General Theory of Finite
Deformation
Kejie Zhao
Instructor: Prof. Zhigang Suo
May.21.2009
Harvard School of Engineering and Applied Sciences
1
Beyond linear theory
Ingredients in linear theory



Deformation geometry
Force balance
Material model
Beyond linear theory
2
Framework of finite deformation

In continuum mechanics, we model the body by a
field of particles, and update the positions of the
particles by using an equation of motion
Equation of motion
Deformation
kinematics
Conservation laws
Product of entropy
Materials model
3
Kinematics of deformation

Name a material particle by the coordinate of the
place occupied by the material particle when the
body is in a reference state: particle X
4
Kinematics…

Field of deformation x  x( X , t )
A central aim of continuum mechanics is to evolve the field of deformation
by developing an equation of motion
5
Kinematics…



Displacement  x  x( X , t   t )  x( X , t )
x( X , t   t )  x( X , t ) x( X , t )

t
t
2
Velocity
v
Acceleration
 x( X , t )
a
t 2
6
Kinematics…

Deformation gradient
xi ( X  dX , t )  xi ( X , t ) xi ( X , t ) •F(X,t) maps the vector between
FiK 

two nearby material particles in
dX K
X K
reference state, dX, to the vector
 dx  F ( X , t )dX
between the same two material
particles in the current state, dx
7
Kinematics…

Polar decomposition: any linear operator can be
written as a product
F  RU
Rotation vector Stretch vector
U 2  C  FT F
C: Green deformation tensor
8
Conservation laws
Conservation of Mass
Conservation of Linear
Momentum
Conservation of Angular
Momentum
Conservation of Energy
9
Conservation of mass

Define the nominal density of mass
mass in current state

volume in reference state

A material particle does not gain or lose mass,
so that the nominal density of mass is time
independent during deformation
  (X )
10
Conservation of linear momentum

It requires that the rate of change of the linear
momentum, in any part of a body, should
equal to the force acting on the part
Linear momentum:
Rate of change:
x( X , t )
 t  ( X )dV ( X )
d x( X , t )
 2 x( X , t )
 ( X )dV ( X )  
 ( X )dV ( X )
2

dt
t
t
11
Forces

Nominal density of body force
force in current state
B( X , t ) 
volume in reference state

Nominal traction
force in current state
T ( X , t) 
area in reference state

Conservation of linear momentum
d
x( X , t )
 T ( X , t )dA   B( X , t )dV  dt   ( X ) t dV
12
Linear momentum…

Conservation of linear momentum

 2 x( X , t ) 
 T ( X , t )dA    B( X , t )   ( X ) t 2  dV 0
Inertial force
13
Linear momentum…

Stress-traction relation
siK N K  Ti
As the volume of the tetrahedron decreases, the ratio of area over
volume becomes large, and the surface traction prevail over the
body force
14
Linear momentum…

Divergence theorem
siK
 Ti dA   siK N K dA   X K dV

Conservation of linear momentum in
differential form
siK
 2 xi ( X , t )
 Bi ( X , t )   ( X )
X K
t 2
15
Conservation of angular momentum

For any part of a body at any time, the momentum
acting on the part equals to the rate of change in the
angular momentum
 x( X , t )  T ( X , t )dA   x( X , t )  B( X , t )dV
d
x( X , t )
  x( X , t ) 
 ( X )dV
dt
t
 siK FjK  s jK FiK or sF =Fs
T
T
The conservation of angular momentum requires that the product
sFT be a symmetric tensor.
16
Conservation of energy

Displacement of a particle
 x  x( X , t   t )  x( X , t )


Work

 2 xi
 Ti xi dA    Bi   t 2
Recall

 xi dV

 FiK  FiK ( X , t   t )  FiK ( X , t ) 
 xi
X K
Ti  siK N K
siK
 2 xi
 Bi   2
X K
t
work in current state

 siK  FiK
volume in reference state
The nominal stress is work-conjugate to the deformation gradient
17
Conservation of energy…

Heat
Q( X , t ) 
energy received up to current state
volume in reference state
energy across up to current state
q( X , t ) 
volume in reference state

Nominal density of internal energy
internal energy in current state
u( X , t ) 
volume in reference state
Conservation of energy requires the work done by the forces
upon the part and the heat transferred into the part equal
to the change in the internal energy
  udV   s
iK
 FiK dV    QdV    qdA
18
Conservation of energy…

q-IK relation

IK NK  q
Divergence theorem
I K
 qdA   I K N K dA   X K dV

Conservation of energy in differential form

 u  siK  FiK   Q 
 IK
X K
Work done by
external forces
Energy received
from reservoirs
Energy due to
net conduction
19
Conservation of energy…

