A measure of basis efficiency at solving the Schrödinger torsion equation. Reaching the variational limit

We analyze the variational limit of one-dimensional next-to-nearest neighbours (NNN) discrete systems as the lattice size tends to zero when the energy densities are of multiwell or Lennard–Jones type. Properly scaling the energies, we study several phenomena as the formation of boundary layers and phase transitions. We also study the presence of local patterns and of anti-phase transitions in the asymptotic behaviour of the ground states of NNN model subject to Dirichlet boundary conditions. We use this information to prove a localization of fracture result in the case of Lennard–Jones type potentials.

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Variational linearization of pure traction problems in incompressible elasticity

Zeitschrift für angewandte Mathematik und Physik ◽

10.1007/s00033-020-01377-7 ◽

2020 ◽

Vol 71 (5) ◽

Author(s):

Edoardo Mainini ◽

Danilo Percivale

Keyword(s):

Finite Elasticity ◽

Linearized Elasticity ◽

Incompressible Elasticity ◽

External Loads ◽

Variational Limit

Abstract We consider pure traction problems, and we show that incompressible linearized elasticity can be obtained as variational limit of incompressible finite elasticity under suitable conditions on external loads.

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QUASI-STATIC EVOLUTIONS IN LINEAR PERFECT PLASTICITY AS A VARIATIONAL LIMIT OF FINITE PLASTICITY: A ONE-DIMENSIONAL CASE

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s0218202513500097 ◽

2013 ◽

Vol 23 (07) ◽

pp. 1275-1308 ◽

Cited By ~ 5

Author(s):

ALESSANDRO GIACOMINI ◽

ALESSANDRO MUSESTI

Keyword(s):

Dimensional Case ◽

Plasticity Model ◽

Perfect Plasticity ◽

One Dimensional ◽

Energetic Approach ◽

Finite Plasticity ◽

Variational Limit

In the framework of the energetic approach to rate-independent evolutions, we show that one-dimensional linear perfect plasticity can be obtained by linearization as a variational limit of a finite plasticity model with hardening proposed by A. Mielke (SIAM J. Math. Anal., 2004).

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Plate theory as the variational limit of the complementary energy functionals of inhomogeneous anisotropic linearly elastic bodies

Mathematics and Mechanics of Solids ◽

10.1177/1081286517707998 ◽

2017 ◽

Vol 23 (8) ◽

pp. 1119-1139

Author(s):

François Murat ◽

Roberto Paroni

Keyword(s):

Dual Problem ◽

Plate Theory ◽

Energy Functionals ◽

Elastic Bodies ◽

Hyper Elastic ◽

Cylindrical Region ◽

Anisotropic Bodies ◽

Γ Convergence ◽

Variational Limit

We consider a sequence of linear hyper-elastic, inhomogeneous and fully anisotropic bodies in a reference configuration occupying a cylindrical region of height [Formula: see text]. We study, by means of Γ-convergence, the asymptotic behavior as [Formula: see text] goes to zero of the sequence of complementary energies. The limit functional is identified as a dual problem for a two-dimensional plate. Our approach gives a direct characterization of the convergence of the equilibrating stress fields.

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Reply of discussion of ‘Study on the action of the active earth pressure by variational limit equilibrium method’

International Journal for Numerical and Analytical Methods in Geomechanics ◽

10.1002/nag.2133 ◽

2012 ◽

Vol 37 (12) ◽

pp. 1911-1912

Author(s):

Xinggao Li ◽

Weining Liu

Keyword(s):

Limit Equilibrium ◽

Earth Pressure ◽

Limit Equilibrium Method ◽

Active Earth Pressure ◽

Equilibrium Method ◽

Variational Limit

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Sharp conditions for the linearization of finite elasticity

Calculus of Variations and Partial Differential Equations ◽

10.1007/s00526-021-02037-y ◽

2021 ◽

Vol 60 (5) ◽

Author(s):

Edoardo Mainini ◽

Danilo Percivale

Keyword(s):

Elastic Energy ◽

Compatibility Condition ◽

Finite Elasticity ◽

Linear Elastic ◽

Approximating Sequence ◽

Standard Linear ◽

Nonlinear Elastic ◽

External Loads ◽

Variational Limit

AbstractWe consider the topic of linearization of finite elasticity for pure traction problems. We characterize the variational limit for the approximating sequence of rescaled nonlinear elastic energies. We show that the limiting minimal value can be strictly lower than the minimal value of the standard linear elastic energy if a strict compatibility condition for external loads does not hold. The results are provided for both the compressible and the incompressible case.

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