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Equivalent Stress Concept for Limit State Analysis pp 195–221Cite as

Generalized Pressure-Sensitive Criteria

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Part of the book series: Advanced Structured Materials ((STRUCTMAT,volume 86))

Abstract

The second necessary condition (Sect. 8.2.1) states that all materials are pressure-sensitive, at least in the tensile region \(I_1>0\). The criteria of pressure-insensitive material behavior (Chap. 9) can only be used in the region \(I_1\le 0\) for some ductile materials, cf. Föppl and Föppl Drang und Zwang: Eine höhere Festigkeitslehre für Ingenieure. R. Oldenbourg, München, 1920 [1], Huber Specific strain work as a measure of material effort Czasopismo Techniczne, Lwów, Organ Towarzystwa Politechnicznego we Lwowie 22:34–40, 49–50, 61–62, 80–81, 1904 [2], von  Mises Mechanik des festen Körpers im plastischen deformablen Zustand Nachrichten der Königlichen Gesellschaft der Wissenschaften Göttingen, Mathematisch-physikalische Klasse 1913:589–592, 1913 [3].

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Notes

  1. 1.

    This substitution results if two points \(Z(1,\,1)\) and \(A_\mathrm {Z}(1/\gamma _1,\,0)\) in the normalized Burzyński-plane \((I_1/\sigma _+,\,\sqrt{3\,I_2'}/\sigma _+)\) are connected with a straight line.

  2. 2.

    The reference values for the hydrostatic nodes \(A_\mathrm {Z}\) and \(A_\mathrm {D}\) are as follows:

    • NSH—the normal stress hypothesis with \(a_+^\mathrm {hyd}=1\) (Sect. 4.1),

    • TT—lower bound for the point \(A_\mathrm {D}\) with respect to the normal stress hypothesis as trigonal trapezohedron Eq. (2.4), \(a_-^\mathrm {hyd}=d\) [12, 13], and

    • TD—lower bound for the point \(A_\mathrm {D}\) with respect to the normal stress hypothesis as triangular dipyramid \(a_-^\mathrm {hyd}=2\,d/3\) [12, 13].

    The hydrostatic nodes have been set so that the point \(A_\mathrm {D}\) is located in the region

    $$\begin{aligned} 1/\gamma _2\in [-3\,d,\,-2\,d]. \end{aligned}$$

    The triangle in the \(\pi \)-plane (Fig. 9.1, shape b) and the point \(A_\mathrm {Z}\) with

    $$\begin{aligned} \gamma _1=1/3 \end{aligned}$$

    were set on the basis of the normal stress hypothesis. The Poisson’s ratio

    $$\begin{aligned} \nu _+^\mathrm {in}=1/2 \end{aligned}$$

    defines a tangent at the point Z parallel to the hydrostatic axis (Sect. 5.4).

  3. 3.

    In this formulation, the shape of the surface in the \(\pi \)-plane is specified with \(d_\pi \) and \(k_\pi \) for the criteria of pressure-insensitive material behavior (Sect. 5.2) (Fig. 9.1, \(d-k\) diagram). Under the condition \(\nu _+^\mathrm {in}=1/2\), it follows \(d=d_\pi \) and \(k=k_\pi \) (5.9).

  4. 4.

    Theory of elasticity with different Young’s moduli \(E_+\ne E_-\) and the elastic Poisson’s ratios \(\nu _+^\mathrm {el}\ne \nu _-^\mathrm {el}\) at tension and compression [44,45,46], cf. (Sect. 2.2.2).

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  1. Fraunhofer Institute for Structural Durability and System Reliability (LBF), Darmstadt, Germany

    Vladimir A. Kolupaev

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Kolupaev, V.A. (2018). Generalized Pressure-Sensitive Criteria. In: Equivalent Stress Concept for Limit State Analysis. Advanced Structured Materials, vol 86. Springer, Cham. https://doi.org/10.1007/978-3-319-73049-3_10

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  • DOI: https://doi.org/10.1007/978-3-319-73049-3_10

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