Mahendaran Uchimali

Constitutively informed Particle Dynamics

The modern frontier of computational mechanics is striving to describe the effect of microstructure on the mechanical response of materials. Atomistic models describe some of these aspects at length and time scales far remote from the macroscale. On the other hand, continuum models smooth out the underlying discreteness and rely on the constitutive description of the microscale effects. The existing discrete models are bottom-up approaches and they describe a very limited range of macroscopic constitutive behaviour.

Hence, I have developed a novel discrete particle method to describe materials with evolving microstructures. I have studied a few paradigmatic problems involving a material undergoing structural phase transformations. The emphasis is on the influence of martensite morphology and ledge on the stress-strain response of shape memory alloys.

A systematic top-down discrete method called constitutively informed particle dynamics (CPD) is presented here in a nutshell. This novel approach is very broad and can be applied to many material behaviours.

In my model, the body of interest B is taken to be composed of N discrete particles which may be arranged uniformly or distributed irregularly through the domain.

The interacting neighbours and their multi-body interactions are defined based on Delauney triangulation.

Multi-body Interaction

Deformation of the body under applied load is shown with emphasis on the specific triangle composed of three particles. The interaction of these particles is defined as a negative gradient of the energy of the triangle.

The energy of the triangle is expressed as volume times free energy of the material through the Lagrangian strain (F is the discrete analog of deformation gradient).

The trajectory of particles is obtained by solving newton's equations of motion with the above novel definition of force.

Material behaviours described using CPD

Thermally induced martensitic transformation

Polyconvex free energy is considered pertinent to the material undergoing structural phase transformations. By reducing the temperature many martensite nucleus forms at random locations. They grow and impinge upon one other to generate the final martensite microstructure.

The observed microstructure shows the alternate arrangement of martensite variants (Blue and red colour in figure) in an equal volume fraction typically seen in SMAs which leads to property referred to as "self accommodation".

Twin boundaries are oriented +45 or -45 to reference austenite phase which results from compatibility conditions.

The microscale features affecting the stress-strain response are,

Number and length of the interface (Twin boundary & macro twin boundary)
Number of mobile ledges and its Velocity

Influence of morphology on stress-strain behaviour

Banded morphology requires higher stress than lamellar morphology during the detwinning process.
The F-d response shows a jerky nature due to a large number of microscale events.

Propagating Ledge

Movement of the twin boundary happens through propagation of ledge along the boundary. Our simulation shows emitting dispersive waves from the moving ledge. The obtained relation between driving force and velocity of the ledge shows stick-slip nature. This is the first attempt in literature to capture dispersive waves in 2D.

Sharp needle like twins

Advancement of phase boundary under temperature gradient
Sharp needle-like twins are formed very similar to experimental observation

Size effect on morphology

Evolution of martensite under thermal gradient
The Size of the sample greatly affects the resulting martensite morphology
Lamellar and Banded martensite morphology are obtained

Influence of grain orientation

Grain orientation influence the morphology due to compatibility condition on the phase boundary
The effect of the sample width is seen clearly in the microstructure formed as well as the shape of the phase boundary.

Shape Memory Effect

Microstructural evolution of shape memory effect (thermal transformation and detwinning). We can observe the,
The evolution of martensite (red and blue are variants) during the cooling process
The evolution of red variant of martensite during detwinning process,
The evolution of austenite (Green colour) in a heating process.

Annihilation of a twin during detwinning

Multiple ledge propagation along both the twin boundaries makes the twin thinner
After critical thickness whole twin transforms by the ledge motion.

Effect of temperature on detwinning

The detwinning stress arises from the resistance of the material to the nucleation and propagation of mobile ledges along the twin boundary
The detwinning stress shows a monotonic decrease with temperature
The CPD model describes the individual ledge motion. Free energy function shows an increase in the energy barrier between variants at lower temperatures. This increase in the energy barrier results in an increase in the plateau stress during detwinning.

Other material behaviours described using CPD

Brittle crack

Brittle cracks described using strain energy-based failure criteria
Mode I and mixed-mode I/II loading crack paths are in good agreement with experiments

Brittle crack branching

Increased rate of applied load is accommodated by creating more new surfaces results in crack branching
Stress distribution in the domain influences the crack path

Heat transfer by conduction

CPD model is extended to describe the conductive heat transfer
Evolution of temperature in the domain is compared with analytical solution

Ceramics under thermal shock

CPD model is used to describe thermal cracks in ceramics
The hierarchical crack pattern formed is quantitatively compared with experiments