Notes on Strain and Deformation Tensors

7/24/2019 Notes on Strain and Deformation Tensors

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Notes on strain and deformation tensors

Marcel FrehnerSeptember 2006

[email protected]

for internal use only

These short notes summarize the basic principles and equations describing deformation and strain.

They do not contain a lot of words but mainly mathematical equations. Most of the descriptions are

taken from the internet site

www.engin.brown.edu/courses/en175/Notes/Kinematics/Kinematics.htm

which provides a more detailed description. A Matlab (Matlab 7.1.0.246) and a Maple (Maple 10)

script were written as well to understand the basic tensors described in this text. They are named

Matlab script: Strain_Tensors.m

Maple script: Strain_Tensors.mw

In the following text all bold letters denote vectors or tensors, normal letters denote scalar values.

1.

Displacement field, displacement gradient tensor and transformation tensor


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Definitions: P= (p1,p2) Point before deformation with coordinates p1and p2

Q= (q1,q2) Point before deformation with coordinates q1and q2

P = (p1,p2) Point after deformation with coordinates p1 and p2

Q = (q1,q2) Point after deformation with coordinates q1 and q2

l length of line segment between points Pand Qbefore deformation

l length of line segment between points P and Q after deformation

m vector of unit length pointing from Pto Qbefore deformation

m vector of unit length pointing from P to Q after deformation

therefore dx = l m vector connecting point Pand Q

dx = l m vector connecting point P and Q

Q = P+ dx

Q = P + dx

The displacement field u(x,y) a vector field describes the displacement for every point. The

displacement gradient tensor H(x,y) describes the spatial change of the displacement field.

( , ) , ( , )

x x

x

y y y

u u

u x yx y x y

u u u

x y

= =

u H (1)

The displacement gradient tensor H can be used to calculate the displacement of point P and Q

assuming that points Pand Qare infinitesimally close to the point of origin where the displacement

gradient tensor is defined here. With this assumption a first order Taylor expansion can be made using

the displacement gradient tensor.

1 2

1 2

' ( , ) ( )

' ( , ) ( )

p p

q q

= + = + = +

= + = + = + = + + +

P P u P HP I H P

Q Q u Q HQ I H Q P dx HP Hdx (2)

where Iis the identity tensor. The coordinate transformation written out:

1 1

2 2

1'

'1

x x

y y

u u

x yp p

u up p

x y

+ = +

F

144424443

(3)

The transformation tensor Fis defined as

+F = I H (4)


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The vector connecting the two points changes according to

' ' 'du = dx - dx = (Q - P ) - (Q - P)

d u = P + d x + H P + H d x - P - H P - P - d x + P

du = Hdx

(5)

or 'dx = dx + H dx = Fdx (6)

Fmaps any undeformed vector into its deformed state (Eqation 6). This vector can also be a position

vector of a point. Therefore Falso maps any point into its new position after deformation (Equation 3).

According to Equation (6) the deformation gradient tensor maps a undeformed vector into its deformed

state. Considering two successive deformations Equation (6) is applied twice.

1

2 2 1

'

'' 'tot

=

= =

dx F dx

dx = F dx F Fdx F dx (7)

where dx represents the vector after two deformation steps. The finite transformation tensor Ftotmaps

the undeformed vector dxdirectly into its state after two deformation steps dx. This procedure can be

applied to any number of deformation steps. The finite transformation tensor Ftot then maps the

undeformed vector directly into its final state. All following considerations can be applied to F, as well

as to Ftot, except formulations for infinitesimal strain.

2. Cauchy-Green strain tensors

Using Equation (6) the length of a vector after the deformation l can be expressed with the length

before the deformation l.

2 2' ' ' ( )T T T T T T l l= = = =dx dx Fdx Fdx dx F Fdx m F Fm (8)

Quadratic extension of a vector is defined as follows. Equation (8) is substituted into the definition.

{

2 2

2 2

' T T T Tl l

l l= = =

C

m F Fmm F F m (9)

where Cis the right Cauchy-Green strain tensor.

TC = F F (10)


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Using Equation (6) the length of a vector before the deformation l can be expressed with the length

after the deformation l.

( )1

2 1 1 1 2( ') ' ' ' ' ' 'T T T T T T l l = = = =dx dx F dx F dx dx F F dx m FF m (11)

Inverse Quadratic extension Iof a vector is defined as follows. Equation (11) is substituted into the

definition.

( )( )

122

1

2 2

' ' '' '

' '

T T

T T

I

ll

l l

= = =

B

m FF mm FF m123

(12)

where Bis the left Cauchy-Green strain tensor.

TB = F F (13)

Both Cauchy-Green strain tensors contain information about the strain, i.e. change of length of a

vector. They are both symmetric and positive definite. They do not contain information about the rigid

body rotation, i.e. rotation of a vector without change of length. The information of the total

deformation, i.e. change of length of a vector AND rigid body rotation is only provided by the

displacement gradient tensor Hor by the transformation tensor F.

3. Lagrange strain tensor

The Lagrange description of strain is similar to the Cauchy-Green description of the quadratic strain

(Equation 9). It only uses a different definition of the quadratic extension, here called the Lagrange

quadratic extension L. Equation (8) is again substituted into the definition.

