Dynamic Growth and Coalescence of Voids under Diffusion and Plasticity

1
Dynamic Growth and Coalescence of Voids under
Diffusion and Plasticity Kinjal Dhruva1, Alberto
Cuitino1 and Michael Ortiz2 1Department of
Mechanical and Aerospace Engineering, Rutgers
University, Piscataway, NJ 08854,USA
2Graduate Aeronautical
Laboratories, California Institute of Technology
INTRODUCTION A numerical scheme based on a
eulerian description of the elasto-plastic and
vacancy concentration fields over a regular
finite difference grid is utilized to study void
growth and coalescence in plastically deforming
solid. The procedure is augmented with a
level-set description of the void interfaces and
immersed boundary approach, which is based on the
ghost method used in fluid problems. This
approach allows us to track the complex
topological changes due to void coalescence
without the need for remeshing.
BACKGROUND Void nucleation and growth are the
most important mechanism of dynamic failure in
ductile metals like Copper.Once nucleated,the
voids grow as the surrounding material strains
plastically to accommodate growing void size. The
void growth is driven by the combined effect of
diffusion of vacancies and plastic
deformation.This void growth and subsequent
coalescence results in the formation of crack
which eventually leads to fracture of material.
Thus an accurate numerical model simulating void
growth is imperative in understanding ductile
fracture

• CONCLUSION OUTLOOK
• The scheme effectively models void growth of
  multiple void and follows the complex topology
  even after coalescence takes place.
• The stress and velocity fields agree with the
  experimental observation in terms of average
  stress values and time taken for the void growth
  to take place.
• The scheme has been extended to solve diffusion
  as well as plasticity problem in higher dimension
  under combined setting
• The scheme offers an efficient way of studying
  the rate of void growth for multiple void
  problem, which is difficult to achieve by
  analytical approach.

RESULTS OF THE NUMERICAL SCHEME
Two Dimensional Void Evolution Under Combined
Setting
Void Growth by Diffusion
Void Growth by Plasticity
Stress Along x-axis (Sx)
Two Dimensional Void Growth Under Combined
Setting
Combined Void Growth (DiffusionPlasticity)
U-velocity
Three Dimensional Void Evolution Driven Under
Combined Setting
GHOST METHOD As we are using a fixed Cartesian
grid over entire computational domain,the
interface doesnt necessarily lie on the grid
points which requires an effective method to
transform the boundary condition given on
interface onto grid nodes.We use ghost nodes
based method to achieve this. (Fedkiw,Tsen
g et al) The idea is to create a ghost node
corresponding to every grid node that has a
neighboring point lying on the interface.The
values of flow variables at ghost nodes are
computed using a local reconstruction scheme.Once
the ghost nodes are defined,standard numerical
schemes can be used over the entire domain that
will comprise only of physical domain and ghost
cell domain.
LEVEL SET REPRESENTATION This method represents
an interface using a level set function
f(X,t) defined at every grid point having the
following properties f(X,t)lt0 Inside the
interface f(X,t)gt0 Outside the interface
f(X,t)0 On the Interface Subsequent
interface motion under the velocity field V(x,t)
is governed by the equation
METHODOLOGY

• The vacancy component of the velocity is
  obtained by solving a diffusion equation over
  entire domain.The solution involves curvature (k)
  dependent boundary condition and provides
  concentration gradients that are proportional to
  the velocity component
• The plasticity component on the other hand is
  obtained by solving elasto-plastic equations of
  solid mechanics.Once the mechanics equations are
  solved using immersed interface approach based on
  Ghost method, the velocity component is derived
  and new geometry is updated accordingly.

IMAGE ANALYSIS The complex topological changes
that take place during void growth and
coalescence are very hard to quantify.To achieve
this we use techniques of image analysis by which
plots of growing void interfaces are converted
into a bitmap image that can be then be analyzed
to study the rate of growth of void and other
associated properties.