Non-Linear Strain Paths (Stress-Based FLCs)

Non-Linear Strain Paths (Stress-Based FLCs)

FLC and FLD

Conventional Forming Limit Curves (FLCs) gained widespread industrial use since being introduced by Dr. Stuart Keeler in the 1960’s. Applications from feasibility analysis to stamping plant troubleshooting use these principles. The strain hardening exponent (n-value) and thickness are inputs into a shortcut to create the curve placement and shape, but this is applicable to only mild steels, conventional High-Strength Steels, and some Advanced High-Strength Steels. Furthermore, this shortcut is an approximation, coming from a best-fit curve generated from data points gathered over multiple grades.

A typical method used in creating most FLCs includes deforming samples of different widths with a 100 mm (4 inch) diameter hemispherical punch – known as the Nakajima method. An alternate approach uses a flat-bottom cylindrical punch, known as the Marciniak method (Figure 1). Independent of the punch shape used, generating FLCs involves measuring the strains resulting from deforming a blank to a formed shape. The conventional FLC plots major strain on the vertical axis against minor strain on the horizontal axis. This FLC applies only to in-plane stretching in linear strain paths, and assumes that there are no through-thickness stress or strain differences. Assessing bendability or cut edge ductility is not possible with this approach.

Figure 1: Punch Shape Used to Create Forming Limit Curves Result in Through-Thickness Strain Differences Which Influence the Shape and Placement of The FLC [Reference 1]

Figure 1: Punch Shape Used to Create FLCs Result in Through-Thickness Strain Differences Which Influence the Shape and Placement of The FLC S-37

 

Figure 2 compares the FLCs generated by deforming DP980 with the three punch shapes highlighted in Figure 1. Note the higher strains associated with the 50 mm diameter hemispherical punch compared with the strains generated from the 100 mm diameter hemispherical punch. This punch curvature difference impacts the magnitude of the strains that develop through the thickness of the sheet. On samples deformed with a hemispherical punch, the selected strain measurement technique (circle/square grid analysis or Digital Image Correlation, for example) directly measures strains on the outer top surface only, with the middle and inner surface having progressively lower strains as a function of the R/T ratio. A punch or feature with small R/T leads to high strains on the outermost surface. Strains exceeding the FLC on only this outer surface will not lead to necks on the formed panel. Exceeding the FLC through the entire thickness – from the inner surface to the outer surface – must occur for the sample to show a neck.T-17

Figure 2: FLCs of the same batch of DP980 Showing Dependence on Punch Shape and Curvature [References 1 and 3]

Figure 2: FLCs of the same batch of DP980 Showing Dependence on Punch Shape and Curvature.S-37, M-15

 

In addition to the through-thickness strain differences from the punch curvature, the metal flow differences resulting from the punch shapes leads to directional changes in the strain path taken by the deforming metal. A channel drawn part with a hat-shaped cross section in which there are no features like embossments is likely to have a linear strain path. Forming every other engineered stamped part geometry involves some degree of a non-linear strain path (NLSP).

The importance of strain path and deformation history comes from the changes in the forming limit that occur once metal deformation starts. The black curve in Figure 3 shows the FLC for an alloy generated in a conventional manner with as-received metal, assuming a linear strain path. The red curve results from testing the same metal that initially stretched to an equal-biaxial plastic pre-strain of 0.07. In this strain path, substantially less deformation can occur before reaching the forming limit. However, the strain path changes if the local part contour is different, and that strain path results in a different amount of subsequent deformation prior to necking. The magnitude and direction of the shift changes based on the strain and the orientation relative to the rolling direction. Citation S-38 highlights these curves and presents more examples of the effects of different strain paths. The important conclusion is that the amount of deformation that a metal is capable of withstanding prior to necking changes throughout the forming process and depends on the local part shape (among other variables), and cannot be discerned by using only the conventional strain based FLC.

Figure 3: Experimental FLCs for a linear strain path (in black) and for a bilinear strain path after 0.07 strain in equal biaxial tension in strain space (in red) [Reference 4]

Figure 3: Experimental FLCs for a linear strain path (in black) and for a bilinear strain path after 0.07 strain in equal biaxial tension in strain space (in red) S-38

 

Figure 4 shows the strain paths associated with the FLCs presented in Figure 2, with along with a magnified portion of one of the curves. This non-linearity is a characteristic of samples formed with a dome, associated with the sample wrapping around the punch during the initial contact and experiencing a combination of biaxial bending and stretching. Citation M-15 presents a method to correct for strain path effects.

