strain to the true thickness strain at a particular value of length strain, indicated in Equation 1. Strains of 15% to 20% are commonly used for determining the r-value of low carbon sheet steel. Like n-value, the ratio will change depending on the chosen reference strain value.

Equation 1

The normal anisotropy ratio (rm) defines the ability of the metal to deform in the thickness direction relative to deformation in the plane of the sheet. rm, also called r-bar (i.e. r average) is the “normal anisotropy parameter” in the formula for Equation 2:

Equation 2

where r0, r90, and r45 are the r-value determined in 0, 90, and 45 degrees relative to the rolling direction.

For rm values greater than one, the sheet metal resists thinning. Values greater than one improve cup drawing (Figure 1), hole expansion, and other forming modes where metal thinning is detrimental. When For rm values are less than one, thinning becomes the preferential metal flow direction, increasing the risk of failure in drawing operations.

Figure 1: Higher r-value is associated with improved deep drawability.

High-strength steels with tensile strength greater than 450 MPa and hot-rolled steels have rm values close to one. Therefore, conventional HSLA and Advanced High-Strength Steels (AHSS) at similar yield strengths perform equally in forming modes influenced by the rm value. However, r-value for higher strength grades of AHSS (800 MPa or higher) can be lower than one and any performance influenced by r-value would be not as good as HSLA of similar strength. For example, AHSS grades may struggle to form part geometries that require a deep draw, including corners where the steel is under circumferential compression on the binder. This is one of the reasons why higher strength level AHSS grades should have part designs and draw dies developed as “open ended”.

A related parameter is Delta r (∆r), the “planar anisotropy parameter” in the formula shown in Equation 3:

Equation 3

Delta r is an indicator of the ability of a material to demonstrate a non-earing behavior. A value of 0 is ideal in can making or the deep drawing of cylinders, as this indicates equal metal flow in all directions, eliminating the need to trim ears prior to subsequent processing.

Testing procedures for determining r-value are covered in Citations A-44, I-15, J-14.

## N-Value, The Strain Hardening Exponent

Metals get stronger with deformation through a process known as strain hardening or work hardening, resulting in the characteristic parabolic shape of a stress-strain curve between the yield strength at the start of plastic deformation and the tensile strength.

Work hardening has both advantages and disadvantages. The additional work hardening in areas of greater deformation reduces the formation of localized strain gradients, shown in Figure 1.

Figure 1: Higher n-value reduces strain gradients, allowing for more complex stampings. Lower n-value concentrates strains, leading to early failure.

Consider a die design where deformation increased in one zone relative to the remainder of the stamping. Without work hardening, this deformation zone would become thinner as the metal stretches to create more surface area. This thinning increases the local surface stress to cause more thinning until the metal reaches its forming limit. With work hardening the reverse occurs. The metal becomes stronger in the higher deformation zone and reduces the tendency for localized thinning. The surface deformation becomes more uniformly distributed.

Although the yield strength, tensile strength, yield/tensile ratio and percent elongation are helpful when assessing sheet metal formability, for most steels it is the n-value along with steel thickness that determines the position of the forming limit curve (FLC) on the forming limit diagram (FLD). The n-value, therefore, is the mechanical property that one should always analyze when global formability concerns exist. That is also why the n-value is one of the key material related inputs used in virtual forming simulations.

Work hardening of sheet steels is commonly determined through the Holloman power law equation:

where
σ is the true flow stress (the strength at the current level of strain),
K is a constant known as the Strength Coefficient, defined as the true strength at a true strain of 1,
ε is the applied strain in true strain units, and
n is the work hardening exponent

Rearranging this equation with some knowledge of advanced algebra shows that n-value is mathematically defined as the slope of the true stress – true strain curve. This calculated slope – and therefore the n-value – is affected by the strain range over which it is calculated. Typically, the selected range starts at 10% elongation at the lower end to the lesser of uniform elongation or 20% elongation as the upper end. This approach works well when n-value does not change with deformation, which is the case with mild steels and conventional high strength steels.

Conversely, many Advanced High-Strength Steel (AHSS) grades have n-values that change as a function of applied strain. For example, Figure 2 compares the instantaneous n-value of DP 350/600 and TRIP 350/600 against a conventional HSLA350/450 grade. The DP steel has a higher n-value at lower strain levels, then drops to a range similar to the conventional HSLA grade after about 7% to 8% strain. The actual strain gradient on parts produced from these two steels will be different due to this initial higher work hardening rate of the dual phase steel: higher n-value minimizes strain localization.

