Research Papers

Using Damage Delocalization to Model Localization Phenomena in Bammann-Chiesa-Johnson Metals

[+] Author and Article Information
Koffi Enakoutsa

e-mail: koffi.enakoutsa@msstate.edu

Fazle R. Ahad

e-mail: fra11@cavs.msstate.edu
Starkville, MS 39759

Kiran N. Solanki

Tempe, AZ 85287
e-mail: kiran.solanki@asu.edu

Yustianto Tjiptowidjojo

e-mail: yusti@cavs.msstate.edu

Douglas J. Bammann

e-mail: bammann@cavs.msstate.edu
Starkville, MS 39759

The Cocks and Ashby [25] void growth model is based on a cylinder containing a spherical void.

The normalization factor B(x) in Eq. (13) decreases near the boundaries because the volume studied and the region where A(xy) meet is very small.

1Corresponding author.

Contributed by the Materials Division of ASME for publication in the JOURNAL OF ENGINEERING MATERIALS AND TECHNOLOGY. Manuscript received February 14, 2012; final manuscript received August 1, 2012; published online September 6, 2012. Assoc. Editor: Irene Beyerlein.

J. Eng. Mater. Technol 134(4), 041014 (Sep 06, 2012) (9 pages) doi:10.1115/1.4007352 History: Received February 14, 2012; Revised August 01, 2012

The Bammann, Chiesa, and Johnson (BCJ) material model predicts unlimited localization of strain and damage, resulting in a zero dissipation energy at failure. This difficulty resolves when the BCJ model is modified to incorporate a nonlocal evolution equation for the damage, as proposed by Pijaudier-Cabot and Bazant (1987, “Nonlocal Damage Theory,” ASCE J. Eng. Mech., 113, pp. 1512–1533.). In this work, we theoretically assess the ability of such a modified BCJ model to prevent unlimited localization of strain and damage. To that end, we investigate two localization problems in nonlocal BCJ metals: appearance of a spatial discontinuity of the velocity gradient in any finite, inhomogeneous body, and localization of the dissipation energy into finite bands. We show that in spite of the softening arising from the damage, no spatial discontinuity occurs in the velocity gradient. Also, we find that the dissipation energy is continuously distributed in nonlocal BCJ metals and therefore cannot localize into zones of vanishing volume. As a result, the appearance of any vanishing width adiabatic shear band is impossible in a nonlocal BCJ metal. Finally, we study the finite element (FE) solution of shear banding in a rectangular plate, deformed in plane strain tension and containing an imperfection, thereby illustrating the effects of imperfections and finite size on the localization of strain and damage.

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Grahic Jump Location
Fig. 1

Two dimensional scheme of nonlocal interaction between material points within a vicinity of length l

Grahic Jump Location
Fig. 2

Discontinuity surface S toward a finite body

Grahic Jump Location
Fig. 3

Deformed mesh for the strain in the vertical direction (displacement magnification factor: 50) for the local BCJ model. Note the presence of two different shear bands due to the presence of the imperfection and oriented at 45 deg with respect to the loading axis. One of them ends on the lateral traction-free surface on both sides, while the other one ends on a free surface on one side and on upper surface subject to the prescribed displacement on the other side. In fact, we choose a rectangular mesh rather than a square one to force one of the shear bands to end on free surfaces on both sides contrary to the other one, in order to generate a dissymmetry between the two shear bands.

Grahic Jump Location
Fig. 4

Representation of the strain on the deformed mesh in the vertical direction for both the local (top) and nonlocal (bottom) models, respectively. Note the influence of the damage delocalization on the width of the shear bands: the shear bands in the nonlocal BCJ material are notably larger and are extended over several elements. Also remarkable from this figure is the predominance of one shear band over the other one. It is except that the system will choose one shear band during the subsequent loading and that the other one will cease developing.

Grahic Jump Location
Fig. 5

Damage distribution in the early stage of the loading for both the local (top) and nonlocal (bottom) BCJ models, respectively. Contrary to the strain, the distribution of the damage is substantially affected by the damage nonlocality. However, the width of the shear band is less than that of two elements (of the size of the characteristic length scale); this leads to the conclusion that the damage process in the body considered may begin by the appearance of shear bands which govern void growth from the larger inhomogeneity.

Grahic Jump Location
Fig. 6

Damage distribution in the later time of the loading for both the local (top) and nonlocal (bottom) BCJ models. Note the dramatic influence of damage delocalization on the localization of the damage in the two shear bands: the damage is now concentrated in much narrower zones and its maximum value is higher in local the BCJ material than in nonlocal one. Also remarkable from this figure is the predominance of one shear band over the other one. The width of the band is extended over almost four elements; thus, the shear band width is proportional to the length scale, which is of the size of two elements in this problem.



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