Lecture 2 Notes Elasticity Theory

Lecture 2 Notes Elasticity Theory - Lecture 2 Notes...

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Lecture 2 Notes: Elasticity Theory How can we connect the atomistic view of oscillating bonds between atoms with elastic response of a material as a whole? What is the relationship between the Young’s modulus and the bond strength? The harmonic analysis described above assumed elastic energy that is proportional to the displacement squared: V u K 2 u 2 The resulting restoring force (acting on the atoms in the lattice) is: F   dV u   Ku du K represents the bond stiffness. In order to compress a solid of cross section A with N bonds within the cross section by the displacement u we need to apply a force: F applied   F restoring N Ku Then the force per unit area or stress is: F A N A Ku If the area occupied by a unit cell is on average a , where a is the equilibrium bond length, then: Na N 2 Ku K a u a E ; E K a , u a Here is the bond displacement per bond length or strain and E is Young’ s modulus or bond stiffness per bond length. This relates (in an oversimplified way) the spring constant (or bond
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