Parameswaran and Shukla (2002) recently presented a fracture mechanics study of nonhomogeneous material behavior related to functionally graded materials. They investigated a two-dimensional plane stress problem where Poisson’s ratio remained constant, but Young’s modulus varied as E(x) = E_oe^ax, where E_o and a are constants. For this case, with zero body forces, show that the governing Airy stress function equation is given by:

  

This result may be compared with the more general case given by relation (14.1.6). Note that, when a = 0, this result reduces to the homogeneous form ∇⁴∅ = 0. The nonhomogeneous result is a challenging equation, and its solution was developed for the limited case near the tip of a crack using asymptotic analysis. 

Equation 14.1.6

Transcribed Image Text:

22 v - 20%+ (V) + a\ - a(1 + v); dx 0