1. Materials Characterization

2. Learning Objectives Identify compressive and tensile forces
Identify brittle and ductile characteristics
Calculate the moment of inertia
Calculate the modulus of elasticity

3. Elasticity
When a material returns to its original shape after removing a stress
Example: rubber bands
Elasticity describes a material that does not lose its shape after a force is removed. Some materials, such as rubber, have extensive elastic properties and can tolerate a large amount of deformation and still return to its original shape. Other products, such as an aluminum can, may only tolerate a small amount of stress before its shape is permanently altered.

4. Elastic Material Properties
Unstressed Wire
Apply Small Stress
The coat hangar shown above demonstrates the elastic property of returning to it’s pre-stress dimensions after removing the stress. This occurs if the stress isn’t too much. Beyond this ‘yield point’ (defined further in the presentation), the wire does not return to its pre-stress dimensions. You can easily demonstrate this during the lecture using a straightened paper clip.
Remove Stress and
Material Returns to
Original Dimensions

5. Inelastic Material Properties
Have a student crush an aluminum can for a real-time demo.
Bottle Undergoing
Compressive
Stress
Unstressed
Bottle
Inelastic
Response

6. Compression Applied stress that squeezes the material
Example: compressive stresses can crush an aluminum can
Students become more involved if you bring in samples that demonstrate the stress. Bring plastic bottles, aluminum cans, sponges, rags, etc for students to participate with the lecture.
This is a fairly intuitive stress that can be easily identified in everyday structures such as bridges or buildings. You might ask students to name building materials in the room that are under compressive stress.

7. Compression Example
Unstressed Sponge
Sponge in Compression

8. Compressive Failure
This paper tube was crushed, leaving an accordion-like failure
This manila file folder tube was compression stress characterized. A load was continually added until the tube crushed.
Characterizations like this are constantly done inside factories to understand how their material will perform in the field. These measurements are then used by the Civil Engineers so they can design structures that will be able to easily withstand expected forces.

9. Tension Applied stress that stretches a material
Example: tensile stresses will cause a rubber band to stretch
Students become more involved if you bring in samples that demonstrate the stress. Bring rubber bands, balloons, etc.
This is a less intuitive stress for students to grasp.

10. Tension Example Steel cables supporting I-Beams are in tension.
This freeway overpass is under construction. The vertical steel supports and horizontal I-Beams support the concrete based roadway until they can be supported by the concrete pillar seen above. The end I-Beam is held in place by support cables pulling in opposite directions. These steel cables are in tension.

11. Tensile Failure Frayed rope Most strands already failed
Prior to catastrophic fail
This frayed rope is a great example of a many strands having failed a tensile stress and about to have a catastrophic failure.

12. Tensile Failure
This magnesium test bar is tensile strained until fracture
Machine characterizes the elastic response
Data verifies manufacturing process control
Visited an industrial site where they do injection molding of magnesium. The tester clamps onto the fatter ends of the dogbone shaped rod, then slowly pulls the material apart. The tester records the amount of force (stress) and measures the amount of stretching (strain) on the test rod. The stress/strain diagram shows the rod’s elastic region, the yield stress point, and finally to failure. The yield stress point is recorded and used for statistical process control and to record experimental results.

13. Force Directions
AXIAL: an applied force along the length or axis of a material
TRANSVERSE: an applied force that causes bending or deflection
Axial and transverse describe force orientations.
An axial force can be in either compression (a vertical post supporting the center span of a bridge) or in tension (a tuned piano wire pulled taut).
A transverse force is easily visualized as a downward force in the center of a beam supported at each end. During this stress, the beam is experiencing both a compressive and tensile stress. See the next slide for an example.

14. Force Direction Examples
The left hand picture depicts an axial stress on the post is due to the trellis load. The stress is along the length, or axis, of the post. This example is of a compressive axial stress.
The right hand picture shows the effects of a transverse stress (weights in the can) on a length of a horizontal aluminum rod. Note a transverse stress will subject the material to both a tensile and compressive stress. The bottom of the rod is in tension (forces are stretching the outer skin material) while the top of the rod is in compression (forces are pushing skin material together).
Transverse Stress on the Horizontal Aluminum Rod
Axial Stress on the Vertical Post

15. Graphical Representation
Force vs. Deflection in the elastic region 5 10 15 20 25
Deflection, y (in x 0.01)
Steel Beam Data
Linear Regression
Materials in the elastic region are characterized by a linear deflection response to applied load force.
The slope of the line will be used in the lab to determine a material’s modulus of elasticity. This line is characterized by the line equation y=mx+b; where y is the Load, P (lbs); m is the line slope; x is the Deflection, y (inx0.01); and b, the y-intercept, is 0 because there is no deflection with no applied load.
A sharp student will ask why the independent variable is on the Y-Axis. The best answer is because the shape of the line becomes too ‘weird’ when characterized beyond the yield point. When the Load is graphed on the Y-Axis like above, the characterization curve will flatten out before finally breaking. If the Load was graphed on the X-Axis, the curve would plot almost straight up which is harder to grasp.

