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