1. Chapter 1: Measuring the Strength of Metals
Issues to Address:
• Definitions of stress and strain
• The stress-strain curve
• Elastic deformation
• The elastic-plastic transition
• “True” versus “Engineering” stress and strain
• Necking in a tensile test

2. Chapter 1: Measuring the Strength of Metals
Take-Away Concepts:
• The stress-strain curve is the tool used by engineers to measure strength.
• At low loads, the force applied to a test specimen (e.g., a tensile test specimen) varies linearly with deflection. This is elastic behavior.
• It is convenient to normalize the applied force in a test by the specimen cross sectional area. This defines stress.
• Similarly, it is convenient to normalize the deflection in a test specimen by the original length.

3. Chapter 1: Measuring the Strength of Metals
Take-Away Concepts (Cont.):
• With increasing application of force, a test specimen will begin to deform plastically, which is characterized by a more rapid increase in strain and a less rapid increase in stress.
• The transition point between elastic and plastic behavior is called the yield point.
• Dividing the force by the original cross sectional area gives the engineering strain. Similarly, dividing the deflection by the original length gives the engineering strain.

4. Chapter 1: Measuring the Strength of Metals
Take-Away Concepts (Cont.):
• True stress and strain are computed by dividing the force and deflection by the current area and length.
• The maximum strain reached in a tensile test is defined by a balance between the increasing load due to strain hardening and the decreasing load due to the area reduction.
• Test temperature and strain rate are common test variables.

5. Defining Strength – Recent Tragic Accident
May 4, Four circus performers seriously injured in a “hair-hanging” act when a metal carabiner clip snapped.
Initial reports stated that the carabiner was thought to be able to withstand several times the applied loads.

6. Defining Strength
Consider a simple truss bridge as shown with two vehicles passing over the ravine.
Unfortunately, the bridge has a weight limit lower than the weight of the truck and several of the beams fail.
In this case, strength is defined by the load that the affected beams can withstand.

7. Defining Strength
Actually applying a load to a structure to determine whether it can withstand that load is termed a “proof test” and is often used.
More often, however, a designer analyzes a structure and determines the size and shape of beams required to withstand the anticipated loads based on the reported strength of the materials of construction (e.g., steel).
The materials engineer in turn performs standard mechanical tests to determine the strength of the materials of construction.

8. Mechanical Test
Shown below is a mechanical test machine configured to pull in tension a test specimen.
Not identified is the “load cell” that measures the force to stretch the specimen.
Moving Crosshead
Grip
Tensile Specimen with Extensometer
Data acquisition system and test machine controls
The “Extensometer” measures the change in length of the specimen as it stretches.

9. Mechanical Test Specimens
Typical tensile test specimens from round (left) and flat (right) stock are shown below.
The reduced cross-section at the center of the specimens stretch when an axial force is applied.

10. Test Grips, Specimen, and Extsometer
Extensometer
Test Specimen
Force

11. Mechanical Testing - Measurements
Once a test specimen is in place, the operator begins to apply a force to one end of the specimen and data recording is initiated.
Measurements include:
Force versus time – with force measured with a load cell.
Extension versus time – with extension measured with an extensometer.
Often, the test temperature is measured as well.

12. Tensile Test - Example Mild steel cylindrical test specimen
Diameter (of reduced cross-section) = 6.35 mm
Length (of reduced cross-section portion) = 25.4 mm
Cross-sectional area:

13. Tensile Test - Example
Assume that the operator programs the cross-head to move at a constant velocity of 0.15 mm/min.
The recorded data would look something like this:

14. Tensile Test - Example
Since both the force and the deflection increase linearly with time, it is common to instead plot force versus deflection.
If the operator were to now unload the specimen by instructing the machine to move in the opposite direction,
the force and deflection would decrease toward zero along the same path.

15. Force Varies Linearly with Deflection
A linear dependence of force on deflection is precisely the behavior of a spring.
Interestingly, metals (as well as ceramics
and ionic materials are springs.
The “spring constant”:
From the data in the Load versus Deflection chart:
Note: k for a spring such as this ~ 10 N / m
Materials act as very stiff springs

16. Tensile Test – Normalizing the Measurements to Remove Specimen Geometry
If the diameter of the specimen were instead 12.7 mm, the load to deflect a given distance would increase by 2X.
Similarly, if the gage length (L) of the specimen were increased to 50.8 mm, the measured displacement at a given load would be 2X that measured in the 25.4 mm long specimen.
Dividing the measured force by the initial cross-sectional area Ao and dividing the measured displacement by the initial gage length Lo removes these geometric factors.

17. Definition of Stress
Stress (s) is defined as the Force divided by the cross-sectional area.
For now, we will use the initial cross-sectional area Ao, and
The units of stress are N / m2 or Pascal (Pa). Since the applied forces in N are typically quite high and the cross-sectional areas in m2 are typically quite low, the typical unit of stress is 106 N/m2 or MPa.
A popular engineering unit for stress is lbf / in2 or psi.

