expectations after today’s lecture know stretch, deformation gradient, and deformation tensor know...

Expectationsafter today’s lecture

• Know stretch, deformation gradient, and deformation tensor• Know the strain descriptions

– Engineering– True– Almansi– Green

• Know how to obtain strain from stretch or displacement• Be able to transform a state of strain from one system of

coordinates to another and find principal strains using:– Direct methods– Mohr’s circle– Eigenvalues and eigenvectors

• Revisit stress for generalize case. (Previously formulated finite descriptions only for cases without shear stresses.)

Important observation:

The part of the brain that’s good for math is different from the part that can communicate math.

“Mathematics has no symbols for confused ideas.”

George Stigler

“Calculus is the language of God.”Richard Feynman

StrainContinuum Mechanics

BME 615

Infinitesimal strain description(with displacements)

Finite Strain

Salvador Dali

If L = 1.01 and L0 = 1.00, or 1% strain

If L = 2 and L0 = 1,

Patterns of deformation• These are cases of uniform strain

(or, linearly changing strain vertically but uniform axially in bending) across the specimen.

• Consider case (a) where L = final length and L0 = initial length. Strain (which is normalized deformation) can be normalized in number of ways.

GreenStrain

AlmansiStrain

EngineeringStrain

Fung YC, Biomechanics ref. p29

1 2

1

8

3e

TrueStrain

0.7t

1D Strain defined by stretch

Define stretch0L

L

Infinitesimal Strain (engineering strain) 10

0

L

LL

Finite Strain (typically used when strain exceeds 10%)

In Lagrangian (material) reference system, define Green (St. Venant) strain

In Eulerian (spatial) reference system, define Almansi (Hamel) strain

Strain defined by stretch - on a differential element

Original shape differential lengths

Now define stretch on a diff. element

In 1D, Lagrangian deformation tensor

dX1

dX2

dX3

Deformed shape differential lengths

dx3

dx2

dx1

2C

In 1D, Green strain

Note: There can also be shear deformations defined by stretch

Strain defined by stretch – continued 1

So that Green strain in 3D is

In 3D, stretch becomes Lagrange deformation gradient F

In 3D, Lagrange deformation tensor C – also called the right Cauchy-Green deformation tensor is

where is the identity matrix (ones on diagonal terms and zeros elsewhere)

I

Strain defined by stretch – continued 2

The 3D Eulerian or Cauchy deformation tensor c is

the Cauchy deformation tensor c in 1D in Eulerian reference system can be used to define Almansi strain e on a differential element

In 3D, becomes the Eulerian deformation gradient f

where B is the left Cauchy-Green deformation tensor

1

Note: •Green strain and Almansi strain are consistent measures of normalized finite deformation in their respective reference systems. •Engineering strain is not! •Engineering strain is a first order approximation that works well when deformations are small (usually < 10%)

Other common measures of strain

Mean normal strain:

Spherical strain:

Note: Constant relating pressure to spherical strain is called the Bulk Modulus

Deviatoric strain:

where δij is Kronecker delta: = 1 if i = j and = 0 if i ≠ j

Volumetric Change

i = L/L0 = 2

V0 = 1

Ai = 4

V = 8

Biaxial Stretch

1

2

1

1

1.5

1.5

Shear Deformation – 45o

1

2

1

1 1

1

Shear Deformation – 45o

1

2

1

1 1

1

v = 0

2

2u

Shear Deformation – 45o

1

2

1

1 1

1

v = 0

2

2u

If thickness into plane remains the same, have we lost volume?

How do you know?

Shear Deformation – 45o

1

2

1

1 1

1

Note: Above is affine mapping of

where

Then derivatives give the same results!Check corners to prove

Definition of “congugates” from Oxford English Dictionary

• Mathematics: Joined in a reciprocal relation• Biology: Fused• Chemistry: Related to…..• Mechanics: Variables that are defined in such

a way they are duals of one another

Energy Conjugates

Kirchhoff Stress S and Green Strain E (Lagrangian reference system)

12

1 2 E

Lagrange Stress T and right Cauchy-Green deformation tensor C

Cauchy Stress s and Almansi Strain e (Eulerian reference system)

A

Fs

Energy Conjugates & SED(assume incompressibility)

