Introduction to Elasticity/Williams asymptotic solution

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Williams' Asymptotic Solution[edit | edit source]

Ref: M.L. Williams, ASME J. Appl. Mech., v. 19 (1952), 526-528.

The Williams' solution
  • Stress concentration at the notch.
  • Singularity at the sharp corner, i.e, .
  • William's solution involves defining the origin at the corner and expanding the stress field as an asymptotic series in powers of r.
  • If the stresses (and strains) vary with as we approach the point , the strain energy is given by

This integral is bounded only if . Hence, singular stress fields are acceptable only if the exponent on the stress components exceeds .

Stresses near the notch corner[edit | edit source]

  • Use a separated-variable series as in equation (3).
  • Each of the terms satisfies the traction-free BCs on the surface of the notch.
  • Relax the requirement that in equation (3) is an integer. Let .

The stresses are

The BCs are at .Hence,

The BCs are at .Hence,

The above equations will have non-trivial solutions only for certain eigenvalues of , one of which is . Using the symmetries of the equations, we can partition the coefficient matrix.

Eigenvalues of [edit | edit source]

Adding equations (9) and (10),

Subtracting equation (10) from (9),

Adding equations (11) and (12),

Subtracting equation (12) from (11),

Therefore, the two independent sets of equations are


Equations (17) have a non-trivial solution only if

Equations (18) have a non-trivial solution only if

  • From equation (4), acceptable singular stress fields must have .Hence, is not acceptable.
  • The term with the smallest eigenvalue of dominates the solution. Hence, this eigenvalue is what we seek.
  • leads to . Unacceptable.
  • We can find the eigenvalues for general wedge angles using graphical methods.

Special case : [edit | edit source]

In this case, the wedge becomes a crack.In this case,

The lowest eigenvalue is . If we use, this value in equation (17), then the two equations will not be linearly independent and we can express them as one equation with the substitutions

where is a constant. The singular stress field at the crack tip is then

where, is the { Mode I Stress Intensity Factor.}

If we use equations (18) we can get the stresses due to a mode II loading.