True Stress - True Strain Curve

The engineering stress-strain curve does not give a true indication of the deformation characteristics of a metal because it is based entirely on the original dimensions of the specimen, and these dimensions change continuously during the test. Also, ductile metal which is pulled in tension becomes unstable and necks down during the course of the test. Because the cross-sectional area of the specimen is decreasing rapidly at this stage in the test, the load required continuing deformation falls off. The average stress based on original area likewise decreases, and this produces the fall-off in the stress-strain curve beyond the point of maximum load. Actually, the metal continues to strain-harden all the way up to fracture, so that the stress required to produce further deformation should also increase. If the true stress, based on the actual cross-sectional area of the specimen, is used, it is found that the stress-strain curve increases continuously up to fracture. If the strain measurement is also based on instantaneous measurements, the curve, which is obtained, is known as a true-stress-true-strain curve. This is also known as a flow curve since it represents the basic plastic-flow characteristics of the material. Any point on the flow curve can be considered the yield stress for a metal strained in tension by the amount shown on the curve. Thus, if the load is removed at this point and then reapplied, the material will behave elastically throughout the entire range of reloading.
The true stress s is expressed in terms of engineering stress s by

(1)

The derivation of Eq. (1) assumes both constancy of volume and a homogenous distribution of strain along the gage length of the tension specimen. Thus, Eq. (1) should only be used until the onset of necking. Beyond maximum load the true stress should be determined from actual measurements of load and cross-sectional area.

(2)

The true strain emay be determined from the engineering or conventional strain e by

(3)

This equation is applicable only to the onset of necking for the reasons discussed above. Beyond maximum load the true strain should be based on actual area or diameter measurements.

(4)

Figure 1 compares the true-stress-true-strain curve with its corresponding engineering stress-strain curve. Note that because of the relatively large plastic strains, the elastic region has been compressed into the y-axis. In agreement with Eqs. (1) and (3), the true-stress-true-strain curve is always to the left of the engineering curve until the maximum load is reached. However, beyond maximum load the high-localized strains in the necked region that are used in Eq. (4) far exceed the engineering strain calculated from Eq. (1). Frequently the flow curve is linear from maximum load to fracture, while in other cases its slope continuously decreases up to fracture. The formation of a necked region or mild notch introduces triaxial stresses, which make it difficult to determine accurately the longitudinal tensile stress on out to fracture.
The following parameters usually are determined from the true-stress-true-strain curve.

True Stress at Maximum Load

The true stress at maximum load corresponds to the true tensile strength. For most materials necking begins at maximum load at a value of strain where the true stress equals the slope of the flow curve. Let s_u and e_u denote the true stress and true strain at maximum load when the cross-sectional area of the specimen is Au. The ultimate tensile strength is given by

Eliminating P_max yields

(5)

True Fracture Stress

The true fracture stress is the load at fracture divided by the cross-sectional area at fracture. This stress should be corrected for the, triaxial state of stress existing in the tensile specimen at fracture. Since the data required for this correction are often not available, true-fracture-stress values are frequently in error.

True Fracture Strain

The true fracture strain e_f is the true strain based on the original area A₀ and the area after fracture A_f

(6)

This parameter represents the maximum true strain that the material can withstand before fracture and is analogous to the total strain to fracture of the engineering stress-strain curve. Since Eq. (3) is not valid beyond the onset of necking, it is not possible to calculate e_f from measured values of e_f. However, for cylindrical tensile specimens the reduction of area q is related to the true fracture strain by the relationship

(7)

True Uniform Strain

The true uniform strain e_u is the true strain based only on the strain up to maximum load. It may be calculated from either the specimen cross-sectional area A_u or the gage length L_u at maximum load. Equation (3) may be used to convert conventional uniform strain to true uniform strain. The uniform strain is often useful in estimating the formability of metals from the results of a tension test.

(8)

True Local Necking Strain

The local necking strain e_n is the strain required to deform the specimen from maximum load to fracture.

(9)

The flow curve of many metals in the region of uniform plastic deformation can be expressed by the simple power curve relation

(10)

where n is the strain-hardening exponent and K is the strength coefficient. A log-log plot of true stress and true strain up to maximum load will result in a straight-line if Eq. (10) is satisfied by the data (Fig. 1). The linear slope of this line is n and K is the true stress at e = 1.0 (corresponds to q = 0.63). The strain-hardening exponent may have values from n = 0 (perfectly plastic solid) to n = 1 (elastic solid) (see Fig. 2). For most metals n has values between 0.10 and 0.50 (see Table 1.).
It is important to note that the rate of strain hardening ds /de, is not identical with the strain-hardening exponent.
From the definition of n

or

(11)

Table 1. Values for n and K for metals at room temperature

Metal	Condition	n	K, psi
0,05% C steel	Annealed	0,26	77000
SAE 4340 steel	Annealed	0,15	93000
0,60% C steel	Quenched and tempered 1000oF	0,10	228000
0,60% C steel	Quenched and tempered 1300oF	0,19	178000
Copper	Annealed	0,54	46400
70/30 brass	Annealed	0,49	130000

There is nothing basic about Eq. (10) and deviations from this relationship frequently are observed, often at low strains (10^-3) or high strains (e»1,0). One common type of deviation is for a log-log plot of Eq. (10) to result in two straight lines with different slopes. Sometimes data which do not plot according to Eq. (10) will yield a straight line according to the relationship

(12)

Datsko has shown how e₀, can be considered to be the amount of strain hardening that the material received prior to the tension test.
Another common variation on Eq. (10) is the Ludwig equation

(13)

where s₀ is the yield stress and K and n are the same constants as in Eq. (10). This equation may be more satisfying than Eq. (10) since the latter implies that at zero true strain the stress is zero. Morrison has shown that s₀ can be obtained from the intercept of the strain-hardening point of the stress-strain curve and the elastic modulus line by

The true-stress-true-strain curve of metals such as austenitic stainless steel, which deviate markedly from Eq. (10) at low strains, can be expressed by

where e^K is approximately equal to the proportional limit and n1 is the slope of the deviation of stress from Eq. (10) plotted against e. Still other expressions for the flow curve have been discussed in the literature.
The true strain term in Eqs.(10) to (13) properly should be the plastic strain

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