Instability conditions in this chapter relate to a maximum force or pressure. However uniform deformation does not generally cease at a maximum force or pressure as it does in a tension test. Reconsider the case of the thin wall sphere under internal pressure.
Although the maximum pressure occurs when ¯ε=(2/3)n, strain localization cannot occur, because localization would cause a decrease of the local radius of curvature, which would decrease the stress in that area below that of the rest of the sphere. Instead the sphere will continue to expand uniformly. The walls are in biaxial tension so the maximum wall force will occur when ¯ε=2n. Strain localization still cannot occur because any localization would decrease the radius of curvature. Figure 4.9shows these points.
Any strain localization under biaxial tension must occur because of local inhomo- geneity. The role of inhomogeneity is developed in Chapter15.
PROBLEMS 49
0.5
0.4
0.3
0 .1 .2 .3 .4 .5
ε
normalized pressure,P 2Kto/ro
Pmax at
ε = (2/3)n
max wall force at ε = 2n
1.0 1.1 1.2 1.3
r/ro
4.9. Variation of pressure with effective strain forn=2.25. Note that the maximum pressure occurs before the maximum wall force.
NOTE OF INTEREST
Professor Zdzislaw Marciniak is a member of the Polish Academy of Sciences and was head of the Faculty of Production Engineering at the Technical University of Warsaw.
For some years he was also acting rector of that university. He has published a number of books (mainly in Polish) on scientific aspects of engineering plasticity and on the technology of metal forming. He developed machines for incremental cold forging including the rocking die and rotary forming processes. His inhomogeneity model of local necking in stretch forming of sheet, published with K. Kuczynski in 1967, has had a profound effect on the understanding of forming limits in sheet metal.
REFERENCES
W. A. Backofen,Deformation Processing, Addison-Wesley, 1972.
W. F. Hosford,Mechanical Behavior of Materials, Cambridge University Press, 2005.
PROBLEMS
4.1. If ¯σ =Kε¯n, the onset of tensile instability occurs when n=εu. Determine the instability strain as a function ofnif
a) σ¯ = A(B+ε¯)n.
b) σ¯ = Aen whereeis the engineering strain.
4.2. Consider a balloon made of a material that shows linear elastic behavior to fracture and has a Poisson’s ratio of 1/2. If the initial diameter is d0, find the diameter,d, at the highest pressure.
4.3. Determine the instability strain in terms ofnfor a material loaded in tension while subjected to a hydrostatic pressureP. Assume ¯σ =Kε¯n.
4.4. A thin-wall tube with closed ends is pressurized internally. Assume that ¯σ = 150¯ε0.25MPa.
a) At what value of effective strain will instability occur with respect to pres- sure?
b) Find the pressure at instability if the tube had an initial diameter of 10 mm, and a wall thickness of 0.5 mm.
4.5. Figure 4.10 shows an aluminum tube fitted over a steel rod. The steel may be considered rigid and the friction between the aluminum and the steel may be neglected. If ¯σ =160¯ε0.25MPa for the tube and it is loaded as indicated, calculate the forceFat instability.
10 cm 1 mm
steel
aluminum
F
4.10.Sketch for Problem 4.5.
4.6. A thin-wall tube with closed ends is subjected to an ever-increasing internal pressure. Find the dimensionsrandtin terms of the original dimensionsroand t0at maximum pressure. Assume ¯σ =500¯ε0.20MPa.
4.7. Consider the internal pressurization of a thin-wall sphere by an ideal gas for whichPV=constant. One may envision an instability condition for which the decrease of pressure with volume, (−dP/dV)gas, due to gas expansion is less than the rate of decrease in pressure that the sphere can withstand, (−dP/dV)sph. For such a condition, catastrophic expansion would occur. If ¯σ =Kε¯n, find ¯ε as a function ofn.
4.8. For rubber stretched under biaxial tensionσx=σy =σ, the stress is given by σ =N kT(λ2−1/λ4) whereλis the stretch ratio,Lx/Lx0= Ly/Ly0. Consider what this equation predicts about how the pressure in a spherical rubber balloon varies during the inflation. For t0 =r0, plotPvs.λand determine the strain λ at which the pressure is a maximum.
4.9. For a material that has a stress–strain relationship of the form ¯σ = A− Bexp(−Cε) whereA, B,andCare constants, find the true strain at the onset of necking and express the tensile strength,Suin terms of the constants.
4.10. A tensile bar was machined with a stepped guage section. The two diameters were 2.0 and 1.9 cm. After some stretching the diameters were found to be 1.893 and 1.698 cm. Findnin the expression ¯σ = Kε¯n and find ¯εas a function ofn.
PROBLEMS 51
4.11. In a rolled sheet, it is not uncommon to find variations of thickness of±1%
from one place to another. Consider a sheet nominally 0.8 mm thick with a
±1% variation of thickness. (Some places are 0.808 mm and others are 0.792 mm thick.) How high wouldnhave to be to ensure that in a tensile specimen every point was strained to at leastε=0.20 before the thinner section necked?
4.12. A material, which undergoes linear strain hardening so that σ =Y +1.35Y, is stretched in tension.
a) At what strain will necking begin?
b) A stepped tensile specimen was made from this material with the diameter of region A being 0.990 times the diameter of region B. What would be the strain in region B when region a reached a strain of 0.20?
5 Temperature and Strain-Rate Dependence
The effects of strain hardening on flow stress were treated in Chapter3. However, flow stress also depends on strain rate and temperature, usually increasing with strain rate and decreasing with temperature. The effect of strain rate at constant temperature will be considered first.