scholarly journals A gradient system with a wiggly energy and relaxed EDP-convergence

2019 ◽  
Vol 25 ◽  
pp. 68 ◽  
Author(s):  
Patrick Dondl ◽  
Thomas Frenzel ◽  
Alexander Mielke
Keyword(s):  
Energy Dissipation ◽  
Gradient System ◽  
Gradient Structure ◽  
Gamma Convergence ◽  
Dissipation Potential ◽  
Kinetic Relation ◽  
Microscopic Scale ◽  
Limit Passage ◽  
Primal And Dual ◽  
Microscopic Energy

For gradient systems depending on a microstructure, it is desirable to derive a macroscopic gradient structure describing the effective behavior of the microscopic scale on the macroscopic evolution. We introduce a notion of evolutionary Gamma-convergence that relates the microscopic energy and the microscopic dissipation potential with their macroscopic limits via Gamma-convergence. This new notion generalizes the concept of EDP-convergence, which was introduced in [26], and is now called relaxed EDP-convergence. Both notions are based on De Giorgi’s energy-dissipation principle (EDP), however the special structure of the dissipation functional in terms of the primal and dual dissipation potential is, in general, not preserved under Gamma-convergence. By using suitable tiltings we study the kinetic relation directly and, thus, are able to derive a unique macroscopic dissipation potential. The wiggly-energy model of Abeyaratne-Chu-James (1996) serves as a prototypical example where this nontrivial limit passage can be fully analyzed.

scholarly journals Exploring families of energy-dissipation landscapes via tilting: three types of EDP convergence

2021 ◽  
Author(s):  
Alexander Mielke ◽  
Alberto Montefusco ◽  
Mark A. Peletier
Keyword(s):  
Energy Dissipation ◽  
Test Problems ◽  
Gradient System ◽  
Gamma Convergence ◽  
Gradient Flows ◽  
Gradient Systems ◽  
New Concepts

AbstractWe introduce two new concepts of convergence of gradient systems $$({\mathbf{Q}}, {{\mathcal {E}}}_\varepsilon ,{{\mathcal {R}}}_\varepsilon )$$ ( Q , E ε , R ε ) to a limiting gradient system $$({\mathbf{Q}},{{\mathcal {E}}}_0,{{\mathcal {R}}}_0)$$ ( Q , E 0 , R 0 ) . These new concepts are called ‘EDP convergence with tilting’ and ‘contact–EDP convergence with tilting.’ Both are based on the energy-dissipation-principle (EDP) formulation of solutions of gradient systems and can be seen as refinements of the Gamma-convergence for gradient flows first introduced by Sandier and Serfaty. The two new concepts are constructed in order to avoid the ‘unnatural’ limiting gradient structures that sometimes arise as limits in EDP convergence. EDP convergence with tilting is a strengthening of EDP convergence by requiring EDP convergence for a full family of ‘tilted’ copies of $$({\mathbf{Q}}, {{\mathcal {E}}}_\varepsilon ,{{\mathcal {R}}}_\varepsilon )$$ ( Q , E ε , R ε ) . It avoids unnatural limiting gradient structures, but many interesting systems are non-convergent according to this concept. Contact–EDP convergence with tilting is a relaxation of EDP convergence with tilting and still avoids unnatural limits but applies to a broader class of sequences $$({\mathbf{Q}}, {{\mathcal {E}}}_\varepsilon ,{{\mathcal {R}}}_\varepsilon )$$ ( Q , E ε , R ε ) . In this paper, we define these concepts, study their properties, and connect them with classical EDP convergence. We illustrate the different concepts on a number of test problems.

scholarly journals Genetic recombination as a generalised gradient flow

2021 ◽  
Author(s):  
Frederic Alberti
Keyword(s):  
Arbitrary Number ◽  
Genetic Recombination ◽  
Reaction Network ◽  
Gradient Flow ◽  
Mass Action ◽  
Gradient System ◽  
Gradient Structure ◽  
Law Of Mass Action ◽  
Reversible Chemical Reaction ◽  
Time Partitioning

AbstractIt is well known that the classical recombination equation for two parent individuals is equivalent to the law of mass action of a strongly reversible chemical reaction network, and can thus be reformulated as a generalised gradient system. Here, this is generalised to the case of an arbitrary number of parents. Furthermore, the gradient structure of the backward-time partitioning process is investigated.

