Beltrami Stress Functions
It can be shown that a complete solution to the equilibrium equations may be written as
Using index notation:
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Engineering notation
where is an arbitrary second-rank tensor field that is continuously differentiable at least four times, and is known as the Beltrami stress tensor. Its components are known as Beltrami stress functions. is the Levi-Civita pseudotensor, with all values equal to zero except those in which the indices are not repeated. For a set of non-repeating indices the component value will be +1 for even permutations of the indices, and -1 for odd permutations. And is the Nabla operator
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