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Chapter 3 Principle of Metal Forming Theory

Chapter 3 Principle of Metal Forming Theory. Chapter 3. Chapter 3: Principle of Metal Forming Theory .

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Chapter 3 Principle of Metal Forming Theory

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  1. Chapter 3Principle of Metal Forming Theory Chapter 3: Principle of Metal Forming Theory - IE252

  2. Chapter 3 Chapter 3: Principle of Metal Forming Theory. 3.1 Experimental Stress-Strain Flow Curve.3.2 Nominal and true stresses and strains. 3.3 Volume constancy phenomena in metal forming.3.4 Plastic tensile instability and necking conditions.3.5 Analytical stress-strain flow curves. 3.6 Yielding criteria . 3.7 Plane strain and plane stress conditions in metal forming processes. 3.8 Work and energy method application in metal forming processes. 3.9 Problems Chapter 3: Principle of Metal Forming Theory - IE252

  3. Fracture point Upper yield point Ultimate tensile strength Type I Type II σ σ σ Lower yield point Type III e e e Fig. 3.1 Common experimental engineering stress-strain flow curves for metals. Chapter 3: Principle of Metal Forming Theory. 3.1 Experimental Stress-Strain Flow Curve. • The relationship between stress and strain is obtainable using the experimental uni-axial tension test, at constant temperature and strain rate (loading speed). • The uni-axial compression planer strain and torsion tests can also be used to obtain the same relationship between stress and strain. • This relationship is obtained by continuous loading rather than by cyclic loading used in fatigue test. This type of test is called quasi-static stress-strain flow curve and run under slow strain rate (nearly 10-3 /sec). • The flow curve of most metals takes either of the three forms (Type I, II or III) shown in Fig. 3.1 Chapter 3: Principle of Metal Forming Theory - IE252

  4. Fracture point Upper yield point Ultimate tensile strength Type I Type II σ σ σ Lower yield point Type III e e e Fig. 3.1 Common experimental engineering stress-strain flow curves for metals. Chapter 3: Principle of Metal Forming Theory. 3.1 Experimental Stress-Strain Flow Curve. • Type I is a linear elastic flow curve, type II is an elastic homogeneous plastic flow curve, while type III is a elastic-plastic flow curve,with discontinuous yielding. • Most metallic materials have flow curves of type II; however, a very common metal like carbon steel has a flow curve of type III. • The main difference between the stress-strain curve used in mechanical design and the one used in metal forming is that in the former, the developed stresses result from small elastic strains, while in the later the stresses developed result from elastic and large plastic strains. Furthermore, strain hardening is involved with large strains, which always happened in metal forming processes. Chapter 3: Principle of Metal Forming Theory - IE252

  5. Chapter 3: Principle of Metal Forming Theory. 3.1 Experimental Stress-Strain Flow Curve. • Power law equations are commonly used to model strain-hardening behavior in analytical stress. Fig. 3.2 shows the two commonly used tension test specimens, for metallic materials. Chapter 3: Principle of Metal Forming Theory - IE252

  6. Chapter 3: Principle of Metal Forming Theory. • 3.2 Nominal and true stresses and strains. • Engineering stresses and strains are commonly used for small deformations (elastic deformation), which is commonly used in structural engineering design. • True stresses and true strains are commonly used when large strains are involved like what is happening in metal forming. Chapter 3: Principle of Metal Forming Theory - IE252

  7. Chapter 3: Principle of Metal Forming Theory. • 3.2 Nominal and true stresses and strains. • The true stresses are calculated by dividing the load, P , by the current or instantaneous cross-section area, Ac or A,at the instant of measuring the load, P • True strain prior to necking, is obtained by referring small incremental change in length to the instantaneous length, l. The true strain is calculated as follows Chapter 3: Principle of Metal Forming Theory - IE252 • True strain also called logarithmic strain, incremental strain or natural strain.

