My Challenge in the Development of a Mixed Variational Method in Solid Mechanics

Abstract

Approximately 130years ago, Josiah Willard Gibbs, the founder of modern thermodynamics, established the following important theorems assuming existence and positive definiteness of the strain energy of a deformable solid under isothermal or adiabatic temperature conditions (1-2): the uniqueness of the solution in elastostatics and dynamics,the minimum potential energy theorem,the minimum complementary energy theorem.Therefore, any structural design should have been made confirming accuracy of the stiffness evaluation by bracketing it between the upper bound solution based on (ii) and lower bound solution based on (iii). The former method is called the Displacement Method, while the latter the Force Method or Equilibrium Method (hereafter they are abbreviated to DM, FM, and EM respectively). Unfortunately, however, FM fell down quickly in popularity with the appearance of the “Direct Stiffness Method” proposed by Turner and his research group of the Boeing Airplane Co. at that time, including Clough, Martin, and Topp in 1956 (3). Subsequent development of DM and its progress up to the present was incredibly rapid, and its function as a tool to explore frontier science and technology has been duly established. World demand, however, is far beyond the capability of the present NASTRAN computer code, because it can give only the upper bound of a true solution. Therefore, development of a new variational formulation that can create the lower bound solution is strongly anticipated.