Problem 3(a) (20 points) Show that, in any solution to a static boundary value problem (no velocity or acceleration) in

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Problem 3 A 20 Points Show That In Any Solution To A Static Boundary Value Problem No Velocity Or Acceleration In 1 (70.84 KiB) Viewed 11 times

Problem 3(a) (20 points) Show that, in any solution to a static boundary value problem (no velocity or acceleration) in an isotropic linear elastic solid, with no body forces present, the first invariant of the stress tensor, given by 1(x) = tro(x) = Okk(x), is a harmonic function of position; that is to say, I satisfies Laplace's equation: 221(x) 2-1(x) 321(x) 0 = ΔΙ1(x) = = (11),mm 022 ar3 2:23 + + Hint: recall the fundamental field equations of isotropic linear elastic boundary value prob- lems : = • strain-displacement relations: 2€(x) = grad u(x) + (grad u(x))", (or 2€ij = Wij + uji); • stress-strain relations: 0(x) = 2ue(x) + Xtre(x) 1, (or Oij = 2€ij + dekkdij); • equilibrium relations: moment balance: 0(x) = o'(x) or (0 ij = 0); – force balance: divo(x) = 0; or (bijj = 0). 0