  8. Chapter 3: Principle of Metal Forming Theory. • 3.2 Nominal and true stresses and strains. • True strain is given the symbol “ε” while engineering strain is given the symbol of “e”. The relationship between true and engineering strains can be drawn as follows: Chapter 3: Principle of Metal Forming Theory - IE252

  9. Chapter 3: Principle of Metal Forming Theory. • 3.2 Nominal and true stresses and strains. • Main features of engineering and true strain : • True strain has the same numerical value in tension and in compression loadings (with negative sign in compression), which is not the case for engineering strain. • For example : • Consider a test specimen the gauge length of which is elongated from 10 to 20 mm or compressed from 20 to 10 mm, the true and engineering strains for both cases are given as follows: Chapter 3: Principle of Metal Forming Theory - IE252

  10. Chapter 3: Principle of Metal Forming Theory. • 3.2 Nominal and true stresses and strains. • Main features of engineering and true strain : • For example 1: • Consider a test specimen the gauge length of which is elongated from 10 to 20 mm or compressed from 20 to 10 mm, the true and engineering strains for both cases are given as follows: Chapter 3: Principle of Metal Forming Theory - IE252

  11. Chapter 3: Principle of Metal Forming Theory. • 3.2 Nominal and true stresses and strains. • Main features of engineering and true strain : • True strain is additive, if done in successive loading. If a specimen had a gauge length, lo,and was elongated to, l1 ,then to, l2 , the total true strain is given as follows: • While engineering strain is given by Chapter 3: Principle of Metal Forming Theory - IE252

  12. Chapter 3: Principle of Metal Forming Theory. • 3.2 Nominal and true stresses and strains. • Main features of engineering and true strain : • Example 2: For example, consider the following loading condition; True and engineering strains are calculated as follows: Chapter 3: Principle of Metal Forming Theory - IE252

  13. Chapter 3: Principle of Metal Forming Theory. • 3.2 Nominal and true stresses and strains. • Main features of engineering and true strain : • Example 2: For example, consider the following loading condition; True and engineering strains are calculated as follows: Chapter 3: Principle of Metal Forming Theory - IE252

  14. Chapter 3: Principle of Metal Forming Theory. • 3.3 Volume constancy phenomena in metal forming. • Main features of engineering and true strain : • Based on experimental evidence it was found for all metals, the volume of material is constant during plastic deformation, which is not the case for elastic deformation (however this volume is very small so the change could be neglected). This can be expressed as follows: Chapter 3: Principle of Metal Forming Theory - IE252

  15. Chapter 3: Principle of Metal Forming Theory. • 3.3 Volume constancy phenomena in metal forming. • Main features of engineering and true strain : Chapter 3: Principle of Metal Forming Theory - IE252

  16. Chapter 3: Principle of Metal Forming Theory. • 3.3 Volume constancy phenomena in metal forming. • Main features of engineering and true strain : • The relationship between nominal and true stress can be drawn using volume constancy principles and given as follows; Chapter 3: Principle of Metal Forming Theory - IE252

  17. Chapter 3: Principle of Metal Forming Theory. • 3.4 Plastic tensile instability and necking conditions. • The Plastic tensile instability starts after yielding point, just before ultimate load. During this period, the increase in load is associated with increased strain. At ultimate load, the specimen elongated without any increase in load. • At this point the material starts behave unstable. This deformation is called Instability condition, under tension load. At this point necking occurs at the weakest points and the deformation changes from being uniform distribution to local necking. However, the change in load becomes zero Eq 3.11 Chapter 3: Principle of Metal Forming Theory - IE252 • For volume constancy: Eq 3.12 Eq 3.13

  18. Chapter 3: Principle of Metal Forming Theory. • 3.4 Plastic tensile instability and necking conditions. • This result indicates that instability occurs when the slope of the stress-strain flow curve (rate of work hardening) is equal to the magnitude of the existing stress (σu in current case, ultimate tensile strength). • Instability loading condition will lead a necking formation and finally specimen fracture. • In metal forming processes, the maximum deformation can be obtained for ductile metals subjected to tension loading without failure through the strain at the instability (εinst –critical true strain). • The critical true stain, εinst at necking can be obtained graphically by plotting versus (ε) on the stress-strain flow curve as shown Chapter 3: Principle of Metal Forming Theory - IE252

  19. ) Chapter 3: Principle of Metal Forming Theory. • 3.5 Analytical stress-strain flow curves. • Numerical modeling of strain hardening behavior may be achieved using one of the following equations: Eq 3.14 Eq 3.15 Chapter 3: Principle of Metal Forming Theory - IE252 Eq 3.16 Where K refers to the strength coefficient, n; the strain hardening exponent, m and m´; strain rate sensitivity indices, and εo reference strain at which strain rate hardening is negligible [3.2]. Note, the strain rate can be defined as the instantaneous deformation velocity divided by the instantaneous length or height of tested specimen.

