Asymptotic analysis of a junction of hyperelastic rods

2021 ◽  
pp. 1-20
Author(s):  
Pedro Hernández-Llanos
Keyword(s):  
Sobolev Space ◽  
Asymptotic Analysis ◽  
Total Energy ◽  
Asymptotic Model ◽  
Elastic Strings ◽  
Γ Convergence

In this article we obtain a 1-dimensional asymptotic model for a junction of thin hyperelastic rods as the thickness goes to zero. We show, under appropriate hypotheses on the loads, that the deformations that minimize the total energy weakly converge in a Sobolev space towards the minimum of a 1 D-dimensional energy for elastic strings by using techniques from Γ-convergence.

ON A NONLOCAL FUNCTIONAL ARISING IN THE STUDY OF THIN-FILM BLISTERING

2010 ◽  
Vol 08 (02) ◽  
pp. 109-123
Author(s):  
N. ANSINI ◽  
V. VALENTE
Keyword(s):  
Thin Film ◽  
Asymptotic Analysis ◽  
Circular Plate ◽  
One Dimensional ◽  
Von Karman ◽  
Nonlocal Functional ◽  
Γ Convergence ◽  
Von Kármán

The energy of a Von Kármán circular plate is described by a nonlocal nonconvex one-dimensional functional depending on the thickness ε. Here we perform the asymptotic analysis via Γ-convergence as the parameter ε goes to zero.

Asymptotic Analysis of Size Dependent Fracture Strength Using Boundary Effect Concept

2008 ◽  
Vol 41-42 ◽  
pp. 125-133
Author(s):  
Kai Duan ◽  
Xiao Zhi Hu
Keyword(s):  
Experimental Data ◽  
Fracture Toughness ◽  
Asymptotic Analysis ◽  
Size Effect ◽  
Boundary Effect ◽  
Damage Zone ◽  
Crack Tip ◽  
Characteristic Size ◽  
Asymptotic Model ◽  
Size Dependent

In this paper, the extensively-reported “size effect” phenomena in fracture mechanics tests are explained using the boundary effect concept. It is pointed out that the widely-observed size effect in fracture, including the dependence of the fracture energy on ligament, strength and fracture toughness on crack and/or ligament and the strength of geometrically similar specimens on characteristic size, is in fact, due to the boundary influence on the crack tip damage zone. Furthermore, the recently-developed asymptotic model is used to demonstrate that the dependence of strength on crack and ligament lengths as well as on the characteristic size of geometrically similar specimens is a result of the dominance of the distance of the crack tip to specimen boundaries on the specimen failure mode. To verify further the boundary effect concept, the asymptotic model is also applied to two sets of selected experimental data available in the literature, and the implications are discussed.

A Γ-convergence result for fluid-filled fracture propagation

2020 ◽  
Vol 54 (3) ◽  
pp. 1003-1023
Author(s):  
Annika Bach ◽  
Liesel Sommer
Keyword(s):  
Asymptotic Analysis ◽  
Numerical Simulations ◽  
Discontinuous Galerkin ◽  
Phase Field ◽  
Field Model ◽  
Phase Field Model ◽  
Fracture Propagation ◽  
Convergence Result ◽  
Elastic Media ◽  
Γ Convergence

In this paper we provide a rigorous asymptotic analysis of a phase-field model used to simulate pressure-driven fracture propagation in poro-elastic media. More precisely, assuming a given pressure p ∈ W 1,∞ (Ω) we show that functionals of the form $$ E(\vec{u})={\int }_{\mathrm{\Omega }} e(\vec{u}):\mathbb{C}e(\vec{u})+p\nabla \cdot \vec{u}+\left\langle \nabla p,\vec{u}\right\rangle\enspace \mathrm{d}x+{\mathcal{H}}^{n-1}({J}_{\vec{u}}),\enspace \vec{u}\in \mathrm{G}{SBD}(\mathrm{\Omega })\cap {L}^1(\mathrm{\Omega };{\mathbb{R}}^n) $$ can be approximated in terms of Γ-convergence by a sequence of phase-field functionals, which are suitable for numerical simulations. The Γ-convergence result is complemented by a numerical example where the phase-field model is implemented using a Discontinuous Galerkin Discretization.

ASYMPTOTIC ANALYSIS OF PERIODICALLY-PERFORATED NONLINEAR MEDIA AT THE CRITICAL EXPONENT

2009 ◽  
Vol 11 (06) ◽  
pp. 1009-1033 ◽  
Author(s):  
LAURA SIGALOTTI
Keyword(s):  
Asymptotic Analysis ◽  
Critical Exponent ◽  
Space Dimension ◽  
Convergence Result ◽  
Nonlinear Media ◽  
Perforated Domains ◽  
Extra Term ◽  
Γ Convergence ◽  
Vector Valued ◽  
Critical Scale

We give a Γ-convergence result for vector-valued nonlinear energies defined on periodically perforated domains. We consider integrands with n-growth where n is the space dimension, showing that there exists a critical scale for the perforations such that the Γ-limit is non-trivial. We prove that the limit extra-term is given by a formula of homogenization type, which simplifies in the case of n-homogeneous energy densities.

scholarly journals Asymptotic variational analysis of incompressible elastic strings

2020 ◽  
pp. 1-28
Author(s):  
Dominik Engl ◽  
Carolin Kreisbeck
Keyword(s):  
Cross Sections ◽  
Three Dimensional ◽  
Differential Constraint ◽  
Energy Functionals ◽  
Elastic Strings ◽  
Non Linear ◽  
Applied Forces ◽  
Linear Differential ◽  
Γ Convergence ◽  
External Forces

Starting from three-dimensional non-linear elasticity under the restriction of incompressibility, we derive reduced models to capture the behaviour of strings in response to external forces. Our Γ-convergence analysis of the constrained energy functionals in the limit of shrinking cross-sections gives rise to explicit one-dimensional limit energies. The latter depend on the scaling of the applied forces. The effect of local volume preservation is reflected either in their energy densities through a constrained minimization over the cross-section variables or in the class of admissible deformations. Interestingly, all scaling regimes allow for compression and/or stretching of the string. The main difficulty in the proof of the Γ-limit is to establish recovery sequences that accommodate the non-linear differential constraint imposed by the incompressibility. To this end, we modify classical constructions in the unconstrained case with the help of an inner perturbation argument tailored for 3d-1d dimension reduction problems.

scholarly journals Asymptotic analysis of the lubrication problem with nonstandard boundary conditions for microrotation

Filomat ◽  
2016 ◽  
Vol 30 (8) ◽  
pp. 2233-2247 ◽  
Author(s):  
Igor Pazanin
Keyword(s):  
Boundary Condition ◽  
Film Thickness ◽  
Asymptotic Analysis ◽  
Boundary Conditions ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Type System ◽  
Effective Model ◽  
Asymptotic Model ◽  
Incompressible Micropolar Fluid

The purpose of this paper is to propose a new effective model describing lubrication process with incompressible micropolar fluid. Instead of usual zero Dirichlet boundary condition for the microrotation, we consider more general (and physically justified) type of boundary condition at the fluid-solid interface, linking the velocity and microrotation through a so-called boundary viscosity. Starting from the linearized micropolar equations, we derive the second-order effective model by means of the asymptotic analysis with respect to the film thickness. The resulting equations, in the form of the Brinkman-type system, clearly show the influence of new boundary conditions on the effective flow. We also discuss the rigorous justification of the obtained asymptotic model.

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