Localization+(more...)


 * Mechanisms and Mechanics of Strain Localization**

Localization in the form of a shear band is observed in vastly different classes of materials


 * metals exhibiting dislocation driven plastic deformations
 * metals exhibiting stress-induced martensitic phase transformations
 * shear yielding polymers
 * metallic glasses
 * granular packing (to be developed…)

A discussion of this topic was presented by Ravi-Chandar; A podcast of the discussion is at: http://online.kitp.ucsb.edu/online/earthq05/ravichandar/


 * Lüder’s bands in** **//a//-iron and steels**

The phenomenon of Lüder’s bands is discussed at great length in the book by Hall; the two images below are taken from that book


 * E.O. Hall, **Yield** **Point Phenomena in Metals and Alloys**, Plenum Press, New York 1970.



When a thin polycrystalline specimen of //a//-iron is stretched in uniaxial tension, the nominal stress (load/original area) vs nominal strain (extension/specimen length) exhibits a variation indicated above.


 * At the peak load (the upper yield point) the deformation localizes in the form of a Lüder’s band at a location of stress concentration (typically at a gripping region in the specimen) or at a local defect in the interior; it can be triggered in an experiment by introducing a defect in the gage section of the specimen.
 * The orientation of the band in thin strip specimens is ~54º
 * Within the band, large strains, called the Lüder’s strain are generated.
 * The nominal stress drops to a plateau level (the lower yield point) and at constant stress the Lüder’s band propagates, enveloping unstrained material as it moves through the specimen. A photograph of a specimen with a propagating Lüder’s band is shown above; the band is made visible since the brittle oxide layer flakes off under the Lüder’s strain.
 * The mechanisms of deformation can be understood in the following terms:
 * Elastic deformation of the lattice in the linear elastic region
 * Release of dislocations pinned at solute atoms, and other defects at the onset of localization
 * Arrest of dislocation motion at forest dislocations, other impurities and grain boundaries (strain hardening)


 * Shear Localization in Phase Transforming Materials**

The phenomenon of localization and propagation of phase transformation is discussed in detail in the following papers.


 * J.Shaw and S. Kyriakides, Thermomechanical Aspects of NiTi, //Journal of the Mechanics and Physics of Solids//, **43**, 1995, 1243-1281. (quasi-static experiments) http://dx.doi.org/10.1016/0022-5096(95)00024-D
 * R. Abeyaratne, K. Bhattacharya and J. K. Knowles, Strain-Energy Functions with Multiple Local Minima: Modeling Phase Transformations using Finite Thermoelasticity, in Nonlinear Elasticity: Theory and Appliations, Y. Fu and R.W. Ogden, Cambridge University Press, 2001, 433-490. (continuum theory)
 * J.A. Shaw and S. Kyriakides, Initiation and propagation of localized deformation in elasto-plastic strips under uniaxial tension, //International Journal of Plasticity//, **13**, 1997, 837-871. (numerical simulations) http://dx.doi.org/10.1016/S0749-6419(97)00062-4
 * J. Niemczura and K. Ravi-Chandar, Dynamic propagation of phase fronts (link to be made)



When a thin polycrystalline specimen of Ni-Ti is stretched in uniaxial tension, the nominal stress (load/original area) vs nominal strain (extension/specimen length) exhibits a variation indicated above by the red line. The strain measured locally at a strain gage is also shown by the black line. The test was done at 95ºC with the specimen in the austenitic phase.


 * At the peak load the material undergoes a diffusionless, stress-induced transformation to the martensitic phase. As in the case of the Lüder’s band this occurs at a location of stress concentration (typically at a gripping region in the specimen) or at a local defect in the interior; it can be triggered in an experiment by introducing a defect in the gage section of the specimen.
 * The orientation of the band in thin strip specimens is ~54º
 * Within the band, large strains, called the transformation strain are generated; for the stress-induced martensitic transformation in NiTi, this strain is about 6%.
 * The nominal stress drops to a plateau level and at constant stress the phase transformation band propagates, enveloping untransformed material as it moves through the specimen. A photograph of a specimen with a propagating phase transformation front is shown above; the band is made visible since the brittle oxide layer flakes off under the transformation strain.
 * During propagation of the front, the strain gage indicates a constant strain until the phase transformation front arrives at the location of the gage; this point then experiences the 6% strain of transformation rapidly.

