2017/09/19 Tuesday, September 19, 2017 8:44 AM

Trace of a second order tensor

Note: trace is a scalar CM_F17 Page 1 Note: trace is a scalar

Inner product of 2nd order tensors:

CM_F17 Page 2 Define a norm from inner product:

FYI: Other norms for tensors Vector-induced norm definition:

CM_F17 Page 3 Basically if we have a definition of a norm for a vector => from which we derive a norm for a 2nd order tensor.

This is a very good definition. If we use L2 norm for the vectors (

Then the norm of a 2nd order tensor with this definition is simply:

The inverse of a tensor:

CM_F17 Page 4 You already showed the following in your HW1:

Orthogonal tensors:

CM_F17 Page 5 Remember Q was an orthogonal tensor

CM_F17 Page 6 For an orthogonal tensor -> rows and columns are orthonormal (i.e. basis vectors of an orthonormal coordinate system)

Orthogonal tensors are nothing but:

1. Rotation (proper orthogonal (det T = 1). 2. (Rotation ?) + Reflection (improper orthogonal): (det T = -1)

CM_F17 Page 7 Examples:

CM_F17 Page 8 If T1, T2, ….. are are rotations

T = T1 T2 T3 … Is also a rotation (det T = 1 and T T = I)

Reflection: Example, reflection w.r.t. x2, x3 plane

CM_F17 Page 9 det T = -1 and T is orthogonal

For any improper orthogonal tensor we have ONE reflection + (possibly some rotations)

Some properties of orthogonal tensors

CM_F17 Page 10 CM_F17 Page 11 Higher order tensors

We will see: - Strain is a second order tensor - Stress is also a second order tensor - We know stress depends on strain (for linearized elasticity) stress is a linear function of strain:

CM_F17 Page 12 function of strain:

CM_F17 Page 13 CM_F17 Page 14 This is how a fourth order tensor components should transform (?)

Think about

Higher order generalization of dyadic product:

CM_F17 Page 15 You can show that it in fact is linear (-> is a tensor)

--- How about generalization of expanding a tensor in terms of its components:

CM_F17 Page 16 Generalization of components of a tensor:

--- Examples of expanding a tensor:

CM_F17 Page 17 Definition of polyads:

CM_F17 Page 18 Coordinate transformation rule for any order tensor:

CM_F17 Page 19