A Theory of Elastic, Plastic, and Creep Deformations of an Initially Isotropic Material Showing Anisotropic Strain-Hardening, Creep Recovery, and Secondary Creep

1958 ◽  
Vol 25 (4) ◽  
pp. 529-536
Author(s):  
J. F. Besseling
Keyword(s):  
Plastic Deformation ◽  
Energy Dissipation ◽  
Strain Hardening ◽  
Isotropic Material ◽  
Creep Recovery ◽  
Secondary Creep ◽  
Stress Strain ◽  
Microscopic Scale ◽  
Anisotropic Strain ◽  
Creep Deformations

Abstract Stress-strain relations are given for an initially isotropic material, which is macroscopically homogeneous, but inhomogeneous on a microscopic scale. An element of volume is considered to be composed of various portions, which can be represented by subelements showing secondary creep and isotropic work-hardening in plastic deformation. If the condition is imposed that all subelements of an element of volume are subjected to the same total strain, it is demonstrated that the inelastic stress-strain relations of the material show anisotropic strain-hardening, creep recovery, and primary and secondary creep due to the nonuniform energy dissipation in deformation of the sub-elements. Only quasi-static deformations under isothermal conditions are considered. The theory is restricted to small total strains.

scholarly journals Mass transport in Fokker–Planck equations with tilted periodic potential

2019 ◽  
Vol 31 (4) ◽  
pp. 709-736
Author(s):  
MICHAEL HERRMANN ◽  
BARBARA NIETHAMMER
Keyword(s):  
Energy Dissipation ◽  
Periodic Potential ◽  
Temporal Dynamics ◽  
Approximation Error ◽  
Gradient Structure ◽  
Subcritical Regime ◽  
Spatio Temporal ◽  
Fokker Planck Equations ◽  
Double Well Potential ◽  
Fokker Planck

We consider Fokker–Planck equations with tilted periodic potential in the subcritical regime and characterise the spatio-temporal dynamics of the partial masses in the limit of vanishing diffusion. Our convergence proof relies on suitably defined substitute masses and bounds the approximation error using the energy-dissipation relation of the underlying Wasserstein gradient structure. In the appendix, we also discuss the case of an asymmetric double-well potential and derive the corresponding limit dynamics in an elementary way.

Coefficients of Restitution in Normal Adhesive Impact between Smooth and Rough Elastic Bodies

2020 ◽  
Vol 1 (1) ◽  
pp. 103-109
Author(s):  
Valentin L. Popov ◽  
Keyword(s):  
Energy Dissipation ◽  
Rough Surface ◽  
Mechanical Energy ◽  
Rolling Friction ◽  
Adhesive Contact ◽  
Dissipated Energy ◽  
Microscopic Scale ◽  
Elastic Bodies ◽  
Rear Part ◽  
The Moment

n 1975, Fuller and Tabor have shown that roughness can destroy macroscopic adhesion. This means that in spite of the presence of adhesion at the microscopic scale, the macrosopic force of adhesion vanishes. The mechanism of vanishing macroscopic adhesion is very simple: during approach of elastic bodies, asperities are elastically deformed so strongly that after unloading they destroy the microscopic adhesive junctions. However, both in the moment of formation of microscopic adhesive junctions in the loading phase and their destruction during unloading, mechanical energy disappears. This means that the microscopic adhesion makes the contact dissipative even if there is no macroscopic force of adhesion. In particular, the force-distance dependency during indentation and pull-off do not coincide with each other showing some "adhesive hysteresis". When a ball rolls on such rough surface, there will be a final energy dissipation due to formation of a new contact at the frontline of the contact and its destruction at the rear part. Thus, microscopic adhesion will lead to appearance of rolling friction in an apparently non-adhesive contact. In the present paper, we calculate the approach and pull-off dependencies of force on distance, the dissipated energy in one loading-unloading cycle and estimate the force of rolling friction due to microscopic adhesion.