F = 1 = L/L0 = 2

A0 = 1

A = ½V = 1

Energy Conjugates & SED

+

+

+

s = 2e = 3/8

= 1 = 1

S = 1/2E = 3/2

W = 3/8

W = 1/2

W = 3/8

0.25 0.5 0.75Strain

Str

ess

1.0 1.25 1.5

0.5

1.0

1.5

2.0

Finite strain descriptions(with displacements)

X

Y

Finite strain from deformations Common notation here can be troubling. Do not confuse deformations with displacements

wherefor i = 1, 2, 3 is the original coordinate vector

is the deformed coordinate vector

is the displacement vector

http://en.wikipedia.org/wiki/Finite_strain_theory

3 things that drive me crazy!

Stretch ≠ strainMechanical behavior ≠ material behavior

Deformation ≠ displacement

Green strain from deformations for i, j = 1, 2, 3 sum on k

Thus, for i = 1, j = 1

Note, for small deformations, higher order terms are not significant.

Note, for 1D we can easily go from previous formula for E to current form.

Almansi strain from deformations

for i, j = 1, 2, 3 sum on k

Second Order Strain Tensors

Any strain formulation in 3D is a 2nd order tensor. Therefore, it has the following properties:1. Transformation methods that we used for stress hold for

principal strain or maximum shear strain or strain on any axis, etc.

2. Mohr’s circle method holds for strain transformations.3. All the same invariants hold to describe dilatational (or

hydrostatic) versus distortional (or deviatoric) strains.4. The same methods for eigenvalues and eigenvectors

hold.

Revisit Cauchy Stress – (s)

• Eulerian reference of deformed state

• Unloaded thickness h0

• Loaded thickness h• Unloaded density ρ0

• Unloaded density ρ

where

Lagrangian stress (T) revisited (1st Piola-Kirchhoff stress tensor)

• Lagrangian reference of undeformed state

• Unloaded thickness h0

• Loaded thickness h• Unloaded density ρ0

• Unloaded density ρ

011 11 211 11 2 3 11

20 0 2 20 0 1

1F F L hT s s

L h L h L h

1 1det

detT F F s or s F T

F

Kirchhoff Stress (S) revisited (2st Piola-Kirchhoff stress tensor)

• Kirchhoff stress references the undeformed state

• Unloaded thickness h0

• Loaded thickness h• Unloaded density ρ0

• Unloaded density ρ

Example Strain Problem

BME 615

1 1 2x X kX

Consider a deformation that is given by:

2 1 2x kX X

Xi (i=1~3) represent original coordinates

k represents displacement gradient

Start with an undeformed unit square and draw the deformation

X1, x1

X2, x2

k

k

1

1

3 3x X

X1, x1

X2, x2

k

k

1

1

1 1 2x X kX

2 1 2x kX X

Evaluate the Lagrangian deformation gradient tensor

1 1 1

1 2 3

2 2 2

1 2 3

3 3 3

1 2 3

1 0

1 0

0 0 1ij

x x x

X X Xk

x x xF k

X X X

x x x

X X X

3 3x X

Deformation tensor

2

2

1 2 01 0 1 0

1 0 1 0 2 1 0

0 0 1 0 0 1 0 0 1

T

k kk k

C F F k k k k

Green strain tensor

22

2 2

1 2 0 1 0 0 2 01 1 1

2 1 0 0 1 0 2 02 2 2

0 0 1 0 0 00 0 1

k k k k

E C I k k k k

Strain transformations

Just like stress, equations from the direct approach can be used for strain

Or these equations can be reformulated with a double angle trig identity

Mohr’s circle approach

Strain gage rosettes

Rectangular rosette

3 equations, 3 unknowns relating 1 2, , , ,A B C to

Delta rosette

Following similar approach, one can obtain principal strains and orientation

Expectationsafter today’s lecture

• Know stretch, deformation gradient, and deformation tensor• Know the strain descriptions

– Engineering– True– Almansi– Green

• Know how to obtain strain from stretch or displacement• Be able to transform a state of strain from one system of

coordinates to another and find principal strains using:– Direct methods– Mohr’s circle– Eigenvalues and eigenvectors

• Revisit stress for generalize case. (Previously formulated finite descriptions only for cases without shear stresses.)