Elasticity/Williams asymptotic solution

Williams' Asymptotic Solution

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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

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  • 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 λ

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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

 

and

 

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 : α = π = 180°

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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.