Summary continuum physics
Jonathan Pilgram
January 25, 2023
1 The stress and strain tensor
All small x are the deformed coordinates and all big X are the material coordinates (before de-
formation).
1.1 Deformation gradient
∂xi
Fij =
∂Xj
1.2 Tangent vectors
∂xi
Gi = e
∂Xj i
With ei being unit base vectors.
1.3 Displacement vector
ui = xi − Xi
1.4 Greens strain tensor
1 ∂ui ∂uj ∂uk ∂uk
Eij = ( + + )
2 ∂Xj ∂Xi ∂Xi ∂Xj
1.5 Infinitesimal strain tensor
Also known as linear or Cauchy strain tensor. Coordinate system is interchangeable as the dif-
ferences are very small.
1 ∂ui ∂uj
ϵij = ( + )
2 ∂Xj ∂Xi
1.5.1 Compatibility equation for 2D infinitesimal strain tensor
∂ 2 ϵ11 ∂ 2 ϵ22 ∂ ∂
2 + −2 ϵ12 = 0
∂x2 ∂x21 ∂x1 ∂x2
1
, 1.5.2 Compatibility equations for 3D infinitesimal strain tensor
∂ 2 ϵ11 ∂ 2 ϵ22 ∂ ∂
2 + −2 ϵ12 = 0
∂x2 ∂x21 ∂x1 ∂x2
∂ 2 ϵ22 ∂ 2 ϵ33 ∂ ∂
2 + −2 ϵ23 = 0
∂x3 ∂x22 ∂x2 ∂x3
∂ 2 ϵ11 ∂ 2 ϵ33 ∂ ∂
2 + 2 −2 ϵ13 = 0
∂x3 ∂x1 ∂x1 ∂x3
∂ ∂ϵ23 ∂ϵ31 ∂ϵ12 ∂ ∂
(− + + )− ϵ11 = 0
∂x1 ∂x1 ∂x2 ∂x3 ∂x2 ∂x3
∂ ∂ϵ23 ∂ϵ31 ∂ϵ12 ∂ ∂
( − + )− ϵ22 = 0
∂x2 ∂x1 ∂x2 ∂x3 ∂x3 ∂x1
∂ ∂ϵ23 ∂ϵ31 ∂ϵ12 ∂ ∂
( + − )− ϵ33 = 0
∂x3 ∂x1 ∂x2 ∂x3 ∂x1 ∂x2
1.6 Velocity field in Eulerian and Lagrangian description
Given a Lagrangian description of a flow field as:
X(x0 , t)
And Eulerian description of a flow field as:
u(x, t)
Then they are related by:
∂X
u(X(x0 , t), t) = (x0 , t)
∂t
1.7 Line segment length
(ds)2 − (dS)2 = Eij xi xj
With ds being the infinitesimal line segment after transformation and dS being infinitesimal line
segment before transformation.
1.8 Principal values stress tensor
The principal values of the stress tensor are the solution to the eigenvalue problem:
|σ − Iλ| = 0
If you order these principal values as as σI > σII > σIII , then the maximum shear stress is given
by:
σIII − σI
σs,max = ±
2
Now one can draw Mohr’s circles by putting down the principal stress values on a horizontal nor-
mal stress axis. The shear stress will then by the vertical axis. Then draw three circles between
the points.
2
Jonathan Pilgram
January 25, 2023
1 The stress and strain tensor
All small x are the deformed coordinates and all big X are the material coordinates (before de-
formation).
1.1 Deformation gradient
∂xi
Fij =
∂Xj
1.2 Tangent vectors
∂xi
Gi = e
∂Xj i
With ei being unit base vectors.
1.3 Displacement vector
ui = xi − Xi
1.4 Greens strain tensor
1 ∂ui ∂uj ∂uk ∂uk
Eij = ( + + )
2 ∂Xj ∂Xi ∂Xi ∂Xj
1.5 Infinitesimal strain tensor
Also known as linear or Cauchy strain tensor. Coordinate system is interchangeable as the dif-
ferences are very small.
1 ∂ui ∂uj
ϵij = ( + )
2 ∂Xj ∂Xi
1.5.1 Compatibility equation for 2D infinitesimal strain tensor
∂ 2 ϵ11 ∂ 2 ϵ22 ∂ ∂
2 + −2 ϵ12 = 0
∂x2 ∂x21 ∂x1 ∂x2
1
, 1.5.2 Compatibility equations for 3D infinitesimal strain tensor
∂ 2 ϵ11 ∂ 2 ϵ22 ∂ ∂
2 + −2 ϵ12 = 0
∂x2 ∂x21 ∂x1 ∂x2
∂ 2 ϵ22 ∂ 2 ϵ33 ∂ ∂
2 + −2 ϵ23 = 0
∂x3 ∂x22 ∂x2 ∂x3
∂ 2 ϵ11 ∂ 2 ϵ33 ∂ ∂
2 + 2 −2 ϵ13 = 0
∂x3 ∂x1 ∂x1 ∂x3
∂ ∂ϵ23 ∂ϵ31 ∂ϵ12 ∂ ∂
(− + + )− ϵ11 = 0
∂x1 ∂x1 ∂x2 ∂x3 ∂x2 ∂x3
∂ ∂ϵ23 ∂ϵ31 ∂ϵ12 ∂ ∂
( − + )− ϵ22 = 0
∂x2 ∂x1 ∂x2 ∂x3 ∂x3 ∂x1
∂ ∂ϵ23 ∂ϵ31 ∂ϵ12 ∂ ∂
( + − )− ϵ33 = 0
∂x3 ∂x1 ∂x2 ∂x3 ∂x1 ∂x2
1.6 Velocity field in Eulerian and Lagrangian description
Given a Lagrangian description of a flow field as:
X(x0 , t)
And Eulerian description of a flow field as:
u(x, t)
Then they are related by:
∂X
u(X(x0 , t), t) = (x0 , t)
∂t
1.7 Line segment length
(ds)2 − (dS)2 = Eij xi xj
With ds being the infinitesimal line segment after transformation and dS being infinitesimal line
segment before transformation.
1.8 Principal values stress tensor
The principal values of the stress tensor are the solution to the eigenvalue problem:
|σ − Iλ| = 0
If you order these principal values as as σI > σII > σIII , then the maximum shear stress is given
by:
σIII − σI
σs,max = ±
2
Now one can draw Mohr’s circles by putting down the principal stress values on a horizontal nor-
mal stress axis. The shear stress will then by the vertical axis. Then draw three circles between
the points.
2
