The (symmetric) Hessian for geometrically nonlinear models in solid mechanics: Intrinsic definition and geometric interpretation

DOI10.1016/0045-7825(92)90131-3zbMath0761.73016OpenAlexW2033169156MaRDI QIDQ1196496

Juan C. Simo

Publication date: 14 December 1992

Published in: Computer Methods in Applied Mechanics and Engineering (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1016/0045-7825(92)90131-3

zbMATH Keywords

Riemannian metricsymmetrizationfinite elementtangent stiffness matrixrotation groupcovariant derivativebi-invariant metricGateaux derivative

Mathematics Subject Classification ID

Applications of local differential geometry to the sciences (53B50) Generalities, axiomatics, foundations of continuum mechanics of solids (74A99)

Related Items (38)

Rotational self-friction problem of elastic rods ⋮ On the differentiation of the Rodrigues formula and its significance for the vector‐like parameterization of Reissner–Simo beam theory ⋮ Asymptotic post-buckling FEM analysis using corotational formulation ⋮ An updated Lagrangian Bézier finite element formulation for the analysis of slender beams ⋮ Consistent tangent operator for an exact Kirchhoff rod model ⋮ A geometrically exact discrete elastic rod model based on improved discrete curvature ⋮ Constitutive and geometric nonlinear models for the seismic analysis of RC structures with energy dissipators ⋮ On a stress resultant geometrically exact shell model. VII: Shell intersections with 5/6-DOF finite element formulations ⋮ Stress resultant geometrically nonlinear shell theory with drilling rotations. II: Computational aspects ⋮ The Cosserat surface as a shell model, theory and finite element formulation ⋮ A kinematically exact space finite strain beam model -- finite element formulation by generalized virtual work principle ⋮ On finite element implementation of geometrically nonlinear Reissner's beam theory: Three-dimensional curved beam elements ⋮ Nonlinear `master-slave' relationships for joints in 3-D beams with large rotations ⋮ Stability of elastic rods with self-contact ⋮ Shell theory versus degeneration—a comparison in large rotation finite element analysis ⋮ An exact finite rotation shell theory, its mixed variational formulation and its finite element implementation ⋮ On the choice of finite rotation parameters ⋮ On the “Symmetry” of Tangent Operators in Nonlinear Mechanics ⋮ A novel four-field mixed FE approximation for Kirchhoff rods using Cartan's moving frames ⋮ A Fully Nonlinear Beam Model of Bernoulli–Euler Type ⋮ A mixed hybrid finite beam element with an interface to arbitrary three-dimensional material models ⋮ Geometrically exact 3D beam theory: Implementation of a strain-invariant finite element for statics and dynamics ⋮ Non-smooth multisurface plasticity and viscoplasticity. Loading/unloading conditions and numerical algorithms ⋮ Formulations of finite elasticity with independent rotations ⋮ On the simultaneous use of simple geometrically exact shear-rigid rod and shell finite elements ⋮ A non-linear symmetric \(\mathrm{G}^1\)-conforming Bézier finite element formulation for the analysis of Kirchhoff beam assemblies ⋮ A drill rotation formulation for geometrically exact shells ⋮ Sensitivity analysis for dynamic mechanical systems with finite rotations ⋮ Stress recovery in geometrically nonlinear membranes by a Hu-Washizu finite element formulation ⋮ Conservative systems, potential operators and tangent stiffness: Reconsideration and generalization ⋮ A co-rotational formulation for 3D beam element using vectorial rotational variables ⋮ A simple finite element for the geometrically exact analysis of Bernoulli-Euler rods ⋮ A geometrical nonlinear eccentric 3D-beam element with arbitrary cross-sections ⋮ Static analysis of beam structures under nonlinear geometric and constitutive behavior ⋮ A mixed co-rotational formulation of 2D beam element using vectorial rotational variables ⋮ Quadratically convergent direct calculation of critical points for 3d structures undergoing finite rotations ⋮ Techniques based on genetic algorithms for large deflection analysis of beams ⋮ On the role of frame invariance in structural mechanics models at finite rotations

Cites Work

• Unnamed Item
• Unnamed Item
• A finite strain beam formulation. The three-dimensional dynamic problem. I
• A three-dimensional finite-strain rod model. II. Computational aspects
• On the dynamics in space of rods undergoing large motions - A geometrically exact approach
• The Hamiltonian structure of nonlinear elasticity: The material and convective representations of solids, rods, and plates
• Nonlinear finite element analysis of elastic systems under nonconservative loading - natural formulation. Part I. Quasistatic problems
• A sequel to: Nonlinear finite element analysis of elastic systems under nonconservative loading - natural formulation. I. Quasistatic problems
• Manifolds, tensor analysis, and applications.
• Finite element method - The natural approach
• An excursion into large rotations
• A consistent co-rotational formulation for nonlinear, three-dimensional, beam-elements
• Pressure Loaded Structures under Large Deformations
• A beam finite element non-linear theory with finite rotations
• Kirchhoff’s problem for nonlinearly elastic rods
• On One‐Dimensional Large‐Displacement Finite‐Strain Beam Theory
• Riemannian geometry

This page was built for publication: The (symmetric) Hessian for geometrically nonlinear models in solid mechanics: Intrinsic definition and geometric interpretation