Dilatational Components of Stress and Strain

A general state of stress at a point or the stress tensor at a point can be separated into two components, one of which results in a change of shape (deviatoric) and one which results in a change of volume (dilatational). Shape changes due to a pure shear stress such as that of a bar in torsion given in Fig. 2,2 are easy to visualize and are shown by the dashed lines in Fig. 2.16(a) (assuming only a horizontal motion takes place). 

Shear Modulus Because only shear stresses and strains exist for the case of pure shear, the shear modulus can easily be determined from a torsion test by measuring the angle of twist over a prescribed length under a known torque, i.e.. 

Bulk Modulus Volume changes are produced only by normal stresses. For example, consider an element loaded with only normal stresses (principal stresses) as shown in Fig. 2.16(b). The change in volume can be shown to be (for small strains). 

Substituting the values of strains from the generalized Hooke s law, Eq. 2.28, gives. 

If Poisson s ratio is v 0.5, the change in volume is zero or the material is incompressible. Here it is important to note that Poisson s ratio for metals and many other materials in the linear elastic range is approximately 0.33 (i.e., V 1/3). However, near and beyond the yield point, Poisson s ratio is approximately 0.5 (i.e., V 1/2). That is, when materials yield, neck or flow, they do so at constant volume. 

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