Equations
The following two equations are both considered to be Hooke's law:
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Hooke's law
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Hooke's law for normal stress and normal strain
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Hooke's law for shear stress and shear strain
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The latter has more use in mechanics of materials. Typically, when Hooke's law is mentioned, reference is being made to the latter. The former is commonly associated with the spring constant.
Nomenclature
Details
Most engineering structures are designed to undergo relatively small deformations, involving only the straight-line portion of a corresponding stress-strain diagram. For that initial portion of the diagram, the stress σ is directly proportional to the strain ε.
Since strain is a dimensionless quantity, the modulus of elasticity is expressed in the same units as the stress.
The largest value of the stress for which Hooke's law can be used for a given material is known as the proportional limit of that material.
The equation:
states that the force applied to a material is proportional to the elongation of the material. Also:
states that the stress applied to a material is proportional to its strain. Note that the above equation corresponds to normal stress and normal strain.
For values of shearing stress which do not exceed the proportional limit in shear, for any homogeneous isotropic material:
This relation is known as Hooke's law for shearing stress and strain. Since the strain γxy was defined as an angle in radians, it is dimensionless, and the modulus G is expressed in the same units as τxy, that is, in pascals or in psi. The modulus of rigidity G of any given material is less than one-half, but more than one-third of the modulus of elasticity E of that material.
Considering now a small element of material subjected to shearing stresses τyz and τzy, the shearing strain γyz is defined as the change in the angle formed by the faces under stress. The shearing strain γzx is defined in a similar way by considering the element subjected to shearing stresses τzx and τxz. For values of the stress which do not exceed the proportional limit, the two additional relations are:
and:
This lesson is associated with Physics, Mechanics of Materials, Machine Design