1. Expectations after 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.)

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

3. “Mathematics has no symbols for confused ideas. ”
“Mathematics has no symbols for confused ideas.” George Stigler “Calculus is the language of God.” Richard Feynman

4. Strain Continuum Mechanics
BME 615
Strain Continuum Mechanics

5. Infinitesimal strain description (with displacements)

6. Finite Strain
Salvador Dali

7. 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.
Fung YC, Biomechanics ref. p29
If L = and L0 = 1.00, or 1% strain
If L = 2 and L0 = 1,
Engineering
Strain
True
Strain
Almansi
Strain
Green
Strain

8. 1D Strain defined by stretch
Define stretch
Infinitesimal Strain (engineering strain)
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

9. Strain defined by stretch - on a differential element
Now define stretch on a diff. element
dX2
dx2
dX3
dx3
dX1
dx1
Original shape
differential lengths
Deformed shape
differential lengths
In 1D, Lagrangian deformation tensor
Note: There can also be shear deformations defined by stretch
In 1D, Green strain

10. Strain defined by stretch – continued 1
In 3D, stretch l becomes Lagrange deformation gradient F
In 3D, Lagrange deformation tensor C
– also called the right Cauchy-Green deformation tensor is
So that Green strain in 3D is
where is the identity matrix
(ones on diagonal terms and zeros elsewhere)

11. Strain defined by stretch – continued 2
the Cauchy deformation tensor c in 1D in Eulerian reference system can be used to define Almansi strain e on a differential element
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%)
In 3D, becomes the Eulerian deformation gradient f
The 3D Eulerian or Cauchy deformation tensor c is
where B is the left Cauchy-Green deformation tensor

12. Other common measures of strain
Spherical strain:
Note: Constant relating pressure to spherical strain is called the Bulk Modulus
Mean normal strain:
Deviatoric strain:
where δij is Kronecker delta: = 1 if i = j and = 0 if i ≠ j

13. Volumetric Change
V0 = 1
li = L/L0 = 2
Ai = 4
V = 8

14. Biaxial Stretch 2 1
1.5 1

15. Shear Deformation – 45o 2 1 1 1 1 1

16. Shear Deformation – 45o 2 1 1
v = 0 1 1 1

17. Shear Deformation – 45o 2 1 1
v = 0 1 1 1
If thickness into plane remains the same, have we lost volume?
How do you know?

18. Shear Deformation – 45o 2 1 Note: Above is affine mapping of where
Check corners to prove
Then derivatives give the same results!

19. 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

20. Energy Conjugates
Lagrange Stress T and right Cauchy-Green deformation tensor C
Kirchhoff Stress S and Green Strain E (Lagrangian reference system)
Cauchy Stress s and Almansi Strain e (Eulerian reference system)

21. Energy Conjugates & SED (assume incompressibility)
V = 1
F = 1
l = L/L0 = 2

22. Energy Conjugates & SED
0.25
0.5
0.75
Strain
Stress
1.0
1.25
1.5
2.0
s = 2
e = 3/8 +
W = 3/8
s = 1
e = 1 +
W = 1/2
S = 1/2
E = 3/2 +
W = 3/8

23. Finite strain descriptions (with displacements)Y X

24. Finite strain from deformations
Common notation here can be troubling. Do not confuse deformations with displacements
for i = 1, 2, 3
where
is the original coordinate vector
is the deformed coordinate vector
is the displacement vector

25. 3 things that drive me crazy!
Stretch ≠ strain Mechanical behavior ≠ material behavior Deformation ≠ displacement

26. 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.

27. Almansi strain from deformations
for i, j = 1, 2, 3 sum on k

28. Second Order Strain Tensors
Any strain formulation in 3D is a 2nd order tensor.
Therefore, it has the following properties:
Transformation methods that we used for stress hold for principal strain or maximum shear strain or strain on any axis, etc.
Mohr’s circle method holds for strain transformations.
All the same invariants hold to describe dilatational (or hydrostatic) versus distortional (or deviatoric) strains.
The same methods for eigenvalues and eigenvectors hold.

29. Revisit Cauchy Stress – (s)
Eulerian reference of deformed state
Unloaded thickness h0
Loaded thickness h
Unloaded density ρ0
Unloaded density ρ
where

30. 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

31. 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

32. Example Strain Problem
BME 615

33. Consider a deformation that is given by:
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 1

34. Evaluate the Lagrangian deformation gradient tensor
X1, x1
X2, x2 k 1
Evaluate the Lagrangian deformation gradient tensor

35. Deformation tensor
Green strain tensor

36. 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

37. Mohr’s circle approach

38. Strain gage rosettes

39. Rectangular rosette
3 equations, 3 unknowns relating

40. Delta rosette
Following similar approach, one can obtain principal strains and orientation

41. Expectations after 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.)