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Research Papers

A Lumped Energy Model for Crack Growth in Shape-Memory Materials

[+] Author and Article Information
Perry H. Leo

Professor
Department of Aerospace
Engineering and Mechanics,
University of Minnesota,
Minneapolis, MN 55455
e-mail: phleo@umn.edu

Thomas W. Shield

Professor
Department of Aerospace
Engineering and Mechanics,
University of Minnesota,
Minneapolis, MN 55455
e-mail: shield@umn.edu

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received October 13, 2015; final manuscript received November 26, 2015; published online December 16, 2015. Assoc. Editor: Daining Fang.

J. Appl. Mech 83(3), 031010 (Dec 16, 2015) (7 pages) Paper No: JAM-15-1554; doi: 10.1115/1.4032114 History: Received October 13, 2015; Revised November 26, 2015

We construct an energy-based model to study crack growth behavior in a shape-memory alloy that undergoes a stress-induced austenite to martensite transformation. The total energy, which is the sum of the elastic energy of the specimen and loading device, the surface energy of the crack, and the energy associated with transforming austenite to martensite, depends on the applied extension, the crack length, and the martensite volume fraction. The crack length and martensite volume fraction are coupled through a transformation criteria at the crack tip. By tracking the progression of equilibrium cracks as extension increases, we show that the transformation leads to a regime of stable crack growth followed by unstable growth. These results are in agreement with experiments on both single crystal and polycrystal shape-memory alloys.

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References

Figures

Grahic Jump Location
Fig. 1

An experimental load–displacement curve such as in Refs. [1,2]. Stable crack propagation begins at point 2. The crack becomes unstable and propagates rapidly at point 5. The crack then stabilizes and grows to point 8 before rapid propagation again occurs. The crack restabilizes at point 9. The specimen is unloaded at point 10.

Grahic Jump Location
Fig. 2

A schematic picture of the geometry of the cracked specimen is shown. The deflection on the machine side of the system, Δ, is related to that on the specimen side, δ, by the machine stiffness K. The dashed lines show the assumed geometries for the transformation band and the transformation zone.

Grahic Jump Location
Fig. 11

The stress σ is plotted against extension Δ for the case where austenite transforms to martensite in a band geometry, and for different values of transformation stress σT. The other normalized parameters are machine stiffness k = 5, surface energy γ=6.67×10−5, and transformation strain εT=0.1. The stress–extension curve for the case with no transformation is also shown up to where the initial crack becomes unstable. Similar results hold for the transformation zone geometry.

Grahic Jump Location
Fig. 3

The elastic, surface, and total energies are shown as functions of crack length a for the case with no transformation. The normalized extension Δ=0.01. The other normalized parameters are machine stiffness k = 5 and surface energy γ=6.67×10−5.

Grahic Jump Location
Fig. 4

The critical crack length a* is plotted versus normalized extension Δ for the case with no transformation. The other normalized parameters are machine stiffness k = 5 and surface energy γ=6.67×10−5. The critical crack length satisfies ∂W(a*)/∂a=0. For values of Δ below about 0.0034 there are no solutions. Fracture occurs when a* reaches the initial crack length a0, here taken as 0.05.

Grahic Jump Location
Fig. 5

The elastic, surface, transformation, and total energies are shown as functions of crack length a for the case where austenite transforms to martensite in a band geometry. The normalized extension Δ=0.01. The other normalized parameters are machine stiffness k = 5, surface energy γ=6.67×10−5, transformation stress σT=0.02, and transformation strain εT=0.1.

Grahic Jump Location
Fig. 6

The partial derivative of the energy with respect to crack length, ∂W(a)/∂a, is plotted against crack length a for the case where austenite transforms to martensite in a band geometry. The normalized extension Δ=0.01. The other normalized parameters are machine stiffness k = 5, surface energy γ=6.67×10−5, transformation stress σT=0.02, and transformation strain εT=0.1. The smaller solution of ∂W(a)/∂a=0 corresponds to a stable crack (∂2W/∂a2>0), while the larger solution corresponds to an unstable crack.

Grahic Jump Location
Fig. 7

The stress σ is plotted against extension Δ for the case where austenite transforms to martensite in a band geometry. The normalized parameters are machine stiffness k = 5, surface energy γ=6.67×10−5, transformation stress σT=0.02, and transformation strain εT=0.1. The value Δ1=0.01037 at which stable crack growth begins is clearly seen as a transition from a roughly linear increasing stress to a roughly constant stress.

Grahic Jump Location
Fig. 8

The martensite volume fraction λ is plotted against extension Δ for the case where austenite transforms to martensite in a band geometry. The normalized parameters are machine stiffness k = 5, surface energy γ=6.67×10−5, transformation stress σT=0.02, and transformation strain εT=0.1. Before stable crack growth (Δ≤Δ1=0.01037), λ increases as stress increases at fixed crack length (dashed line). During stable crack growth (Δ>Δ1), λ increases as crack length increases at roughly constant stress (solid line).

Grahic Jump Location
Fig. 9

The stress σ plotted against extension Δ for the cases where austenite transforms to martensite in both the band geometry and the zone geometry. In both cases, the normalized parameters are machine stiffness k = 5, surface energy γ=6.67×10−5, transformation stress σT=0.02, and transformation strain εT=0.1.

Grahic Jump Location
Fig. 10

The stress σ is plotted against extension Δ for the case where austenite transforms to martensite in a band geometry, and for different values of machine stiffness k. The other normalized parameters are surface energy γ=6.67×10−5, transformation stress σT=0.02, and transformation strain εT=0.1. Similar results hold for the transformation zone geometry.

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