美国麻省理工大学：《结构分析与设计技术》教学资源（讲义）CQ11

Concept Question Which of the following expressions most ac curately and simply represents the comple mentary strain energy density Uc of a linear elastic material? 0 ∈i None of the above Eqn. (1) and Eqn.(4 Eqn、(2) 4. ean DKDU

� � � � ++++ ++++ ++++ ++++ ++++ 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 Uc = σij�ij − σijd�ij (1) 0 � = 1 Uc Cijkl�ij�kl (2) 2 � σij Uc = �ijdσij (3) 0 � �ij Uc = U − σijd�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 o22≠0 due to Poisson’ s effect Eqn.(?? )is correct, 633+0 due to the mposed displacement,∈11≠0 due to pois sons effect and E22=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?=0

The right answer is (4) ++++ . • Eqn. (??) is wrong because σ22 �= 0 due to Poisson’s effect. • Eqn. (??) is correct, �33 �= 0 due to the imposed displacement, �11 �= 0 due to Pois￾son’s effect 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 u2 = 0