Materials Characterization: Understanding Forces and Properties in Elastic Behavior

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

4. Elastic Material Properties Unstressed Wire Apply Small Stress Remove Stress and Material Returns to Original Dimensions

5. Inelastic Material Properties Bottle Undergoing Compressive Stress Unstressed Bottle Inelastic Response

6. Compression • Applied stress that squeezes the material • Example: compressive stresses can crush an aluminum can

7. Compression Example Unstressed Sponge Sponge in Compression

8. Compressive Failure • This paper tube was crushed, leaving an accordion-like failure

9. Tension • Applied stress that stretches a material • Example: tensile stresses will cause a rubber band to stretch

10. Tension Example • Steel cables supporting I-Beams are in tension.

11. Tensile Failure • Frayed rope • Most strands already failed • Prior to catastrophic fail

12. Tensile Failure • This magnesium test bar is tensile strained until fracture • Machine characterizes the elastic response • Data verifies manufacturing process control

13. Force Directions • AXIAL: an applied force along the length or axis of a material • TRANSVERSE: an applied force that causes bending or deflection

14. Force Direction Examples Transverse Stress on the Horizontal Aluminum Rod Axial Stress on the Vertical Post

15. 25 20 15 10 Steel Beam Data 5 Linear Regression 0 0 5 10 15 20 Deflection, y (in x 0.01) Graphical Representation • Force vs. Deflection in the elastic region

16. Yield Stress • The stress point where a member cannot take any more loading without failure or large amounts of deformation.

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

18. Ductile Example Unstressed Coat Hangar After Applied Transverse Stress Beyond the Yield Stress Point

19. Brittle Response • Just beyond the yield stress point, the material immediately fails • Example: plastics and wood

20. Brittle Example Unstressed Stick Brittle Failure After Applied Stress Beyond the Yield Stress Point

21. 25 20 15 Ductile Response 10 Brittle Response 5 Failure 0 0 15 30 45 60 Deflection, y Brittle and Ductile Response Graphs

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

23. Moment of Inertia forCommon Cross Sections • Rectangle with height ‘h’ and length ‘b’ • I = (in4 or mm4) • Circle with radius ‘r’ • I = (in4 or mm4) h   bh3 ____  b  12  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

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)

26. 25 20 15 10 Steel Beam Data 5 Linear Regression 0 0 5 10 15 20 Deflection, y (in x 0.01) Calculating the Modulus of Elasticity • Slope = • Measure L • Calculate I • Solve for E 48EI _______ L3 Slope is 1.342 lb/in

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.