1. Show that
(a) ( ) ( ) ( )
Where the quadratic invariant of deviator stresses and principal normal stresses
(b) ( ) ( ) ( )
Where the quadratic invariant of deviator strains and principal normal strains
2. (3.1) A polycrystalline metal has a plastic stress-strain curve that obeys Hollomon's equation, Determine n, knowing that the flow stresses of this material at 2% and 10% plastic deformation (offset) are equal to 175 and 185 MPa, respectively. (n=0.0345)
3. (3.2) You are traveling in an airplane. The engineer who designed it is, casually, on your side. He tells you that the wings were designed using the von Mises criterion. Would you feel safer if he had told you that the Tresca criterion had been used? Why?
4. (3.4) A steel with a yield stress of 300 MPa is tested under a state of stress where and . What are the stresses at which yielding occurs if it is assumed that:
(a) The maximum-normal-stress criterion holds? (300, 150, 0 MPa)
(b) The maximum-shear-stress criterion holds? (300, 150, 0 MPa)
(c) The distortion-energy criterion holds? (346.4, 173.2, 0 MPa)
5. (3.5) Determine the maximum pressure that a cylindrical gas reservoir can withstand, using the three flow criteria. Use the following information:
Material: AISI 304 stainless steel - hot finished and annealed,
Thickness: 25 mm
Diameter: 500 mm
Length: 1 mm
Hint: Determine the longitudinal and circumferential (hoop) stresses by the method of sections. (Tresca: 20.5 MPa, von Mises: 23.67 Mpa)
6. (3.9) The stress-strain curve of a 70-30 brass is described by the equation:
Until the onset of plastic instability.
(a) Find the 0.2% offset yield stress. (68.2 MPa)
(b) Applying Considere's criterion, find the real and engineering stress at the onset of necking. ( )
7. (3.14) What is the strain-rate sensitivity of AISI 1040 steel at a strain of 0.02 and a strain of 0.05 (Obtain your data from Figure 3.12(a)) (0.035, 0.016, the values are approximation)
8. The plastic deformation of a material obeys the following law: ( ) . What is the value of the tensile strength of the material? (15000 psi)
9. Determine the condition for necking in a thin-walled cylinder under internal pressure in terms of the effective strain and the strain-hardening exponent, n. The deformation of the materials obeys the following law: where is the applied true.