Darrell Wallace

1 Stress-Strain Curves:

Stress-strain curves are perhaps the single most widely used material test for metals. The reason for this isthat the many predictions can be made about the behavior of a large piece of metal under various loadingand deformation conditions based solely on the results obtained from a simple tensile test. While this willnot cover the test or evaluation procedures in-depth,

1.1 Where Do Stress-Strain Curves Come From?A stress-strain curve (or when referring to the true stress-strain curve, a “flow-stress” curve) couldtheoretically come from a number of metal deformation processes. However, the most common source ofthis kind of material data is derived from a standard tensile test. The details of such a test will not bediscussed here. However, it should be noted that very detailed standards (well over 100 standards fortensile tests alone) are put forth by the American Society for Testing and Materials (ASTM) regarding theproper method to conduct a tensile test for a given material.

1.2 The Basic Idea of a Tensile TestIn a tensile test, a specimen, such as that shown in Figure 1, is pulled from both ends. Load cells measurethe force applied to the specimen throughout the stroke. Likewise, a device called an extensometermeasures the change in length of the gate region. From this test a graph of Load vs. Elongation (in inches,not %) will be obtained, Figure 2.

Figure 1Typical tensile test specimen

Width(W )0

Leng

th(L

) 0

Extensometer

GageRegion

Review of Selected Materials Topics

Figure 2 Typical output from a tensile test

1.3 Engineering Stress and StrainTo interpret the result of the test for any random part geometry, the load-elongation data (or load-strokedata as it is often called) must be converted to stresses and strains. Given the initial geometry of thespecimen, any point on the load-stroke curve can be converted using the following equations:

Engineering Stress Engineering Strain

σ e A 0e

L 0

A typical engineering stress-strain curve is shown in Figure 3. Key features of this figure are discussedbelow.

Elongation (in)

F L-L0

Figure 3 Typical engineering stress-strain curve with key points labeled

1.4 Important Features of the Engineering Stress-Strain Diagram

1.4.1 Young’s Modulus, EYoung’s Modulus, denoted as E, is the slope of the engineering stress strain curve in the elastic region.The elastic region is the linear portion at the beginning of the curve (marked in the figure as “elasticregion”). E, also called the modulus of elasticity, has units of pressure (psi or Mpa). This is because strainis unitless (in/in).

1.4.2 Yield Strength, YThe yield strength is the engineering stress at which the material begins to undergo permanent plasticdeformation. When a lower stress is applied, the material will deform under load, but will return to itsoriginal geometry when the load is removed. This point is observed as the departure of the stress-straincurve from a perfectly linear relationship. Because this point is difficult to determine accurately, a rulecalled the 0.2% criterion is used.

According to the 0.2% criterion, the yield strength, Y, occurs at the point where the stress-strain curvedeviates from a straight line by 0.2% (0.002 strain). To find this point, a line is drawn parallel to the elasticregion of the curve (slope = E), intersecting the x-axis at a strain of 0.002 (0.2%). This line intersects thestress-strain curve at the yield point. The stress at this point is Y.

1.4.3 Tensile StrengthThe tensile strength of the material, TS is sometimes also referred to as the Ultimate Tensile Strength(UTS). This is the highest stress which the material can undergo before the onset of necking. Necking islocalized deformation which is the first step toward fracture. When necking occurs, the specimen begins todeform locally, hence the observed drop in load and engineering stress toward the end of the curve. Theonset of necking occurs at the highest measured load or engineering stress (just before the curve turnsdownward). The tensile strength, TS, is the engineering stress at this point.

e (%)

Y

TSFracture

slope =E (Young's Modulus)

0.2%

elas

t ic re

gion

EL

1.4.4 % Elongation (At Failure)Used as a measure of ductility, the %Elongation at failure, EL, is the % elongation of the specimen whenfracture occurs. An alternative measurement of ductility is the %Area Reduction, AR. However, the ELmeasurement is often preferred because of the ease of determining the value from a tensile test.

