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General Theory of Finite Deformation

Kejie Zhao

Instructor: Prof. Zhigang SuoMay.21.2009

Harvard School of Engineering and Applied Sciences

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Beyond linear theory

Ingredients in linear theory Deformation geometry Force balance Material model

Beyond linear theory

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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

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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

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Kinematics…

Field of deformation ( , )x x X tA central aim of continuum mechanics is to evolve the field of deformationby developing an equation of motion

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Kinematics… Displacement

Velocity

Acceleration

( , ) ( , )x x X t t x X t ( , ) ( , ) ( , )x X t t x X t x X t

vt t

2

2

( , )x X ta

t

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Kinematics…

Deformation gradient( , ) ( , ) ( , )i i i

iKK K

x X dX t x X t x X tF

dX X

( , )dx F X t dX

•F(X,t) maps the vector betweentwo nearby material particles inreference state, dX, to the vectorbetween the same two materialparticles in the current state, dx

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Kinematics…

Polar decomposition: any linear operator can be written as a product

F RU

Rotation vector Stretch vector

2 TU C F F C: Green deformation tensor

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Conservation laws

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Conservation of mass

Define the nominal density of mass

A material particle does not gain or lose mass, so that the nominal density of mass is time independent during deformation

mass in current state

volume in reference state

( )X

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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 tX dV X

t

2

2

( , ) ( , )( ) ( ) ( ) ( )

d x X t x X tX dV X X dV X

dt t t

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Forces

Nominal density of body force

Nominal traction

Conservation of linear momentum

force in current state( , )

volume in reference stateB X t

force in current state( , )

area in reference stateT X t

( , )( , ) ( , ) ( )

d x X tT X t dA B X t dV X dV

dt t

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Linear momentum…

Conservation of linear momentum2

2

( , )( , ) ( , ) ( ) 0

x X tT X t dA B X t X dV

t

Inertial force

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Linear momentum…

Stress-traction relation

As the volume of the tetrahedron decreases, the ratio of area over volume becomes large, and the surface traction prevail over the body force

iK K is N T

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Linear momentum…

Divergence theorem

Conservation of linear momentum in differential form

iKi iK K

K

sT dA s N dA dV

X

2

2

( , )( , ) ( )iK ii

K

s x X tB X t X

X t

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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

The conservation of angular momentum requires that the product

be a symmetric tensor.

( , ) ( , ) ( , ) ( , )

( , )( , ) ( )

x X t T X t dA x X t B X t dV

d x X tx X t X dV

dt t

T T or sF =FsiK jK jK iKs F s F

TsF

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Conservation of energy Displacement of a particle

Work

Recall

2

2i

i i i i

xT x dA B x dV

t

2

2

( , ) ( , ) iiK iK iK

K

i iK K

iK ii

K

xF F X t t F X t

X

T s N

s xB

X t

( , ) ( , )x x X t t x X t

work in current state

volume in reference state iK iKs F

The nominal stress is work-conjugate to the deformation gradient

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Conservation of energy…

Heat

Nominal density of internal energy

energy received up to current state( , )

volume in reference stateQ X t

energy across up to current state( , )

volume in reference stateq X t

internal energy in current state( , )

volume in reference stateu X t

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

iK iKudV s F dV QdV qdA

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Conservation of energy…

q-IK relation

Divergence theorem

Conservation of energy in differential form

K KI N q

KK K

K

IqdA I N dA dV

X

iK iK KK

u s F Q IX

Work done by external forces

Energy received from reservoirs

Energy due tonet conduction

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Conservation of energy…

When the body undergoes rigid-body rotation

Free energy is unchanged

“…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

2

2

( , )( , ) ( , ) ( ) 0

x X tT X t dA B X t X dV

t

iK jK jK iKs F s F

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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

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Entropy…

Nominal density of entropy

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

entropy in current state( , )

volume in reference stateX t

( , )u F

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Entropy…

An isolated system produces entropy by varying the internal variables

A list of internal variables:

Three types of constraints Deformation kinematics Conservation laws Materials model

0R R

Q qdV dV dA

, , s, x, F, I, Q, qu

( , )u F

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Entropy…

Differential form of

Deformation gradient

Entropy production of the composite

Conservation of energy( , ) ( , )

iKiK

u F u Fu F

u F

( , )iiK

K

x X tF

X

K K

KiK iK

K

I N q

Iu s F Q

X

1 10

iiK K

iK K K

R R

xs dV I dV

F u X X u

QdV qdAu u

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Entropy…

The variation of the independent internal variables are:

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

1 10

iiK K

iK K K

R R

xs dV I dV

F u X X u

QdV qdAu u

x, I, Q, q

Material model?

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Material model…

Thermodynamic equilibrium: isothermal deformation of an elastic body

( , ) 1

( , ) ( )

( , )R

iK

iK

u F

uX t t

su F

F

The model of isothermal deformation of elastic body is specified by the following equations

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2

( , )

( , )( , ) ( )

( , ) 1 ( , ),

( , ) ( )

iiK

K

iK K i

iK ii

K

iK

iK

R

x X tF

X

s N T

s x X tB X t X

X t

su F u F

u F

X t t

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Material model… Inverting the nominal density of entropy

In differential form:

Nominal density of Helmholtz free energy

As a material model, we assume the free energy is a function of temperature and deformation gradient

( , )u u F

( , ) ( , ), , iK iK iK

iK

u F u Fu s F s

F

, iK iKW u W s F

( , )W W F ( , ) ( , ), iK

iK

W F W Fs

F

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Equation of motion

Given free-energy function W(F), the field equations

Boundary conditions

Initial conditions

2

2

( , )

( , )( , ) ( )

( , )

iiK

K

iK ii

K

iKiK

x X tF

X

s x X tB X t X

X t

W Fs

F

t

u

( , ) ( ) prescribed, for X s

( , ) prescribed, for X siK Ks X t N X

x X t

0 0( , ), ( , )x X t V X t

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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

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Research interest

Coupled diffusion and creep deformation of Li-ion battery electrode Stress level induced by lithium ion insertion

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Research interest… For insertion processes, the deformation of the

host material may be assumed to be linear with the volume of ions inserted

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!!!

1 1(1 )

3 2ij ij

ij kk ij ij

d sCv v

dt dt dt

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Thanks!