  20. ) Chapter 3: Principle of Metal Forming Theory. • 3.5 Analytical stress-strain flow curves. • The most commonly used analytical model is the power law Eq. 3.14, which do not include the strain rate (idealized stress-strain flow curve) and called Lidwik-Hollomon equation Eq 3.14 Eq 3.13 Chapter 3: Principle of Metal Forming Theory - IE252 Using Eq 3.13 and 3.14 Eq 3.17

  21. Chapter 3: Principle of Metal Forming Theory. • 3.5 Analytical stress-strain flow curves. Eq 3.17 • This means that necking occurs when ε = n [3.3]. From Eq. 3.17, it can be concluded that true strain at instability is equal to the strain hardening exponent (n). This means that (n) is a measure of the ability of the metal to undergo plastic deformation without failure. • Table below, shows K and n values for two different metals 1100-O aluminum and 18-8 stainless steel. It is clear from the table that n value for stainless steel is higher than that of aluminum. This also means that stainless steel has more ability for elongation (just before instability) than aluminum has. Chapter 3: Principle of Metal Forming Theory - IE252

  22. Chapter 3: Principle of Metal Forming Theory. • 3.5 Analytical stress-strain flow curves. • Therefore, it is important to be able to predict the stress and strain at the on-set of instability, by adjusting process parameters, to avoid failure. • It is worth noting that in metal forming processes the state of stress or loading conditions are more complex than the tensile loading condition. For example, in tension load testing, yielding takes place when (where σ1 is the stress along loading direction) at which plastic deformation initiated. However, in metal forming processes the deformation takes place under more complex state of stresses i.e. . Hence, a criterion is required to predict the yielding under this complex state of stresses, i.e. ( ). Hence, a criterion is required to predict the yielding under this complex state of stresses. Chapter 3: Principle of Metal Forming Theory - IE252

  23. Chapter 3: Principle of Metal Forming Theory. • 3.5 Analytical stress-strain flow curves. What are the roles of K and n values Chapter 3: Principle of Metal Forming Theory - IE252

  24. Chapter 3: Principle of Metal Forming Theory. • 3.5 Analytical stress-strain flow curves. What are the roles of K and n values Chapter 3: Principle of Metal Forming Theory - IE252

  25. Chapter 3: Principle of Metal Forming Theory. • 3.6 Yielding criteria . • Assumptions: • Metals are homogeneous, continuous and isotropic. • Same yielding strength in tensile and compression loading (i.e. ductile metals). • The volume is constant during plastic deformation, and the sum of the plastic strain increments is zero i.e. ( ). • Strain rate and temperature effects are not considered. • Two yielding criteria are commonly used to predict when yielding starts for complex state of stresses, namely; Von Mises and Tresca criterions. Chapter 3: Principle of Metal Forming Theory - IE252

  26. Chapter 3: Principle of Metal Forming Theory. • 3.6 Yielding criteria . • What are those principal stresses ? • There are nine components of stresses acting on an infinitesimal element. • Definition of Principal stresses: It is always possible to find a set of axes along which the shear stress terms vanish. In this case σ1, σ2, and σ3 are called the principal stresses. Chapter 3: Principle of Metal Forming Theory - IE252

  27. Chapter 3: Principle of Metal Forming Theory. • 3.6 Yielding criteria . Chapter 3: Principle of Metal Forming Theory - IE252

  28. Chapter 3: Principle of Metal Forming Theory. • 3.6 Yielding criteria . Chapter 3: Principle of Metal Forming Theory - IE252

  29. Chapter 3: Principle of Metal Forming Theory. • 3.6 Yielding criteria . Chapter 3: Principle of Metal Forming Theory - IE252

