? ? ? ? ++++ ++++ ++++ ++++ ++++ Concept Question ++++ ++++ ++++ ++++ ++++ Which of the following expressions most ac- curately and simply represents the comple- mentary strain energy density ? U c of a linear elastic material? ? ? ij U c = σ ij ? ij ? σ ij d? ij (1) 0 ? = 1 C ijkl ? ij ? kl (2)U c 2 ? σ ij U c = ? ij dσ ij (3) 0 ? ? ij U c = U ? σ ij d? ij (4) 0 1. ++++ None of the above 2. ++++ Eqn.(1) and Eqn.(4) 3. ++++ Eqn.(2) 4. ++++ Eqn.(3) 5. ++++ DKDU The right answer is (4) ++++ . ? Eqn. (??) is wrong because σ 22 ?= 0 due to Poisson’s e?ect. ? Eqn. (??) is correct, ? 33 ?= 0 due to the imposed displacement, ? 11 ?= 0 due to Pois- son’s e?ect and ? 22 = 0 because of the plane strain condition. ? Eqn. (??) is wrong, it contradicts Eqn. (??). ? Eqn. (??) is correct, the plane strain condi- tions are actually materialized via u 2 = 0.