Research by T. W. Shield

Needles in Martensites

Research conducted with R. D. James and R. Kohn of the Courant Institute and presented at the ICOMAT-95 conference in Lausanne Switzerland, 1995.

Needles are a common microstructure observed in martensites involving compund twins. The picture above shows a set of horizontal needles ending at a vertical interface. The different colors show the two variants of the martensite involved. These are visible using polarized light which detects the different cyrstal orientation in each variant. The two variants are compound twins, that is, they are symmetry related. Addtional experimental pictures of twins have been taken by Dr. C. Chu in Professor James' laboratory.

As the loading on the specimen is changed the volume fraction (area percentage) of the two variants changes as the loading causes one variant to be favored over another. When the microstructure starts out with very few thin needles and the load is changed so that the needles become thicker, a phenonema called tip splitting occurs. The photo above shows a needle tip shortly after it has split. The complete process can be viewed in this MPEG movie of tip splitting. Future work will consider the mechanics of this process.

Needle Microstructures

The needle microstructure forms when a region of twinned martensite layers (the right side of the image above) meets a pure variant (the wide dark band on the lefthand side). Althought the dark material is all the same variant, the rotation in the needles is different from the rotation in the pure variant to the left. Thus it is not kinematically possible for the two dark regions (A below) to touch without elastic deformations occurring. The light orange colored region, variant B, is kinematically compatible with both regions of variant A across horizontal and vertical planes (the twin planes). This situation is sketched below.

Thus a horizontal band of variant A cannot meet the vertical band of variant A without some elastic deformation. If such a meeting must occur then the needles pictured above are the most energetically favorable microstructure, and this is what occurs.

The image above and other experimental photos show two key features of this microstructure:

  1. The needles are bent.
  2. The vertical interface is wavy.

There are two possible explanations for each of these features:

  1. That is the shape of the interface in the reference configuration.
  2. The shape is due to elastic deformations of the interfaces.

Professors James and Kohn postulated that the bent shape of the needles was due to elastic strains that are required to make the microstructure compatible. A further part of their theory is that these types of interfaces contain a large fraction of the energy in the microstructure and thus understanding these microstructures is key to modeling the behavior of the entire material. This theory can be tested by calculation of the equilibrium shapes of these microstructures. If a guess of the shape of a microstructure can be made that is topologically equivlent to the actual microstructure, then a process of relaxation can be used to determine the actual shapes of the microstructure. This method requires that a solution for the stresses and strains in the microstructure be obtained. If the strains (both elastic and transformation) are assumed to be small so that linear kinematics can be applied, then conventional Finite Element Methods (FEM) can be used to calculate the solution. For an interface to be in equilibrium the driving traction, f, given by

where W is the elastic strain energy and sigma and epsilson are the stress and strain respectively. The ||.|| operator means jump across the interface. By calculating the driving traction, f, at each point on each interface and then assuming that the motion of the interface is proportional to the driving traction, the new shape for the interface can be found. This proceedure is repeated until the driving traction is approximately zero at all points on the interfaces. The resulting shape is the relaxed shape for the microstructure.

FEM Calculations

FEM calculations were performed using the orthotropic elastic constants of the two variants and using thermal strains (with anisotropic thermal expansion coeeficients) to model the transformation strains. A typical mesh used for this calculation is shown below. The two variants are different colors.

Initially a guess at the needle shape (a single exponential) from the experimental photos and a straight vertical interface were used. Then the relaxation prodeedure, described above, was performed. It turned out that the largest driving tractions were on the vertical interface and these were relaxed by changing the shape of the vertical interface. The plot below shows the reduction in the driving force along the vertical interface. The peaks are at the locations where the needle tips intersect the vertical interface.

The largest driving forces are for when the interface is straight and the smallest is for the interface shown in the mesh above. I used the intermediate result to make a guess (linearly) at the amount to move the interface for the third calculation.

The final results of these calculations are shown below. As shown in the mesh figure above the shapes of the interfaces in the reference configurations have the following features:

  1. The needles are symmetric (and thus the bending is due to elastic strains).
  2. The vertical interface is wavy.

The images below show the values of the shear strain as a function of position as false color values on the deformed geometry. Note that the scales for the two variants are not the same. Variant A has all negative strains while variant B has all positive strains. Also note that the needles are bent in the deformed configuration.

Variant A shear strains range from -0.02 to -.0185.

Variant B shear strains range from 0.0185 to 0.02.

A complete paper on this work is in preparation.