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Studying stress transformation is essential in understanding how stress components within a material, like a cube under plane stress, change with rotation. This change is analyzed by considering a prismatic element within the cube. As the element rotates, the stress components acting on it—both normal and shearing stresses—change in magnitude and orientation. This change is quantified using trigonometric functions of the rotation angle, relating the forces acting on the rotated element's...
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Stress concentration is when stress intensifies near discontinuities such as holes or abrupt cross-sectional changes in a structural member. This localized stress can often surpass the average stress within the member. The stress distribution in flat bars, either with a circular hole or varying widths connected by fillets, can be determined experimentally using a photoelastic method. The results are based on ratios of geometric parameters like the ratio of the hole's radius to the smaller...
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The concept of stress concentration is crucial for understanding how materials respond under bending stresses, particularly when there are irregularities or discontinuities in the material's geometry. Normally, stress in a symmetric member subjected to pure bending is assumed to be uniformly distributed across the entire cross-section. However, this assumption does not hold when there are variations in the cross-sectional geometry or the presence of notches and holes.
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To grasp the intricacy of real-world conditions where multiple loads are applied simultaneously to a structure, one might visualize a section passing through a specific point within a body, aligned parallel to the xy plane. This section is subjected to various forces, including original loads, normal forces, and shearing forces.
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Understanding stress on an oblique plane under axial loading is pivotal in material mechanics. This analysis offers insight into a material's durability and strength, which is crucial for civil engineering and structural design. Axial loading refers to force application along the material's central axis, causing compression or elongation and leading to normal stress. Normal stress occurs when a force acts perpendicularly to the material's area, resulting in compressive or tensile...
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The graphical depiction of normal and shearing stress equations is represented by a circle, demonstrating the interplay between these stresses under different angular conditions. The center of this circle C, located on the vertical axis, represents the average normal stress, while its radius shows the range of stress variations. At points A and B, where the circle intersects the horizontal axis, the maximum and minimum normal stresses are observed, occurring without shearing stress. These...
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Transition to stress focusing for locally curved sheets.

Thomas Barois1, Ilyes Jalisse1, Loïc Tadrist2

  • 1Univ. Bordeaux, CNRS, LOMA, UMR 5798, F-33400 Talence, France.

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|August 20, 2021
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Summary

This study investigates the buckling of narrow elastic sheets under compression. A critical width transition to buckling with stress focusing is observed in thin sheets, while thick sheets exhibit no buckling.

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Area of Science:

  • Solid mechanics
  • Materials science
  • Nonlinear dynamics

Background:

  • Elastic sheets are fundamental in engineering and biological systems.
  • Understanding buckling phenomena is crucial for predicting material failure and designing stable structures.
  • Previous research has explored buckling in various geometries, but the specific case of narrow rectangular sheets under point compression requires further investigation.

Purpose of the Study:

  • To investigate the buckling behavior of rectangular elastic sheets subjected to point compression.
  • To identify the critical conditions and scaling laws governing the transition to buckling in thin sheets.
  • To explore the buckling behavior in the thick sheet limit and establish a stability criterion for curved sheets.

Main Methods:

  • Deformation of a rectangular thin elastic sheet by applying contact forces at two points.
  • Analytical and numerical solutions for a spring network model to analyze thick sheets.
  • Experimental validation of theoretical predictions for thin sheet buckling.

Main Results:

  • A transition to buckling with stress focusing occurs for sufficiently narrow thin sheets.
  • The critical width for buckling is proportional to the sheet length with an exponent of 2/3 in the small thickness limit.
  • Buckling does not occur for the thickest sheets, and a stability criterion for curved sheets was established.

Conclusions:

  • The study reveals a distinct buckling transition in narrow thin elastic sheets.
  • The findings provide a critical width criterion for buckling, dependent on sheet dimensions and thickness.
  • A stability criterion for curved sheets was developed, contributing to the understanding of elastic sheet mechanics.