16. Yield Stress
The stress point where a member cannot take any more loading without failure or large amounts of deformation.
This is the point where the material characteristics leave the linear elastic region. Beyond the yield stress point, a material will either fail catastrophically or deform a large amount with incrementally increased loading.

17. Ductile Response
Beyond the yield stress point, the material responds in a non-linear fashion with lots of deformation with little applied force
Example: metal beams
Metal bends but doesn’t break making it attractive for building material. If stresses are in the metal’s elastic region such as small earthquakes, the building structure will survive with no major damage. If the stress goes beyond the elastic region, as in a major earthquake, the building structure is damaged but it doesn’t fall. Also by becoming permanently changed, it is easy to identify where the structural damage is located.

18. Ductile Example Unstressed Coat Hangar After Applied Transverse
Stress Beyond the Yield
Stress Point
The applied transverse stress exceeded the yield point and caused permanent deformation. Any additional small stress will cause a lot more deformation.

19. Brittle Response
Just beyond the yield stress point, the material immediately fails
Example: plastics and wood
Though these materials have many redeeming values, they fail in a brittle mode. Engineers have found ways to keep utilizing these materials in a safe manner by improving other aspects, such as design improvements like diagonal bracing.

20. Brittle Failure After Applied Stress Beyond the Yield Stress Point
Brittle Example
Unstressed Stick
This wooden beam has a small elastic region but broke when the applied stress was beyond the yield stress point.
Brittle Failure After Applied Stress Beyond the Yield Stress Point

21. Brittle and Ductile Response Graphs5 10 15 20 25 30 45 60
Deflection, y
Ductile Response
Brittle Response
Failure
This graph has a lot of information.
The red squares are the linear load vs. deflection of a brittle material, say a plastic sample. The stress point and the failing point are nearly identical.
The blue triangles represent the load vs. deflection of a ductile material, say a piece of steel. The yield stress point is near 20 pounds. Note by adding a small amount of additional load of four pounds results in a great amount of deflection. Eventually the material reaches its failing point.

22. Moment of Inertia Quantifies the resistance to bending or buckling
Function of the cross-sectional area
Formulas can be found in literature
Units are in length4 (in4 or mm4)
Symbol: I
Following page has equations to calculate the moment of inertia (I) for a materials cross sectional shape. The derivations of these equations requires calculus and is beyond the scope of this lesson. For the lab we recommend using rectangular bar-stock or cylindrical rods.

23. Moment of Inertia for Common Cross Sections
Rectangle with height ‘h’ and length ‘b’
I = (in4 or mm4)
Circle with radius ‘r’ h
 
bh3
____
 b  12
The moment of inertia is a geometric property, based on the second moment of the area above the neutral axis. This quantity is fully understood using calculus, but that is beyond the scope of this experiment. For our lab, we will use the rectangular bar-stock and cylindrical rods shown above.
 2r 
π r4
____ 4

24. Modulus of Elasticity
Quantifies a material’s resistance to deformation
Constant for a material, independent of the material’s shape.
Units are in force / area. (PSI or N/m2)
Symbol: E
Each material has its own characteristic modulus of elasticity. Designers select material with the appropriate E and cost tradeoff.
Modulus of elasticity can be found in literature and will be measured in this week’s lab. The slope of the load/deflection plot is a function of length, moment of inertia, and modulus of elasticity. Use the linear regression function in the Excel program to calculate slope, the length can be measured, the moment of inertia can be calculated from the material’s cross sectional measurements, and then solve for the modulus of elasticity.

25. Flexural Rigidity Quantifies the stiffness of a material
Higher flexural rigidity = stiffer material
Product of the Modulus of Elasticity times the Moment of Inertia (E*I)
This is another parameter for Civil Engineers to design their solution. If a design fails to meet the requirements because the flexural rigidity isn’t high enough, the designer can change the shape of the material (effecting the moment of inertia) resulting in a satisfactory flexural rigidity that does meet the design requirements.

26. Calculating the Modulus of Elasticity
48EI
_______
Slope =
Measure L
Calculate I
Solve for E L3 5 10 15 20 25
Deflection, y (in x 0.01)
Steel Beam Data
Linear Regression
Slope is lb/in
The slope equation is explained in ‘W2 Beam Paper’.
Students can use the spreadsheet to calculate the slope of the linear regression and solve for E. This equation will be examined closely in the lab session.

27. Acknowledgements
Many terms and the laboratory are based a paper titled A Simple Beam Test: Motivating High School Teachers to Develop Pre-Engineering Curricula, by Eric E. Matsumoto, John R. Johnson, Edward E. Dammel, and S.K. Ramesh of California State University, Sacramento.
This paper can be found in the documents directory, titled Material Characterization Paper W2.pdf.