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Modelling the Voids Growth in Ductile Fracture

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Multiscale Modelling in Sheet Metal Forming

Part of the book series: ESAFORM Bookseries on Material Forming ((EBMF))

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Abstract

Ductile fracture in metals and metallic alloys is due to the evolution of microscopic voids during plastic deformation. Voids nucleate around foreign inclusions or at grain boundaries and grow in regions with large triaxial stresses. Larger voids promote the formation of bands of localized deformation where new small voids are nucleated, thus forming a macroscopic crack. The microscopic dimples present on ductile fracture faces are a direct proof for such a mechanism.

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Notes

  1. 1.

    We anticipate here the use of Tabanov’s natural coordinates \( r,\theta ,\phi \) for the spheroidal and ellipsoidal coordinates, with a pseudo-radial and two angular coordinates; see Sect. 4.3.

  2. 2.

    There is also a similar result in Lewis (1996) for the gradient of \( F. \)

  3. 3.

    For arbitrary values of the exponent, this integral is expressible as a hypergeometric function.

  4. 4.

    Hereafter we use the usual convention of summing over repeated indices.

  5. 5.

    Multiplied by some leading terms involving square roots.

  6. 6.

    These polynomials are also homogeneous only in the spherical case.

  7. 7.

    We use the same notation for the basis on M and N as there is no confusion possible.

  8. 8.

    The entire argument applies equally well to another second order operator—the Laplacian written as div grad. There is no reason an arbitrary diffeomorphism will preserve harmonic functions. But transformed harmonic functions are still harmonic is we redefine the Laplacian using the transformed metric. This simple idea is the key to all recent work on cloaking transformations.

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Authors and Affiliations

  1. Technical University of Cluj Napoca, 28 Memorandumului, 400114, Cluj Napoca, Romania

    Mihai Gologanu, Dan-Sorin Comsa & Dorel Banabic

  2. Amirkabir University of Technology, 424 Hafez Avenue, Tehran, Iran

    Abdolvahed Kami

Authors
  1. Mihai Gologanu
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  2. Dan-Sorin Comsa
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  3. Abdolvahed Kami
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  4. Dorel Banabic
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Correspondence to Mihai Gologanu .

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  1. CERTETA Research Center, Technical University of Cluj Napoca, Cluj Napoca, Romania

    Dorel Banabic

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Gologanu, M., Comsa, DS., Kami, A., Banabic, D. (2016). Modelling the Voids Growth in Ductile Fracture. In: Banabic, D. (eds) Multiscale Modelling in Sheet Metal Forming. ESAFORM Bookseries on Material Forming. Springer, Cham. https://doi.org/10.1007/978-3-319-44070-5_4

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