Measuring the Plastic Strain Ratio of Sheet Metals

        Measuring the Plastic Strain Ratio of Sheet Metals

Drawing metal successfully relies, in part, on understanding precisely how the metal reacts to tensile forces. When subjected to tensile forces, a flat section of sheet material becomes thinner because of dimensional changes in its width and thickness. The ratio of the changes in width and thickness make up the plastic strain strain ratio.

Sheet metal forming operations vary from simple to difficult; at one end of the spectrum is bending; in the middle is stretching; and at the other end is deep drawing of complex parts. Regardless of the forming operation, the sheet material’s mechanical properties greatly influence its formability, which is a measure of the amount of deformation a material can withstand before excessive thinning or fracture occurs.
Determining how much a material can deform is necessary for designing a reproducible forming operation. Testing the incoming sheet material is also essential because material properties vary from coil to coil and affect the part quality and scrap rate.
>> Plastic Strain Ratio
The plastic strain ratio, r, is considered a direct measure of sheet metal’s drawability and is useful for evaluating materials intended for forming shapes by deep drawing. The r value is the ratio of the true strain in the width direction to the true strain in the thickness direction when a sheet material is pulled in uniaxial tension beyond its elastic limit (see the following figure).

Determining the plastic strain ratio is governed by ASTM E517 Standard Test Method for Plastic Strain Ratio r for Sheet Metal. The plastic strain ratio is calculated as shown in Equation 1:
r=e_w/e_t
Where:
True width strain e_w = ln(w_f/w_o)
True thickness strain e_t = ln(t_f/t_o)
w_f = Final width
w_o = Original width
t_f = Final thickness
t_o = Original thickness
Equation 1 shows that the r value is dependent on the ratio of width and thickness changes as the sample is pulled in tension. The word plastic in the phrase plastic strain ratio implies that you have exceeded the specimen’s elastic limit and that only the strain that induces plastic flow is considered in the calculation.
Because it is difficult to measure thickness changes accurately, it is assumed the volume of the specimen remains constant and the thickness strain is expressed as e_t = ln(L_ow_o/L_fw_f).
After substituting e_t into Equation 1 and inverting it to eliminate negative values, the plastic strain ratio is given by Equation 2
r = ln(w_o/w_f)/ln(L_fw_f/L_ow_o)
Where
L_f = Final length
L_o = Original length
Equation 2 enables you to calculate the plastic strain ratio either manually with a set of calipers or automatically with the use of two extensometers – one to measure the change in axial gauge length and the other to measure the change in width (see the following figure).

If you use the manual approach, it is necessary to measure with calipers the specimen width and the distance between gauge marks before testing. You pull the specimen to a strain less than maximum force (point D in the following figure), unload it, and measure the final width and gauge length.
 
If you use the automatic method, you can pull the specimen until it fractures (see the following figure). This enables you to determine the ultimate strength, yield strength, and elongation in the same pull, which saves time and money. To calculate the plastic strains using the automatic method, you must calculate and subtract the elastic strains from the measured strains.


>> Errors in Determining the Plastic Strain Ratio
If you were to perform an error analysis on Equation 2, you would find that the r value is much more sensitive to errors in width measurement than errors in length measurement. R values that are off by more than 40 percent are not unheard of. Furthermore, the reported values are always greater than the true value. The two primary sources of errors in width strain measurement are caused by:

• Edge curling (the specimen’s edges curl along the length of the specimen as it is pulled)
• Concentrated stresses (the sharp, knifelike edges on the extensometer create highly concentrated stresses that result in increased localized straining at the point of measurement).

Both sources of error result in greater width strains and higher r values
After each test you need to inspect the specimen to determine if it flat. Errors in the r value persist unless you compensate for the curling. Errors associated with sharp knife edges are easily eliminated by installing knife edges with rounded or flat surfaces at the point of contact.

>> Other Points to Consider
For many materials, the r value remains constant over the range of plastic strains up to the maximum force applied to the specimen. For some sheet materials, however, the r value varies with the applied axial strain. For such materials, you should report eh as-tested strain level.
Because rolled sheet metals develop planar anisotropy (characteristics that are directional), sample orientation can be significant to the measurement of the plastic strain ratio. Therefore, you must cut test specimens 0 degrees, 45 degrees, and 90 degrees respective to the rolling direction, and you must report the cut direction with each result.