1.5 True Stress and StrainBecause the specimens dimensions change continually throughout the test process, the engineering stress-strain curve does not exactly represent the “true” stresses and strains within the material. (See Figure 4.) Ina true stress-strain curve, the stress can never decrease, thus it is not possible to locate the onset of neckingby looking at a true stress-strain diagram. The true stress-strain diagram conforms to an exponentialequation of the form: σ=Kεn.

1.5.1 True StrainEngineering strain is based from the initial gage length. However, as the material is strained, eachincremental change in length acts over the entire length of the specimen which becomes progressivelylonger as the test continues. In other words, if we look stepwise for a change of length of ∆L, we see thatthe true strain must account for the continuous change in length of the specimen.

L i Li 1

∆ L

L 0 Gage

L 1 L 0 ∆ L

L 2 L 1 ∆ L L 1.2 ∆ L

...L n L 0 ∆ L

We find that the incremental strain is

StrainL n

Ln 1

The true strain is the sum of the incremental strains as ∆L approaches 0. To convert from engineering totrue strain we use the following relation:

ε ln( )1 e

1.5.2 True StressMuch as with strain, the engineering value is based on the assumption of constant cross-sectional area.However, the reality is that as the specimen is elongated volume constancy requires that the area decrease.To take this into account, engineering stress values may be converted to true stress values according to thefollowing relation:

σ=σe(1+e)

1.5.3 Strain Hardening Exponent, nThe exponent in the exponential equation which models the true stress-strain curve is called the strain-hardening exponent. This value determines the rate at which the material hardens. The greater theexponent, the greater the effect of strain on material strength and hardness. It should be noted that thisvalue is equal to the true strain at the onset of necking (found on the engineering stress-strain diagram).

Figure 4 True stress-strain diagram (flow stress)

ε (%)

Kσ ε=K n

n=ε

True Strain at the onset ofnecking. This point must be found from the -e curvethen converted to .

σε

e

Example Calculations:

If you are given:A0=0.10 in 2

L0=2.0 inand:

Point Load (lbf) Elongation (in)A1 3500 0.00230B1 4000 0.00667C1 6500 0.352D1 4600 0.560

From this data, you should be able to fill in the data for the engineering and true stress strain curves usingthe equations discussed.

For the engineering values we find:

Point σe (ksi) e(%)A2 35 0.12%B2 40 0.33%C2 65 17.6%D2 46 28%

For the true values we find:

Point σ (ksi) ε(%)A3 35.04 0.12%B3 40.13 0.33%C3 76.44 16.2%D3 58.88 24.7%

We can also find (by looking at the curves and the calculated data):

Tensile Strength, TS = 65ksi

%Elongation at Failure, EL = 28%

Yield Strength, Y = 40ksi

Elongation (in)

A1A3

B1B3

C1C3

D1

D

e (%)0.2%

A2

B2

C2D2

3

ε%

We can also calculate:

Young’s Modulus, E:E=slope in linear region= (35 ksi / 0.0012)=29.2 x 106 psi

K and n:n:We know that n is the true strain at the onset of necking. From the engineering curve we see thatnecking onsets at point C. The true strain at point c is 0.162. Therefor, n=0.162.

K:K can be found, in this case, by selecting a point on the true stress-strain curve and solving for K

in the exponential equationσ=Kεn

76.44ksi = K (0.162)0.162

K=102.65ksi

2 Time-Temperature Transformation Diagram (TTT)

The TTT diagram is used to determine the structures of a metal that will be present based on the timehistory of its cooling. Below are given 4 examples of cooling curves and the structures which would beexpected in the final material at room temperature:

For the four cooling trajectories shown, A-D, the expected compositions below 200C are:

A: MartensiteB: Martensite + PearliteC: Austenite + PearliteD: Pearlite

3 Phase Diagrams / the Lever RuleAttached is an excerpt from: Van Vlack, L.H., Elements of Materials Science and Engineering,” 1989,Addison Wesley, Reading, Massachusett. This is a better explanation of the “Lever Rule” with an example.