Aspects of Plastic Flow

In this chapter we show that k = Oy/2, and use k to relate the hardness to the yield strength of a solid. We then examine tensile instabilities which appear in the drawing of metals and polymers. 

A tensile stress applied to a piece of material will create a shear stress at an angle to the tensile stress. Let us examine the stresses in more detail. Resolving forces in Fig. 11.1 gives the shearing force as 

The area over which this force acts in shear is 

If we plot this against 0 as in Fig. 11.2 we find a maximum t at 0 = 45° to the tensile axis. This means that the highest value of the shear stress is found at 45° to the tensile axis, and has a value of cr/2. 

Example Approximate calculation of the hardness of solids. This concept of shear yielding - where we ignore the details of the grains in our polycrystal and treat the material as a continuum - is useful in many respects. For example, we can use it to calculate the loads that would make our material yield for all sorts of quite complicated geometries. 

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