Liu Cheng-qun. Note on the Critical Variational State in Elasticity Theory[J]. Applied Mathematics and Mechanics, 1984, 5(6): 895-901.
 Citation: Liu Cheng-qun. Note on the Critical Variational State in Elasticity Theory[J]. Applied Mathematics and Mechanics, 1984, 5(6): 895-901.

# Note on the Critical Variational State in Elasticity Theory

• Publish Date: 1984-12-15
• Recently Prof, Chien Wei-zang[1] pointed out that certain cases, by means of ordinary Lagrange multiplier method, some of undetermined Lagrange multipliers may turn out to be zero during variation.This is a critical state of variation.In this critical state, the corresponding variational constraint can not be eliminated by means of simple Lagrange multiplier method. This is indeed the case when one tries to eliminate the constraint condition of stress-strain relation in variational principle of minimum complementary energy by the method of Lagrange multiplier. By means of Lagrange multiplier method, one can only derive, from minimum complementary energy principle, the Hellinger-Reissner principle[2,3], in which only two types of independent variables, stresses and displacements, exist in the new functional.Hence Prof, Chien Wei-zang introduced the high-order Laaranae multiplier method by addine the quadratic terms
Aifk1(eij-biimnσmn)(eki-bk1pqσpq)
to the original functionals.The purpose of this paper is to show that by adding the quadratic terms
Aifk1(eij-biimnσmn)(eki-1/2uk2-1/2u1:k)
to original functionals one can also eliminate the constraint condition of strain-stress by the high-order Lagrange multiplier method. With this method, we find more general form of generalized variational principle ever known to us from Helliager-Reissner principle, In particular, this more general form of functional can be, reduced into all known functionals of eaisting generalized variational principles in elasticity. Similarly, we can also find snore general form of functional by Hu-Washizu principle[4,5].
•  [1] 钱伟长,高阶拉氏乘子法和弹性理论中更一般的广义变分原理.应用数学和力学,4.2(1983),137-150. [2] Hellinger,E.,Der Allgemeine Ansatz der Meshanik der Kontinua,Encyclopadio der Mothematishen Wissenshaften,4,4(1914).602-694. [3] Reissner,E.,On a variational theorem in elasticity,Journal of Mathematics and Phgsics,yg,2(1950).90-95. [4] 胡海昌,弹塑性理论中的一些变分原理,中国科学,4.1(1955),33-54. [5] Washizu,K.,On the variational principles of elasticity and plasticity,Aeroelastic and Structures Research Laboratory,Massachusettes Institute of Technology,Technical Report 25-18.(1955). [6] 梁国平、傅子智,混合杂交罚函数有限元法及其应用,在大连举行的国际混合杂交元研究讨论班上的报告(1982.8).11-28

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