Symmetry and bifurcation in three-dimensional elasticity. I

DOI10.1007/BF00253119zbMath0509.73018MaRDI QIDQ1838345

D. R. J. Chillingworth, Jerrold E. Marsden, Yieh-Hei Wan

Publication date: 1982

Published in: Archive for Rational Mechanics and Analysis (Search for Journal in Brave)

zbMATH Keywords

geometrybifurcationdeformationsLie group theorytraction boundary value problemthree-dimensional nonlinear elasticityaction of SO(3)natural stateall three are equal and non-zeroastatic load matrixastatic load matrix has three distinct eigenvaluesaxis of equilibriumexactly one is stableexactly two eigenvalues are equal and non-zeroexistence of four solutionsgenerate five casesmore complex behaviourno axis of equilibriumstatic equilibrium loadsStopelli's local existence and uniqueness theoremsubcase (type 0)subcases (types 1 and 2)

Mathematics Subject Classification ID

Nonlinear elasticity (74B20) Bifurcation and buckling (74G60) Dynamical problems in solid mechanics (74H99) Generalities, axiomatics, foundations of continuum mechanics of solids (74A99) Equilibrium (steady-state) problems in solid mechanics (74G99)

Related Items

Variational methods for elastic crystals, Structure of the set of equilibrated loads in nonlinear elasticity and applications to existence and nonexistence, Almost global attitude stabilization of a rigid body for both internal and external actuation schemes, Attitude synchronization of satellites with internal actuation, Resolution of a fixed point problem by an incremental method and application in nonlinear elasticity, Incremental methods in nonlinear, three-dimensional, incompressible elasticity, The Traction Problem For Incompressible Materials, On the hydrodynamics of biaxial nematic liquid crystals. I: General theory, Linearization stability and Signorini Series for the traction problem in elastostatics, Minimization in nonlinear elasticity theory for bodies reinforced with inextensible cords, Heteroclinic cycles in dynamical systems with broken spherical symmetry, Elastostatics in the presence of a temperature distribution or inhomogeneity, Bifurcation and symmetry breaking, Critical points in the energy of hyperelastic materials, Abstract forced symmetry breaking and forced frequency locking of modulated waves, The flip side of buckling, Existence theorems for nonlinear elastic plates with periodic boundary conditions, On the steady motion of a coupled system solid-liquid, Symmetry and bifurcation in three-dimensional elasticity. II, Symmetry and bifurcation in three-dimensional elasticity, Asymptotic behavior of constrained local minimizers in finite elasticity, Quasiconvexity and uniqueness of equilibrium solutions in nonlinear elasticity

Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Sull'esistenza di soluzioni delle equazioni dell'elastostatica isoterma nel caso di sollecitazioni dotate di assi di equilibrio. I-III
• Imperfect bifurcation in the presence of symmetry
• On Signorini's perturbation method in finite elasticity
• Convexity conditions and existence theorems in nonlinear elasticity
• The elastic instability of a director rod
• Stability of \(C^\infty\) mappings. II: Infinitesimal stability implies stability
• Morse theory of certain symmetric spaces
• Stability of critical points under small perturbations. II: Analytic theory
• The Morse Lemma in Infinite Dimensions via Singularity Theory
• Bifurcation near degenerate families
• Almost-Riemannian Structures on Banach Manifolds: The Morse Lemma and the Darboux Theorem
• Inequalities associated with the inversion of elastic stress-deformation relations and their implications
• A theory for imperfect bifurcation via singularity theory