When the body undergoes rigid-body rotation

x 2 ( X , t ) 
  T ( X , t )dA    B( X , t )   ( X )
dV 0

2
t



Free energy is unchanged
 siK FjK  s jK FiK
“…knowing the law of conservation of energy and the formulae
for calculating the energy, we can understand other laws. In other
words many other laws are not independent, but are simply
secret ways of talking about the conservation of energy. The
simplest is the law of the level”
---Richard Feynman
20
Product of entropy

To apply the fundamental postulate, we need to
construct an isolated system, and identify the
internal variables.

The body

A field of reservoirs in
volume thermal contact

A field of reservoirs in
surface thermal contact

All the mechanical forces
The mechanical forces do not contribute to the entropy
21
Entropy…

Nominal density of entropy
 ( X , t) 

entropy in current state
volume in reference state
To construct thermodynamics model of the
material, we assume the system has two
independent variables: the nominal density of
energy u, and the deformation gradient F
   (u, F )
22
Entropy…

An isolated system produces entropy by varying
the internal variables
Q
q
    dV  
dV   dA  0
R
R

A list of internal variables:
 , u, s, x, F, I, Q, q

Three types of constraints



Deformation kinematics
Conservation laws
Materials model    (u, F )
23
Entropy…

Differential form of 
 (u , F )
 (u , F )
 
u 
 FiK
u
FiK

Deformation gradient
xi ( X , t )
FiK 
X K


Conservation of energy
IK NK  q
 I K
 u  siK  FiK   Q 
X K
Entropy production of the composite
  
  xi

  FiK  u siK  X K dV   X K
  

  I K dV
 u 
  1 
 1  
 
   QdV    
  qdA  0
 u  R 
  R u 
24
Entropy…
  
  xi

  FiK  u siK  X K dV   X K


  

  I K dV
 u 
  1 
 1  
 
   QdV    
  qdA  0
 u  R 
  R u 
The variation of the independent internal variables are:
x, I, Q, q
The inequality consists of contributions to the entropy product
due to three distinct processes: the deformation of the body,
the heat conduction in the body, and the heat transfer between
the body and reservoirs
Material model?
25
Material model…

Thermodynamic

equilibrium: isothermal
deformation of an elastic
body
The model of isothermal
deformation of elastic
body is specified by the
following equations
 (u , F ) 1

u

 ( X , t )   R (t )
FiK 
siK
 (u , F )

FiK

xi ( X , t )
X K
siK N K  Ti
siK
 2 xi ( X , t )
 Bi ( X , t )   ( X )
X K
t 2
siK
 (u , F ) 1  (u , F )
 ,

u

FiK

 ( X , t )   R (t )
26
Material model…

Inverting the nominal density of entropy

u  u ( , F )
In differential form:


u ( , F )
u ( , F )
 u    siK  FiK ,  
, siK 

FiK
Nominal density of Helmholtz free energy
W  u   ,  W    siK  FiK
As a material model, we assume the free energy is a
function of temperature and deformation gradient
W  W ( , F )   W ( , F ) , s  W ( , F )
iK

FiK
27
Equation of motion

Given free-energy function W(F), the field equations
xi ( X , t )
FiK 
X K
siK
 2 xi ( X , t )
 Bi ( X , t )   ( X )
X K
t 2
siK 

W ( , F )
FiK
Boundary conditions
siK ( X , t ) N K ( X )  prescribed, for X  s t
x( X , t )  prescribed, for X  s u

Initial conditions x( X , t0 ), V ( X , t0 )
28
Summary


Kinematics of deformation
Conservation laws






Conservation of mass
Conservation of linear momentum
Conservation of angular momentum
Conservation of energy
Product of entropy
Material model
Equation of motion
29
Research interest

Coupled diffusion and creep deformation of
Li-ion battery electrode

Stress level induced by lithium ion insertion
30
Research interest…

For insertion processes, the deformation of the
host material may be assumed to be linear with
the volume of ions inserted
d  ij 1
1  C  sij
 (1  v) ij  v kk ij   
 ij  
dt
dt
dt  3
 2
 Assume lithium ion is much more mobile than
the host particles
Coupled partial differential equations of
concentration and stress field
With material law, appropriate boundary
conditions, it’s solvable!!!
31
Thanks!
32