Hooke’s law describes the behavior of a bar in the elastic region in the case of tensional stress of a uniform bar (stress-strain curve). In this region, the elongation of the bar is directly proportional to the tensile force and the length of the bar and inversely proportional to the cross-sectional area and the modulus of elasticity. Up to limiting stress, a body will be able to recover its dimensions on the removal of the load. The applied stresses cause the atoms in a crystal to move from their equilibrium position, and all the atoms are displaced the same amount and maintain their relative geometry. When the stresses are removed, all the atoms return to their original positions, and no permanent deformation occurs. According to Hooke’s law, the stress is proportional to the strain (in the elastic region), and the slope is Young’s modulus.
We can extend the same idea of relating stress to strain to shear applications in the linear region, relating shear stress to shear strain to create Hooke’s law for shear stress:
For isotropic materials within the elastic region, you can relate Poisson’s ratio (ν), Young’s modulus of elasticity (E), and the shear modulus of elasticity (G):
Shear Modulus of Elasticity
The shear modulus, or the modulus of rigidity, is derived from the torsion of a cylindrical test piece. It describes the material’s response to shear stress. Its symbol is G. The shear modulus is one of several quantities for measuring the stiffness of materials, and it arises in the generalized Hooke’s law.
Shear stress. Shear stress exists when two parts of material tend to slide across each other in any typical plane of shear upon application of force parallel to that plane. Torsion is a variation of pure shear in which a structural member is twisted. Torsional forces produce a rotational motion about the longitudinal axis of one end of the member relative to the other. Shear stress is also of great importance in nature, intimately related to the downslope movement of earth materials (as in the case of avalanches).- Shear strains. A shear strain results from shear stress, and it is a strain computed from relative displacements that are measured parallel to two reference planes. Shear strains measure the relative parallel movement of one reference plane for another. The symbol for shear strain is usually the lowercase Greek symbol gamma (γ ).
In many situations, the yield strength is used to identify the allowable stress to which a material can be subjected. This criterion is not adequate for components that have to withstand high pressures, such as those used in pressurized water reactors (PWRs). The maximum shear stress theory of failure has been incorporated into the ASME (The American Society of Mechanical Engineers) Boiler and Pressure Vessel Code, Section III, Rules for Construction of Nuclear Pressure Vessels to cover these situations. This theory states that failure of a piping component occurs when the maximum shear stress exceeds the shear stress at the yield point in a tensile test.
