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Computational materials science

Which is the correct way of thinking when using the finite element method to calculate the elastic stresses in a component with holes or notches?

  1. ABecause rounding errors accumulate as the number of elements increases, the error grows, so elements should be made as coarse as possible, even where stress concentrates
  2. BIn a linear elastic calculation, even the stress at a sharp crack tip converges to a definite finite value if the elements are fine enough, so that value can be compared with the tensile strength
  3. CIt is an approximate method, so elements are refined where stress concentrates and it is checked that the result hardly changes with element size (convergence)
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Answer: C. It is an approximate method, so elements are refined where stress concentrates and it is checked that the result hardly changes with element size (convergence)

In the finite element method, the structure is divided into small elements such as triangles and quadrilaterals, and the displacement within each element is interpolated from the nodal displacements with simple functions. The global stiffness equation [K]{u} = {F} is solved for the nodal displacements, from which strains and stresses are calculated. Because the interpolation is approximate, the basic practice is to refine the elements near holes and notches where the stress changes sharply, and to check that the values stop changing as the mesh is refined. Coarser elements normally give larger errors. On the other hand, in linear elasticity the stress at the tip of a sharp crack is theoretically infinite, so the value keeps growing as the mesh is refined and never converges. Cracks are therefore evaluated not by stress but by the stress intensity factor or the J-integral, with singular elements or similar used at the tip.

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