The mechanisms of deformation can be understood in the following terms:


 * Elastic deformation of the austenite (bcc) lattice
 * Stress-induce-martensite (monoclinic) forms in various domains, along with the associated strain change
 * Further strain is limited by the stiffness of the martensite and leads to band propagation


 * Lüder’s bands in polycarbonate**

Many thermoplastic polymers also exhibit strain localization in the form of a shear band. A detailed investigation of this phenomenon in polycarbonate has been explored in the following references:


 * G. Buisson and K. Ravi-Chandar, On the constitutive behaviour of polycarbonate under large deformation, //Polymer//, **31**, 1990, 2071-2076. http://dx.doi.org/10.1016/0032-3861(90)90078-D
 * J. Lu and K. Ravi-Chandar, Inelastic deformation and localization in polycarbonate under tension, //International Journal of Solids and Structures//, **36**, 1999, 391-425. http://dx.doi.org/10.1016/S0020-7683(98)00004-3



When a thin polycarbonate specimen is stretched in uniaxial tension, the nominal stress (load/original area) vs nominal strain (extension/specimen length) exhibits a variation indicated above; the figure on the right shows greater detail for the first 12% engineering strain


 * At the peak load the deformation localizes in the form of a Lüder’s band at a location of stress concentration (typically at a gripping region in the specimen) or at a local defect in the interior; it can be triggered in an experiment by introducing a defect in the gage section of the specimen.
 * The orientation of the band is in thin strip specimens ~54º
 * Within the band, large strains, called the Lüder’s strain are generated; for the polycarbonate this corresponds to a very large stretch of about 1.7 times its initial length; this is associated with a molecular reorientation.
 * The nominal stress drops to a plateau level and at constant stress the Lüder’s band propagates, enveloping unstrained material as it moves through the specimen. In contrast to the metallic materials, the large stretch generates a spreading of the band into what is typically called a “neck” and this propagates along the length of the specimen.

A series of images of the specimen taken at different points during the stretching of the specimen is shown above.



The mechanisms of deformation can be understood in the following terms:


 * Elastic deformation from stretching of weak bonds
 * Breaking of chain entanglements, viscoelastic effects?
 * Microshear banding at free volume defects?
 * Stretch limits arise from the limited extensibility of polymer chains

Localization in a plane-strain compression test in polystyrene:


 * The orientation of the band is ~45º




 * R. Quinson, J. Perez, M. Rink, and A. Pavan, Yield criteria for amorphous glassy polymers, //Journal of Materials Science//, **32**, 1997, 1371 – 1379,


 * Essential Ingredients in Triggering a Propagating Shear Localization**


 * Threshold stress for triggering deformation mechanism; plastic flow by dislocation motion, phase transformation or molecular reorientation)
 * Volume preserving deformation mechanism; this dictates the orientation of the band observed;

if the material is pressure-dependent, appropriate modifications to the plasticity theory are required and the band orientation is affected.


 * Limit to the amount of strain generated by the deformation mechanism; in the absence of such a mechanism, the band cannot propagate; further deformation occurs at the localized band leading to failure or additional bands are nucleated elsewhere

The orientation of the band with respect to the loading direction is dictated by the volume constraint. For plane-stress conditions, the orientation is 54º while for plane-strain conditions it is 45º, precisely as observed in the experiments. These are the characteristics in the rigid-plastic theory (see Kachanov, Foundations of the Theory of Plasticity, North-Holland Pub. Co, Amsterdam, 1971).