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Section 04_Elastic_Properties

# Section 04_Elastic_Properties - Physics 927 E.Y.Tsymbal...

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Physics 927 E.Y.Tsymbal 1 Section 4: Elastic Properties Elastic constants Elastic properties of solids are determined by interatomic forces acting on atoms when they are displaced from the equilibrium positions. At small deformations these forces are proportional to the displacements of atoms. As an example, consider a 1D solid. A typical binding curve has a minimum at the equilibrium interatomic distance R 0 : Expanding the energy at the minimum in the Taylor series we find: 0 0 2 2 0 0 0 2 ( ) ( ) ( ) ... R R U U U R U R R R R R R = + + + (4.1) At equilibrium 0 0 R U R = , so that 2 0 1 ( ) 2 U R U ku = + , (4.2) where we defined 0 2 2 1 2 R U k R = and 0 u R R = is the displacement of an atom from equilibrium position R 0 . Differentiating Eq.(4.2), U F R = , we obtain force F acting on an atom : F ku = . (4.3) The constant k is an interatomic force constant. Eq.(4.3) represents the simplest expression for the Hooke’s law showing that the force acting on an atom, F , is proportional to the displacement u . This law is valid only for small displacements and characterizes a linear region in which the restoring force is linear with respect to the displacement of atoms. The elastic properties are described by considering a crystal as a homogeneous continuum medium rather than a periodic array of atoms. In a general case the problem is formulated as follows: (i) Apply forces, which are described in terms of stress σ , and determine displacements of atoms which are described in terms of strain ε . (ii) Define elastic constants C relating stress and strain , so that = C . U R R 0 U 0

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Physics 927 E.Y.Tsymbal 2 Example : In 1D case, F ku = , where u is a change in the crystal length under applied force F . We can therefore write F kL u C A A L σ ε  = = =   , (4.4) where A is the area of the cross section, and L is the equilibrium length of the 1D crystal. The stress is defined as the force per unit area and the strain is the dimensionless constant which describes the relative displacement (deformation). In a general case of a 3D crystal the stress and the strain are tensors which are defined as follows.
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Section 04_Elastic_Properties - Physics 927 E.Y.Tsymbal...

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