Figure 2: Instantaneous n-values versus strain for DP 350/600, TRIP 350/600 and HSLA 350/450 steels.K-1

As a result of this unique characteristic of certain AHSS grades with respect to n-value, many steel specifications for these grades have two n-value requirements: the conventional minimum n-value determined from 10% strain to the end of uniform elongation, and a second requirement of greater n-value determined using a 4% to 6% strain range.

Plots of n-value against strain define instantaneous n-values, and are helpful in characterizing the stretchability of these newer steels. Work hardening also plays an important role in determining the amount of total stretchability as measured by various deformation limits like Forming Limit Curves.

Higher n-value at lower strains is a characteristic of Dual Phase (DP) steels and TRIP steels. DP steels exhibit the greatest initial work hardening rate at strains below 8%. Whereas DP steels perform well under global formability conditions, TRIP steels offer additional advantages derived from a unique, multiphase microstructure that also adds retained austenite and bainite to the DP microstructure. During deformation, the retained austenite is transformed into martensite which increases strength through the TRIP effect.  This transformation continues with additional deformation as long as there is sufficient retained austenite, allowing TRIP steel to maintain very high n-value of 0.23 to 0.25 throughout the entire deformation process (Figure 2). This characteristic allows for the forming of more complex geometries, potentially at reduced thickness achieving mass reduction. After the part is formed, additional retained austenite remaining in the microstructure can subsequently transform into martensite in the event of a crash, making TRIP steels a good candidate for parts in crush zones on a vehicle.

Necking failure is related to global formability limitations, where the n-value plays an important role in the amount of allowable deformation at failure. Mild steels and conventional higher strength steels, such as HSLA grades, have an n-value which stays relatively constant with deformation. The n-value is strongly related to the yield strength of the conventional steels (Figure 3).

Figure 3: Experimental relationship between n-value and engineering yield stress for a wide range of mild and conventional HSS types and grades.K-2

N-value influences two specific modes of stretch forming:

1. Increasing n-value suppresses the highly localized deformation found in strain gradients (Figure 1).
A stress concentration created by character lines, embossments, or other small features can trigger a strain gradient. Usually formed in the plane strain mode, the major (peak) strain can climb rapidly as the thickness of the steel within the gradient becomes thinner. This peak strain can increase more rapidly than the general deformation in the stamping, causing failure early in the press stroke. Prior to failure, the gradient has increased sensitivity to variations in process inputs. The change in peak strains causes variations in elastic stresses, which can cause dimensional variations in the stamping. The corresponding thinning at the gradient site can reduce corrosion life, fatigue life, crash management and stiffness.  As the gradient begins to form, low n-value metal within the gradient undergoes less work hardening, accelerating the peak strain growth within the gradient – leading to early failure. In contrast, higher n-values create greater work hardening, thereby keeping the peak strain low and well below the forming limits. This allows the stamping to reach completion.
2. The n-value determines the allowable biaxial stretch within the stamping as defined by the forming limit curve (FLC).
The traditional n-value measurements over the strain range of 10% – 20% would show no difference between the DP 350/600 and HSLA 350/450 steels in Figure 2. The approximately constant n-value plateau extending beyond the 10% strain range provides the terminal or high strain n-value of approximately 0.17. This terminal n-value is a significant input in determining the maximum allowable strain in stretching as defined by the forming limit curve. Experimental FLC curves (Figure 4) for the two steels show this overlap.

Figure 4: Experimentally determined Forming Limit Curves for mils steel, HSLA 350/450, and DP 350/600, each with a thickness of 1.2mm.K-1

Whereas the terminal n-value for DP 350/600 and HSLA 350/450 are both around 0.17, the terminal n-value for TRIP 350/600 is approximately 0.23 – which is comparable to values for deep drawing steels (DDS). This is not to say that TRIP steels and DDS grades necessarily have similar Forming Limit Curves. The terminal n-value of TRIP grades depends strongly on the different chemistries and processing routes used by different steelmakers. In addition, the terminal n-value is a function of the strain history of the stamping that influences the transformation of retained austenite to martensite. Since different locations in a stamping follow different strain paths with varying amounts of deformation, the terminal n-value for TRIP steel could vary with both part design and location within the part. The modified microstructures of the AHSS allow different property relationships to tailor each steel type and grade to specific application needs.

Methods to calculate n-value are described in Citations A-43, I-14, J-13.

