FOR SOFT TISSUES MECHANICS BASICS OF CONTINUUM

BASICS OF CONTINUUM

MECHANICS

FOR SOFT TISSUES

Review: deformation

Deformation Mapping

Eulerian/Lagrangian descriptions of motion

Deformation Gradient

x

u(x)

e₃

e₁ e₂

Original Configuration

Deformed Configuration

y

1 2 3

( , , , )

i i i

y  x u x x x t

const

( , ) ( , )

i

i i

i i i j i j

x

y u

y x u x t v x t

t _ t

 

   

 

2

const 2 const

( , ) ( , ) ( , )

i i

i i

i i i j i j i j

x x

y y

y x u y t v y t a y t

t _ t _

 

   

 

2

const 2 const

const const

( , ) ( , )

i i

i i

i k i i i i

ik i j k j

k x y x y k

u y u y v v

a y t v y t

y t t t t y



 

 

      

    

       

 

 

 

( )

or ⁱ _{i i ij} ⁱ _ij

j j j

y u

x u F

x x  x

    

  

    

  

y x u x F

x u(x)

e₃

e₁ e₂

dx dy

u(x+dx)

Original Configuration

Deformed Configuration

i ik k

d d

dy F dx

 



y F x

2

Review

Sequence of deformations

u⁽¹⁾(x)

e₃

e₁ e₂

dx dy

Original Configuration

After first deformation

dz u⁽²⁾(y)

y z

After second deformation

(2) (1) (2) (1)

with or

d z   F d x F  F  F dz  F dx F  F F

Lagrange Strain

e₃

e₁ e₂

l₀

Original Configuration

Deformed Configuration

m

l

1 1

( ) or ( )

2 2

T

ij ki kj ij

E F F 

    

E F F I

 

2 2

0

2 2

0 0 0

2 2

ij i j

l l l l

E m m

l l l

 

      

m E m

Review

 

0

dV det

dV  F  J

Volume Changes

dv

dw

e₃

e₁ e₂

Undeformed

Deformed

dV₀ dV

e₃

e₁ e₂

Original Configuration

Deformed Configuration

t

dA₀

dA n n₀

x

u(x)

dP⁽ⁿ⁾ dP₀⁽ⁿ⁾

Area Elements





1 1

or

2 2

j

T i

ij

j i

u u

x x

 ^{  }

        

ε u u

Infinitesimal Strain

y

x x

y

l₀ l₀





xx yy

l₀ l₀

e₁ e₂

xy

O A

B C

O A

B C



1 1

2 2

j j

i k k i

ij ij

j i j i j i

u u

u u u u

E  ^^x ^x ^x ^x ^ ^^x ^x ^

Approximates L-strain

Related to ‘Engineering Strains’

11 22

12 21 / 2 / 2

xx yy

xy yx

 

 

   





  

4

Review

Principal values/directions of Infinitesimal Strain

e₃

e₁ e₂

Original Configuration

n⁽¹⁾

n⁽²⁾ n⁽²⁾

n⁽¹⁾



( ) ( )

( ) ( )

or

i i

i

i i

kl l i l

e

n e n



 



ε n n





1 1

or

2 2

T i j

ij

j i

u u

w x x

   

        

w u u

Infinitesimal rotation

e3

e₁ e₂

Original

Configuration Deformed

Configuration n⁽¹⁾

n⁽²⁾ I+w

n⁽²⁾

n⁽¹⁾

Decomposition of infinitesimal

motion

i ij ij

j

u w

x



  



Left and Right stretch tensors, rotation tensor

  F R U

  F V R

(1) (1) (2) (2) (3) (3)

1 2 3

  

     

U u u u u u u

(1) (1) (2) (2) (3) (3)

1 2 3

  

     

V v v v v v v

U,V symmetric, so



principal stretches

e₃

e₁ e₂

Original Configuration

Deformed Configuration u⁽¹⁾

u⁽²⁾ v⁽²⁾

v⁽¹⁾

U R

u⁽²⁾

u⁽¹⁾ _ _

_ u⁽²⁾

u⁽¹⁾ u⁽³⁾

u⁽¹⁾

1 1

1

u⁽²⁾

u⁽³⁾

Review

Left and Right Cauchy-Green Tensors

2 T

T

  