18. Definition of Strain
Strain (e) is defined as the deflection divided by the initial gage length of the specimen, Lo.
The units of L and Lo are the same (e.g., mm). Thus, strain is a dimensionless quantity.
With these definitions of stress and strain, specimen size effects have been eliminated.
Instead of plotting load versus deflection, we will plot stress versus strain.

19. Tensile Test – Example For our mild steel tensile test:
The slope of this curve has been shown to relate (by geometric factors) to the spring constant, but in materials, this slope takes on special meaning.
The slope E is defined as the elastic modulus – or the Young’s modulus. In mild steel:

20. Tensile Tests on Different Metals
The example test shown was for a mild steel
test specimen. The curves to the right are the expected results for four other metals.
Immediately evident is that the slope – the elastic modulus varies
with the value being that in tungsten and the lowest value being that in 6061 aluminum.
The elastic modulus is a physical property that varies from metal to metal (and from ceramic to ceramic).

21. Compression Tests
It is often convenient or necessary to perform a test in compression rather than one in tension.
In a compression test, a lubricant is used to minimize friction at the loaded surfaces. F L
Top Platten
Bottom Platten

22. Compression Tests
In a compression test, note that the deflections are negative, and by convention, the force is negative, giving the following stress versus strain curve in our mild steel.
However, engineers often prefer to plot positive values rather than negative value. This becomes important later!

23. The Elastic Plastic Transition
Overstretching a spring will render the spring in a condition where it will not return to the original length or shape when the force is removed.
The same is true in the tensile test.
With continued application of force, the strain begins to deviate from the elastic line.
When force is ultimately released, the stress decreases along the elastic line and a permanent, plastic strain is found at zero force (zero stress).

24. The Elastic Plastic Transition
The point of transition between elastic and plastic behavior is the “yield” point. In turn, the corresponding stress is the “yield stress”.
Definition of this point of transition can be tricky.
One definition is the “proportional limit” or point where the first departure from elastic behavior occurs.
By convention, a useful definition of the yield point is at the strain offset.

25. The Elastic Plastic Transition - Example
Below is a compression test in a 1018 steel cylinder.
Note that the initial behavior follows the elastic line but at a stress of ~ -260 MPa, the curve abruptly deviates.
Reference: G. T. Gray III and S. R. Chen, Los Alamos National Laboratory, LA_CP and LA-CP

26. The Elastic Plastic Transition - Example
This test was actually taken to much higher strains.
And, at an intermediate strain the sample was partially unloaded, then reloaded.
A closer inspection of the region in the vicinity of the unload / reload verifies that these paths follow the elastic curve.

27. True Stress – True Strain
At high strains the stress-strain curve shows a curious increase in (negative) stress with straining. This is a result of our definitions of stress and strain based on the original dimensions.
At constant volume:
Dividing the force by an area that is too small leads to an over predicted stress.

28. True Stress – True Strain
Application of a little calculus enables a “truer” definition of stress and strain.
Define et as the “true” strain.
That is, the true strain is the
natural logarithm of (1 + e).
Up to this chart, the strain used was defined as
This strain is referred to as the “engineering strain” in comparison to the “true strain” defined above.

29. True Stress – True Strain
Also, note that the sign of the strain becomes crucial.
In a tensile test, e is positive, but in a compression test, e is negative.
This highlights the risk of omitting the sign (as is commonplace) when performing compression tests.
True stress: Define st as the true stress
Again, the sign of the strain is important.
s is now referred to as the “engineering stress”

30. True Stress – True Strain
Applying these equations to the 1018 steel compression test gives:
Note the maximum strain now rises to
~ –0.44 and the maximum stress decreases ~ –900 MPa.

31. Instability in a Tensile Test
Even a very ductile metal will ultimately form a neck (reduced cross section) and will fail.
Note the shape of the plastic portion of the 1018 steel stress strain curve.
With continued straining the stress increases (in a negative direction). This is referred to as “hardening”.
With increasing strain, a balance exists between the requirement for an increased force due to hardening and the requirement for a decreased force due to the decreased cross sectional area.

32. Instability in a Tensile Test
Mathematically:
At the point of instability,
dF = 0
From constancy of volume:
At the point of instability.
This is referred to as the Considère Criterion.

33. Instability in a Tensile Test – Example
Tensile test in 304L stainless steel at 160oF
Reference: B. R. Antoun, Sandia National Laboratories, Sandia Report, SAND , November, 2004.
Predicts that the point of instability is ~ et = 0.4
This corresponds roughly to highest strain recorded.

34. Mechanical Testing – Test Variables
The tensile test in 304L stainless steel in the previous chart illustrates one test parameter, which is temperature. This test was performed at 160oF.
Another test variable is strain rate,
The units for strain rate are s-1.
Some testing machines operate at a constant cross head velocity. In this case the engineering strain rate is constant, but the true strain rate is not.
The effect of strain rate on strength is small; thus, the difference between engineering and true strain rates are often of second order importance.