1. Kinematics
Deformation
Flow
Kinematics
Finite Strains

2. Deformation and Flow Solid continua deform and fluid continua flow
Deformation is measured by the strain tensor, which describes change of shape
Flow is measured by the rate-of-deformation tensor which is related to velocity gradients.
This lecture focuses on deformation
Feb 2001 Added some extra slides and animated
Jan2000 Fixed typos on slides 4, 11, 12, 15
Feb2000 This has been shortened and slightly edited from BE250A topic 4

3. Kinematics u = x ­ X X x Reference State Current State
(“undeformed”) (“deformed”)
Time: t=0 t=t
Region: R0 R P x
u = x ­ X O P X
undeformed (or “material”) coordinates
current (or “spatial”) coordinates

4. Lagrangian & Eulerian Descriptions
Lagrangian or “material” description of motion
Motion as seen by the convecting material particle
Undeformed coordinates label material points
Most useful for solid mechanics problems
Eulerian or “spatial” description of motion X x = ( , ) t
Motion as seen by a fixed spatial observer
Current spatial coordinates of material points change
Most useful for fluid flow problems

5. Displacement Vector
The displacement vector, u

6. Deformation Gradient Tensor

7. Displacement Gradient Tensor
The displacement vector, u

8. Polar Decomposition Theorem

9. (Finite) Strain Tensors
Strain is a measure of change of shape independent of rotation. Change of shape corresponds to change of length (i.e. stretch)
Lagrangian Green’s Strain Tensor
Eulerian Almansi’s Strain Tensor

10. Infinitesimal (Cauchy) Strain
The Green’s and Alamansi strain tensors are exact measures of shape change for any finite deformation, but they are nonlinear. In terms of the displacement gradients, the finite strains are quadratic, e.g.
When the displacement gradients are small enough (<1%), we may linearize the finite strains to obtain the infinitesimal Cauchy strain tensor:

11. Strain is Change in Length
Consider the squared length elements:
Hence, E RS
is a measure of squared length-change ds dS dX 2 - = R S

12. Simple extension
Uniform extension

13. Simple Shear X2 Simple Shear x X = + tan γ 𝛾𝛾 X1 tan γ3 = +
tan γ 𝛾𝛾 X1
tan 2 γ
vanishingly small for small strain

14. Pure Torsion of a Cylinder= Θ + θ α Z z α
is the twist per unit length of the tube

15. Length, Area and Volume Change
The stretch and strain tensors U, V, C, B, E and e all describe how material elements of length change from the undeformed dL to deformed dl states, e.g.
Nanson’s formula relates elements of area in the deformed da and undeformed dA states:
Elements of volume in the deformed dv and undeformed dV states are related by detF:
dv = detF dV

16. Measurement Considerations
To overcome problem of defining natural state in an experimental setting, reference dimensions in uniaxial tension tests are often recorded at a finite extension, λ*
Measured or Nominal stress
where A = reference (undeformed) area
Cauchy or true stress
a = deformed area, λ = extension (stretch) ratio