   C F F U B F F V

6

Review

Generalized strain measures

3 ( ) ( )

i=1

3 ( ) ( )

i=1

Lagrangian Nominal strain: ( 1)

Lagrangian Logarithmic strain: log( )

i i

i

i i

i





 







u u

u u

3 ( ) ( )

i=1

3 ( ) ( )

i=1

Eulerian Nominal strain: ( 1)

Eulerian Logarithmic strain: log( )

i i

i

i i

i





 







v v

v v

* 1 1 * 1 1 1

( ) or ( )

2 2

T Eij ij Fki Fkj

   

    

E I F F

Eulerian strain

8

 x

  X , t x  

 X

Lagrangian description: Eulerian description

 

  



F  

 

 

F

dM  ₀  ₀ ^t

 



 



J 

 



 J X ,t d d

     

C ^t

 



 divV x,t d d 

 

V 

 

V grad M

d  

 





d  ^t

2 1





Rate of deformation:

 

   

t t , t , X x

V 



 

 

div 

 

grad 

dA

dP

0

lim

dA

d

 dA

 P

t

dV

dP

0

lim

dV

d

 dV



P

b

t

R

t

e₃ e₂

e₁

n T(n)

T(-n)

( ) -n

0

( ) lim

dA

d

 dA

 P ⁿ

T n

dA

dP

Review: Kinetics

e3

e₁ e₂

Original Configuration

Deformed Configuration

S

R R₀

S₀ b

t

Surface traction

Body Force

Internal Traction

( )

A V

dA









P T n b ^e2

S

R R₀

S₀

b

t

n

V T(n)

Resultant force on a volume

Restrictions on internal traction vector

e₁

e₃ e₂

T(n) n

T(-e₁) -e₁

dA₁

dA⁽ⁿ⁾

dA₁ dA⁽ⁿ⁾

Review: Kinetics

n -n

T(n)

T(-n)

(  ) ( ) T n T n

Newton II

( )   or ( ) T

 n



T n n σ n

Newton II&III

Cauchy Stress Tensor

e₁ e₃

e₂

₁₁

₁₂

₁₃

₂₁

₂₂

₂₃

₃₁

₃₂

₃₃

10

Other Stress Measures

ij ij

J  J 

 

τ σ

e3

e₁ e₂

Original Configuration

Deformed Configuration

S

R R₀

S₀ b

i t

ij ij

j

F u

 ^x

    

F I u 

det( )

J  F

Kirchhoff

Nominal/ 1

Piola-Kirchhoff

Material/2

Piola-Kirchhoff

1 1

ij ik kj

J ^

S

  

S F

σ

1 T 1 1

ij ik kl jl

J ^{ } JF^  F^

 



Σ F

σ

( ) 0

0 i ij

dPjⁿ  dA n S

e₃

e₁ e₂

Original Configuration

Deformed Configuration

t

dA₀

dA n n₀

x

u(x)

dP⁽ⁿ⁾ dP₀⁽ⁿ⁾



Review: Thermodynamics

e3

e₁ e₂

Original Configuration

Deformed Configuration

S

R R₀

S₀ b

t

Specific Internal Energy

Specific Helmholtz free energy     s Temperature 

Heat flux vector q External heat flux ^q

First Law of Thermodynamics d ( )

KE Q W dt    

ij ij i const i

D q q

t y

  





   

 _x 

Second Law of Thermodynamics ^dS_dt ^{ d}_dt^{ 0} Specific entropy s

( / )

i 0

i

s q q

t y

 



 

  

 

1 0

ij ij i

i

D q s

y t t

  

 



   

        12

Constitutive Laws

General Assumptions:

1. Local homogeneity of deformation

(a deformation gradient can always be calculated) 2. Principle of local action

(stress at a point depends on deformation in

a vanishingly small material element surrounding the point)

Restrictions on constitutive relations:

1. Material Frame Indifference – stress-strain relations must transform consistently under a change of observer

2. Constitutive law must always satisfy the second law of

thermodynamics for any possible deformation/temperature history.

Equations relating internal force measures to deformation measures are known as Constitutive Relations

e₃

e₁ e₂

Original Configuration

Deformed Configuration

1 0

ij ij i

i

D q s

y t t

  

 



    

        