Figure 4: Strain Path for FLCs shown in Figure 2. A) 100mm diameter flat punch; B) 100mm diameter hemispherical punch; C) 50mm diameter hemispherical punch; and D) Magnified portion of one curve from Figure 4B showing the non-linearity of the strain path [References 1 and 3]

Figure 4: Strain Path for FLCs shown in Figure 2. A) 100 mm diameter flat punch; B) 100 mm diameter hemispherical punch; C) 50 mm diameter hemispherical punch; and D) Magnified portion of one curve from Figure 4B showing the non-linearity of the strain path.S-37, M-15

 

Accounting for tool contact pressure is critical as well, since pressure through the sheet thickness suppresses the onset of necking. Applying this compensated FLC in simulation or in hands-on analysis parts analysis requires modification for the unique characteristics of each part, with appropriate adjustments for local curvature, contact pressure and deformation history. Citations S-37 and M-15 detail methods to compensate for the effects of strain path, curvature, and tool pressure. Figure 5 shows that after incorporating these corrections, the curves condense to one shape independent of the variables used.

Figure 5: As-generated FLCs compared with FLCs after strain path, curvature, and tool contact pressure corrections [References 1 and 3]

Figure 5: As-generated FLCs compared with FLCs after strain path, curvature, and tool contact pressure corrections.S-37, M-15

 

In summary, FLCs generated from relatively similar simple tools are sensitive to small differences in R/T ratio, incorporation of tool contact pressure, and deviations from a linear strain path. By comparison, engineered stampings require substantially more complex tool shapes with differing degrees of curvature, tool contact pressure, and strain paths all within one part. These complex part shapes contribute to an even wider variation in the yield surface and hardening mechanisms important for simulation, and impacts predictions of formability, springback, and stress analysis.

A common requirement during tooling buyoff – where all strains need to be below the FLC by at least a certain amount called the safety margin – magnifies these challenges. AHSS grades already have low FLCs relative to their lower strength counterparts, so it is critical that the chosen FLC does not further reduce efficient application of these grades. Minimizing sensitivity to the changes in strain path occurring across a complex part requires using a different approach – a FLC with the axes in stress-space rather than the conventional strain-space.

This discussion has centered on conventional strain-based FLCs, which incorporate an assumption of a linear strain path as a flat sheet deforms to the final shape. Stress-based Forming Limit Curves (sFLC or FLSC) are insensitive to deformation history and can be adjusted to reflect the differences in local tool geometry or contact pressure across the stamping. Forming analysis software readily converts conventional FLCs into stress-based units. Figure 6 converts the two strain paths presented in Figure 3 into stress-space, and shows the two experimental stress FLCs generated with different strain paths are independent of the loading history and essentially overlap.  Citations S-38, S-39, S-40 and S-41  contain information about stress-based FLCs, as well as their generation and usage.

Figure 6: After converting the conventional FLCs in Figure 3 to stress-space, the experimental stress-based FLCs show no significant differences.S-38

Figure 6: After converting the conventional FLCs in Figure 3 to stress-space, the experimental stress-based FLCs show no significant differences.S-38

 

Citation H-20 presents a related method to transition from strain-based to stress-based Forming Limit Curves. The proposed stress-based failure criterion postulates that localized necking occurs when a critical normal stress condition is met. This approach adequately describe the experimental strain-based forming limit data in most evaluated materials, failing only with a 3rd Generation AHSS alloy containing a high percentage of retained austenite. For this grade, the authors speculate that a material model more advanced than the one employed in this study will improve correlation.

Accurate simulation requires accurate and complete inputs, including the full range of metal properties, with correct material flow and hardening models, and an understanding of the conditions that will produce failure. Any shortcuts taken increases the likelihood that simulation will not fully match reality for all materials, part shapes, and production processes. A conventional strain-based FLC assumes no effect of part geometry, tool contact pressure, and deformation history – all of which occur on engineered stampings to differing degrees. Analysts should incorporate stress-based FLCs into their simulation with appropriate adjustments to address local geometry and contact pressure to ensure an accurate representation of the metal’s forming characteristics.

For use in the die shop or stamping plant, a growing number of optical systems have built-in features to map strain measurements on to an sFLC. Use caution when employing this approach since these systems measure only the final net strain, and not the strain history as the panel deforms. Proper application involves capturing metal flow from individual breakdown panels and adjusting the FLC accordingly as the panel gets closer to the home position.

 

Special thanks to Dr. Thomas Stoughton, Technical Fellow, General Motors Research & Development, for assistance in preparing this information.

Non-Linear Strain Paths (Stress-Based FLCs)

N-Value

Mechanical Properties

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.

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

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

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

    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.