2014/02/24 by David Yang Gao, Gao, David Y
Engineering · #35Q74 #49S05 #74B20 #Analysis of PDEs (math.AP) #Composite Structure Analysis and Optimization #Elasticity and Material Modeling #FOS: Mathematics #Numerical methods in engineering #Optimization and Control (math.OC)
paper · pdf · doi:10.48550/arxiv.1402.6025
openalex publication_date 2014/02/24 · openalex created_date 2022/10/06 · openalex updated_date 2026/07/28
This paper presents a pure complementary energy variational method for\nsolving anti-plane shear problem in finite elasticity. Based on the canonical\nduality-triality theory developed by the author, the nonlinear/nonconex partial\ndifferential equation for the large deformation problem is converted into an\nalgebraic equation in dual space, which can, in principle, be solved to obtain\na complete set of stress solutions. Therefore, a general analytical solution\nform of the deformation is obtained subjected to a compatibility condition.\nApplications are illustrated by examples with both convex and nonconvex stored\nstrain energies governed by quadratic-exponential and power-law material\nmodels, respectively. Results show that the nonconvex variational problem could\nhave multiple solutions at each material point, the complementary gap function\nand the triality theory can be used to identify both global and local extremal\nsolutions, while the popular (poly-, quasi-, and rank-one) convexities provide\nonly local minimal criteria, the Legendre-Hadamard condition does not guarantee\nuniqueness of solutions. This paper demonstrates again that the pure\ncomplementary energy principle and the triality theory play important roles in\nfinite deformation theory and nonconvex analysis.\n