Non-Linear Finite Element Analysis of Isotropic and Composite Shells by a Total Lagrangian Decomposition Scheme

Total-Lagrangian Formulation and Finite-Element Analysis of Highly Flexible Plates and Shells

“…Hereafter, unless otherwise stated, repeated subscript indices imply summations. We note that the curvatures in (18) do not represent real curvatures because the deformed dx 5 dy6 is not along the i 1 (i 2 ) direction because 61 2 0 ( 62 2 06. If 61 2 62 2 0, the curvatures are normalized curvatures because the differentiations in (18) are taken with respect to the undeformed lengths dx and dy, instead of the deformed lengths 51 3 e 1 6 dx and 51 3 e 2 6 dy.…”

“…We note that the curvatures in (18) do not represent real curvatures because the deformed dx 5 dy6 is not along the i 1 (i 2 ) direction because 61 2 0 ( 62 2 06. If 61 2 62 2 0, the curvatures are normalized curvatures because the differentiations in (18) are taken with respect to the undeformed lengths dx and dy, instead of the deformed lengths 51 3 e 1 6 dx and 51 3 e 2 6 dy. When 61 2 62 2 e 1 2 e 2 2 0, k 1 is the bending curvature of the 1 axis with respect to the 2 axis1 k 2 is the bending curvature of the 2 axis with respect to the 51 axis1 k 61 and k 62 are the twisting curvatures of the axes 51 and 2, respectively1 k 4 is the spiral (or drilling) curvature of the 2 axis with respect to the 3 axis1 and k 5 is the spiral curvature of the 1 axis with respect to the 3 axis.…”

“…Greer and Palazotto [18] showed that the total-Lagrangian finite-element shell model of Pai and Palazotto [17] is accurate in predicting large deformations but the computation is very time-consuming because of the use of a layerwise higher-order shear-deformation theory. Since highly flexible 2D structures have very small thickness and hence transverse shear effects are usually small, we simplify the formulation of Pai and Palazotto [17] here by using an energy-consistent first-order shear-deformation theory [19].…”

“…Moreover, i are along three perpendicular directions and hence they are mutually independent. Taking the variations of curvatures defined in (18) and using ( 16), (17), and ( 25) one can show that [11,12,15] 5k 61…”

“…where f i is the force vector acting on the deformed surface of the undeformed area dx m dx n 5i 2 m 2 n6 of an undeformed infinitesimal cube dx 1 dx 2 dx 3 52 dV 0 6. Using the polar decomposition Jaumann stresses can be proved to be [16] [18]. It follows from (37), ( 19)-( 21), ( 23), (24a,b), and ( 26)-( 29) that the variation of the strain vector 67 can be written as…”