## Tensile Testing

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Tensile testing characterizes the forming and structural behavior of sheet metals. The test involves loading a sample with a well-defined shape along the axis in tension, generally to fracture, and recording the resultant load and displacement to calculate several mechanical properties. Global standardsI-7, A-24, D-19, J-15 prescribe the conditions under which tests must occur.

## Sample Size and Shape

Full-size samples for tensile testing of metal sheets have a rectangular section at the edges for gripping by the test machine. Reducing the width in the central area promotes fracture in the monitored region. These geometrical features result in a sample shape which resembles a dogbone, leading to a descriptive term applied to test samples.

Dimensions of the dogbone samples are associated with tensile test standard from which they apply. ISO I, II, and III (described in Citation I-7) corresponds to the ASTMA-24, DIND-19, and JISJ-15 shapes, respectively. Figure 1 shows the dogbone shapes, highlighting the critical dimensions of width and gauge length. Refer to the Test Standards for other dimensions, tolerances, and other requirements.

Figure 1: Full-size tensile sample shapes for ISO I (ASTM), ISO II (DIN), and ISO III (JIS) standards.I-7, A-24, D-19, J-15

Significant differences exist in the width and gauge length of these tensile bar shapes. Although the ASTM and JIS bars have similar gauge length, the width of the JIS bar is twice that the ASTM bar. The ASTM and DIN bars have a 4:1 ratio of gauge length to width, where the JIS bar has a 2:1 ratio.

These shape differences mean that the calculated elongation changes depending on the test-sample standard used, even when testing identical material. With the combination of the shortest gauge length and widest sample, elongation from JIS bars typically are higher than what would be generated from the other shapes.

Yield strength and tensile strength are not a function of the shape of the tensile bar. Strength is defined as the load divided by the cross-sectional area. Even though each of the bars specify a different sample width (and therefore different cross-section), the load is normalized by this value, which negates differences from sample shape.

Shearing or punching during sample preparation may work-harden the edges of the tensile bar, which may lead to generating an inaccurate representation of the mechanical properties of the sheet metal. Test Standards require subsequent machining or other methods to remove edge damage created during sample preparation. Milling or grinding the dogbone samples minimizes the effects sample preparation may have on the results.

## Tensile Test Procedure

The gauge length is the reference length used in the elongation calculations. Depending on the test standard, the gauge length is either 2 inches, 80 mm, or 50 mm. Multiplying the width and thickness within the gauge length determines the initial cross-sectional area before testing.

Grips tightly clamp the edges of the sample at opposite ends. As the test progresses, the grips move away from each other at a prescribed rate or in response to the restraining load. A load cell within the grips or load frame monitors force. An extensometer tracks displacement within the gauge length. Samples are typically tested until fracture.

During the tensile test, the sample width and thickness shrink as the length of the test sample increases. However, these dimensional changes are not considered in determining the engineering stress, which is determined by dividing the load at any time during the test by the starting cross-sectional area. Engineering strain is the increase in length within the gauge length relative to the starting gauge length. (Incorporating the dimensional changes occurring during testing requires calculating true stress and strain. The differences between engineering and true stress/strain are covered elsewhere (hyperlink to 2.3.2.1-Engineering/True)

A graph showing stress on the vertical axis and strain on the horizontal axis is the familiar engineering stress-strain curve, Figure 2. From the stress-strain curve, numerous parameters important for sheet metal forming appear, including:

Figure 2: Engineering stress-strain curve from which mechanical properties are derived.

## Influence of Test Speed

Conventional tensile testing is done at strain rates slow enough to be called “quasi-static.” These rates are several orders of magnitude slower than the deformation rates during stamping, which itself is several orders of magnitude slower than what is experienced during a crash event.

Stress-strain curves change with test speed, typically getting stronger as the speed increases. The magnitude of these changes varies with grade. Significant challenges exist when attempting to characterize the tensile response at higher strain rates. Improved equipment and data collection capabilities are among the required upgrades.

## Influence of Tensile Test Equipment

Advanced High-Strength Steels (AHSS) may challenge older test equipment. The load and displacement response must reflect only the contributions of the sheet metal, and not be influenced by the load frame and other testing equipment. In much the same way that insufficiently stiff press crowns deflect when stamping AHSS parts, tensile test load frames may similarly deflect, resulting in inaccuracies in the load-displacement measurements.

Grip strength also becomes critical when testing AHSS samples. The high strength of the metal sheets requires more grip pressure to prevent sample slippage through the grips. Pneumatic grips and even some mechanical grips may not generate the necessary pressure. Hydraulically actuated grips may be necessary as the strength increases.

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