Modeling of Void Growth And Coalescence In Plastically Deforming Solids

1
Modeling of Void Growth And Coalescence In
Plastically Deforming Solids 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.

• CONCLUSION
• .The scheme effectively models void growth of
  multiple void and follows the complex topology
  even after coalescence takes place.
• The Stress and Velocity fields obtained confirm
  to the experimental observation in terms of
  stress values and time taken for the void growth
  to take place.
• The scheme offers an efficient way of studying
  the rate of void growth which is dependent on
  volume fraction that can easily be obtained from
  numerical simulation.

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.
RESULTS OF THE NUMERICAL SCHEME
Stress Along x-axis (Sx)
U-velocity
Effective Plastic Deformation
Void Growth by Diffusion
Void Growth by Plasticity
Combined Void Growth
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
Diffusion Equation
Elasto-Plastic Equations
Elasto-Plastic Equations in Conservative Form
Vacancy Diffusion -Surface Bulk
diffusion -Diffusivity of material -Curvature and
gradients

• Dynamic Elasto-Plasticity
• Mechanics Equations
• Hardening Laws
• Flow Rule

• The vacancy component of the velocity is obtained
  by solving a diffusion equation over entire
  domain.The solution involves curvature (k)
  dependent boundary conditions with the velocity
  being proportional Concentration Gradients
• 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.

Mie-Gruneisen e.o.s
(Cuitino 02)
Equations governing plastic deformation
(Sethian,Aslam)