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Elasticity is the ability of an object to withstand the effects of distortion and to return to its original size and shape once the forces causing deformation are removed. When an elastic material deforms under the action of an external force, it experiences internal resistance to the deformation. However, if no external force is applied, it returns to its original state.
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Upon subjecting concrete to moderate or high uniaxial compressive or tensile stresses, the strain response is non-linear relative to the stress applied. As the stress is removed, the resulting stress-strain curve deviates from the original path traced during loading, creating a hysteresis loop, indicative of the concrete's non-linear and non-elastic properties. Typically, a material's modulus of elasticity, which is a measure of the material's stiffness, is inferred from the linear...
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Strain energy quantifies the energy stored within a material due to deformation under loading conditions, a fundamental concept in materials science and engineering. The strain energy can be modeled when a material is subjected to axial loading with uniformly distributed stress. In this scenario, the stress experienced by the material is the internal force divided by the cross-sectional area, and the strain induced is directly proportional to this stress through the modulus of elasticity.
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Deformation occurs in axial and transverse directions when an axial load is applied to a slender bar. This deformation impacts the cubic element within the bar, transforming it into either a rectangular parallelepiped or a rhombus, contingent on its orientation. This transformation process induces shearing strain. Axial loading elicits both shearing and normal strains. Applying an axial load instigates equal normal and shearing stresses on elements oriented at a 45° angle to the load axis.
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As discussed in previous lessons, strain energy in a material is the energy stored when it is elastically deformed, a concept crucial in materials science and mechanical engineering. This energy results from the internal work done against the cohesive forces within the material. When a material undergoes shearing stress and corresponding shearing strain, the strain energy density, which is the energy stored per unit volume, is calculated. Within the elastic limit, where the stress is...
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The Mechanics of Poro-Elastic Contractile Actomyosin Networks As a Model System of the Cell Cytoskeleton
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Degenerate Elastic Networks.

Giacomo Del Nin1, Alessandra Pluda2, Marco Pozzetta2

  • 1Mathematics Institute, University of Warwick, Zeeman Building, Coventry, CV4 7HP UK.

Journal of Geometric Analysis
|November 1, 2021
PubMed
Summary
This summary is machine-generated.

This study introduces degenerate elastic networks to analyze curve networks by minimizing length and curvature energy. A new characterization is provided for relaxed problems, simplifying analysis with a combinatorial definition.

Keywords:
Elastic energyNetworksRelaxationSingular structures

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Area of Science:

  • Geometric measure theory
  • Calculus of variations
  • Differential geometry

Background:

  • Minimizing geometric functionals like length and curvature is crucial in various scientific fields.
  • Analyzing networks with fixed topological properties presents challenges due to lack of compactness.
  • Existing methods struggle with the limits of sequences of networks with bounded energy.

Purpose of the Study:

  • To minimize a linear combination of length and curvature-norm for networks within a specified class.
  • To characterize the set of limits for energy-bounded network sequences.
  • To introduce and analyze the concept of degenerate elastic networks.

Main Methods:

  • Minimization of a combined length and curvature energy functional.
  • Characterization of relaxed problems using the concept of degenerate elastic networks.
  • Development of a combinatorial definition for degenerate elastic networks in the 2D case.

Main Results:

  • An explicit representation of the relaxed problem is provided.
  • Degenerate elastic networks are defined, surprisingly independent of curvature.
  • A combinatorial definition for 2D degenerate elastic networks is established, verifiable by a finite algorithm.

Conclusions:

  • The study offers a novel approach to analyzing complex network structures.
  • The concept of degenerate elastic networks simplifies the understanding of relaxed variational problems.
  • The findings provide sharp characterizations and efficient validation methods for network analysis.