√Problem 2

ME 5463-001 Fracture Mechanics Exam #1 — David Wagner 2009/10/11 14:16

2. Verify that the general Westergaard function

F(z) = Re hat{hat{Z}}(z) + y (Im hat{Z}(z) + Im hat{Y}(z))

satisfies the biharmonic equation, ∇^4 F(x,y) = ∇^2 ∇^2 F(x,y) = 0.

First, note that the generalized Westergaard formulation requires,

  • Z(z)=Z(x+iy) and Y(z)=Y(x+iy) are analytic over some domain; and
  • Y(z) ∈ ℜ ≡ ImY(z)=ImY(x)=0, when y=0.

Also note that {d hat{hat{Z}}} / {dz} = hat{Z}, {d hat{Z}} / {dz} = Z, {d hat{Y}} / {dz} = Y, and {dF}/{dz} = F prime.

F(x,y) = Re hat{hat{Z}}(x+iy) + y Im hat{Z}(x+iy) + y Im hat{Y}(x+iy).

Partial derivatives with respect to x:

  1. {partial/{partial x}} F = Re hat{Z} + y Im Z + y Im Y.
  2. {{partial^2}/{partial x^2}} F = Re Z + y Im Z prime + y Im Y prime.
  3. {{partial^3}/{partial x^3}} F = Re Z prime + y Im Z prime prime + y Im Y prime prime.
  4. {{partial^4}/{partial x^4}} F = Re Z prime prime + y Im Z prime prime prime + y Im Y prime prime prime.

Partial derivatives with respect to y:

  1. {partial/{partial y}} F = - Im hat{Z} + y Re Z +Im hat{Z} + y Re Y + Im hat{Y} = y Re Z + y Re Y + Im hat{Y}.
  2. {{partial^2}/{partial y^2}} F = - y Im Z prime +Re Z - y Im Y prime + Re Y + Re Y = - y Im Z prime +Re Z - y Im Y prime + 2Re Y.
  3. {{partial^3}/{partial y^3}} = -Im Z prime -y Re Z prime prime - Im Z prime -Im Y prime -y Re Y prime prime -2 Im Y prime = -2 Im Z prime -y Re Z prime prime -y Re Y prime prime -3 Im Y prime.
  4. {{partial^4}/{partial y^4}} = -2 Re Z prime prime - Re Z prime prime +y Im Z prime prime prime - Re Y prime prime + y Im Y prime prime prime -3 Re Y prime prime = -3 Re Z prime prime +y Im Z prime prime prime - 4 Re Y prime prime + y Im Y prime prime prime.

Mixed partials.

  1. {{partial^2}/{partial x^2 partial y}} = -Im Z prime + Im Z prime +y Re Z prime prime + Im Y prime + y Re Y prime prime = y Re Z prime prime + Im Y prime + y Re Y prime prime.
  2. {{partial^2}/{partial x^2 partial y^2}} = Re Z prime prime -y Im Z prime prime prime +Re Y prime prime + Re Y prime prime -y Im Y prime prime prime = Re Z prime prime -y Im Z prime prime prime +2 Re Y prime prime -y Im Y prime prime prime.

∇^4 F(x,y) = ∇^2 ∇^2 F(x,y) = 0 = ({{partial^2}/{partial x^2}} + {{partial^2}/{partial y^2}}) · ({{partial^2}/{partial x^2}} + {{partial^2}/{partial y^2}}) F = {{partial^4 F}/{partial x^4}} +2 {{partial^4 F}/{partial x^2 partial y^2}} +{{partial^4 F}/{partial y^4}} =0.

= Re Z prime prime + y Im Z prime prime prime + y Im Y prime prime prime +2(Re Z prime prime -y Im Z prime prime prime +2 Re Y prime prime -y Im Y prime prime prime) -3 Re Z prime prime +y Im Z prime prime prime - 4 Re Y prime prime + y Im Y prime prime prime = Re Z prime prime + y Im Z prime prime prime + y Im Y prime prime prime +2Re Z prime prime -2y Im Z prime prime prime +4 Re Y prime prime -2y Im Y prime prime prime -3 Re Z prime prime +y Im Z prime prime prime - 4 Re Y prime prime + y Im Y prime prime prime = Re Z prime prime +2 Re Z prime prime -3 Re Z prime prime + 4 Re Y prime prime - 4 Re Y prime prime + y Im Z prime prime prime + y Im Z prime prime prime -2y Im Z prime prime prime + y Im Y prime prime prime -2y Im Y prime prime prime + y Im Y prime prime prime = 0.


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