On the adjoint constitutive law in nonlinear elastic optimal design

This work is concerned with a structural optimization problem, formulated in quite general terms, involving an elastic nonlinear isotropic three-dimensional continuum. The resolution is approached through a variational technique (Lagrange multiplier method); this technique yields, in addition to the optimality condition, a set of equations which can be interpreted as governing equations of another structural problem, “adjoint” to the given one. The constitutive law of the adjoint problem is studied in detail and the differences between the real and the adjoint constitutive laws are pointed out. In particular, it is shown that the material of the adjoint problem is orthotropic and its principal directions of orthotropy are determined. Finally. the results obtained are specialized to a «compliance» optimization problem for elastic nonlinear plates in bending; the difference between the present case and the already known case where the plate is linearly elastic is discussed.