Fission with microscopic energy dissipation

1980 ◽  
Vol 297 (4) ◽  
pp. 289-294 ◽  
Author(s):  
Gerhard Sch�tte ◽  
Peter M�ller ◽  
J. Rayford Nix ◽  
Arnold J. Sierk
Keyword(s):  
Energy Dissipation ◽  
Microscopic Energy

scholarly journals Reassessing the Plastic Hinge Model for Energy Dissipation of Axially Loaded Columns

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
R. M. Korol ◽  
K. S. Sivakumaran
Keyword(s):  
Energy Dissipation ◽  
Plastic Hinge ◽  
Local Buckling ◽  
Energy Estimates ◽  
Experimental Program ◽  
Dissipation Potential ◽  
Column Collapse ◽  
Axially Loaded ◽  
Hinge Model ◽  
Overall Buckling

This paper investigates the energy dissipation potential of axially loaded columns and evaluates the use of a plastic hinge model for analysis of hi-rise building column collapse under extreme loading conditions. The experimental program considered seven axially loaded H-shaped extruded aluminum structural section columns having slenderness ratios that would be typical of floor-to-ceiling heights in buildings. All seven test specimens initially experienced minor-axis overall buckling followed by formation of a plastic hinge at the mid-height region, leading to local buckling of the flanges on the compression side of the plastic hinge, and eventual folding of the compression flanges. The experimental energy absorption, based on load-displacement relations, was compared to the energy estimates based on section plastic moment resistance based on measured yield stress and based on measured hinge rotations. It was found that the theoretical plastic hinge model underestimates a column’s actual ability to absorb energy by a factor in the range of 3 to 4 below that obtained from tests. It was also noted that the realizable hinge rotation is less than 180°. The above observations are based, of course, on actual columns being able to sustain high tensile strains at hinge locations without fracturing.

scholarly journals Adhesive Hysteresis and Rolling Friction in Rough "Non-Adhesive" Contacts

2020 ◽  
Author(s):  
Valentin L. Popov
Keyword(s):  
Energy Dissipation ◽  
Mechanical Energy ◽  
Rolling Friction ◽  
Adhesive Contact ◽  
Dissipated Energy ◽  
Microscopic Scale ◽  
Elastic Bodies ◽  
Rear Part ◽  
The Moment ◽  
Adhesive Contacts

In 1975, Fuller and Tabor have shown that roughness can destroy macroscopic adhesion. This means that in spite of the presence of adhesion at the microscopic scale, the macrosopic force of adhesion vanishes. The mechanism of vanishing macroscopic adhesion is very simple: during approach of elastic bodies, asperities are elastically deformed so strongly that after unloading they destroy the microscopic adhesive junctions. However, both in the moment of formation of microscopic adhesive junctions in the loading phase and their destruction during unloading, mechanical energy disappears. This means that the microscopic adhesion makes the contact dissipative even if there is no macroscopic force of adhesion. In particular, the force-distance dependency during indentation and pull-off do not coincide with each other showing some "adhesive hysteresis". When a ball rolls on such rough surface, there will be a final energy dissipation due to formation of a new contact at the frontline of the contact and its destruction at the rear part. Thus, microscopic adhesion will lead to appearance of rolling friction in an apparently non-adhesive contact. In the present paper, we calculate the approach and pull-off dependencies of force on distance, the dissipated energy in one loading-unloading cycle and estimate the force of rolling friction due to microscopic adhesion.

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