In light of Exercise 2.14 , the compatibility equations (2.6.2) can be expressed as n_ij = ε_ikl ε_jmpe_lp,km = 0, where n_ij is sometimes referred to as the incompatibility tensor (Asaro and Lubarda, 2006). It is observed that hij is symmetric, but its components are not independent from one another. Since the divergence of a curl vanishes, show that they are related through the differential Bianchi relations n_ij,j = 0, which can be expanded to:

Thus, we see again that the six compatibility relations are not all independent.

Data from exercise 2.14

Using relation (1.3.5), show that the compatibility relations (2.6.1) with l = k can be expressed by ŋ_ij = ε_iklε_jmpe_lp,km = 0, which can also be written in vector notation as ∇ × e × ∇ = 0.

Equation 1.3.5

Equation 2.6.1

Equation 2.6.2

Transcribed Image Text:

M11,1 +12,2 + 713,3 = 0 721,1 + 122,2 + 23,3 = 0 131,1 + 132,2+ 133,3 = 0