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Research Papers

The Effect of Rate Dependence on Localization of Deformation and Failure in Softening Solids

[+] Author and Article Information
Alan Needleman

Fellow ASME
Professor
Department of Materials Science
and Engineering,
University of North Texas,
Denton, TX 76207

Contributed by the Applied Mechanics of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received October 17, 2014; final manuscript received November 15, 2014; accepted manuscript posted November 20, 2014; published online December 11, 2014. Editor: Yonggang Huang.

J. Appl. Mech 82(2), 021002 (Feb 01, 2015) (7 pages) Paper No: JAM-14-1489; doi: 10.1115/1.4029180 History: Received October 17, 2014; Revised November 15, 2014; Accepted November 20, 2014; Online December 11, 2014

Localization of deformation and failure, a complete loss of stress carrying capacity, is studied for two rate dependent constitutive relations: (i) a Kelvin–Voigt solid and (ii) a viscoplastic solid. A planar block infinite in one direction is subjected to monotonically increasing shear displacements at a fixed rate. Geometry changes are neglected and attention is confined to quasi-static loading conditions. For the Kelvin–Voigt solid, localization precedes failure if there is hardening outside the band and softening inside the band while failure precedes localization if there is softening both inside and outside the band. For the viscoplastic solid, localization precedes failure when there is softening inside the band regardless of the sign of the hardening parameter outside band. For the Kelvin–Voigt solid, it is found that the localization time (or strain) varies logarithmically with the band thickness for small values of band thickness while the time (or strain) to a complete loss of stress carrying capacity has, in general, a different scaling with band thickness. For the viscoplastic solid, with plastic dissipation outside the band as well as inside the band, the strain and the total plastic dissipation to failure are nearly independent of band thickness for sufficiently small thickness values, with what is sufficiently small decreasing with decreasing rate sensitivity. Possible implications for grid based modeling of localization and failure are discussed.

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References

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Figures

Grahic Jump Location
Fig. 1

Variation of strain rate, γ·A, and shear stress, τ, with time for a homogeneous Kelvin–Voigt solid with bA = -10.0

Grahic Jump Location
Fig. 2

Variation of strain rate outside the band, γ·A, and shear stress, τ, with time for a calculation with β = 0.1, bA = -10.0, and bB = -11.0 so that failure, τ = 0, precedes localization

Grahic Jump Location
Fig. 3

Variation of strain rate outside the band, γ·A, and shear stress, τ, with time far a calculation with β = 0.1, bA = 1.0, and bB = -1.0 so that localization, γ·A = 0 precedes failure

Grahic Jump Location
Fig. 4

Comparison of exact and approximate scaling with β on a semilog plot

Grahic Jump Location
Fig. 5

Variation of strain rate outside the band, γ·A, and shear stress, τ, with shear strain Γ for a homogeneous viscoplastic solid with m = 0.2, hA = hB = -10

Grahic Jump Location
Fig. 6

Variation of strain rate outside the band, γ·A, and shear stress, τ, with shear strain Γ for a calculation with β = 0.1, m = 0.2, hA = -10,and hB = -10.1

Grahic Jump Location
Fig. 7

Variation of strain rate outside the band, γ·A, and shear stress, τ, with overall strain Γ for a calculation with β = 0.1, m = 0.2, hA = 1, and hB = -1

Grahic Jump Location
Fig. 8

Variation of strain rate outside the band, γ·A, and shear stress, τ, with overall strain Γ for a calculation with β = 0.1, m = 1.0, hA = 1, and hB = -1

Grahic Jump Location
Fig. 9

Stress–strain curves for three values of β with m = 0.2, hA = 1, and hB = -1

Grahic Jump Location
Fig. 10

Stress–strain curves for three values of β with m = 0.2, hA = 0, and hB = -1

Grahic Jump Location
Fig. 11

Variation of the strain to failure, Γf, with band thickness, β, for four cases

Grahic Jump Location
Fig. 12

Variation of the plastic dissipation to failure, Wf, with band thickness, β, for four cases

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