Engineering Application
A scalar valued tensor function f that depends merely on the three invariants of a symmetric 3×3 tensor is objective, i.e., independent from rotations of the coordinate system. Moreover, every objective tensor function depends only on the tensor's invariants. Thus, objectivity of a tensor function is fulfilled if, and only if, for some function we have
A common application to this is the evaluation of the potential energy as function of the strain tensor, within the framework of linear elasticity. Exhausting the above theorem the free energy of the system reduces to a function of 3 scalar parameters rather than 6. Within linear elasticity the free energy has to be quadratic in the tensor's elements, which eliminates an additional scalar. Thus, for an isotropic material only two independent parameters are needed to describe the elastic properties, known as the Lame coefficients. Consequently, experimental fits and computational efforts may be eased significantly.
Read more about this topic: Invariants Of Tensors
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