Small deformations. Hookes law

Let a solid body occupy the domain fl C with the smooth boundary T. The solid particle coincides with the point x = xi,X2,xs) G fl. An elastic solid is described by the following functions  

To formulate a model of the solid body, one needs the constitutive law ij ski), the geometrical equation Sij = Sij u) and equilibrium or motion equations. Let / = (/i,/2,/3) be a given function describing an external 

We assume that the formula for Sij u) is provided by the Cauchy law of small deformations 

These are linear equations which give the symmetry of the strain tensor Sij = Sji- In the general case, the strain tensor is nonlinear, 

The constitutive law Jij ski) has a principal meaning for the definition of solid models. The classical Hooke law 

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