Plasticity and Superplasticity – Theory and Applications

Abstract

This chapter first gives an outline of plasticity using both design and manufacturing approach. Plastic deformation in ductile materials is explained by discussing stress-strain curve for mild steel. The main topics include: plastic instability and the mathematical analysis of the condition of plastic instability; the Bauschinger effect; bending of beam, its applications, and engineering analysis; the application of plasticity to sheet metal forming; hydrostatic stress, and deviatoric stresses; superplasticity and its applications; and the mathematical analysis to calculate the strain rate sensitivity index for superplastic forming (SPF). This chapter contains 11 diagrams/figures, 23 worked examples, 41 formulae, 6 MCQs, and 7 problems.

Questions and Problems

• 9.1. What is the importance of plasticity to a: (a) designer, (b) material processing engineer?

• 9.2. Draw a stress-strain curve for mild steel, and explain the plasticity region of the curve.

• 9.3. Why does Bauschinger effect have an adverse affect on the mechanical behavior of engineering components (e.g. automotive suspension springs)?

• 9.4. State the condition of plastic, mathematically prove it.

• 9.5. Define the term “beam”. Give three real-life examples of bending of beams.

• 9.6. Prove that:$\mathrm{Linear}\ \mathrm{strain}=-\frac{deflection}{radius\kern0.17em of\kern0.17em curvature}$

• 9.7. Derive an expression to calculate the strain rate sensitivity index for SPF.

• 9.8. In a plane-stress sheet-metal forming process, the principal strain in direction 1 is 0.21 and the strain ratio is 0.58. Calculate the principal strains in directions 2 and 3.

• 9.9. In symmetrical bending of a beam, the neutral plane IG is deflected 1.2 mm in y-direction to the length ST (Fig. 9.5). The central angle is 3° and the radius of curvature is 35 mm. The length $\overline{IG}$ is 2.6 mm. Calculate the: (a) length $\overline{ST}$, (b) change in the length ST due to deflection of the beam, (c) linear strain in the element ST.

• 9.10. A simply supported 7-mlong beam AB with h = 300 mm is acted upon by bending moments M_o, as shown in Fig. 9.11. The bending moment cause bending of beam into an arc with strain, ε_x =  − 0.034. (a) Is the beam bent above or below the neutral axis? (b) Calculate the curvature and the radius of curvature.

• 9.11. A square element 10mm × 10mm in an un-deformed metal sheet of 1.2-mm-thickness becomes a rectangle, 9mm × 13mm after forming. Assume that the stress normal to the sheet is zero. Calculate the: (a) final thickness of the sheet after the sheet forming process, (b) principal strains in the forming process, (c) strain ratio.

• 9.12. The principal normal stresses in directions 1, 2, and 3 are 270 MPa (tensile), 200 MPa (compr.), and 170 MPa (tens.). Calculate the hydrostatic stress and the deviatoric stresses.

• 9.13. A series of plasticity experiments were conducted on the MgAZ31 alloy at various stresses and strain rates at a temperature of 425 °C. Two of the data points on the graphical plot are as follows. A true stress of 10 MPa was applied at strain rate of 0.001 s ⁻¹ while a true stress of 2 MPa was applied at a strain rate of 0.0001 s ⁻¹. Calculate the strain rate sensitivity index (m) for the deformation process. Is your computed m value lies in the range for SPF?

• 9.14. A square element 12mm × 12mm in an un-deformed metal sheet of 0.9-mm-thickness becomes a rectangle, 8mm × 15mm after forming. Assume that the stress normal to the sheet is zero. The material obeys the following stress-strain law:

$$ \overline{\sigma}=600{\left(0.008+\overline{\varepsilon}\;\right)}^{0.22}\mathrm{MPa}. $$

Calculate the: (a) effective strain, (b) effective stress, (c) membrane stresses, (d) hydrostatic stress, and (e) deviatoric stresses.

• 9.15. (MCQs). Encircle the most appropriate answers for the following statements.

  • (a) Which phenomenon refers to the lowering of yield stress during reverse loading?

    • (i) plastic instability, (ii) Bauschinger effect, (iii) superplasticity, (iv) bending.

  • (b) Which phenomenon refers to the growth of “neck” of material under uniaxial loading?

    • (i) bending, (ii) Bauschinger effect, (iii) superplasticity, (iv) plastic instability.

  • (c) Which phenomenon refers to the extra-ordinary high strain at high temperatures?

    • (i) plastic instability, (ii) Bauschinger effect, (iii) superplasticity, (iv) bending.

  • (d) Which phenomenon involves the curvature and the radius of curvature?

    • (i) bending, (ii) Bauschinger effect, (iii) superplasticity, (iv) plastic instability.

  • (e) In which condition, the true stress becomes equal to the ultimate tensile strength?

    • (i) plastic instability, (ii) Bauschinger effect, (iii) superplasticity, (iv) bending.

  • (f) Which stress refers to the average of the principal normal stresses?

    • (i) membrane stress, (ii) deviatoric stress, (iii) hydrostatic stress, (iv) normal stress.