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Effective behavior of polycrystals that

undergo martensitic phase

transformation

Kaushik Bhattacharya, Robert V. Kohn

Kaushik Bhattacharya, Robert V. Kohn, "Effective behavior of polycrystals

that undergo martensitic phase transformation," Proc. SPIE 1919, Smart

Structures and Materials 1993: Mathematics in Smart Structures, (22 July

1993); doi: 10.1117/12.148412

Event: 1993 North American Conference on Smart Structures and Materials,

1993, Albuquerque, NM, United States

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Effective behavior of polycrystals that undergo martensitic phase transformation

Kaushik Bhattacharya and Robert V. Kohn

Courant Institute of Mathematical Sciences

Mercer Street, New York, NY 10012

ABSTRACT

The shape-memory effect is the ability of a material to recover, on heating, apparently plastic

deformations that it suffers below a critical temperature. These apparently plastic strains are not caused

by slip or dislocation, but by deformation twinning and the formation of other coherent microstructures

by the symmetry-related variants of martensite. In single crystals, these strains depend on the

transformation strain and can be quite large. However, in polycrystals made up of a large number of

randomly oriented grains, the various grains may not deform cooperatively. Consequently, these

recoverable strains depend on the texture and may be severely reduced or even eliminated. Thus, the

shape-memory behavior of polycrystals may be significantly different from that of a single crystal. We

address this issue by studying some model problems in the setting of anti-plane shear.

1. INTRODUCTION

The shape-memory effect is the ability of a material to recover on heating, apparently plastic

deformations that it suffers at low temperatures. At the heart of this phenomenon is a reversible

martensitic phase transformation where the crystal lattice of the high temperature phase (austenite) has

greater symmetry than that of the low temperature phase (martensite). As a result of the change in

crystalline symmetry, there are several variants of martensite. Variants are identical lattices of

martensite that are oriented differently with respect to the austenite. Thus, below the transformation

temperature, a shape-memory material consists of a mixture of different martensitic variants, typically

arranged in a fine-scale rnicrostructure. When loads are applied, the material reduces its energy by

converting from one variant to another, rearranging itself into a new microstructure and consequently

suffering large apparently plastic deformations. On heating, each variant transforms back to the unique

austenite, thereby recovering all deformation. Thus, the deformations that are recoverable by the shape-

memory effect are exactly the deformations that a material can undergo by rearranging its martensitic

variants.

For example, a single crystal consisting of only one grain subjected to shear can deform

apparently plastically by deformation twinning. On heating, both twin variants transform to the

austenite and all the shear strain is recovered. The maximum magnitude of the shear is determined by

the transformation strains associated with the variants. The direction of shear is determined by the

twinning plane and direction of twinning shear. In a polycrystal consisting of a large number of

randomly oriented grains, the shear directions of the various grains do not coincide. Therefore, the

recoverable shear that a polycrystal can undergo will at best be some average of the shears suffered by

single crystals of different orientations. In fact, it could be worse. Since each grain is constrained by

the surrounding grains, it can not deform freely. This suggests that the deformation of the polycrystal

may be limited by the deformation of the worst-oriented grain. On the other hand, there is a possibility

of finding a "percolating path of suitably oriented grains". This will increase the deformation of the

polycrystal. The goal of this paper is to explore these competing effects. In particular, we wish to

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estimate the amount of recoverable strain in a polycrystal, and its dependence on polycrystalline texture,

the change of crystalline symmetry and the transformation strain.

Unfortunately, the problem is

extremely difficult to address in any generality. In order to explore the interesting issues and the guiding

principles, we decided to study some model problems in the setting of anti-plane shear. In this paper, we

state the preliminary results of our investigations.

Experimentally, there are some alloys which display good shape-memory effect as single

crystals, but only partial memory as polycrystals. Many alloys suffer inter-granular fracture at relatively

low strains or after just a few shape-memory cycles. On the other hand, there are materials like Ni-

5Oat%Ti which show excellent shape-memory effect even as polycrystals. They can undergo a large

number of cycles and their recoverable strains are close to their transformation strains. We should

mention here that our knowledge of experimental results is sketchy at best19. One of our motivations

for presenting our partial results to a wide audience is the hope that knowledgeable readers will educate

us about experiments.

In the last few years, there has been considerable theoretical work on materials that undergo

martensitic transformations1019. An important idea is a stored energy density function that depends on

the deformation gradient and the temperature. Below the critical temperature, the energy density has

multiple wells corresponding to the different martensitic variants. Minimizing this energy leads to fine-

scale microstructure. The macroscopic behavior of the crystal is described by a relaxed or effective

energy which takes into account the fine-scale microstructure. The recoverable deformations or strains

of the shape-memory effect correspond to flat or degenerate regions of this effective energy. Up to now,

the investigation of this model has been limited to single crystals. On the other hand, a lot is known

. .

20-24

about

polycrystals made from matenals that do not undergo martensitic transformation

The

goal of

this work is to link these two areas of investigation.

2. MODEL PROBLEMS IN ANTI-PLANE SHEAR

2.1. Preliminaries

We consider a planar domain in two-dimensions and limit our attention to "out-of-plane"

deformations. The position vector is x =

{x1,

x2}and the deformation i(x) is a real-valued function. We

describe the orientation of the grains in a polycrystal by a spatially dependent rotation R(x) which gives

the orientation of the grain situated at position x. For a typical polycrystal, R(x) is piecewise constant.

For a single crystal, the orientation function R(x) is equal to the identity matrix, ,

everywhere.

We fix the temperature at a value well below the transformation temperature and assume that

there is a stored energy function

that depends only on the deformation gradient, which is the vector

Vi =

{ - }. The

total energy of a polycrystal with orientation function R(x) subjected to a

deformation r is given by 5

4(R(x)V

(x)) dx. Our starting point is the hypothesis that the crystal

assumes the configuration that minimizes the total energy for a given boundary condition. It is well-

known that this minimization problem gives rise to fine-scale microstructure.

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For a given polycrystal, we define the effective energy as the function (f which gives the

spatially averaged energy of the polycrystal when the average or macroscopic strain is f. Formally,

4(f)

inf

-j-j 5p(Rx Vii(x)) dx

ril=f.x I

takes into account the effect of microstructure as well as polycrystalline texture. It describes the

macroscopic or effective behavior of the polycrystal.

We say that a polycrystal is isotropic if (f) =

4(Qf)

for all rotations Q

and

for all vectors f.

Figure 1. The stored energy density of the two variant material.

Figure 2. The effective energy of a single crystal of the two variant material.

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