Most polycrystalline materials have within their elastic range an almost constant relationship between stress and strain. In 1678 an English scientist named **Robert Hooke** ran experiments that provided data that showed that in the elastic range of a material, **strain is proportional to stress**. Robert Hooke concluded that the force F in any spring is proportional to the extension (the deformation from the relaxed state) x as follows:

**F = k · x**

where the term **k** is the** stiffness** of the spring and x is small compared to the total possible deformation of the spring. It must eventually fail once the forces exceed some limit, since no material can be compressed beyond a certain minimum size, or stretched beyond a maximum size, without some permanent deformation or change of state.

In case of tensional stress of a uniform bar (stress-strain curve), the **Hooke’s law** describes behaviour of a bar in the elastic region. 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 a limiting stress, a body will be able to recover its dimensions on removal of the load. The applied stresses cause the atoms in a crystal to move from their equilibrium position. All the atoms are displaced the same amount and still maintain their relative geometry. When the stresses are removed, all the atoms return to their original positions and no permanent deformation occurs. According to the** 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):

The elastic moduli relevant to polycrystalline materials:

**Young’s Modulus of Elasticity.**The Young’s modulus of elasticity is the elastic modulus for tensile and compressive stress in the linear elasticity regime of a uniaxial deformation and is usually assessed by tensile tests.**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.**Bulk Modulus of Elasticity.**The bulk modulus of elasticity is describes volumetric elasticity, or the tendency of an object to deform in all directions when uniformly loaded in all directions. For example, it describes the elastic response to hydrostatic pressure and equilateral tension (like the pressure at the bottom of the ocean or a deep swimming pool). It is also the property of a material that determines the elastic response to the application of stress. For a fluid, only the bulk modulus is meaningful.

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