( )

( )

22 2 2 2

2 2 2

1'

2 2 2

1 1

2 2

T TT T

L

T T

T TL

ll l l l

l l l

= = =

= =

E

m F Fmm F Fm

m F Fmm F F I m14243

(14)

The Lagrange strain tensor Eis defined as

( )1

2

T= E F F I (15)


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4. Eulerian strain tensor

The Eulerian description of strain is similar to the Cauchy-Green description of the inverse quadratic

strain (Equation 12). It only uses a different definition of the inverse quadratic extension, here called

the Eulerian quadratic extension E. Equation (11) is again substituted into the definition.

( ) ( )( )

( )( )( )

*

11 22 22 2

2 2 2

1

1

' 1 ' '' ' ' ''

2 ' 2 ' 2 '

1 ' ' 1' '

2 2

T TT T

E

T T

T T

E

ll ll l

l l l

= = =

= =

E

m FF mm FF m

m FF mm I FF m1442443

(16)

The Eulerian strain tensor E*is defined as

( )( )1

* 1

2

T

= E I FF (17)

Both Lagrange and Eulerian strain tensors contain information about the strain, i.e. change of length of

a vector. They are both symmetric and positive definite. They do not contain information about the

rigid body rotation, i.e. rotation of a vector without change of length. The information of the total

deformation, i.e. change of length of a vector AND rigid body rotation is only provided by the

displacement gradient tensor Hor by the transformation tensor F.

5. Infinitesimal strain tensor and infinitesimal rotation tensor

The general Lagrange strain tensor Ewritten out is

( )

22

111 01 1

0 12 21 1

21

2

x xyx

T

y y yx

y y y yx x x x x

yx x x

u uuu

x yx x

u u uu

y y x y

u u u uu u u u u

x x x y x y x y x

uu u u

y x y x

++ = = + +

+ + + + + =

+ +

E F F I

E2 2

2y y y y x

u u u u u

y x y y y

+ + +

(18)


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For infinitesimal strain, i.e. all terms are much smaller than 1, the quadratic terms in Equation (18) are

negligible. The infinitesimal strain tensor results

21

22

yx x

y yx

uu u

x y x

u uu

y x y

+

=

+

E (19)

This is equal to ( )1

2

T= + H H (20)

The infinitesimal strain tensor is only an approximation of the more general Lagrange strain tensor for

small strains. It contains information about the strain, i.e. change of length of a vector. It is symmetric

and positive definite. It does not contain information about the rigid body rotation, i.e. rotation of a

vector without change of length. The information of the total deformation, i.e. change of length of a

vector AND rigid body rotation is only provided by the displacement gradient tensor H or by the

transformation tensor F.

Equivalent to the infinitesimal strain tensor the infinitesimal rotation tensor can be defined. It is

also an approximation for a more general rotation tensor Rfor small rotations. It describes the rigid

body rotation, i.e. rotation of a vector without change of length.Ris not described here.

( )12

T= H H (21)

Only the combination of the infinitesimal strain tensor and the infinitesimal rotation tensor

provides information about the total deformation.

= +H (22)

6. Principal values and directions of strain

There are several equivalent ways to calculate the principal stretching values e1and e2.

1.

With the eigenvalues Cof the right Cauchy-Green strain tensor C

1,2 1c Ce = (23)

2. With the eigenvalues Bof the left Cauchy-Green strain tensor B

1,2 1b Be = (24)

3. With the eigenvalues Eof the Lagrange strain tensor E

1,2 1 2 1e Ee = + (25)


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4. With the eigenvalues E*or the Eulerian strain tensor E*

* *

1,2 1 2 1e Ee = (26)

5.

With the eigenvalues

of the infinitesimal strain tensor . This method provides only anapproximation for small strains.

1,2e

(27)

The principal stretching values 1and 2describe the stretching of the two vectors that are stretched

maximally and minimally in a certain displacement field. The length change of these vectors is

( )1,2' 1l e l= + (28)

or

( )

( )

1

max

2

min

'1

'1

le

l

le

l

= +

= +

(29)

The orientation of these two vectors can either be defined before or after the deformation.

1. Before deformation: The orientation of a vector that WILL BE stretched maximally or

minimally in a certain displacement field is given by

the normalized eigenvectors vC1and vC

2of the right Cauchy-Green strain tensor C

the normalized eigenvectors vE1and vE

2of the Lagrange strain tensor E

the normalized eigenvectors v1and v

2of the infinitesimal strain tensor .These

eigenvectors are an approximation for small strains.

2. After deformation: The orientation of a vector that WAS stretched maximally or minimally

in a certain displacement field is given by

the normalized eigenvectors vB1and vB

2of the left Cauchy-Green strain tensor B

the normalized eigenvectors vE*1and vE*

2of the Eulerian strain tensor E*


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7. Strain ellipse

The strain ellipse is a graphical display of the state of strain. Before deformation a circle is assumed

which is then deformed in a certain displacement field. The resulting ellipse shows the direction and

length of the two principal stretching directions. So the vectors of the major and minor axes of the

strain ellipse are

major 1 1 1 1 1 1

* *

minor 2 2 2 2

(1 ) (1 ) (1 )

(1 ) (1 )

C C E E

B B E E

e e e

e e

= + = + +

= + = +

a v v v

a v v (30)

A finite strain ellipse, i.e. a deformed circle after several deformation steps can be calculated the same

way but using the finite transformation tensor Ftotaccording to Equation (7).

8. Some helpful tensor rules

There are some tensor rules that may help with the handling of tensors.

( )A B +C = AB + AC

( ) ( )A BC = AB C

( )T T T=AB B A

( ) ( )1

1 T

T T = =A A A

( )1 T

T T =A B B A

( )1 1 1 =AB B A

Notes on Strain and Deformation Tensors

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