  30. Chapter 3: Principle of Metal Forming Theory. • 3.6 Yielding criteria . Chapter 3: Principle of Metal Forming Theory - IE252

  31. Chapter 3: Principle of Metal Forming Theory. • 3.6 Yielding criteria . Note: The Von-Misses criterion is most commonly used in metal forming processes to predict the initial yielding conditions Chapter 3: Principle of Metal Forming Theory - IE252

  32. Chapter 3: Principle of Metal Forming Theory. • 3.7 Plane strain and plane stress conditions in metal forming processes. Chapter 3: Principle of Metal Forming Theory - IE252 Specimen

  33. Chapter 3: Principle of Metal Forming Theory. • 3.7 Plane strain and plane stress conditions in metal forming processes. Using plane strain condition, Von Misses effective stresses, volume constancy principle, and effective strain equation (as shown below), it is possible to simplify the calculations of the effective strain and stress for different metal forming processes Chapter 3: Principle of Metal Forming Theory - IE252

  34. Chapter 3: Principle of Metal Forming Theory. • 3.7 Plane stress conditions in metal forming processes. Chapter 3: Principle of Metal Forming Theory - IE252

  35. Chapter 3: Principle of Metal Forming Theory. • 3.8 Work and energy method application in metal forming processes. Chapter 3: Principle of Metal Forming Theory - IE252

  36. Chapter 3: Principle of Metal Forming Theory. • 3.8 Work and energy method application in metal forming processes. Chapter 3: Principle of Metal Forming Theory - IE252

  37. Chapter 3: Principle of Metal Forming Theory. • 3.8 Work and energy method application in metal forming processes. Chapter 3: Principle of Metal Forming Theory - IE252

  38. Chapter 3: Principle of Metal Forming Theory. • 3.8 Work and energy method application in metal forming processes. Chapter 3: Principle of Metal Forming Theory - IE252

  39. Chapter 3: Principle of Metal Forming Theory. • 3.8 Work and energy method application in metal forming processes. Chapter 3: Principle of Metal Forming Theory - IE252

  40. Chapter 3: Principle of Metal Forming Theory. • 3.8 Work and energy method application in metal forming processes. Chapter 3: Principle of Metal Forming Theory - IE252

  41. Chapter 3: Principle of Metal Forming Theory. • Example 3.5: • The following data were obtained in a tensile test using circular cross-section specimen having a wire gaug diameter of 15 mm of mild-steel material and wire gaug length of 50 mm;(d0=15mm, lo=50mm). • Plot the experimental stress-strain curve in terms of ; Nominal stress and strain,True stress and strain. • What is the value of K and n ? • The plastic work required to stretch the specimen at maximum load ? Chapter 3: Principle of Metal Forming Theory - IE252

  42. Example 3.5 solution • Using XLS spread sheet, calculate engineering and true stress and strain values Chapter 3: Principle of Metal Forming Theory - IE252

  43. Example 3.5 solution • Two methods used to get the K and n values (1st Approximate,2nd Accurate methods) Chapter 3: Principle of Metal Forming Theory - IE252 Approximate Method

  44. 1 σ1 • Example 3.5 solution • Two methods used to get the K and n values (1st Approximate,2nd Accurate methods) Δσ 2 σ2 Δε ε1 ε2 Linear curve fit 2nd Accurate Method Calcuate n and K n=(logσ1-logσ2)/(log ε1-log ε2) Calculate σ1 and σ2 at strain ε1=0.3 and ε2=0.1 get the log values using linear fit curve. Calcuatestrain hardening value n=0.305. For K value, for strain equal ε=1, get the value of stress σ(at ε=1)=738.2(1)+339.2=1077.4 N/mm2=K Chapter 3: Principle of Metal Forming Theory - IE252

  45. Chapter 3: Principle of Metal Forming Theory - IE252

  46. Chapter 3: Principle of Metal Forming Theory - IE252

  47. Chapter 3: Principle of Metal Forming Theory - IE252

  48. Chapter 3: Principle of Metal Forming Theory - IE252

  49. End Chapter 3 Chapter 3: Principle of Metal Forming Theory - IE252

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