Polar decomposition and appropriate strains and stresses for nonlinear structural analyses

Buckling analysis of thin-walled open members—A complementary energy variational principle

“…In buckling problems, this characteristic makes the Jaumann stress tensor a more suitable stress measure to adopt than the second Piola Kirchoff tensor, as it is able to represent the force boundary conditions of the problem. According to Pai and Palazotto [47] and Pai et al [48], the Jaumann stresses can be related to the right extensional strains through the constitutive relations of linear elasticity.…”

Thermal post-bunkling analyses of functionally graded material rod

“…s(x) is the arc length function of the deformed axial line, R(x) is the extension rate of the axial line, and θ(x) is the angle between the tangent line of deflection curve and x-axis. Then based on the large deformation theory for extensible rod [1,2] , the geometrical relationships are obtained as…”

“…The non-linear stability analysis of these structures subjected to mechanical and thermal loads has received intensive attention. Post-buckling behaviors of extensible homogeneous rods subjected to mechanical load were already studied by many researchers, such as Stemple [1] , Pai [2] , Cummadi [3] , Sampaio [4] , Filipich [5] . The thermal post-buckling properties of elastic homogeneous rods were investigated by Li [6,7] , Coffin [8] .…”

An incremental Lagrangian formulation to the analysis of piezoelectric bodies subjected to geometric non‐linearities

“…The energetic conjugacy between the second Piola-Kirchhoff stress tensor and the Green strain tensor is well established in the literature [12], so it is omitted in this work. To verify the energetic conjugacy between the Lagrangian electric measures t 0 D and t 0 , we can use the internal energy of a dielectric body [17] in the deformed configuration…”

“…When considering high-order terms, the measures for stress, strain, electric field and electric displacement must be carefully chosen. The mechanical quantities used in finite problems are well-known [12], but their electrical counterparts are very difficult to find in the literature (see Reference [13] for some results related to this topic). In the present paper, electric Lagrangian measures are derived by analogy with their mechanical counterparts, proving their energy conjugacy and frame invariance.…”