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» ひずみ解析用モール円

ひずみ解析用モール円

1900
Mohr's Circle diagram for strain analysis on an engineer's desk with drafting tools.

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The principles of モールの円 can also be directly applied to analyze the two-dimensional state of strain at a point. By replacing normal ストレス ([latex]\sigma[/latex]) with normal strain ([latex]\epsilon[/latex]) and せん断応力 ([latex]\tau[/latex]) with half the shear strain ([latex]\gamma/2[/latex]), an analogous circle can be constructed. This graphical tool helps determine principal strains and the maximum shear strain.

The mathematical structure of the transformation equations for plane strain is identical to that for plane stress. This analogy allows for the use of Mohr’s circle for strain analysis. The horizontal axis represents the normal strain, [latex]epsilon_n[/latex], and the vertical axis represents half the engineering shear strain, [latex]gamma_{nt}/2[/latex]. The use of [latex]gamma/2[/latex] (tensorial shear strain) instead of [latex]gamma[/latex] (engineering shear strain) is necessary to maintain the circular form of the locus.

Given a state of strain defined by [latex]epsilon_x[/latex], [latex]epsilon_y[/latex], and the shear strain [latex]gamma_{xy}[/latex], the circle is constructed with a center at [latex]C = (epsilon_{avg}, 0)[/latex], where [latex]epsilon_{avg} = (epsilon_x + epsilon_y)/2[/latex], and a radius [latex]R = sqrt{left(frac{epsilon_x – epsilon_y}{2}right)^2 + left(frac{gamma_{xy}}{2}right)^2}[/latex]. The intersections with the horizontal axis give the principal strains, [latex]epsilon_1[/latex] and [latex]epsilon_2[/latex]. The maximum in-plane shear strain is twice the radius of the circle, [latex]gamma_{max} = 2R[/latex]. This tool is invaluable in experimental mechanics, where strains are often measured directly using strain gauge rosettes. The circle provides a quick graphical method to convert these measured strains into principal strains and their orientations.

UNESCO Nomenclature: 2203
古典力学

タイプ

抽象システム

混乱

増分

使用法

広く普及している

前駆物質

  • Mohr’s circle for stress
  • Cauchy’s strain tensor theory
  • Hooke’s law relating stress and strain
  • Development of the strain gauge

アプリケーション

  • experimental stress analysis using strain gauges
  • materials testing to determine properties like young’s modulus and poisson’s ratio
  • structural health monitoring
  • geodesy for measuring crustal deformation

特許:

NA

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Related to: Mohr’s circle, strain analysis, principal strain, shear strain, strain gauge, experimental mechanics, elasticity, deformation, materials testing, solid mechanics.

歴史的背景

ひずみ解析用モール円

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1900-12-14

(日付が不明または関連性がない場合、例えば「流体力学」などでは、その注目すべき出現時期の概算値が提示されます。)

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