Linearized elasticity as Mosco limit of finite elasticity in the presence of cracks

2020 ◽  
Vol 13 (1) ◽  
pp. 33-52
Author(s):  
Pascal Gussmann ◽  
Alexander Mielke
Keyword(s):  
Elastic Energy ◽  
Finite Elasticity ◽  
Small Deformation ◽  
Local Constraint ◽  
Deformation Limit ◽  
Linearized Elasticity ◽  
Global Injectivity

AbstractThe small-deformation limit of finite elasticity is considered in presence of a given crack. The rescaled finite energies with the constraint of global injectivity are shown to Γ-converge to the linearized elastic energy with a local constraint of non-interpenetration along the crack.

Studying novel 1D potential via the AIM

2021 ◽  
pp. 2150141
Author(s):  
A. J. Sous
Keyword(s):  
Iteration Method ◽  
Bound States ◽  
Small Deformation ◽  
Asymptotic Iteration Method ◽  
Finite Spectrum ◽  
Deformation Limit ◽  
Potential Parameters

In this work, we would like to apply the asymptotic iteration method (AIM) to a newly proposed Morse-like deformed potential introduced recently by Assi, Alhaidari and Bahlouli.[Formula: see text] This interesting potential can support bound states and/or resonances. However, in this work, we are only interested in bound states. We considered several choices of the potential parameters and obtained the associated spectrum. Finally, we study the small deformation limit at which this finite spectrum system will transition to infinite spectrum size.

Elastic Energy of Surfaces and Residually Stressed Solids: An Energy Approach for the Mechanics of Nanostructures

2015 ◽  
Vol 82 (1) ◽  
Author(s):  
Xiang Gao ◽  
Daining Fang
Keyword(s):  
Surface Energy ◽  
Elastic Energy ◽  
Surface Stress ◽  
Strain Energy ◽  
Surface Effect ◽  
Surface Elasticity ◽  
Small Deformation ◽  
Energy Approach ◽  
Residual Surface Stress ◽  
The Impact

The surface energy plays a significant role in solids and structures at the small scales, and an explicit expression for surface energy is prerequisite for studying the nanostructures via energy methods. In this study, a general formula for surface energy at finite deformation is constructed, which has simple forms and clearly physical meanings. Next, the strain energy formulas both for isotropic and anisotropic surfaces under small deformation are derived. It is demonstrated that the surface elastic energy is also dependent on the nonlinear Green strain due to the impact of residual surface stress. Then, the strain energy formula for residually stressed elastic solids is given. These results are instrumental to the energy approach for nanomechanics. Finally, the proposed results are applied to investigate the elastic stability and natural frequency of nanowires. A deep analysis of these two examples reveals two length scales characterizing the significance of surface energy. One is the critical length of nanostructures for self-buckling; the other reflects the competition between residual surface stress and surface elasticity, indicating that the surface effect does not always strengthen the stiffness of nanostructures. These results are conducive to shed light on the importance of the residual surface stress and the initial stress in the bulk solids.

scholarly journals Variational linearization of pure traction problems in incompressible elasticity

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.

scholarly journals Sharp conditions for the linearization of finite elasticity

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.

THE DYNAMICAL NOETHER SYMMETRIES OF A BOSONIC q-OSCILLATOR

1999 ◽  
Vol 14 (22) ◽  
pp. 3543-3563
Author(s):  
TIAGO J. M. SIMÕES
Keyword(s):  
Phase Space ◽  
Classical Model ◽  
Small Deformation ◽  
First Integrals ◽  
Noether Symmetries ◽  
Oscillator System ◽  
Dynamical Phase ◽  
Deformation Limit ◽  
Highly Nonlinear ◽  
Dynamical Algebra

The classical dynamical (phase space) Noether symmetries which correspond to the quantum, one-dimensional, bosonic, deformed "Biedenharn–Macfarlane q-oscillator" as defined by V. I. Man'ko and others, are given for small values of the parameter q by considering the model as a nondeformed theory with a highly nonlinear but of small strength interaction. For this nonconstrained one-degree of freedom system and by applying Noether's procedure in the form established by Katzin and Levine for velocity dependent transformations, we found the corresponding two functionally independent phase space first integrals. These classical integrals, as we explicitly prove, lead to a finitely generated infinite Poisson bracket dynamical algebra of first integrals which generalizes a recently obtained Noether dynamical algebra of the nondeformed harmonic oscillator system. We also show that a subalgebra of that infinite dynamical algebra, after quantization of the small-q classical model here proposed, corresponds exactly to the small deformation limit of the deformed quantum spectrum generating algebra su q(1,1) previously obtained for the q-oscillator system, on purely quantum grounds, by Kulish and Damaskinsky.

scholarly journals $$ T\overline{T} $$-deformed fermionic theories revisited

2021 ◽  
Vol 2021 (7) ◽  
Author(s):  
Kyung-Sun Lee ◽  
Piljin Yi ◽  
Junggi Yoon
Keyword(s):  
Degrees Of Freedom ◽  
Small Deformation ◽  
Dirac Bracket ◽  
World Sheet ◽  
Deformation Limit

Abstract We revisit $$ T\overline{T} $$ T T ¯ deformations of d = 2 theories with fermions with a view toward the quantization. As a simple illustration, we compute the deformed Dirac bracket for a Majorana doublet and confirm the known eigenvalue flows perturbatively. We mostly consider those $$ T\overline{T} $$ T T ¯ theories that can be reconstructed from string-like theories upon integrating out the worldsheet metric. After a quick overview of how this works when we add NSR-like or GS-like fermions, we obtain a known non-supersymmetric $$ T\overline{T} $$ T T ¯ deformation of a $$ \mathcal{N} $$ N = (1, 1) theory from the latter, based on the Noether energy-momentum. This world- sheet reconstruction implies that the latter is actually a supersymmetric subsector of a d = 3 GS-like model, implying hidden supercharges, which we do construct explicitly. This brings us to ask about different $$ T\overline{T} $$ T T ¯ deformations, such as manifestly supersymmetric $$ T\overline{T} $$ T T ¯ and also more generally via the symmetric energy-momentum. We show that, for theories with fermions, such choices often lead us to doubling of degrees of freedom, with potential unitarity issues. We show that the extra sector develops a divergent gap in the “small deformation” limit and decouples in the infrared, although it remains uncertain in what sense these can be considered a deformation.

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